Nonlinear Analysis, Geometry and Applications: Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, June 24–28, 2019 3030573354, 9783030573355

Citation preview

Trends in Mathematics

Diaraf Seck Kinvi Kangni Philibert Nang Marie Salomon Sambou Editors

Nonlinear Analysis, Geometry and Applications Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, June 24–28, 2019

Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference.

More information about this series at http://www.springer.com/series/4961

Diaraf Seck • Kinvi Kangni • Philibert Nang • Marie Salomon Sambou Editors

Nonlinear Analysis, Geometry and Applications Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, June 24–28, 2019

Editors Diaraf Seck Department of Mathematics of Decision FASEG University Cheikh Anta Diop of Dakar Dakar, Senegal Philibert Nang Mathematics Laboratory Ecole Normale Supérieure de Libreville Libreville, Gabon

Kinvi Kangni Department of Mathematics and Computer Science Felix Houphouet Boigny University Abidjan, Côte d’Ivoire Marie Salomon Sambou University Assane Seck of Ziguinchor Ziguinchor, Senegal

ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISBN 978-3-030-57335-5 ISBN 978-3-030-57336-2 (eBook) https://doi.org/10.1007/978-3-030-57336-2 Mathematics Subject Classification: 49Q10, 65K10, 90C26, 90C27, 58E25, 49J15, 49J20, 34A45, 32A50, 58B20, 20G05, 11S82, 57R12, 32J25, 34C08, 43-XX, 22EXX © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com, by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This volume contains the proceedings of the first Biennial International Research Symposium of the Nonlinear Analysis, Geometry and Applications Project (NLAGA-BIRS), which was held at the Cheikh Anta Diop University, in Dakar, Senegal, on June 24–28, 2019. The symposium brought together African experts, as well as their international partners, to present and discuss new mathematical results in various fields. This book gathers the best nineteen papers presented at this 4-day symposium. It addresses a range of topics related to partial differential equations, geometrical analysis of optimal shapes, geometric structures, optimization and optimal transportation, control theory, and mathematical modeling. The main goal of the NLAGA project is to advance and consolidate the development of nonlinear analysis, geometry, and their applications in West and Central Africa with a focus on solving real-world problems such as coastal erosion, pollution, and urban network and population dynamics problems. (For more details, please visit http://nlaga-simons.ucad.sn.) The contributions in this volume offer insight into the cutting-edge research and thought leadership of mathematical sciences at both African and international levels. Controllability theory, optimal mass transportation, stochastic approximation, and optimization are discussed in the first five papers. The next three papers focus on modeling and stability in ordinary differential equations. They are followed by four papers that are devoted to partial differential equations and two that focus on geometrical analysis of optimal shapes. The final five papers cover topics such as complex analysis in several variables, geometry, and real algebraic geometry. The NLAGA-BIRS Symposium was supported by the NLAGA Grant funded by Simons Foundation, Senegal National Academy of Sciences and Techniques, Cheikh Anta Diop University, the African Center of Excellence in Mathematics and Computer Sciences at the Gaston Berger University (Saint Louis, Senegal), the Alioune Diop University of Bambey (Bambey, Senegal) as well as CREFDES, IFACE, and FASEG (respectively, Institutes and Faculty for Economics and Management of UCAD).

v

vi

Preface

We wish to thank all the contributors of this symposium for making it an outstanding scientific and intellectual event. Finally, we thank deeply all the referees who took their valuable time to review all the submitted papers. Dakar, Senegal Abidjan, Côte d’Ivoire Libreville, Gabon Ziguinchor, Sénegal

Diaraf Seck Kinvi Kangni Philibert Nang Marie Salomon Sambou

Contents

Null Controllability of a Nonlinear Population Dynamics with Age Structuring and Spatial Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Yacouba Simporé

1

Null Controllability of a System of Degenerate Nonlinear Coupled Equations Derived from Population Dynamics . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Mamadou Birba and Oumar Traoré

35

Optimal Mass Transport for Activities Location Problem . . . . . . . . . . . . . . . . . . Mamadou Koné, Babacar Mbaye Ndiaye, and Diaraf Seck Cut-off Phenomenon for Converging Processes in the Sense of α-Divergence Measures .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B. Don Bosco Diatta and Papa Ngom

67

97

Stochastic Optimization in Population Dynamics: The Case of Multi-site Fisheries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 119 Sidy Ly and Diaraf Seck A Hurwitz Like Characterization of GUAS Planar Switched Systems . . . . 147 Daouda Niang Diatta, Moussa Balde, and Aminata D. T. Keita OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak . . . . . . . . . . . . 165 Coura Baldé, Mountaga Lam, and Samuel Bowong A Scalable Engineering Combination Therapies for Evolutionary Dynamic of Macrophages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 201 Moctar Kande, Raphaël M. Jungers, Diaraf Seck, and Moussa Balde Exact Steady Solutions for a Fifteen Velocity Model of Gas.. . . . . . . . . . . . . . . . 231 Amah d’Almeida

vii

viii

Contents

Monotony and Comparison Principle in Non Autonomous Size Structured Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 263 Mamadou Lamine Diagne, Mamadou Moustapha Mbaye, and Ousmane Seydi A Boundary Value Problem of Sand Transport Equations: An Existence and Homogenization Results . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 299 B. K. Thiam, M. A. M. T. Baldé, I. Faye, and D. Seck The Role of the Mean Curvature in a Mixed Hardy-Sobolev Trace Inequality . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 339 El Hadji Abdoulaye Thiam Coupling Between Shape Gradient and Topological Derivative in 2D Incompressible Navier-Stokes Flows . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 359 Aliou Seck, Alassane Sy, and Diaraf Seck Shape Reconstruction in a Non-linear Problem . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 379 Guillaume Itbadio Sadio and Diaraf Seck ¯ The ∂ ∂-Problem for the Differential Forms with Boundary Value in Currents Sense Defined in a Contractible Completely Strictly Pseudoconvex Domain of a Complex Manifold . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 399 Salomon Sambou and Souhaibou Sambou Introduction to the Resolution of d  d  for the Supercurrents in the Non-Archimedean Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 411 Ibrahima Hamidine and Salomon Sambou Minimal Graphs on Three-Dimensional Walker Manifolds .. . . . . . . . . . . . . . . . 425 Abdoul Salam Diallo, Ameth Ndiaye, and Athoumane Niang Quantitative Result on the Deviation of a Real Algebraic Curve from Its Vertical Tangents.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 439 Daouda Niang Diatta, Sény Diatta, and Marie-Françoise Roy Algebraic Points of Degree at Most 2 on the Affine Curve y 11 = x 2 (x − 1)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 459 Chérif Mamina Coly and Oumar Sall

Null Controllability of a Nonlinear Population Dynamics with Age Structuring and Spatial Diffusion Yacouba Simporé

Abstract In this paper, we investigate the null controllability for a nonlinear population dynamics model. This system is such that the nonlinearity is at the level of births. We show, that for a control localized in the space variable as well as with respect to the age, there is a time T dependent on the constraints on the age from which the system is null controllable. We first establish an observability inequality useful for the proof of the null controllability of an auxiliary system. We also apply the Schauder’s fixed point in the proof of the null controllability of the nonlinear system. For illustration, numerical simulations are provided.

1 Introduction and Mains Results In this paper, we study the null-controllability of an infinite dimensional nonlinear system describing the dynamics of age-structured population with spatial position. Let y be a solution of the following system: ∂y ∂y + − y + μy = χ v in  × (0, A) × (0, T ), ∂t ∂a

(1)

with the initial condition y(x, a, 0) = y0 (x, a) ∈  × (0, A)

(2)

where, y0 is given in K = L2 (0, A; L2()),

Y. Simporé () Laboratoire LAMI, Université Ouaga 1 Professeur Joseph Ki-Zerbo, Ouagadougou, Burkina Faso DeustoTech, Fundación Deusto Avda. Universidades, Bilbao, Spain © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_1

1

2

Y. Simporé

the boundary condition on x given by ∂y = 0 on ∂ × (0, A) × (0, T ) ∂ν

(3)

where A is the maximal age. We denote by β the positive function describing the fertility rate that depends only on the age. The birth law ensures the survival of the species. In several works, A one denotes by 0 β(a)yda the number of newborn individuals at position x at time t. If we consider an oviparous species, that is to say whose birth process goes A through egg-laying, we see that 0 β(a)yda is the number of eggs laid at position x, and at time t. Since all the eggs do not reach maturity, we introduce a function giving the proportion of eggs who will arrive there. Let be this function. Then we have   A β(a)yda , (4) y(x, 0, t) = F 0

the distribution of newborn individuals at time t and location x with F (x) = x (x). We further assume that F is a globally Lipschitz function. Moreover, the positive function μ denotes the natural mortality rate of individuals of age a, supposed to be independent of the times t and the space variable x. The control function is u, depending on the time t the spatial position x and the age a, and χ is the characteristic function. We assume that the fertility rate β and the mortality rate μ satisfy the demographic properties:  (H1 ) =

μ∈

μ ≥ 0 a.e in (0, A), A , 0 μ(a)da = +∞

L1loc ([0, A))

and ⎧ ⎨ β ∈ C 1 ([0, A]) and β(a) ≥ 0 ∀a ∈ [0, A], (H2 ) = β(a) = 0 ∀a ∈ (0, a) ˆ where 0 < aˆ < A, ⎩ βμ ∈ L1 (0, A). For more details about the modelling of such system and the biological significance of the hypotheses, we refer to Glenn Webb [9]. Now, we assume that F (0) = 0. Hence, one obtains the system considered in [7] where existence of solution was studied. From now, we set Q =  × (0, A) × (0, T )  = ω × (a1 , a2 ) × (0, T ) where 0 ≤ a1 < a2 ≤ A and ω a open subset of , QA =  × (0, A), QT =  × (0, T ) and  = ∂ × (0, A) × (0, T ).

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

3

We also make some assumptions about the function F (H3 ) : F positive and differentiable at 0. We are now ready to state the main theorem. Theorem 1.1 Let us assume the assumptions (H1 ) − (H2 ) − (H3 ). For every times T > A − a2 + a1 , and for every y0 ∈ L2 (QA ), there exists a control v ∈ L2 () such that the associated solution of the system (1)–(4) verifies: y(x, a, T ) = 0 a.e

x ∈  a ∈ (0, A).

(5)

Many versions of the age-size structured model, both linear and nonlinear, have been investigated, and seminal treatments of such models are given by Metz and Diekmann [5] and Tucker and Zimmerman [8]. And many results of null controllability of the Lotka–McKendrick system without or with spatial diffusion have been obtained by several authors. The first result was obtained by Ainseba and Anita [1]. They proved that the Lotka–Mckendrick system with spatial diffusion can be driven to a steady state in any arbitrary time T > 0 keeping the positivity of the trajectory, provided the initial data is close to the steady state and the control acts in a spatial subdomain ω ⊂  but for all ages. Bedr’Eddine Ainseba and Mimmo Iannelli, in [2], have worked on the controllability of two nonlinear dynamic model problems, the non linearity being on mortality and fertility. In the first model the control corresponds to a supply of individuals on a small age interval. The second one is age and space structured and the control corresponds to a supply of individuals on a small subdomain ω of the whole space domain . In [6], O.Traoré proved that the system (1)–(2)–(4) with Dirichlet boundary condition is null controllable except for a small interval of ages near zero where the controls is localized with respect to the space variable but active for all ages. The authors D. Maity, M. Tucsnak and E. Zuazua in [4] solved a problem of null controllability of the Lotka–Mckendrick system with spatial diffusion where the control is localized in the space variable as well as with respect to the age. The method combines final-state observability estimates with the use of characteristics and the associated semigroup. This paper is devoted to improve the observation domain of the null controllability of this system. Indeed, in the work of Oumar Traoré [7], the control acts for all the ages and we dont have necessarily the extinction on the small interval of ages near zero. • Here, we establish the null controllability of the model where the control localized in the age variable as well as with respect to the space. The estimation of the control time depends on the limits of the sub-age interval. By controlling for all ages, we have a similar result of [7] except that here we have the total extinction of the population. The only compromise is that the kernel β must be zero for the small age and the lower bound of support in age must be in the part where the fertility is zero.

4

Y. Simporé

• Moreover, to establish the observability inequality necessary for the controllability of the linear auxiliary system, we combine final-state observability estimates with the use of characteristics and with L∞ estimates of the associated semigroup (see [4]). The remaining part of this work is organized as follows: In Sect. 2, we establish the observability inequality of the adjoint system of an auxiliary linear system. This allows us to prove its approximate null controllability. The Sect. 3 is devoted to the proof of the main result of this paper. In the Sect. 4 we give a numerical simulation of the control. We give the description of possible extensions and open questions in the Sect. 5.

2 Approximate Null Controllability of an Auxiliary System In this part, we assume that the assumptions (H1 ) − (H2 ) hold. Therefore, we consider the following linearized auxiliary system: ⎧ ∂y ∂y ⎪ ⎪ = χ u ⎪ ∂t + ∂a − y + μ(a)y ⎪ ⎪ ⎨ ∂y =0 ∂ν ⎪ A ⎪ ⎪ y (x, 0, t) = G 0 β(a)yda ⎪ ⎪ ⎩ y (x, a, 0) = y0

in Q, on ,

(6)

in QT , in QA ,

where G ∈ L∞ (QT ) For G ∈ L∞ (QT ), the above system admits a unique solution and we have the following estimate.  yL2 (Q) ≤ C y0 L2 (Ω) + uL2 (Θ) , where C is independent of u and y0 . For the approximate null controllability of the auxiliary system, we define his corresponding adjoint system given by: ⎧ ∂q ∂q ⎪ ⎪ − − − q + μ(a)q − Gβ(a)q(x, 0, t) = 0 in Q, ⎪ ⎪ ∂a ⎪ ⎨ ∂t ∂q = 0 on , ∂ν ⎪ ⎪ ⎪ q (x, A, t) = 0 in QT , ⎪ ⎪ ⎩ q (x, a, T ) = qT in QA . where qT ∈ L2 (QA ).

(7)

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

5

Under the assumption (H1 ) and (H2 ), the system (7) admits a unique solution q. Moreover integrating along the characteristic lines, the solution q of (7) is given by: ⎧ π(a + T − t) ⎪ e(T −t) qT (x, a + T − t) ⎪ ⎪ ⎪ π(a) ⎪ ⎪ ⎪  ⎪  T ⎨ π(a − t + s) (s−t) e β(a + s − t)G(x, s)q(x, 0, s) ds if T − t ≤ A − a, q(t) = + π(a) ⎪ t ⎪ ⎪ ⎪ ⎪ t+A−a   ⎪ ⎪ π(a − t + s) (s−t) ⎪ ⎩ e β(a + s − t)G(x, s)q(x, 0, s) ds if A − a < T − t. π(a) t

(8) a

where π(a) = e− 0 μ(s)ds . We have the following result: Theorem 2.1 Let assume the assumptions (H1 ) − (H2 ). For T > A − a2 + a1 , y0 ∈ L2 (QA ), and for every  > 0, there exists a control u such that the solution y of the system (6) verifies y(., a, T )L2 (0,A;L2()) ≤  .

(9)

The proof of the Theorem 2.1 is based on the following observability inequality.

2.1 Observability Inequality It has been recognized that the null controllability of linear system is equivalent to the observability inequality of this adjoint system. Therefore, for the proof of Theorem 2.1 we establish the following result: Theorem 2.2 Under the assumption of the Theorem 2.1, the solution q of the system (7) is final-state observable for every T > a1 + A − a2 . In other words, for every T > a1 + A − a2 , and for every qT ∈ L2 (QA ) there exists KT > 0 such that the solution q of (7) satisfies: 

A 0



T

q (x, a, 0)dxda ≤ KT 2



0



a2 a1

 q 2 (x, a, t)dxdadt

(10)

ω

For the proof, we need the following estimations: Proposition 2.1 Let us assume true the assumption (H1 ) − (H2 ). For a1 < aˆ and a1 < η < T , there exists a constant C > 0 such that for every qT ∈ L2 (QA ), the

6

Y. Simporé

solutions q of the system (7) verifies the following inequality: 





T −η 0



T

q 2 (x, 0, t)dxdt ≤ C 0



a2

 q 2 (x, a, t)dxdadt.

a1

(11)

ω

and Proposition 2.2 Let us assume true the assumption (H1 ) − (H2 ). For a1 < aˆ and a1 < a0 < aˆ there exists CT > 0 such that the solution q of the system (7) verifies the following inequality 

a0



0



T

q 2 (x, a, 0)dxda ≤ CT 0





a2



a1

q 2 (x, a, t)dxdadt

(12)

ω

For the proof of the Propositions 2.1 and 2.2, we first recall the following observability inequality for parabolic equation (see, for instance, Imanuvilov and Fursikov [3]): Proposition 2.3 Let T > 0, t0 and t1 such that 0 < t0 < t1 < T . Then for every w0 ∈ L2 (), the solution w of the initial and boundary problem ⎧ ∂w(x, λ) ⎪ ⎪ − w(x, λ) = 0 in (t0 , T ) × , ⎪ ⎨ ∂λ ∂w =0 on (t0 , T ) × ∂, ⎪ ⎪ ∂ν ⎪ ⎩ w(x, t0 ) = w0 (x) in .

(13)

satisfies the estimate 

c2  w2 (x, t1 )dx ≤ c1 e t1 − t0

 w (T , x)dx ≤ 2





t1

t0

 w2 (x, λ)dxdλ, ω

where the constant c1 and c2 depend on T and . Proof of the Proposition 2.1 For a ∈ (0, a) ˆ we have β(a) = 0, therefore the system (7) can be rewritten by ⎧ ∂q ∂q ⎪ ⎪ =0 in  × (0, a) ˆ × (0, T ), ⎪ ⎨ − ∂t − ∂a − q + μq ∂q =0 on ∂ × (0, a) ˆ × (0, T ), ⎪ ⎪ ∂ν ⎪ ⎩ q(x, a, T ) = qT (x, a) in  × (0, a). ˆ We denote by q(x, ˜ a, t) = q(x, a, t)e−

a 0

μ(α)dα

(14)

. Then, q˜ satisfies

∂ q˜ ∂ q˜ + + q˜ = 0 in  × (0, a) ˆ × (0, T ). ∂t ∂a

(15)

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

7

Proving the inequality (11) lead also to show that, there exits a constant C > 0 such that the solution q˜ of (15) satisfies 

T −η 0





T

q˜ 2 (x, 0, t)dxdt ≤ C 0





a2 a1

 q(x, ˜ a, t)dxdadt.

(16)

ω

Indeed, we have 

T −η 0



T −η 

 q 2 (x, 0, t)dxdt = 0



 0





T

≤C

T

q˜ 2 (x, 0, t)dxdt ≤ C



0

a2 a1



a2



a1

q˜ 2 (x, a, t)dxdadt ω

 q 2 (x, a, t)dxdadt ω

We consider the following characteristics trajectory γ (λ) = (T − λ, T + t − λ). If T − λ = 0 the backward characteristics starting from (0, t). If T < a1 the trajectory γ (λ) never reaches the observation region (a1 , a2 ) (see Fig. 1). So we choose T > a1 . Moreover, we estimate q(x, 0, t) on (0, T − η) where η > a1 , because, all the characteristics starting at (0, t) with t ∈ (T − a1 , T ) never intersects the observation domain (see Fig. 1). Without loss the generality, let us assume here η < a2 = aˆ < T . The proof will done in two steps: Step 1: Estimation for t ∈ (0, T − a) ˆ We denote by: w(λ) = q(x, ˜ T − λ, T + t − λ) ; (λ ∈ (0, T − a) ˆ Fig. 1 An illustration of the estimate of q(x, 0, t). Here ˆ we have chosen a2 = a. Since t ∈ (0, T − a1 ) all the backward characteristics starting from (0, t) enters the observation domain

and

x ∈ ).

T T

a1

0

a1

a2

A

8

Y. Simporé

Then, w satisfies: ⎧ ∂w(λ) ⎪ ⎪ =0 in (T − a, ˆ T ), ⎪ ⎨ ∂λ − w(λ) ∂w =0 on ∂ × (0, T ), , ⎪ ⎪ ∂ν ⎪ ⎩ w(0) = q(x, ˜ T , T + t) in .

(17)

Using the Proposition 2.3 with 0 < t0 < t1 < T we obtain: c2  t w (t1 )dx ≤ c1 e 1 − t0



 w (T )dx ≤ 2



t1

2



t0

 w2 (λ)dxdλ. ω

That is equivalent to c2  t 1 q˜ (x, 0, t)dx ≤ c1 e − t0

 



t1

2

q˜ 2 (x, T − λ, t + T − λ)dxdλ

t0

c2  t = c1 e 1 − t0

ω



T −t0

q˜ 2 (x, a, t + a)dxda

T −t1

ω

Then, for t0 = T − aˆ and t1 = T − a1 , we obtain 



aˆ

q˜ 2 (x, 0, t)dxdt ≤ C

 q˜ 2 (x, a, t + a)dxda.

a1



ω

Integrating with respect t over (0, T − a) ˆ we get 

T −aˆ





0

q˜ (x, 0, t)dxdt ≤ C 2

aˆ



a1



T −a+a ˆ a

 q˜ 2 (x, a, t)dxdtda. ω

Finally, 

T −aˆ





0

T

q˜ 2 (x, 0, t)dxdt ≤ C 

0



a2 a1

 q˜ 2 (x, a, t)dxdadt. ω

Step 2: Estimation for t ∈ (T − a, ˆ T − η)

where

ˆ η ∈ (a1 , a)

We denote by: w(λ) = q(x, ˜ T − λ, T + t − λ) ; (λ ∈ (T − a, ˆ T )).

(18)

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

Then, w satisfies: ⎧ ∂w(λ) ⎪ ⎪ =0 in (T − a, ˆ T ), ⎪ ⎨ ∂λ − w(λ, x) ∂w =0 on ∂ × (0, T ), ⎪ ⎪ ∂ν ⎪ ⎩ w(T − a) ˆ = q(x, ˜ a, ˆ t + a) ˆ in .

9

(19)

Using the Proposition 2.3 with T − aˆ < t0 < t1 < T we obtain: 

c2  w (t1 )dx ≤ c1 e t1 − t0

 w (T )dx ≤ 2



2



t1

 w2 (λ)dxdλ.

t0

ω

That is equivalent to c2  t q˜ (x, 0, t)dx ≤ c1 e 1 − t0



2



t1

 q(x, ˜ T − λ, T + t − λ)dxdλ.

t0

ω

Then, for t0 = T − η and t1 = T − a1 , we obtain: 





η

q˜ (x, 0, t)dx ≤ C(η, a1 ) 2

q(x, ˜ a, t + a)dxda.

a1



ω

( lim C(η, a1 ) = +∞.) η−→a1

Integrating with respect t over (T − a, ˆ T − η) we get: 

T −η



 q˜ 2 (x, 0, t)dxdt ≤ C(η, a1 )

T −aˆ

η



a+T −aˆ

a1



a+T −η

 q˜ 2 (x, α, t)dxdtda. ω

Finally, 

T −η T −aˆ



 q˜ 2 (x, 0, t)dxdt ≤ C(η, a1 ) 

0

T



a2 a1

 q˜ 2 (x, a, t)dxdadt.

(20)

ω

Combining (18) and (20), we obtain the result of the Proposition 2.1.



Proof of the Proposition 2.2 We consider in this proof the characteristics γ (λ) = (a + λ, λ). For λ = 0 the characteristics starting from (a, 0).

10

Y. Simporé

We have three cases. ˆ Case 1: T < a2 and a2 ≤ a. Two situations can arise: • b0 = a2 − T < a1 < a0 in this situation we split the interval (0, a0 ) as (0, a0 ) = (0, b0 ) ∪ (b0 , a1 ) ∪ (a1 , a0 ).

(21)

• a1 < b0 < a0 , in this situation we split the interval (0, a0 ) as (0, a0 ) = (0, a1 ) ∪ (a1 , a0 ). ˆ Case 2: T ≥ a2 and a2 ≤ a. In this case we split the interval (0, a1 ) as (0, a0 ) = (0, a1 ) ∪ (a1 , a0 ). ˆ In this case we proof similarly the observability in (a1 , a) ˆ and to Case 3: a2 > a: expand to (a1 , a2 ). Here we give the proof in the only situation where T < a2 , a0 ∈ (a1 , a2 ) and b0 = a2 − T < a1 .

(22)

  In the remaining part of the proof we give upper bounds for I  q˜ 2 (x, a, 0)dxda where I is successively each one of the intervals appearing in the decomposition (21) (Fig. 2). Upper Bound on (0, b0 ) For a ∈ (0, b0 ) we first set w(x, λ) = q(x, ˜ T +a−λ, T −λ) (λ ∈ (0, T ) and x ∈ ) a − 0 μ0 (α)dα where q˜ = e q. Then, w verifies ⎧ ∂w(x, λ) ⎪ ⎪ − w(x, λ) =0 in (0, T ), ⎪ ⎨ ∂λ ∂w (23) =0 on ∂ × (0, T ) , ⎪ ⎪ ∂ν ⎪ ⎩ w(T ) = q(x, ˜ a + T,T ) in . By applying the Proposition 2.3 with t0 = 0 and t1 = a + T − a1 , we obtain: c2  a + T − a 1 w (x, T )dx ≤ c1 e



a+T −a1

2



0

 w2 (x, λ)dxdλ. ω

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

11

a0

T

0 b0 a1

a2

A

Fig. 2 An illustration of the estimation of q(x, a, 0) between 0 and a0 for the case 1: green region corresponds to the interval (0, b0 ); black corresponds to the interval (b0 , a1 ); llue corresponds to the interval (a1 , a0 )

Then, we have: c2  a + T − a1 q˜ (x, a, 0)dx ≤ c1 e





a+T

2

q(x, ˜ α, α − a)dxdλ

a1





a+T

=C

a1



ω



a2

q˜ (x, α, α − a)dxdα ≤ C 2

ω

 q(x, ˜ α, α − a)dxdα.

a1

ω

Integrating with respect a over (0, b0 ) we get: 

b0





b0

q˜ 2 (x, a, 0)dxda ≤ C

0



0



a2

 q˜ 2 (x, α, α − a)dxdαda.

a1

ω

As 

b0 0



a2

a1



 q˜ (x, α, α − a)dxdαda = 2

ω

a2



b0

 q˜ 2 (x, α, α − a)dxdadα,

0

a1

ω

then, 

b0 0



 q˜ (x, a, 0)dxda ≤ C 2



a2 a1



α α−b0

 q˜ 2 (x, α, t)dxdtdα. ω

12

Y. Simporé

Finally, 





b0

T

q˜ (x, a, 0)dxda ≤ C 2

0



0





a2

q˜ 2 (x, a, t)dxdadt

a1

(24)

ω

Upper Bound (b0 , a1 ) For a ∈ (b0 , a1 ) we consider always the system (23) but λ ∈ (T + a − a2 , T ). Applying the Proposition 2.3 with t0 = a + T − a2 and t1 = a + T − a1 , we obtain  a2   2 q˜ (x, a, 0)dx ≤ C q˜ 2 (x, α, α − a)dxdα. 

a1

ω

And as before, we get: 





a1

T

q˜ 2 (x, a, 0)dxda ≤ C

b0



0





a2

q˜ 2 (x, a, t)dxdadt

a1

(25)

ω

Upper bound (a1 , a0 ) As above, we obtain for t0 = T + a − a2 and t1 = T this inequality: 

a0





T

q(x, ˜ a, 0)dxda ≤ C

a1



0



a1

 q˜ 2 (x, a, t)dxdadt

a1

(26)

ω

Consequently, combining (24), (25), and (26) we obtain: 

a0 0



 q˜ 2 (x, a, 0)dxda ≤ C

0



T



a2

a1

 q(x, ˜ a, t)dxdadt.



ω

2.2 Proof of the Observability Inequality We also need this Lemma for the proof of the observability inequality. Lemma 2.1 Let us suppose that T > A − a2 + a1 . Then, there exists a0 ∈ (a1 , a) ˆ and κ > 0 such that T > T − (a1 + κ) > A − a0 > A − a

f or

all

a ∈ (a0 , A).

Therefore  q(x, a, 0) = 0

A−a



 π(a + s) s e β(a + s)G(x, s)q(x, 0, s) ds for all a ∈ (a0 , A). π(a)

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

13

Proof Without loose the generality, we suppose that a2 = a. ˆ Suppose that T > a1 + A − a2 ⇔ T − a1 > A − a2 . Then there exists κ > 0 (we choose κ such that 2κ < a2 − a1 it reassures that a1 + κ < a2 − κ) such that T − a1 − 2κ > A − a2 ⇔ T − (a1 + κ) > A − (a2 − κ). We denote by a0 = a2 − κ. And as A − a0 > A − a f or all a ∈ (a0 , A), then, T > T − (a1 + κ) > A − a0 > A − a f or all a ∈ (a0 , A). Moreover, for (a, t) such that T − t > A − a, we have 

t +A−a

q(x, a, t) = t



 π(a − t + s) (s−t ) e β(a + s − t)G(x, s)q(x, 0, s) ds π(a)

and as for t = 0 and a ∈ (a0 , A), we have T − 0 > A − a0 > A − a, then, 

A−a

q(x, a, 0) =



0

 π(a + s) s e β(a + s)G(x, s)q(x, 0, s) ds. π(a)



We are in position to prove the Observability inequality. Proof of the Theorem 2.2 Let a0 ∈ (a1 , a) ˆ as in the previous Lemma 2.1. We have 

A

 0

a0

q 2 (x, a, 0)dxda = 0





 q 2 (x, a, 0)dxda + 

A

q(x, a, 0)dxda a0



From the representation (8) and using the result of the Lemma 2.1 there exists a0 ∈ (a1 , a) ˆ such that  q(x, a, 0) =

A−a

0



 π(a + s) s e β(a + s)G(x, s)q(x, 0, s) ds π(a)

f or

all

a ∈ (a0 , A)

and then A



a0



T −(a1 +κ) 

q 2 (x, a, 0)dxda ≤ K2 0



q 2 (x, 0, t)dxdt. 

Finally with the result of the Proposition 2.1, we obtain: 

A a0

 q 2 (x, a, 0)dxda ≤ K3 

0

T



a2

a1

 q 2 (x, a, t)dxdadt..

(27)

ω

Combining the result of the Proposition 2.2 and the inequality (27), we obtain the result of the Theorem 2.2 (see Fig. 3). 

14

Y. Simporé

Fig. 3 The backward characteristics starting from (a, 0) with a ∈ (a0 , A) (green lines) hits the line (a = A), gets renewed by the renewal condition β(a)q(x, 0, t) and then enters the observation domain

{T

a0

T

t=A

a}

a1

T

0

a1

a2

A

The three figures below illustrate the different stages of the proof of Theorem 2.2: Illustration of Observability Inequality

2.3 Proof of the Approximate Null Controllability of the Auxiliary System Proof of the Theorem 2.1 For  > 0, we consider the functional J define by: 1 J (u) = 2



T 0



a2

a1



1 u dxdadt + 2 ω

A



2

0

y 2 (x, a, T )dxda

(28)



where y is the solution of the following system ⎧ ∂y ∂y ⎪ ⎪ + − y + μy = χ u ⎪ ⎪ ∂t ∂a ⎪ ⎨ ∂y =0 ∂ν ⎪ A ⎪ ⎪ y (x, 0, t) = G 0 βyda ⎪ ⎪ ⎩ y (x, a, 0) = y0

in Q, on ,

(29)

in QT , in QA . 

Lemma 2.2 The functional J is continuous, strictly convex and coercive. Consequently J reaches its minimum at a point u ∈ L2 (). Moreover, setting y the associated solution of (29) and q the solution (6) with 1 q (x, a, T ) = − y (x, a, T ), one has χ u = χ q and there exist C1 > 0 and 

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

15

C2 > 0 independent of  and y0 , such that 



T 0

a2



a1



ω

q2 dxdadt ≤ C1

A 0



y02 dxda

and A

 0

 

y2 (x, a, T )dxda ≤ C2

A 0



y02 dxda

Proof of Lemma 2.2 It is easy to check that J is coercive, continuous, and strictly convex. Then it admits a unique minimiser u . The maximum principle gives that χ  u = χ  q  . Multiplying the first equation of (29) with χ u = χ q by q where 1 q (x, a, T ) = − y (x, a, T ), and integrating with respect to Q, we get:  

T



0

a2



a1

ω

q2 dxdadt 

1 + 

A

 0



y2 (x, a, T )dxda

A

=

y0 q (x, a, 0)dxda. 0



Using the inequality of Young, we obtain for any δ > 0 

T



0

δ ≤ 2

a2 a1

 ω

q2 dxdadt +

A

 0



y02 dxda

1 

1 + 2δ

A

 0



A

 0



y2 (x, a, T )dxda

q2 (x, a, 0)dxda.

Using the observability inequality (10) and choosing KT = δ where KT is given in (10), we obtain 1 2



T 0



a2 a1



1 q dxdadt +  ω

A

 0



y2 (x, a, T )dxda

KT ≤ 2

A

 0



y02 dxda.

This inequality gives us the result of the Lemma. By asking



A

 = C2 0



y02 dxda,

we get the approximate null controllability result.



16

Y. Simporé

3 Null Controllability of the Nonlinear System In this part, we denote by G(l) =

F (l) if l = 0 and l

F (0) if l = 0 l ∈ L2 (QT )

Let be  be an operator defined on L2 (QT ) by 

A

(l) =

βy (l)da 0

where y (l) is the solution of the following system: ⎧ ∂y (l) ∂y (l) ⎪ ⎪ + − y (ł) + μy (l) = χ q (l) ⎪ ⎪ ∂a ⎪ ⎨ ∂t ∂y (l) =0 ∂ν ⎪ A ⎪ ⎪ y (l) (x, 0, t) = G(l) 0 βy (l)da ⎪ ⎪ ⎩ = y0 y (l) (x, a, 0)

in Q, on ,

(30)

in QT , in QA .

and q (l) the minimizer of J defined in the Lemma 2.2 where G is replaced by G(l). We have the following result. Proposition 3.1 The operator  is continuous, bounded, and compact of L2 (QT ). Then  admits a fixed point. For the Proof of the previous Proposition 3.1 we starts the following Lemma: Lemma 3.1 Under the assumption (H3 ), there exist C > 0 and K > 0 independent of , l and y0 , such that the solution y (l) of (30) verifies ∇y (l)2L2 (Q) + y (l)2L2 (Q) ≤ Ky0 2L2 (Q

A)

and 

T 0



A 0



μy2 (l)dxdadt ≤ Cy0 2L2 (Q ) .

Proof (Proof of the Lemma) We denote by p (l) = e−λ0 t y (l) where λ0 > 0.

A

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

17

The function p (l) verifies ⎧ ∂p (l) ∂p (l) ⎪ ⎪ + − p (l) + (λ0 + μ(a))p (l) = χ e−λ0 t q (l) in Q ⎪ ⎪ ∂t ∂a ⎪ ⎨ ∂p (l) =0 on  ∂ν ⎪ A ⎪ ⎪ p (l)(x, 0, t) = 0 Gβ(a)p (l)da in QT ⎪ ⎪ ⎩ p (l)(x, a, 0) = y0 in QA . (31) Multiplying the first equation of (31) by p (l) and integrating with respect to space, the age a, and the times t, we obtain:  A  T  A p2 (l)(x, a, T )dxda + ∇p (l)2L2 () dadt 0





T

+ 0

0





A 0

0

(μ + λ0 )p2 (l)dxdadt =



 + y0 2L2 (Q

A

+ )

T

A 0

χ p e−λ0 t q (l)dxdadt 

2

A

G2 (l)

0



T 0







β(a)p (l)da

dxda.

(32)

0



We have 



T 0

1 ≤ 2





T 0

0

A 0

A



χ p e−λ0 t q (l)dxdadt 

p2 (l)dxdadt

1 + 2



T



0

a2



a1

ω

q2 (l)dxdadt.

Moreover 

T





0

2

A

G2 (l)

β(a)y (l)da 0



dxda ≤ G2∞ β2∞ p (l)2L2 (Q).

Then,  0

 + 0

≤



A  T



p2 (l)(x, a, T )dxda + A

0

1 1 p (l)2L2 (Q) + 2 2

T



0

A 0

 

 0

μ(a)p2 (l)dxdadt + λ0

T



a2 

a1

ω

|∇p (l)|2 dxdadt

 T 0



A

0



p2 (l)dxdadt

q2 (l)dxdadt +y0 2L2 (Q ) +G2∞ β2∞ p (l)2L2 (Q) . A

(33)

18

Y. Simporé

Combining (33) and the result of the Lemma 2.2, we obtain: 

A

 0



 + 0

T

p2 (l)(x, a, T )dxda



A 0

T

+ 0



A 0

 

μp2 (l)dxdadt + λ0

≤ Cy0 2L2 (Q

|∇p (l)|2 dxdadt

 T

0



A 0



p2 (l)dxdadt

 + G2∞ β2∞ + 1/2 p (l)2L2 (Q).

A)

(34)

Finally choosing λ0 such that:

 λ0 > G2∞ β2∞ + 3/2 , we obtain: ∇p (l)2L2 (Q) + p (l)2L2 (Q) ≤ Cy0 2L2 (Q ) , A



T 0



A 0



μp2 (l)dxdadt ≤ Cy0 2L2 (Q ) . A

And finally, we have the result of the Lemma 3.1.



Proof (Proposition 3.1) The proof will be done in two steps as follows: Step 1: Boundedness and Compactness of  Let 

A

P (l) =

βy (l)da 0

and denote QT =  × (0, T ). It easy to prove that P (l) is solution of the following system: ⎧ A ∂P (l) ⎪ ⎪ − P (l) + 0 μβy (l)da = Y (l) in QT , ⎪ ⎨ ∂t ∂P (l) =0 on ∂ × (0, T ), ⎪ ⎪ ∂ν ⎪ A ⎩ P (l)(x, 0) = 0 βy0 da in  × (0, A). where  Y (l) = 0

A

β y (l) + χ q (l)y (l) da

(35)

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

19

From the results of the previous Lemma 3.1, there exists K > 0 such that Y (l)L2 (×(0,T ) ≤ Ky0 L2 (QA ) . Boundness of

A 0

μβy (l)da in L2 (QT ) 

 

T

2

A

μβy (l)da 

T

=

0



 

A

(μβ) 0

dxdt

0

1 2

 2 1 (μβ) 2 y (l) da dxdt.

0



Using the Cauchy Swartz inequality we obtain 

 

T

2

A

μβy (l)da 0





A

≤

T



A

μβda 0

dxdt

0



0

0

Ω

μβy2 dxdadt.

The Lemma 3.1 and the fact that β ∈ C([0, A]), give that 

T

0

   0

A

μβy2 (l)dadxdt < R1 y0 L2 (QA )

where R1 > 0 is independent of , y0 and l. Moreover, as βμ ∈ L1 (0, A), then A 2 0 βμy (l)da is bounded in L (QT ) independently of l and . 2 We have also P (., 0) ∈ L (). Then, P (l) is bounded in L2 (0, T ; H 1 ()) and ∂P (l) is bounded in L2 (0, T ; H −1()). ∂t Hence, using the Lions–Aubin Lemma we conclude that  is bounded and compact in L2 (QT ). Step2: Continuity of  Let ln −→ l strongly in L2 (QT ). Then we can show that the sequence P (ln ) converge to P (l) strongly. For all ln ∈ L2 (QT ) the function 

A

Y (ln ),

μβy (lnk )da

and

P (lnk )

0

are bounded in L2 (QT ) independently of ln . Then we can extract the subsequence   X(lnk ) = P (lnk ),



A

μβy (lnk )da, Y (lnk ) 0

20

Y. Simporé

such that   P (lnk ),

A



  μβy (lnk )da, Y (lnk )  P (l),

0



A

μβy (l), Y (l) 0

weakly in (L2 (QT ))3 . Then P (l) is the solution of the system (35). We deduce that A P (lnk ) −→ P (l) = 0 βy (l)da strongly. Moreover y (l) verify (9). Consequently  is continuous. Since the operator  is continuous, bounded, and compact on L2 (QT ) onto 2 L (QT ), Schauder’s fixed-point theorem implies that  admits a fixed point.  We have 

A

(l) = l =

βy da 0

and y (l) solves ⎧ ∂y ∂y ⎪ ⎪ + − y + μy = χ q  ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎨ ∂y =0 ∂ν 

 ⎪ A ⎪ ⎪ y βy da 0, t) = F (x, ⎪   0 ⎪ ⎪ ⎩ y (x, a, 0) = y0

in Q, on ,

(36)

in QT , in QA .

Moreover we have 

T 0



a2 a1

 ω

q2 dxdadt ≤ C1 y0 L2 (QA )

and

y (., ., T )L2 (QA ) ≤ C2 y0 L2 (QA ) .

From the foregoing, we have that q and y are bounded in L2 (Q). Then we can extract a subsequence of (q , y ) still denoted by ((qn , yn ), such that weakly in L2 (Q),

yn  y 



A

A

βyn da  0

βyda

weakly in L2 (QT )

0

and χ qn  χ v

weakly in L2 (Q).

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

21

So 

A



A

βyn da −→

0

βyda

strongly

in L2 (QT ).

0

Therefore, there exists a subsequence 

A

A 0

βyk da such that



A

βyk da −→

0

βyda

a.e in QT .

0

Now since F is continuous then    A βyk da −→ F F 0



A

βyda

a.e in QT .

0

Therefore, one derives that y solves the following system ⎧ ∂y ∂y ⎪ ⎪ + − y + μy = χ v ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎨ ∂y =0 ∂ν

  ⎪ A ⎪ ⎪ y 0, t) = F βyda (x, ⎪ 0 ⎪ ⎪ ⎩ y (x, a, 0) = y0

in Q, on ,

(37)

in QT , in QA .

Finally, we have also  −→ 0; y (., ., T ) = 0

a.e

in  × (0, A).



4 Numerical Simulations 4.1 Discretization and Simulation of Uncontrolled System In this part, the idea in this part is to highlight the numerical simulation of the nonlinear problem. The first parts is to reduce the PDE to the finite dimensional system of the form U˙ l = Al Ul + vectorF + BVl where Al and B are matrices, and   Ul (t) Vl (t)

22

Y. Simporé

is the finite dimensional state vector. Here, vectorF is the contribution of the nonlinear part, which comes from births. So let’s consider the following system ⎧ ∂y ⎪ ⎪ + Ay + μy = χ V ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂y =0 ∂ν

  ⎪ A ⎪ ⎪ y 0, t) = F β(a)yda (x, ⎪ 0 ⎪ ⎪ ⎩ y (x, a, 0) = y0

in Q, on ,

(38)

in QT , in QA .

2 y, where K is the diffusion coefficient. where Ay = ∂a y − K∂xx To solve (38), the space (dimension 1) and age discretization is performed with finite difference method on rectangular grid on [0, L] × [0, A]. For a given rectangular grid T with vertex (xi , aj ) 1 ≤ i ≤ N, 1 ≤ j ≤ M and uniform step size (without loss of generality) x in x−direction and a in a−direction, we denote by the diameter of the grid. The finite difference approximation of the diffusion term of the operator A is given by

y(xi+1 , aj , t) − 2y(xi , aj , t) + y(xi−1 , aj , t) ∂ 2y (xi , aj , t) = . 2 ∂x (x)2 The finite difference approximation of the aging term of the operator A is given by y(xi , aj , t) − y(xi , aj −1 , t) ∂y (xi , aj , t) = . ∂a a Let yi,j (t) be the approximation of y(xi , aj , t) and Ul (t) = (yi,j (t))1≤i≤N, 1≤j ≤M , where yi,j is at the position i +j ∗(N −1) and Al , (Fig. 4) the matrix of the mortality approximation, the diffusion approximation and the aging approximation. Take into Account the Newborns We denote by β(a), the fertility rate, the newborn is given by:  y(x, 0, t) = F

A

 β(a)y(x, a, t)da .

0

We approximate  0

A

A 0

β(a)y(x, a, t)da by

β(a)y(x, a, t)da = (A/n)

n−1  j=1

β(aj )y(x, aj , t) +

β(0)y(x, 0, t) + β(A)y(xi , A, t) . 2

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

23

Fig. 4 Matrix Al (Matrix of differential Operator)

Then yi,0 = A/n

n−1 

β(ak )y(xi , ak , t) +

k=1

β(0)y(xi , 0, t) + β(A)y(xi , A, t) 2

f or

i = 1 : n.

But as yi,1 (t) − yi,0 (t) ∂yi,1 (t) = ∂a a

i = 1 : n,

then −A/n ∂yi,1 (t) = ∂a

n−1 k=1

β(ak )yi,k (t) + yi,1 (t) + a

i = 1 : n,

β(0)y(xi , 0, t) + β(A)y(xi , A, t) 2 ,

24

Y. Simporé

Fig. 5 Matrix Nl (Bird Matrix)

We create also the nonlinear vectorF from the births. Indeed, if we denote by (Nl )1≤i≤n∗n;1≤j ≤n∗n , the matrix of the births, ⎛ Pl = Nl Ul

where

⎜ ⎝

A

n−1 k=1

β(ak )yi,k (t ) +

β(0)y(xi , 0, t ) + β(A)y(xi , A, t ) ⎞ ⎟ 2 ⎠ na 1≤i≤n

= (Pl )1≤i≤n and (Pl )n+1≤i≤n∗n = 0

and the vectorF is given by (Fig. 5) vectorF = F (Nl Ul ). The corresponding discrete system is given for a continuous initial solution y0 by dUl = Al Ul + vectorF Ul (0) = (y0 (xi , aj ))1≤i≤N,1≤j ≤M dt

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

25

Example 4.1 For the simulation, we take L = 2, A = 2 and x = 1/10 a = 1/10, T = 2 and it k = 0.25. Moreover we choose F (t) = 0.11t 2 which is globally Lipchitz on a compact 2 2 which is our case with the initial condition y0 (x, a) = e−0.3(5(a/10−7) +0.5(x/10−7) ) . The fertility β is given by:

β(a) =

⎧ ⎪ ⎪ ⎨0

if a < 0.4

7(10a−4)4e−0.52(10a−4) (5) ⎪

⎪ ⎩0

if 0.4 ≤ a ≤ 1.7 if a > 1.7.

and the mortality rate μ by μ(a) =

1 10(A − a)

We get the following simulations (Figs. 6, 7, 8, and 9).

Fig. 6 Initial condition

26

Fig. 7 Solution at time t=0.2

Fig. 8 Solution at time t=1

Y. Simporé

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

27

Fig. 9 Solution at time t=2

4.2 Construction of the Control and Numerical Simulation We construct the control problem, which consists in minimizing the functional and we choose the classical Hum functional and the control matrix B = χ where  = (x1 , x2 ) × (a1 , a2 ) × (0, T ), and we suppose that UT is the desired state: J (y, V ) =

1 

A L

 0

0

0

T



a2

a1



x2

V 2 dxdadt

x1

 1  2 2 yi,j (T ).A + BVi,j,k .v.  N

=

 (y(., ., T ) − UT )2 dxda +

N

i=1 j =1

Nt

N

N

k=1 i=1 j =1

Here UT = 0, A = ax and v = xat. The approximate null controllability become the minimization of the functional J, where (Ul , Vl ) represented in the previous equality by the pair  yi,j (tk ), Vi,j (tk ) , verify the following system dUl = Al Ul + vectorF + BVl dt

Ul (0) = (y0 (xi , aj ))1≤i≤N,1≤j ≤M .

28

Y. Simporé

Fig. 10 Uncotrolled final solution

In this part the fertility β is given by:

β(a) =

⎧ ⎪ ⎪ ⎨0

if a < 0.4,

⎪ ⎩0

if a > 1.7.

7(10a−4)4e−0.52(10a−4) (5) ⎪

if 0.4 ≤ a ≤ 1.7,

Example 4.2 For the simulation, we take L = 2, A = 2, x = 0.02, a = 0.02,  = 0.0001 and  = (0, 2) × (0, 2) × (0, 2) and t = 20 dUl = Al Ul + 0.11 ∗ Nl Ul .2 + BVl dt

Ul (0) = (y0 (xi , aj ))1≤i≤N,1≤j ≤M .

In this example we are going back to the data from Example 4.1. We get the following simulations (Figs. 10, 11, 12, and 13): Example 4.3 Here, we take L = 1, A = 1 and x = 0.02 a = 0.02, T = 1,  = 0.0001 and it k = 0.25 and we change the initial condition by y0 (x, a) = 2 2 e−(75(a−5/10) +100(x−6/10) ) . The control support  = (0, 0.8) × (0, 0.8) × (0, 1)

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

Fig. 11 Control V at time t=1

Fig. 12 Control V at final time T=2

29

30

Y. Simporé

Fig. 13 Controlled solution at final time T=2

The fertility is also given by:

β(a) =

⎧ ⎪ ⎪ ⎨0

if a < 0.1

⎪ ⎩0

if a > 0.8.

7(10a−4)0.3e−0.52(10a−4) (5) ⎪

if 0.1 ≤ a ≤ 0.8

Moreover, here too F (t) = 0.11t 2. The following numerical results were obtained (Figs. 14, 15, 16, and 17): We notice that we have better results with positivity of the state, if the support of the control covers the whole domain (space, age). It should however be noted that we could not take into account the positivity constraint in our simulations. It should be noted that we have much better approximations when we do not consider the cost of control. We use the CaSadi toolbox for simulate the control (By the minimization of functional: Hum Method), and ODE 45 for the simulation of the uncontrolled system.

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

Fig. 14 Uncotrolled final solution

Fig. 15 Control V at time t=0.5

31

32

Fig. 16 Control V at final time T=1

Fig. 17 Controlled solution at final time T=1

Y. Simporé

Null Controllability of a Nonlinear Population Dynamics with Age Structuring. . .

33

5 Conclusion Considering a nonlinear Population dynamics with age and diffusion. We have proved a null controllability where control is localized in the space variable as well as with respect to the age. Some open issues and generalization remain to be investigated. They are in order: • Other model of population dynamics: We will be interested in other types of nonlinear population dynamics models, such as the age-size-dependent model. • Numerical implementation: For a given fertility rate β, the mortality rate μ, the initial condition y0 and a positive parameter  > 0, how to determine a numerical algorithm allowing to determine the -approximate null control function h? Acknowledgments The author want to thank Jesus Oroya (DeustoTech, University of Deusto, Bilbao, Spain) for his contribution to the simulations in Section (4). Funding This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 694126-Dycon).

References 1. B. Ainseba, S. Anita, Local exact controllability of the age-dependent population dynamics with diffusion. Abstr. Appl. Anal. 6, 357 (2001) 2. B. Ainseba, M. Iannelli, Exact controllability of a nonlinear population-dynamics problem. Differ. Integral Equ. 16(11), 1369–1384 (2003) 3. A.V Fursikov, O.Y Imanuvilov, Controllability of evolution equations, in Lecture Notes Series, vol. 34 (Seoul National University/Research Institute of Mathematics/Global Analysis Research Center, Seoul, 1996) 4. D. Maity, M. Tucsnak, E. Zuazua, Controllability and positivity constraints in population dynamics with age structuring and diffusion. J. Math. Pures Appl. 129, 153–179 (2019). https:// doi.org/10.1016/j.matpur.2018.12.006 5. J. Metz, O. Diekmann, The Dynamics of Physiologically Structured Populations (Springer Lecture Notes in Biomathematics 68, New York, 1986) 6. O. Traore, Null controllability of a nonlinear population dynamics problem. Int. J. Math. Math. Sci. 20, Art. ID 49279 (2006) 7. O.Traoré, A. Ouedraogo, Sur un probleme de dynamique des populations. IMHOTEP J. Africain de Mathématiques Pures et Appliqués 4(1), 15–23 (2003) 8. S. Tucker, S. Zimmerman, A nonlinear model of population-dynamics containing an arbitrary number of continuous structure variables. SIAM J. Appl. Math. 48, 549–591 (1998) 9. G.F. Webb, Theory of nonlinear age-dependent population dynamics, in Monographs and Textbooks in Pure and Applied Mathematics, vol. 89 (Marcel Dekker Inc., New York, 1985)

Null Controllability of a System of Degenerate Nonlinear Coupled Equations Derived from Population Dynamics Mamadou Birba and Oumar Traoré

Abstract In this paper, we study the null controllability property of a nonlinear coupled model with degenerate diffusion term. Firstly, we establish a Carleman type inequality for the adjoint system of an intermediate model. From this inequality, we derive our observability inequality. Next, by a fixed point argument, we prove the null controllability result with an internal control acting on a small subset of the domain. Keywords Population dynamics · Null controllability · Coupled system · Carleman inequality · Observability inequality · Degenerate parabolic system · Fixed point 2010 Mathematics Subject Classification 35K65, 93B05, 93B07, 93C05, 93C20

M. Birba () Laboratoire LAMI, Université Joseph Ki-Zerbo, Ouagadougou, Burkina Faso O. Traoré Laboratoire LAMI, Université Joseph Ki-Zerbo, Ouagadougou, Burkina Faso Département de Mathématiques de la Décision, Université Ouaga2, Ouagadougou, Burkina Faso e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_2

35

36

M. Birba and O. Traoré

1 Introduction We consider the following nonlinar model of population dynamics: ⎧ yt + ya − (k(x)yx )x + μ1 (t, a, x)y = h1ω ⎪ ⎪ ⎪ ⎪ ⎪ zt + za − (k(x)zx )x + μ2 (t, a, x)z = y1O ⎪ ⎪ ⎪ ⎪ ⎨ y(t, a, 1) = y(t, a, 0) = 0 z(t, a, 1) = z(t, a, 0) = 0 ⎪ ⎪ ⎪ ⎪ y(0, a, x) = y0 (a, x), z(0, a, x) = z0 (a, x) ⎪ ⎪ ⎪ ⎪ y(t, 0, x) = F∗ (y) ⎪ ⎩ z(t, 0, x) = G∗ (z)

in Q, in Q, on (0, T ) × (0, A), on (0, T ) × (0, A), in QA , in QT , in QT .

(1.1)

Here, T > 0, A > 0, Q = (0, T ) × (0, A) × (0, 1), QA = (0, A) × (0, 1), QT = (0, T ) × (0, 1), ω and O are nonempty open subsets of the spatial domain (0, 1). In the system (1.1), y and z are the distributions of individuals of age a at time t and location x. Thus, A is a maximal life expectancy. The functions β1 and β2 , μ1 and μ2 are respectively the natural fertility and mortality rates of individuals. We have denoted by k the diffusion coefficient, by h the control term and by 1ω the characteristic function of the subdomain ω of (0, 1). Finally, for any functions F, G : R × R × R −→ R, we have denoted by F∗ and G∗ , the operators defined respectively by ⎧    A ⎪ ⎪ β1 (t, a, x)y(t, a, x)da , ⎨ F∗ (y)(t, x) = F t, x,   0 A ⎪ ⎪ ⎩ G∗ (z)(t, x) = G t, x, β2 (t, a, x)z(t, a, x)da . 0

The functions F and G govern the distribution of newborns individuals. In some diffusion models, the variable x represents some gene type. See [8] and the references therein. The question of null controllability of population dynamics models is investigated in many papers. Among, them we can cite [1, 10]. But, in these papers, the diffusion coefficient k is a constant or is not degenerated. In [8], the authors examine a linear system where the diffusion coefficient k degenerates at the left hand of the domain i.e. when k(0) = 0. In our paper, we generalize this study for a nonlinear coupled model. More exactly, we show that for all y0 , z0 ∈ L2 (QA ) and any δ ∈ (0, A), there exists a control h ∈ L2 ((0, T ) × (0, A) × ω) such that the associated solution (y, z) of the system (1.1) verifies z(T , a, x) = 0 a.e. in (δ, A) × (0, 1). The control h depends on the parameter δ.

(1.2)

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

37

The system (1.1) models the interaction between a group of infected individuals and a group of suceptible individuals. Here, the infected are incurable and occurs in the domain O. Moreover, any infected individuals can only gives birth to infected one. So, y is the density of the suceptible individuals whereas z is the density of the infected ones. Thus we act on the sensitive individuals through the control h, in order to eliminate the infected individuals at a time T . Note also that in this model, the susceptible individuals are infected as soon as they are in the domain O. The remainder of this paper is organized as follows: in Sect. 2, we give the functional framework in which the system (1.1) is wellposed and we give the main result of this paper. In Sect. 3, we prove our Carleman inequality for an intermediate adjoint system. With the help of this inequality, we establish the observability inequality and show the null controllability of the intermediate system. Using a generalization of the Laray–Schauder fixed point theorem, we deduce in the last section the proof of the main result.

2 Well-Posedness of the Problem of Population Dynamic In this paper, we make the following hypotheses: ⎧ ⎨ i) The coefficient k is such that k ∈ C([0, 1]) ∩ C 1 ((0, 1]), (H1 ) − k > 0 in (0, 1], k(0) = 0, ⎩ i) there exists λ ∈ [0, 1) such that xk (x) ≤ λk(x), ∀x ∈ [0, 1]. ⎧ i) The function μi , i = 1, 2 μi (t, a, x) ≥ 0 a.e. in Q, ⎪ ⎪ ⎪ ⎪ ⎨ ii) μi (t, a, x) = μi (t, a, x) + μ0i (a) where μi ∈ L∞ (Q), 0 (H2 ) − μ0i ∈ L1loc (0,  aA) and μi ≥ 0 and μi (t, a, x) ≥ 0 a.e. in Q, ⎪ ⎪ ⎪ ⎪ ⎩ iii) lim μ0 (s)ds = +∞. a→A 0

i

⎧ i) The function βi , i = 1, 2 satisfies βi ∈ C 2 (Q), ⎪ ⎪ ⎨ βi (t, a, x) ≥ 0 a.e. in Q, (H3 ) − ⎪ ii) ∃0 < a0 < a1 < A, such that βi (t, a, x) = 0, ⎪ ⎩ in [0, T ] × ((0, a0 ) ∪ (a1 , A)) × .

38

M. Birba and O. Traoré

We also assume that F and G satisfy : ⎧ i) the functions (t, x, s) −→ F (t, x, s), (t, x, s) −→ G(t, x, s) ⎪ ⎪ ⎪ ⎪ ⎪ are positive, continuous and there exist positive constants ⎪ ⎪ ⎪ ⎪ C ⎨ 0 and C1 such that (H4 ) − F (t, x, s) ≤ C0 + C1 |s|, G(t, x, s) ≤ C0 + C1 |s| for all s ∈ R, ⎪ ⎪ ⎪ ii) for all (t, x) ∈ R2 , the functions s −→ F (t, x, s), ⎪ ⎪ ⎪ ⎪ s −→ G(t, x, s) are globally Lipschitz, ⎪ ⎪ ⎩ iii) F (t, x, 0) = G(t, x, 0) = 0, for all (t, x) ∈ R2 . Remark 2.1 The assumptions of (H1 ) are technical and they are used in the proof of the Carlman’s inequality. Like μi and βi (i = 1, 2) the assumptions i), ii) of (H2 ) and i) of (H3 ) are natural. The assumption ii) of (H3 ) is also natural, since it means that older and younger individuals are not fertile. The assumption iii) in (H2 ) is also a standard one, it means that all individual dies before the age A. The positivity of the functions F and G in the hypothesis (H4 ) is natural. Let  = (0, 1). To prove the well-posedness of the system (1.1), we introduce the following Sobolev spaces: Hk1 () = {u ∈ L2 (); u abs. cont. in [0, 1], √ kux ∈ L2 (), u(0) = u(1) = 0}, Hk2 () = {u ∈ Hk1 (); k(x)ux ∈ H 1 ()}, equipped respectively with the following norms  u 2H 1 () = u 2L2 () +  k

√

kux 2L2 () ,

 u 2H 2 () = u 2H 1 () +  (kux )x 2L2 () , k

k

∀u ∈ Hk1 (); ∀u ∈ Hk2 ().

Consider the unbounded operator A : D(A) = Hk2 () → L2 () defined by Au = (k(x)ux )x , u ∈ D(A). We recall from [6, 7] that, the operator is closed, symmetric, self-adjoint, negative and its domain is dense in L2 (). Therefore, it is an infinitesimal generator of a strong continuous semi-group. Using the properties of the operator A, and a fixed point argument on the functions F and G, one can show, the existence of a unique solution of the model (1.1). Moreover, this solution has some additional time, age and spatial regularity. More precisely, the following well-posedness result holds: Theorem 2.1 Under the assumptions (H1 )–(H4 ) and for all h ∈ L2 (q) and y0 , z0 ∈ L2 (QA ), the system (1.1) admits a unique solution (y, z). This solution belongs to E := (C([0, T ], L2 ((0, A) × )) ∩ C([0, A], L2 ((0, T ) × )) ∩

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

39

L2 ((0, T ) × (0, A), Hk1 ()))2 . Moreover, (y, z) satisfies the following inequality sup y(t, ·, ·)2L2 (Q

t ∈[0,T ]

A)

+ sup z(t, ·, ·)2L2 (Q t ∈[0,T ]

+ sup z(·, a, ·)2L2 (Q a∈[0,A]



≤C

T

+ )

 h2 dxdadt +

q

QA

A)

+ sup y(·, a, ·)2L2 (Q a∈[0,A]

T)

 √  √ ( kyx )2 + ( kzx )2 dxdadt Q

 (y02 + z02 )dxda ,

(2.1)

where q = (0, T ) × (0, A) × ω. Let us state now the main result of this paper. Let L2πi (QA ) = {g ∈ L2 (QA ); e

A 0

μi (s)ds

g ∈ L2 (QA )}.

Theorem 2.2 Assume that the assumptions (H1 ) to (H4 ) are satisfied, and that ω∩O = ∅. For any γ > 0 assumed to be small enough, for all (y0 , z0 ) ∈ L2π1 (QA )× L2π2 (QA ) and T > 0, there exists a control h ∈ L2 (q), such that the corresponding solution of the system (1.1) verifies (1.2).

3 Null Controllability of an Intermediate System 3.1 Intermediate System  a For α ≥ 0, we set πα (t, a) = exp −αt + 0 μ0i (s)ds . Let us denote by λ0 a positive constant which will be fixed later. We make the following standard changes: 1 = π −1 (t, 0)β1 ,  y = πλ0 (t, a)y,  z = πλ0 (t, a)z,  h = πλ0 (t, a)h, β λ0 −1 2 = π (t, 0)β2 ,  y (a, x) = π (t, a)y (a, x) and  z (a, x) = πλ0 (t, a)z0 (a, x). β 0 λ0 0 0 λ0 Then, the system (1.1) becomes: ⎧ ⎪  yt +  ya − (k(x) yx )x + (λ0 + μ1 ) y = h1ω in Q, ⎪ ⎪ ⎪ ⎪ zt + za − (k(x) zx )x + (λ0 + μ2 ) z= y 1O in Q, ⎨ (3.1)  y (t, a, σ ) =  z(t, a, σ ) = 0 on , ⎪ ⎪ ⎪ z(0, a, x) =  z0 (a, x) in QA ,  y (0, a, x) =  y0 (a, x),  ⎪ ⎪ ⎩ ∗ ( ∗ (  y (t, 0, x) = F y ),  z(t, 0, x) = G z) in QT , where  = (0, T ) × (0, A) × {0, 1} and   ⎧  A ⎪ −λ t λ t 0 0   ⎪ F∗ ( y )(t, x) = e F t, x, e y (t, a, x)da , β1 (t, a, x) ⎪ ⎪ ⎨ 0   ⎪ ⎪ ⎪ ⎪ ∗ ( ⎩ G z)(t, x) = e−λ0 t G t, x, eλ0 t

0

A

 2 (t, a, x) z(t, a, x)da . β

40

M. Birba and O. Traoré

Note that these changes allow us to have a system with bounded coefficients. Thus, Carleman’s inequality can be applied to the adjoint system of (3.1). Then, the null controllability problem of (1.1) is now reduced to find  h ∈ L2 (q) such that the associated solution ( y , z) of the system (3.1) verifies (1.2). In the following, for a sake of simplicity, we will use the system (3.1) without hats and in addition we will write μi instead of λ0 + μi (i = 1, 2). We consider the following linear coupled system in which a, b ∈ L2 (QT ) ⎧ ⎪ yt + ya − (k(x)yx )x + μ1 y = h1ω ⎪ ⎪ ⎪ ⎪ ⎨ zt + za − (k(x)zx )x + μ2 z = y1O y(t, a, σ ) = z(t, a, σ ) = 0 ⎪ ⎪ ⎪ y(0, a, x) = y0 (a, x), z(0, a, x) = z0 (a, x) ⎪ ⎪ ⎩ y(t, 0, x) = a(t, x), z(t, 0, x) = b(t, x)

in Q, in Q, on , in QA , in QT .

(3.2)

Let us state a Carleman estimate for the adjoint system of (3.2).

3.2 Carleman Inequalities Consider the adjoint system of (3.2) that is: ⎧ ⎪ −wt − wa − (k(x)wx )x + μ1 w = v1O ⎪ ⎪ ⎪ ⎪ ⎨ vt + va + (k(x)vx )x − μ2 v = 0 w(t, a, σ ) = v(t, a, σ ) = 0 ⎪ ⎪ ⎪ w(T , a, x) = wT (a, x), v(T , a, x) = vT (a, x) ⎪ ⎪ ⎩ w(t, A, x) = v(t, A, x) = 0

in Q, in Q, on , in QA , in QT .

(3.3)

x1 + 2x2 2x1 + x2 ,β = be fixed. 3 3 We assume that the coefficient of diffusion k satisfies (H1 ), μi satisfies (H2 ) for i = 1, 2, wT , vT ∈ L2 ((0, A) × ). It is well known (see [9]), that for any open subset ω0 ⊂⊂ ω, there exists a function σ which verifies σ ∈ C 2 ([0, 1]), σ > 0 in , σ = 0 on {0, 1} and σx (x) = 0 in [0, 1] \ ω0 . We also define:

  x r θ (t, a) = (t (T −t1 ))4 a 4 ∀(t, a) ∈ (0, T ) × (0, A), ψ(x) = c1 0 k(r) dr − c2 , Let ω = (x1 , x2 ) and α =

(x) = erσ (x) − e2rσ ∞ , ϕ(t, a, x) = θ (t, a)ψ(x), η(t, a, x) = θ (t, a)erσ (x) and (t, a, x) = θ (t, a)(x), where r ≥ 4σln(2) ∞ ,   2rσ   2rσ   ∞ ∞ −erσ ∞ k(1)(2−λ) e −1 4 e 5 c2 > k(1)(2−λ) . , and c1 ∈ , c2 k(1)(2−λ)−1 3c2

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

41

Lemma 3.1 Under the hypothesis (H1 ), we have:     k(1)(2−λ) e2rσ ∞ −1 4 e2rσ ∞ −erσ ∞ = ∅; , (i) c2 k(1)(2−λ)−1 3c2  x 1 r (ii) 43 ≤ ϕ ≤ , 2 − ϕ ≤ 0 and 0 k(r) dr ≤ k(1)(2−λ) . Proof (i) We have   4 e2rσ ∞ − erσ ∞ k(1)(2 − λ) e2rσ ∞ − 1 k(1)(2 − λ) = + − 3c2 c2 k(1)(2 − λ) − 1 c2 k(1)(2 − λ) − 1

 c2 k(1)(2−λ)−4 erσ ∞ erσ ∞ 4c − 1 2 k(1)(2−λ)−4 . 3c2 [c2 k(1)(2 − λ) − 1] 4 ln(2) 5 σ ∞ and c2 > k(1)(2−λ) , we c2 k(1)(2−λ)−4 1 4c2 k(1)(2−λ)−4 < 16 . c Thus, erσ ∞ 4c22k(1)(2−λ)−4 k(1)(2−λ)−4 > 1 and therefore,

Now, for r ≥

have erσ ∞ ≥ 16 and the previous difference is

positive.

 k(1)(2−λ) e2rσ ∞ −1 , we have ϕ(t, a, x) ≤ (t, a, x) and for c2 k(1)(2−λ)−1  2rσ  rσ  ∞ ∞ 4 e −e , we have 43 (t, a, x) ≤ ϕ(t, a, x) for all (t, a, x) ∈ Q. 3c2

(ii) For c1 ≥

c1 ≤ Moreover, 2 − ϕ = 4 − 3ϕ + 2(ϕ − ) < 0 Now, using the assomptions rλ on the function k, the function r −→ k(r) is nondecreasing on . x r  1 1−λ 1 1 Thus, 0 k(r) dr ≤ k(1) 0 r dr = k(1)(2−λ) .  Theorem 3.1 Consider the following system with g ∈ L2 (Q), ⎧ wt + wa + (k(x)wx )x − μ1 w = g, ⎪ ⎪ ⎨ w(t, a, 1) = w(t, a, 0) = 0, ⎪ w(T , a, x) = wT (a, x), ⎪ ⎩ w(t, A, x) = 0.

(3.4)

Under the hypotheses (H1 ), (H2 ), for all T > 0 and l ∈ N, there exist two constants C > 0 and s0 > 0 such that for all s ≥ s0 , any solution w of the system (3.4), satisfies the following inequality   x2 |w|2 e2sϕ dtdadx s l+1 θ l+1 k(x)|wx |2 e2sϕ dtdadx + s l+3 θ l+3 k(x) Q Q  l θ l e2s |g|2 dtdadx ≤ Cs Q



+ Cs

θ l+3 e2s |w|2 dtdadx,

l+3 q

where q = (0, T ) × (0, A) × ω

(3.5)

42

M. Birba and O. Traoré

For the proof, we need the following results: Proposition 3.1 Under the hypotheses of Theorem 3.1, there exist two constants C > 0, and s0 such that for any s ≥ s0 , and all solution w of the system (3.4), we have the following inequality : 

 θ l+1 k(x)|wx |2 e2sϕ dtdadx + s l+3

s l+1 Q

θ l+3 Q



≤ Cs l

x2 |w|2 e2sϕ dtdadx k(x)

θ l e2sϕ |g|2 dtdadx Q



A T

+ Ck(1)s l+1 0

θ l+1 e2sϕ(t,a;1)|wx (t, a, 1)|2dtda,

(3.6)

0

where ϕ(t, a, x) = θ (t, a)ψ(x). Proposition 3.2 (Classic Carleman Inequality) If k ∈ C 1 ([0, 1]) is a strictly positive function, for any l ∈ N, there exist two constants C > 0 and s0 > 0 such that for all s ≥ s0 and any solution z of the following system: ⎧ zt + za + (k(x)zx )x − μ1 z = g ⎪ ⎪ ⎨ z(t, a, α) = z(t, a, β) = 0 ⎪ z(T , a, x) = zT (a, x) ⎪ ⎩ z(t, A, x) = 0

Q = (0, T ) × (0, A) × (α, β), on (0, T ) × (0, A), Q A = (0, A) × (α, β), Q T = (0, T ) × (α, β),

we have:   (sη)l+1 |zx |2 + (sη)l+3 |z|2 e2s dtdadx Q

    l l 2s 2 l+3 l+3 2s 2 η e |g| dtdadx + s η e |z| dtdadx , ≤C s Q

(3.7)

q

where (t, a, x) = θ (t, a)(x), η(t, a, x) = θ (t, a)erσ (x), r > 0 and μ1 ∈ L∞ (Q ) . Remark 3.1 For the proof of Proposition 3.2, one can use the technique developed in [9], since μ1 ∈ L∞ (Q) and k ∈ C 1 ([0, 1]). 1 , for all (t, a) ∈]0, T [×]0, A[. Then, for all at (T − t) n, m ∈ N∗ , with m ≤ n, one has θ1m (t, a) ≤ (AT 2 )(n−m) θ1n (t, a), for all (t, a) ∈ ]0, T [×]0, A[. Lemma 3.2 Let θ1 (t, a) =

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

43

Proof For all n, m ∈ N∗ , with m ≤ n and for all (t, a) ∈]0, T [×]0, A[, we have n − m ≥ 0 and θ1m (t, a) = (at (T − t))(n−m) ≤ (AT 2 )(n−m) . θ1n (t, a) Thus, θ1m (t, a) ≤ (AT 2 )(n−m) θ1n (t, a), for all (t, a) ∈]0, T [×]0, A[.



Proof (Proposition 3.1) Consider the following system with g1 ∈ L2 (Q), ⎧ wt + wa + (k(x)wx )x = g1 , ⎪ ⎪ ⎨ w(t, a, 1) = w(t, a, 0) = 0, ⎪ w(T , a, x) = wT (a, x), ⎪ ⎩ w(t, A, x) = 0.

(3.8)

We are going to prove the following inequality    x2 |w|2 e2sϕ dtdadx s l+1 θ l+1 k(x)|wx |2 + s l+3 θ l+3 k(x) Q  θ l e2sϕ |g1 |2 dtdadx ≤ Cs l Q



+ Ck(1)s l+1 0

A T

θ l+1 e2sϕ(t,a;1)|wx (t, a, 1)|2 dtda,

(3.9)

0

for all w solution of (3.8). We consider the function u(t, a, x) = (sθ )l/2 esϕ w(t, a, x). Then, we have ⎧ + P u + Ps− u = (sθ )l/2 esϕ g1 , ⎪ ⎪ ⎨ s u(t, a, 1) = u(t, a, 0) = 0, ⎪ u(T , a, x) = u(0, a, x) = 0, ⎪ ⎩ u(t, A, x) = u(t, 0, x) = 0, where Ps+ u = (k(x)ux )x − s(ϕt + ϕa )u + s 2 k(x)ϕx2u, l Ps− u = ut + ua − s(k(x)ϕx )x u − 2sk(x)ϕx ux − θ −1 (θt + θa ) u. 2

(3.10)

44

M. Birba and O. Traoré

Let us prove that there exist two constants m > 0 and m > 0 such that l 2





2

2

(sθ ) e g1  ≥ ms sϕ

θ k(x)|ux | dtdadx + ms Q

− m k(1)s



T

θ3 Q



0

3

A

x2 |u|2 dtdadx k(x)

θ (t, a)|ux (t, a, 1)|2dtda.

(3.11)

0

Indeed, we have 2Ps+ u, Ps− u =



s (θaa + 2θat + θt t ) ψ(x)|u|2 dtdadx Q 2  θ (2k(x) − xk (x))|ux |2 dtdadx + sc1 Q

   x2 l |u|2 dtdadx − c12 s 2 2+ θ (θa + θt ) 2 k(x) Q  l θ −1 (θt + θa ) k(x)|ux |2 dtdadx + 2 Q    x 2 + c13 s 3 θ3 (2k(x) − xk (x))|u|2dtdadx k(x) Q  sl θ −1 (θa + θt )2 ψ(x)|u|2 dtdadx + 2 Q  T A θ (t, a)|ux (t, a, 1)|2 dtda. − c1 sk(1) 0

0

c1 + c1 c2 . k(1)(2 − λ) Using Lemma 3.2, there exists a constant C > 0 such that |θ −1 (θt + θa )| ≤ Cθ , −1 |θ (θt + θa )2 | ≤ Cθ 3 , |θ (θt + θa )| ≤ Cθ 3 and |(θt t + 2θt a + θaa )| ≤ Cθ 3/2 . Using now, the conditions (H1 ) on k, the Hölder’s, Young’s and Hardy–Poincaré’s type inequalities, we deduce that there exist two constants s0 > 0 and m > 0 such that for all s ≥ s0 , one has Note also that |ψ(x)| ≤ C2 =

2Ps+ u, Ps− u

   2 2 3 3 x 2 |u| dtdadx sθ k(x)|ux | + s θ ≥m k(x) Q  A T − 2c1 sk(1) θ |ux (t, a, 1)|2 dtda. 0

l

0

As (sθ ) 2 esϕ g1 2 ≥ 2Ps+ u, Ps− u, then one deduces the inequality (3.11).

(3.12)

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

45

Recalling now that, w = (sθ )−l/2 e−sϕ u, one obtains (sθ )l+1 e2sϕ |wx |2 ≤ 2sθ |ux |2 + 2s 3 θ ϕx2 |u|2 . Multiplying this inequality by m and integrating on Q, we get 

 θ l+1 k(x)e2sϕ |wx |2 dtdadx ≤ 2m

s l+1 Q

sθ k(x)|ux |2 dtdadx Q

+ 2ms

 3

θ3 Q

x2 |u|2 dtdadx. k(x)

(3.13)

We also have (sθ )l+1 e2sϕ |wx (t, a, 1)|2 = sθ |ux (t, a, 1)|2, since u(t, a, 1) = 0.

(3.14)

The inequalities (3.11), (3.13) and the equality (3.14), give the estimation (3.6) for any solution of the system (3.8). Now, we apply (3.9) with the function g1 = g + μ1 w. Hence, there are two positive constants C and s0 such that, for all s ≥ s0 , the following inequality holds    2 l+1 l+1 2 l+3 l+3 x 2 |w| e2sϕ dtdadx s θ k(x)|wx | + s θ k(x) Q  l θ l e2sϕ |g1 |2 dtdadx ≤ Cs Q

+ Ck(1)s



A T

l+1 0

θ l+1 e2sϕ(t,a;1)|wx (t, a, 1)|2dtda.

(3.15)

0

On the other hand, we have    θ l e2sϕ |g1 |2 dtdadx ≤ 2 θ l e2sϕ |g|2 dtdadx + 2μ1 ∞ θ l e2sϕ |w|2 dtdadx. Q

Q

Q

Now, applying Hardy–Poincaré inequality to the function esϕ w and the fact that x2 x −→ k(x) is increasing, we obtain  l 2sϕ

θe Q

 esϕ |w| 2 θ k(x) dtdadx. x Q   2 C1 μ1 ∞ ≤2 θ l k(x) esϕ w x dtdadx. k(1) Q

μ ∞ |w| dtdadx ≤ 2 1 k(1) 2





l

46

M. Birba and O. Traoré

≤2 ≤2

C1 μ1 ∞ k(1) C1 μ1 ∞ k(1)

 

Q

Q

θ l k(x)e2sϕ (s 2 ϕx2 w2 + wx2 )dtdadx. θ l e2sϕ (s 2 θ 2 c12

x2 2 w + k(x)wx2 )dtdadx, k(x) (3.16)

where C1 > 0. Thus,   sl θ l e2sϕ |g1 |2 dtdadx ≤ 2s l θ l e2sϕ |g|2 dtdadx Q

Q

   2 l+2 x 2 l 2 w + (sθ ) k(x)wx e2sϕ dtdadx, + C2 (sθ ) k(x) Q

  C1 c12 μ1 ∞ 1 ∞ with C2 = max 2 C1 μ , 2 . k(1) k(1) For s ≥ s0 + C2 (AT 2 )4 , (3.15) becomes  s



l+1

l+1

θ Q l

k(x)|wx | e

2 2sϕ

dtdadx + s

l+3

θ l+3 Q



≤Cs

l 2sϕ

θ e





|g| dtdadx + C k(1)s 2

A T

l+1 0

Q

x2 |w|2 e2sϕ dtdadx k(x) θ l+1 e2sϕ(t,a;1)dtda,

0

with C > 0.



We also need the following lemma in the proof of Theorem 3.1: Lemma 3.3 ([2, 8] Caccioppoli’s Type Inequality) Let ω ⊂⊂ ω. Then there exists a positive constant C such that: 

 |wx | e

2 2sϕ

q





dtdadx ≤ C

|g| e

2 2sϕ

q

dtdadx + Cs

θ |w| e

3

3

2 2sϕ

dtdadx ,

q

where q = (0, T ) × (0, A) × ω and q = (0, T ) × (0, A) × ω . Now, using Propositions 3.1, 3.2, and Lemma 3.3, we are going to prove Theorem 3.1. Proof (Theorem 3.1) Let w be the solution of the system (3.4). Set W = ρ(x)w, Z = (1 − ρ(x))w, where ρ ∈ C ∞ (R) such that 0 ≤ ρ(x) ≤ 1, ∀x ∈ , ρ(x) = 1, ∀x ∈ (0, α), and ρ(x) = 0, ∀x ∈ (β, 1).

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

47

Thus, using the definition of the function ρ, W solves: ⎧ Wt + Wa + (k(x)Wx )x − μ1 W = G, ⎪ ⎪ ⎨ W (t, a, 1) = W (t, a, 0) = 0, ⎪ W (T , a, x) = WT (a, x), ⎪ ⎩ W (t, A, x) = 0

(3.17)

with G = ρ(x)g + 2ρ(x)x k(x)wx + (ρ(x)x k(x))x w, and Z solves ⎧ Zt + Za + (k(x)Zx )x − μ1 Z = G , ⎪ ⎪ ⎨ Z(t, a, 1) = Z(t, a, α) = 0, ⎪ Z(T , a, x) = ZT (a, x), ⎪ ⎩ Z(t, A, x) = 0

(3.18)

with G = (1 − ρ(x))g + 2(1 − ρ(x))x k(x)wx + [(1 − ρ(x))x k(x)]x w. Applying Proposition 3.1 to the system (3.17), we get    x2 |W |2 e2sϕ dtdadx s l+1 θ l+1 k(x)|Wx |2 + s l+3 θ l+3 k(x) Q  ≤ Cs l θ l e2sϕ |G|2 dtdadx Q



+ k(1)s l+1 0

A



T

θ l+1 e2sϕ(t,a;1)|Wx (t, a, 1)|2dtda.

(3.19)

0

By the definition of the function ρ, we have Wx (t, a, 1) = 0. Moreover, by Caccioppoli’s type inequality ((α, β) ⊂⊂ ω) and taking into account that W is supported on [0, T ] × [0, A] × [0, α], (3.19), becomes    2 l+1 l+1 2 l+3 l+3 x 2 |W | e2sϕ dtdadx s θ k(x)|Wx | + s θ k(x) Q   l l 2sϕ 2 l+3 θ e |g| dtdadx + C1 s θ l+3 e2sϕ w2 dtdadx. ≤ C1 s Q

(3.20)

q

Now, as k is a strictly positive function on (α, 1), we can use Proposition 3.2, with system (3.18). We get, 

 (sη)l+1 |Zx |2 + (sη)l+3 |Z|2 e2s dtdadx Qα

  ≤ C3 s l

ηl e2s |G |2 dtdadx + s l+3 Qα

 q

 ηl+3 e2s |Z|2 dtdadx ,

(3.21)

48

M. Birba and O. Traoré

where ω ⊂⊂ (α, 1). By the definition of the function ρ and the Caccioppoli’s type inequality (ω ⊂⊂ ω) and taking into account that Z is supported on [0, T ] × [0, A] × [α, 1], (3.21), becomes   (sη)l+1 |Zx |2 + (sη)l+3 |Z|2 e2s dtdadx Q

    l l 2s 2 l+3 l+3 2s 2 ≤ C4 s η e |g| dtdadx + s η e |w| dtdadx . q

Q  k(1)(2−λ) e2rσ ∞ −1 , one c2 k(1)(2−λ)−1 x 2 2sϕ 2s 2s ≤ C5 e and e2sϕ C5 e , k(x) e

For c1 ≥

(3.22)

has the following inequalities: k(x)e2sϕ ≤ ≤ C5 e2s .

Thus, (3.22) becomes    x2 |Z|2 e2sϕ dtdadx (sθ )l+1 k(x)|Zx |2 + (sθ )l+3 k(x) Q     ηl e2s |g|2 dtdadx + s l+3 ηl+3 e2s |w|2 dtdadx . ≤ C4 C5 s l q

Q

(3.23)

Using the inequalities (3.20), (3.23), and Lemma 3.1, we obtain    2 l+1 l+1 2 2 l+3 l+3 x 2 2 (|W | + |Z| ) e2sϕ dtdadx s θ k(x)(|Wx | + |Zx | ) + s θ k(x) Q     θ l e2s |g|2 + s l+3 θ l+3 e2s |w|2 dtdadx, ≤ C6 s l (3.24) Q

q

 for all s ≥ (AT 2 )4 and C6 = (1 + C5 )max C1 , C4 C5 e(l+3)σ ∞ . Notice that w = W + Z. Thus, we have |w|2 ≤ 2(|W |2 + |Z|2 ) and |wx |2 ≤ 2(|Wx |2 + |Zx |2 ). Consequently, (3.24) becomes 

 s l+1 θ l+1 k(x)|wx |2 e2sϕ dtdadx + s l+3 Q

θ l+3 Q



≤ 2C6 s l

θ l e2s |g|2 dtdadx Q

+

2C6 s

x2 |w|2 e2sϕ dtdadx k(x)

 θ l+3 e2s |w|2 dtdadx,

l+3 q

for all s ≥ (AT 2 )4 . This completes the proof of Theorem 3.1 for the any solution of (3.4).



Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

49

Let us give now, the following result useful for the proof of our observability inequality. Theorem 3.2 We assume that (H1 ), (H2 ) hold true and ω ∩ O = ∅. For any subset ω such that ω ⊂⊂ ω ∩ O, there exist two positive constants C and s0 , such that for any solution (w, v) of the system (3.2) and for every s ≥ s0 , one has :    2 2 2 3 3 x 2 4 4 2 6 6 x 2 |v| + s θ k(x)|wx | + s θ |w| e2sϕ dtdadx sθ k(x)|vx | + s θ k(x) k(x) Q   A T ≤C |w|2 dtdadx. (3.25) ω

0

0

The following known lemma is useful for the proof of Theorem 3.2. Lemma 3.4 ([5]) Let g ∈ C() such that −g(x) ≥ m0 > 0, ∀x ∈  and let m, n ∈ N. Let (t, a, x) = θ (t, a)g(x) (here g(x) = (x)), for (t, a, x) ∈ Q and m0 = min(−g). Then 

i) for any n ≥ m, (sθ (t, a))m ≤ C(sθ (t, a))n , ∀s ≥ CT 8 A4 ,   7 7 7A4 T 8 7 7 2s ii) s θ (t, a) e ≤ for any s > 9 and (t, a, x) ∈ Q. em0 2 m0

(3.26) (3.27)

Proof We have for all s ≥ 1, s 7 θ (t, a)7 e2s ≤ s 7 θ (t, a)7 e−2sm0 θ . 4T 8 ≥ Let f be the function s −→ f (s) = s 7 θ (t, a)7 e−2sm0θ . For all s ≥ 7A 29 m0 

7

7 7 7 ≤ 2m0 θ , the function f is decreasing. So, we have f (s) ≤ f 2m0 θ = 2em0 7

7 .  em0 Proof (Proof of Theorem 3.2) Let O ⊂⊂ ω ∩ O. Applying the Theorem 3.1 and taking l = 3 for the first equation of the system (3.3) and l = 0 for the second equation of the same system; we find respectively the following inequalities :  

 x2 |w|2 e2s dtdadx k(x) Q     T θ 3 e2s |v1O |2 dtdadx + s 6 θ 6 e2s |w|2 dtdadx , ≤ C s3 s 4 θ 4 k(x)|wx |2 + s 6 θ 6

Q

0

O

(3.28)

50

M. Birba and O. Traoré

for all s ≥ s1 ; and   sθ k(x)|vx |2 + s 3 θ 3 Q



≤ Cs

T

3 0

 x2 |v|2 e2s dtdadx k(x)

 θ 3 e2s |v|2 dtdadx,

O

for all s ≥ s2 .

(3.29)

Adding both inequalities and using the fact that 

 O

0

A T

  θ 3 e2s |v|2 dtdadx ≤

A T

O 0

0

θ 3 e2s |v|2 dtdadx,

0

one obtains for every s ≥ s1 + s2 ,    x2 x2 |v|2 + s 4 θ 4 k(x)|wx |2 + s 6 θ 6 |w|2 e2s dtdadx sθ k(x)|vx |2 + s 3 θ 3 k(x) k(x) Q  T  T θ 3 e2s |v|2 dtdadx + Cs 6 θ 6 e2s |w|2 dtdadx. (3.30) ≤ Cs 3 0

O

0

O

Now, let O ⊂⊂ ω ⊂⊂ ω ∩ O, we define the function ζ ∈ C ∞ () as follows ⎧ ⎨ 0 ≤ ζ(x) ≤ 1 ζ(x) = 1 if ⎩ ζ(x) = 0 if

∀x ∈ , x ∈ O , x ∈  \ ω .

In addition it is assumed that ζx ∈ L∞ () ζ 1/2

and

ζxx ∈ L∞ (). ζ 1/2

(3.31)

Indeed, one can just take ζ = ζ04 where ζ0 ∈ C ∞ (), to satisfy the two previous conditions on ζ . We set χ = s 3 θ 3 e2s and multiply the first equation of the system (3.3) by ζ χv and integrate the result on Q. We obtain 

 v 1O ζ χdtdadx = 2

 k(x)vx wx ζ χdtdadx −

2

Q

Q

Q

 +

Q

(μ1 + μ2 )vwζ χdtdadx +

vw (k(x)(ζ χ)x )x dtdadx  vwζ(χt + χa )dtdadx. Q

There exists a positive constant C1 > 0 such that |χt + χa | ≤ C1 s 4 θ 5 e2s

and

|[k(ζ χ)x ]x | ≤ C1 s 5 θ 5 ζ 1/2e2s

(3.32)

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

51

and, using Hölder’s and Young’s inequalities, one obtains  I1 = 2s 3

k(x)θ vx wx e2s ζ dtdadx Q



k(x)θ |vx |2 e2sϕ dtdadx

≤ 2δ0 s Q

+

1 5 s 2δ0

 k(x)θ 5|wx |2 e2s(2 −ϕ)ζ 2 dtdadx, δ0 > 0.

(3.33)

Q

Using Hölder’s and Young’s inequalities and (3.32), we get  I2 = Q

[(μ1 + μ2 )χ + (χt + χa ) − [k(x)(ζ χ)x ]x ]ζ vwdtdadx 

≤ C2

(sθ )5 e2s ζ vwdtdadx Q

 √    3/2 sϕ x 7/2 k s(2 −ϕ) ζ we ≤ dtdadx (sθ ) e √ v C2 (sθ ) x k Q  x2 |v|2 e2sϕ dtdadx θ3 ≤ 2δ1s 3 k(x) Q  C22 7 k(x) + s θ 7 2 |w|2 e2s(2 −ϕ)ζ 2 dtdadx, 8δ1 x Q

(3.34)

where δ1 > 0. Now, we look to bound the term  1 5 s k(x)θ 5 |wx |2 e2s(2 −ϕ)ζ 2 dtdadx. 2δ0 Q To do this, we are going to multiply the first equation of the system (3.3) by s 5 θ 5 e2s(2 −ϕ)ζ 2 w, and integrate over Q. In order to simplify the calculation, we set χ = s 5 θ 5 e2s(2 −ϕ)ζ 2 . Then, 

 Q

v1O wχ = − =

1 2

 wwt χ −



Q

Q

 wwa χ −

Q

w2 (χ t + χ a ) +

 Q

 (wχ ) (k(x)wx )x +

Q

k(x)χ wx2 −

1 2

 Q

aχw2 Q

w2 (k(χ )x )x +

 Q

μ1 χ w2 .

52

M. Birba and O. Traoré

Therefore, we get  k(x)χζ |wx |2 dtdadx ≤ Q

   1 v1O wχ + |w|2 |χ t + χ a | dtdadx 2 Q     1 2  |w| (k(χ)x )x  + |μ1 χ||w|2 dtdadx. + Q 2

Using Hölder–Young’s inequalities and observing that there exists a positive constant K > 0 such that |χ t + χ a | ≤ Ks 7 θ 7 e2s(2 −ϕ)ζ 2

and

|(kχ x )x | ≤ Ks 7 θ 7 ζ 1/2 e2s(2 −ϕ),

we get 1 J1 = 2



 Q

  |χ t + χ a | + (k(χ)x )x  + 2|μ1 χ| |w|2 dtdadx

 θ 7 e2s(2 −ϕ)|w|2 ζ 1/2dtdadx

≤ K1 s 7

(3.35)

Q

and  J2 = s

5 Q

θ 5 e2s(2 −ϕ)ζ 2 |wv1O |dtdadx



≤ γs

3

θ3 Q

1 7 s + 4γ



x 2 2sϕ 2 e |v| dtdadx k(x) θ7

Q

k(x) 2s(4 −3ϕ) 4 2 e ζ |w| dtdadx, x2

(3.36)

where γ > 0. From (3.35) and (3.36), we infer that  s



5

5

θ k(x)ζ e Q

+

2 2s(2 −ϕ)

1 7 s 4γ

3

θ3 Q

 θ7 

|wx | dtdadx ≤ γ s 2

Q

k(x) 2s(4 −3ϕ) 4 2 e ζ |w| dtdadx x2

θ 7 e2s(2 −ϕ)|w|2 ζ 1/2dtdadx.

+ K1 s 7 Q

x 2 2sϕ 2 e |v| dtdadx k(x)

(3.37)

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

53

Finally, combining inequalities (3.33), (3.34), and (3.37), one gets :  s

θ |v| e

3

3

2 2s

Q

   x2 γ |v|2 e2sϕ dtdadx ζ dtdadx ≤ 2δ1 + s3θ 3 2δ k(x) 0 Q  2 C k(x) + 2 s7 θ 7 2 e2s(2 −ϕ)|w|2 ζ 2 dtdadx 8δ1 x Q  k(x) K1 7 s θ 7 2 e2s(2 −ϕ)|w|2 ζ 1/2dtdadx + 2δ0 x Q  k(x) 1 7 s θ 7 2 e2s(4 −3ϕ)|w|2 ζ 4 dtdadx + 8δ0 γ x Q  2δ0 sθ |vx |2 e2sϕ dtdadx (3.38) + Q

We now set δ1 = δ20 , γ = 2δ02 , using the fact that suppζ ⊂ O ⊂⊂ ω, the function k(x) is bounded on ω and (3.38), we deduce that x2  s



3

3

2 2s

θ |v| e

ζ dtdadx ≤ 2δ0 K2 s

3

Q

 + Kδ0 s 7

Taking now δ0 =

1 4K2 ,

θ 3 |v|2 e2s dtdadx

θ 7 e2s(2 −ϕ)|w|2 dtdadx 

+ Kδ0 s 7

q

q

θ 7 e2s(4 −3ϕ)|w|2 ζ 4 dtdadx. Q

we find





s3 Q

θ 3 |v|2 e2s ζ dtdadx ≤ 2Kδ0 s 7 + 2Kδ0 s

θ 7 e2s(2 −ϕ)|w|2 dtdadx q



7

θ 7 e2s(4 −3ϕ)|w|2 ζ 4 dtdadx.

(3.39)

Q

Taking s > CT 8 A4 , using Lemmas 3.1 and 3.4, and multiplying (3.39) by C and using inequalities (3.30), we obtain (3.25). 

54

M. Birba and O. Traoré

3.3 An Observability Inequality Result This paragraph is devoted to the proof of the observability inequality of the adjoint system (3.3) of (3.2). This inequality is obtained by using the Carleman estimate (3.25) and Hardy–Poincaré’s inequality. Proposition 3.3 We assume that (H1 ), (H2 ) hold and that there exists a real γ > 0 such that wT (a, x) = vT (a, x) = 0 a.e. in (0, γ ) × .

(3.40)

Then, there exists a positive constant Cγ > 0 such that the following inequality holds:     w2 (0, a, x) + v 2 (0, a, x) dadx + w2 (t, 0, x) + v 2 (t, 0, x) dtdx Q A

Q T



≤ Cγ

w2 (t, a, x)dtdadx

(3.41)

q

for all solution (w, v) of the adjoint system (3.3), where Q A = (0, A − γ /2) × , Q T = (0, T − γ /2) ×  and q = (0, T ) × (0, A) × ω. For the proof, we need to show a crucial technical result. For this, consider the following subsets (see [10]) and (see Fig. 1 below) N1 = {(t, a) ∈ (0, T ) × (0, A); t ≥ a + T − γ } , N2 = {(t, a) ∈ (0, T ) × (0, A); t ≤ a + γ − A},  T − γ2 a+T − D1 = (t, a) ∈ (0, T ) × (0, A); t ≥ − A − γ2  A − γ2 t +A− D2 = (t, a) ∈ (0, T ) × (0, A); a ≥ − T − γ2

 γ , 2

(3.42)

 γ (γ − 2A) , 2(2T − γ )

/ N1 ∪ N2 }. D3 = (0, T ) × (0, A) − (D1 ∪ D2 ), D4 = {(t, a) ∈ D3 ; (t, a) ∈ Lemma 3.5 Suppose that (3.40) holds. Then, any solution of the system (3.3) verifies w(t, a, x) = v(t, a, x) = 0 a.e. in (N1 ∪ N2 ) × .

(3.43)

Proof Let (t0 , a0 ) ∈ N1 . Then, we have t0 = a0 + T − γ + s, with 0 ≤ s < γ . Therefore a0 < γ − s. Let Ss = {(t0 + r, a0 + r); r ∈ (0, γ − s − a0 )} be a characteristic line in N1 . Now, setting w(r, x) = w(t0 + r, a0 + r, x),  μ1 (r, x) = μ1 (t0 +r, a0 +r, x), v(r, x) = v(t0 +r, a0 +r, x) and  μ2 (r, x) = μ2 (t0 +r, a0 +r, x),

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

55

t

Fig. 1 Decomposition of the region (0, T ) × (0, A)

T N1 D2 T− δ D4

δ D1 N2 A− δ

0

A

a

where (w, v) is the solution of the system (3.3). Then, (w, v) solves ⎧ ⎪ −wr − (k(x)wx )x +  μ1 (r, x)w = v1O ⎪ ⎪ ⎪ ⎪ v + (k(x)v ) − μ  (r, x)v = 0 r x x 2 ⎪ ⎪ ⎨ w(r, 1) = w(r, 0) = 0 ⎪ v(r, 1) = v(r, 0) = 0 ⎪ ⎪ ⎪ ⎪ w(γ − a0 − s, x) = wT (γ − s, x) ⎪ ⎪ ⎩ v(γ − s − a0 , x) = vT (γ − s, x)

in (0, γ − s − a0 ) × , in (0, γ − s − a0 ) × , on (0, γ − s − a0 ), on (0, γ − s − a0 ), in , in .

(3.44)

Then, from (3.40) for almost all s ∈ (0, γ ), standard results of the heat equation imply that v = 0. Thus, for a.e. s ∈ (0, γ ), v = 0 on Ss . Therefore, v = 0 in N1 × . Now, as for all λ ∈ (0, r), with r ∈ (0, γ − s − a0 ), v(λ, .) = 0, we deduce by the same manner that w = 0. Then, w = 0 in N1 × . The same argument and the fact that w(t, A, x) = v(t, A, x) = 0 in (0, T ) ×  allow us to prove that w = v = 0 in N2 × .  Proof (Proposition 3.3) Consider a smooth cut-off function η ∈ C0∞ (R2 , [0, 1]) stated as follows η(t, a) = 1, (t, a) ∈ D1 , η(t, a) = 0, (t, a) ∈ D2 and η(t, a) > 0, (t, a) ∈ D3 . Setting w  = ηw and  v = ηv, it follows that ( w, v ) solves, ⎧ ⎪ − wt − w a − (k(x) wx )x + μ1 w = v 1O − (ηt + ηa )w ⎪ ⎪ ⎪ ⎪ va − (k(x) vx )x + μ2 v = −(ηt + ηa )v vt −  ⎨ − w (t, a, σ ) =  v (t, a, σ ) = 0 ⎪ ⎪ ⎪ w (T , a, x) = 0,  v (T , a, x) = 0 ⎪ ⎪ ⎩ w (t, A, x) =  v (t, A, x) = 0

in Q, in Q, on , in QA , in QT .

(3.45)

56

M. Birba and O. Traoré

Multiplying the first and the second equation of the system (3.45) by w  and by  v respectively and integrating over Q, using the definition of η and Lemma (3.5), we obtain    w2 (0, a, x)dadx + w2 (t, 0, x)dtdx ≤ −2  vw 1O dtdadx Q A

Q T

−2



Q



η(ηt + ηa )w2 dtdadx

Q

and 





2

Q A

v (0, a, x)dadx +

2

Q T

η(ηt + ηa )v 2 dtdadx.

v (t, 0, x)dtdx ≤ −2 Q

Adding the two last inequalities, we get  Q A



 2 2 w (0, a, x) + v (0, a, x) dadx + 

Q T

 w2 (t, 0, x) + v 2 (t, 0, x) dtdx



≤ −2

η(ηt + ηa )(w + v )dtdadx − 2 2

2

Q

Q



 vw 1O dtdadx

(w2 + v 2 )dtdadx.

≤ 4Mγ

(3.46)

×D4

Thanks to Hardy–Poincaré’s inequality and the fact that θ is bounded in D4 , we infer that for all s ≥ 1,     2 2 w (0, a, x) + v (0, a, x) dadx + w2 (t, 0, x) + v 2 (t, 0, x) dtdx Q A

Q T



≤ Cγ ×D4

k(x)(s 4 θ 4 wx2 + sθ vx2 )e2sϕ dtdadx,

(3.47)

4CMγ . Taking s large enough and thanks to the Carleman inequality k(1) stated in Theorem (3.2), we obtain the observability inequality of the adjoint system (3.3).  where Cγ =

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

57

3.4 Null Controllability of the Intermediate System This paragraph is devoted to the study of the null controllability property of the system (3.2). Thus, let ε > 0 and consider the following cost function defined on L2 (q). 1 Jε (h) = 2ε

1 A

 0

γ

1 z (T , a, x)dadx + 2



2

h2 (t, a, x)dtdadx.

(3.48)

q

One can prove that Jε is continuous, convex and coercive. Then, it admits at least one minimizer hε . Furthermore, the maximum principle (see [4]) gives hε (t, a, x) = −wε (t, a, x)1ω (x) in Q

(3.49)

with (wε , vε ), the solution to the following system ⎧ −(wε )t − (wε )a − (k(x)(wε )x )x + μ1 wε = vε 1O ⎪ ⎪ ⎪ ⎪ ⎪ (vε )t + (vε )a + (k(x)(vε )x )x − μ2 vε = 0 ⎪ ⎪ ⎪ ⎨ wε = vε = 0 wε (T , a, x) = 0, ⎪ ⎪ ⎪ 1 ⎪ ⎪ vε (T , a, x) = − zε (T , a, x)1(γ ,A)(a) ⎪ ⎪ ε ⎪ ⎩ wε (t, A, x) = vε (t, A, x) = 0

in Q, in Q, on , in QA ,

(3.50)

in QA , in QT ,

with zε coupled to yε and associated to the control hε solves the system (3.2). Multiplying the first and the second equation of the system (3.50) by yε and by zε respectively, integrating over Q, using (3.49) and Young’s inequality, we obtain that 1 ε

   γ

A

 zε2 (T , a, x)dadx



+ q

h2ε (t, a, x)dtdadx 

+ 2Cγ

+

1 2Cγ

QT

b2 (t, x)dtdx QT

1 + 2Cγ

a 2 (t, x)dtdx

≤ 2Cγ

  

QA

QT

+ 2Cγ QA



wε2 (0, a, x) + vε2 (0, a, x) dadx 

wε2 (t, 0, x) + vε2 (t, 0, x) dtdx

 y02 (a, x) + z02 (a, x) dadx,

58

M. Birba and O. Traoré

with Cγ the constant given in Proposition 3.3. Hence, by the observability inequality (3.41) and (3.49), we find that 1 ε

 

A

 γ

zε2 (T , a, x)dadx

3 + 2



 q

h2ε (t, a, x)dtdadx 

≤ 2Cγ

a 2 (t, x)dtdx QT

b2(t, x)dtdx

+ 2Cγ 

QT

 y02 (a, x) + z02 (a, x) dadx,

+ 2Cγ QA

and this yields  

A

 γ

  zε2 (T , a, x)dadx ≤ 2Cγ ε

T

(a 2 (t, x) + b 2 (t, x))dtdx

 0

  + 2Cγ ε

A

 y02 (a, x) + z02 (a, x) dadx,

 0

(3.51)

and  q

h2ε (t, a, x)dtdadx

4 ≤ Cγ 3 4 + Cγ 3

 

T

(a 2 (t, x) + b 2 (t, x))dtdx

 0

 

 0

A

 y02 (a, x) + z02 (a, x) dadx.

(3.52)

Then, we can extract two subsequences of (yε , zε ) and (hε ) still denoted (yε , zε )  2 and (hε ) that converge weakly towards (y, z) and h in L2 ((0, T )×(0, A); Hk1()) and in L2 (q) respectively. Furthermore, (y, z) is the unique solution of the system (3.2) that satisfies (1.2). We have then shown the following proposition. Proposition 3.4 For any γ > 0 assumed to be small enough, for all y0 , z0 ∈ L2 (QA ), there exists a control h ∈ L2 (q) such that the associated solution of the system (3.2) verifies (1.2).

4 Proof of the Main Result We will prove the null controllability property of the system (1.1). For any (R, S) ∈ M = L2 (QT ) × L2 (QT ), let 

a = e−λ0 t F (t, x, eλ0 t R), b = e−λ0 t G(t, x, eλ0 t S).

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

59

Consider now, the following system ⎧ ⎪ yt + ya − (k(x)yx )x + μ1 y = h1ω ⎪ ⎪ ⎪ ⎪ ⎨ zt + za − (k(x)zx )x + μ2 z = y1O y(t, a, σ ) = z(t, a, σ ) = 0 ⎪ ⎪ ⎪ y(0, a, x) = y0 (a, x), z(0, a, x) = z0 (a, x) ⎪ ⎪ ⎩ y(t, 0, x) = a(t, x), z(t, 0, x) = b(t, x)

in Q, in Q, on , in QA , in QT .

(4.1)

Using (H4 ), we have a, b ∈ L2 (QT ). Thus, from Proposition 3.4 that there exists a control h that verifies (3.52) so that the corresponding solution of (4.1) verifies (1.2). So, for all (R, S) ∈ M = L2 (QT ) × L2 (QT ) one defines the set : Aγ (R, S) = {h ∈ L2 (q), (y, z) solution of (4.1) and z satisfies (1.2) and h satisfies (3.52)}. Thus, we can introduce the multi-valued mapping : γ : (R, S) ∈ M −→ γ (R, S) ∈ 2M , defined by 



A

γ (R, S) =

β1 yda, 0



A

β2 zda

, (yh , zh ) is solution of (4.1)

0

with h ∈ Aγ (R, S) . We will prove that this mapping admits at least one fixed point (R, S) by using a generalization of the Leray-Schauder fixed point theorem stated in [3]. So, we introduce the set ! Nγ = (R, S) ∈ M : (∃)ρ ∈]0, 1[, (R, S) ∈ ργ (R, S).

(4.2)

The existence of a fixed point of the multi-valued mapping γ will be an immediate consequence of the following proposition. Proposition 4.1 (i) (ii) (iii) (iv)

Nγ is bounded in M. For all (R, S) ∈ M, γ (R, S) is a closed and convex set of M. γ : M −→ 2M , is a compact multivalued mapping. γ (R, S) is upper semi-continuous on M.

60

M. Birba and O. Traoré

Proof We recall that M = L2 (QT ) × L2 (QT ). The proof will be done in four steps. (i) We are going to prove that Nγ is a bounded subset of M. 1 Let (R, S) ∈ Nγ then, there exists ρ ∈  such that (R, S) ∈ (R, S). ρ As a consequence, there exists a pair (y, z) ∈ X2 , where X2 = L2 ((0, T ) × (0, A); Hk1())) × L2 ((0, T ) × (0, A); Hk1())) associated to the control h ∈  A  A 2 L (q) such that R = ρ β1 yda, S = ρ β2 zda, where (y, z) is the 0

0

solution of the system (4.1) associated to the control h which satisfies (3.52). Then, R2L2 (Q

T)

≤ C 2 (β1 , A)y2L2 (Q) and S2L2 (Q

T)

≤ C 2 (β2 , A)z2L2 (Q) ,

and adding these last inequalities, we obtain R2L2 (Q

T)

+ S2L2 (Q

T)

≤ C(β1 , β1 , A)(y2L2 (Q) + z2L2 (Q) ),

(4.3)

with C(β1 , β2 , A) = C 2 (β1 , A) + C 2 (β2 , A). From (3.52) and using (H4 ), we have

 4 4 Cγ C(C0 , , T ) + Cγ C12 R2L2 (Q ) + S2L2 (Q ) T T 3 3

 4 + Cγ y0 2L2 (Q ) + z0 2L2 (Q ) . (4.4) A A 3

h2L2 (q) ≤

Now, multiplying the first and the second equation of the system (4.1) by y and by z respectively, integrating over Q and using the Young’s inequality and (H4 ), one obtains   λ0 1 2 k(x)yx dtdadx + y 2 dtdadx ≤ (C(C0 , , T )+  R2L2 (Q ) ) T 2 Q 2 Q  1 1 + h2 dtdadx + y0 2L2 (Q ) (4.5) A 2λ0 q 2 and  Q

k(x)zx2 dtdadx

λ0 + 2 +



1 2λ0

z2 dtdadx ≤ Q



1 (C(C0 , , T ) + S2L2 (Q ) ) T 2

1 y 2 dtdadx + z0 2L2 (Q ) . A 2 q

(4.6)

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

61

Adding (4.5) and (4.6), using (4.3), (4.4) and taking λ0 ≥ 2, we obtain    1 λ0 Cγ + 1 (y 2 + z2 )dtdadx ≤ L0 2 Q 3   1 Cγ + (y0 2L2 (Q ) + z0 2L2 (Q ) ) + A A 2 3 " " ##  1 1 Cγ C12 + +C + (y 2 + z2 )dtdadx 4 2 3 Q



k(x)(yx2 + zx2 )dtdadx +

Q

where"C = C(β1 , #β2 , A) and L0 = C(C0 , , T ). Thus, we find, setting 1 Cγ C12 L1 = + 2 3  

 1 λ0 − − CL1 (y 2 + z2 )dtdadx ≤ + 2 4 Q     1 1 Cγ Cγ + 1 + + (y0 2L2 (Q ) + z0 2L2 (Q ) ) L0 A A 3 2 3

 Q

k(x)(yx2

+ zx2 )dtdadx

  1 Now, taking λ0 > max 2, + 2CL1 , we obtain 2 

 Q

k(x)(yx2 + zx2 )dtdadx + + $

where L2 = max L0

1 3 Cγ

 + 1 + 12 ,

Therefore, y2X + z2X ≤

(y 2 + z2 )dtdadx ≤ Q

L2 L3

L2 (y0 2L2 (Q ) + z0 2L2 (Q ) ) A A L3 Cγ 3

%

and L3 =

1 . 1 C1 − − CL1 min 1, 2 4 

 L2 1 + (y0 2L2 (Q ) + z0 2L2 (Q ) ) . A A L3 (4.7)

Using (4.7) and the fact that X ⊆ L2 (Q) we infer that there exist two positive constants C2 > 0 and C3 > 0 such that (4.3) and (4.4) give : R2L2 (Q

T)

+ S2L2 (Q

T)

≤ C2 1 + (y0 2L2 (Q

A)

+ z0 2L2 (Q ) ) A

 (4.8)

62

M. Birba and O. Traoré

and

h2L2 (Q) ≤ C3 1 + (y0 2L2 (Q where C2 =

L2 C L3

and C3 =

A

+ z0 2L2 (Q ) ) )

 (4.9)

A

 4 Cγ C12 C2 + max {1, C} . Thus, we deduce that 3

Nγ is a bounded set of M. (ii) We prove now that for all (R, S) ∈ M, γ (R, S) is a nonempty closed and convex subset of M. First, for all (R, S) ∈ M, γ (R, S) is nonempty because the system (4.1) admits a solution. Moreover, the corresponding control h satisfies (3.52). Since the mapping (R, S) −→ (y, z) is affine, thanks to its definition, it follows that the subset γ (R, S) is convex. Let ((ρn , ηn ))n ⊂ γ (R, S) be such that, (ρn , ηn ) −→ (ρ, η) in M. We have to prove that (ρ, η) ∈ γ (R, S). For all n, there exists hn ,  A  A β1 yn da and ηn = β2 zn da that verifies (3.52), such that ρn = 0

0

where, (yn , zn ) is the corresponding solution of the system (4.1) and (yn , zn ) verifing (4.7). Then, from (4.3), (4.4), and (4.7), we deduce that one can extract subsequences still denoted by ((yn , zn ))n and (hn )n that converges weakly to (y, z) and h, in X2 and L2 (q) respectively. Standard device implies that  A  A β1 yda and η = β2 zda. In addition, (y, z) is the solution of the ρ = 0

0

system (4.1). Moreover, h verifies (3.52) and z verifies (1.2). Therefore, the definition of γ (R, S) yields that (ρ, η) ∈ γ (R, S). (iii) Let (R, S) ∈ K, a bounded subset of M. Then, there exist r > 0 and s > 0, such that RL2 (QT ) ≤ r and SL2 (QT ) ≤ s. Consider now ((ρn , ηn ))n ⊂ γ (R, S). From the definition of γ , for all n, there exists a pair (yn , zn ) and  A 2 2 β1 yn da and ηn = hn in X and in L (q) respectively such that ρn = 0  A β2 zn da where, (yn , zn ) is the corresponding solution of the system (4.1) 0

and hn , verifies (3.52). Now, using (3.52) we get  q

h2n (t, a, x)dtdadx ≤

 4 Cγ L0 + C12 (r 2 + s 2 ) 3

+ y0 2L2 (Q

A)

+ z0 2L2 (Q ) . A

(4.10)

Hence, (hn ) is bounded in L2 (q). Thus, there exists a subsequence of (hn ) denoted by (hnk ) which converges weakly towards some h in L2 (q). On the other hand, multiplying the first and the second equation of the system (4.1) by yn and zn respectively, integrating over Q and using Young inequality’s we obtain (4.5) and (4.6). Adding (4.5) and (4.6) and using (4.10), we find for all

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

63

  1 λ0 > max 2, + 2CL1 , that 2   k(x)((yn )2x + (zn )2x ) + ((yn )2 + (zn )2 ) dtdadx ≤ C4 (L0 + (r 2 + s 2 )) Q

+ C4 (z0 2L2 (Q "

with C4 =

A)

+ y0 2L2 (Q ) ),

1 ! 1+ Cγ max C12 ,1 3 $ % λ min 1, 20 + 14

(4.11)

A

#

. Therefore, (yn , zn ) is bounded in X2 . Hence,

we can extract a subsequence of ((yn , zn ))n denoted by ((ynk , znk )) which  A β1 ynk da and converges weakly to a pair (y, z). Now, consider ρnk = 0  A β2 znk da. Using (H2 ), we conclude that (ρnk , ηnk ) solves the ηnk = 0

following system ⎧  A ⎪ ⎪ ⎪ (ρnk )t − (k(x)(ρnk )x )x + β1 μ1 ynk da = unk ⎪ ⎪ ⎪ ⎪  0A ⎪ ⎪ ⎪ ⎪ β2 μ2 znk da = vnk (ηnk )t − (k(x)(ηnk )x )x + ⎪ ⎪ ⎪ 0 ⎪ ⎨ ρnk (t, 0) = ρnk (t, 1) = 0 ⎪ ηnk (t, 0) = ηnk (t, 1) = 0 ⎪ ⎪  A ⎪ ⎪ ⎪ ⎪ ρ β1 (0, a, x)y0(a, x)da ⎪ nk (0, x) = ⎪ ⎪ 0 ⎪  ⎪ A ⎪ ⎪ ⎪ β2 (0, a, x)z0(a, x)da ⎩ ηnk (0, x) =

in QT , in QT , on (0, T ), on (0, T ), in , in ,

0

(4.12) with  unk =

A 0



A

− 0

 β1 hnk 1ω da +

A 0

((β1 )t + (β1 )a )) ynk da 

[(k(x)(β1)x )x ] ynk da − 2

A

k(x)(β1)x (ynk )x da,

0

and  vnk = 

A 0 A

− 0

 β2 ynk 1O da +

A 0

((β2 )t + (β2 )a )) znk da

[(k(x)(β2)x )x ] znk da − 2

 0

A

k(x)(β2 )x (znk )x da.

64

M. Birba and O. Traoré

Taking into account the assumptions on k, using Hardy–Poincaré and Minkowski’s inequalities, we infer that there exist two constants C(A, β1 ) > 0 and C(A, β2 ) > 0, such that  unk 2L2 (Q

T)



≤ C(A, β1 ) Q

(k(x)|(ynk )x |2 + |ynk |2 )dtdadx

+ C(A, β1 )hnk 2L2 (q)

(4.13)

and  vnk 2L2 (Q ) T





≤ C(A, β2 )

|ynk 1O | + 2

Q

 + C(A, β2 )

Q

(k(x)|(znk )x | + |znk | ) dtdadx 2

Q

2

|ynk 1O |2 dtdadx

(4.14)

Now, using the inequalities (4.10) and (4.11), we deduce that unk 2L2 (Q

T)

+ vnk 2L2 (Q

T)

 ≤ C5 L0 + r 2 + s 2

 + C5 y0 2L2 (Q ) + z0 2L2 (Q ) , A

A

(4.15)

where 

! C5 = max {2, C(C(A, β1 ), C(A, β2 )} C4 + 43 Cγ max 1, C12 > 0. Multiplying the first and the second equation of the system (4.12) by ρnk and ηnk respectively, integrating over QT and using Young inequality, we obtain  QT

k(x)(ρnk )2x dtdx +

λ0 2

 QT

(ρnk )2 dtdx ≤

+ C(μ1 )(y0 2L2 (Q

A)

1 2λ0

 QT

u2nk dtdx

+ r 2 ),

(4.16)

and  QT

k(x)(ηnk )2x dtdx +

λ0 2 +

 QT

(ηnk )2 dtdx ≤

C(μ2 )(z0 2L2 (Q ) A

1 2λ0

 QT

+ s ), 2

vn2k dtdx (4.17)

where C(μ1 ) and C(μ2 ),$ are strictly positive constants. % 1 Taking λ0 > max 2, 2 + 2CL1 in (4.16), (4.17) and using (4.11) and (4.15), we can conclude that the subsequences (ρnk ) and (ηnk ) are bounded in L2 ((0, T ), Hk1 ()). Now, using Corollary 3.4.6 (see [11]), we infer that the subsequences ((ρnk )t )n and ((ηnk )t )n are bounded in L2 ((0, T ); Hk−1 ()).

Null Controllability of a System of Degenerate Nonlinear Coupled Equations. . .

65

Since Hk1 () is compactly embedded in L2 (), we conclude by Aubin– Lions lemma the existence of subsequences of (ρnk ) and (ηnk ) denoted by (ρni ) and (ηni ) respectively that converge strongly towards ρ and η respectively in L2 (QT ). This implies that (ρnk ) and (ηnk ) converge weakly towards ρ and η respectively in L2 (QT ). Thus, ⎧  ⎪ ⎪ gρ dtdx −→ gρdtdx, ⎨ ni  QT QT ⎪ ⎪ gηni dtdx −→ gηdtdx, ⎩ QT

∀g ∈ L2 (QT ), ∀g ∈ L2 (QT ).

(4.18)

QT

On the other hand, ((ynk , znk )) converges weakly to (y, z) in X2 . Then, ((ynk , znk )) converges weakly towards (y, z) in L2 (Q) (because X ⊆ L2 (Q)). Subsequently, ((yni , zni )) the subsequence of ((ynk , znk )) associated to ((ρni , ηni )) converges weakly towards (y, z) as well. The fact that gβj ∈ L2 (QT ), for all g ∈ L2 (QT ) and for all j = 1, 2, implies that  A  ⎧  ⎪ ⎪ gρni dtdx −→ g β1 yda dtdx, ⎨  QT QT 0 A ⎪ ⎪ ⎩ gηni dtdx −→ g β2 zda dtdx. QT

QT

(4.19)

0

Accordingly, by (4.18) and (4.19), we infer that   ⎧  A ⎪ ⎪ g ρ− β1 yda dtdx = 0, ∀g ∈ L2 (QT ) ⎨ 0 Q T     A ⎪ ⎪ ⎩ g η− β2 zda dtdx = 0, ∀g ∈ L2 (QT ), QT

(4.20)

0

Therefore, ⎧  A ⎪ ⎪ ⎨ρ = β1 yda, 0 A ⎪ ⎪ ⎩η = β2 zda,

a.e. (t, x) ∈ QT (4.21) a.e. (t, x) ∈ QT

0

Furthermore, we can check by a standard argument that (y, z) solves the system (4.1) and z satisfies (1.2). Moreover, h ∈ L2 (q) verifies (3.52). This completes the proof of iii). (iv) Notice that γ , is upper semi-continuous on M if and only if for each closed subset K ⊂ M, −1 γ (K) is closed in M. Thus, let K a closed subset of −1 M. Let the sequence ((Rn , Sn ))n ⊂ −1 γ (K). Notice also that, γ (K) = {(R, S) ∈ K; γ (R, S) ∩ K = ∅}. Assumes that (Rn , Sn ) −→ (R, S) in M. Then, (Rn , Sn ) is bounded. There exists a sequence ((ρn , ηn ))n ∈ K,

66

M. Birba and O. Traoré

such that for n ≥ 1, (ρn , ηn ) ∈ γ (Rn , Sn ). From the definition of γ , there  A 2 2 exist a pair (yn , zn ) ∈ X and hn ∈ L (q), such that ρn = β1 yn da and 0  A ηn = β2 zn da, hn verifies (3.52) and (yn , zn ) satisfies (3.43) and solves 0

the system (4.1) and zn verifies (1.2). From the inequalities (4.10) and (4.11), we deduce that (hn ) and (yn , zn ) are bounded in L2 (q) and in X2 respectively. Thus, there exist a subsequences still denoted by ((hn ))n and ((yn , zn ))n , that converge weakly to h and (y, z) in L2 (q) and in X2 respectively. Now, by a  A  A β1 yda, η = β2 zda standard device, we see that h verifies (3.52), ρ = 0

0

and (y, z) solves the system (4.1) and z verifies (1.2). This implies that (ρ, η) ∈ γ (R, S).

(4.22)

On the other hand, thanks to (4.15), (4.16), and (4.17) and Lions-Aubin lemma once again, we can extract a subsequence also denoted by ((ρn , ηn ))n that converges strongly towards (ρ, η) in M. Since, K is closed, we deduce that (ρ, η) ∈ K. Finally, from (4.22), we deduce that (R, S) ∈ −1 γ (K). 

References 1. B. Ainseba, S. Anita, Internal exact controllability of the linear population dynamics with diffusion. Electron. J. Differ. Equations 2004(112), 1–11 (2004) 2. E.M. Ait ben hassi, F. Ammar Khodja, A. Hajjaj, L. Maniar, Null controllability of degenerate parabolic cascade systems. Port. Math. 68, 345–367 (2011) 3. C. Avramescu, A fixed point theorem for multivalued mappings. Electron. J. Qual. Theory Differ. Equations 2004, 1–10 (2004) 4. V. Barbu, T. Precupanu, Convexity and Optimization in Banach Spaces (D. Reidel Publishing Company, Dordrecht, 1986). 5. M. Birba, O. Traoré; Controllability of nonlinear degenerate parabolic cascade system. Electron. J. Differ. Equations 2016(219), 1–25 (2016) 6. M. Campiti, G. Metafune, D. Pallara Degenerate self-adjoint evolution equations on the unit interval. Semigroup Forum. 57, 1–36 (1998) 7. P. Cannarsa, P. Martinez, J. Vancostenoble, Null controllability of degenerate heat equations. Adv. Differ. Equations 10, 153–190 (2005) 8. Y. Echarroudi, L. Maniar, Null controllability of a model in population dynamics. Electron. J. Differ. Equations 2014(240), 1–20 (2014) 9. A. Fursikov, O.Yu. Imanuvilov, Controllability of evolution equations, in Lecture Notes, Research Institute of Mathematics (Seoul National University, Korea, 1996) 10. O. Traore, Null controllability of a nonlinear population dynamics problem. Int. J. Math. Sci. 2006, 1–20 (2006) 11. M. Tucsnak, G. Weiss, Observation and control for operator semigroups, in Birkhuser Advanced Texts Basler Lehrbucher Series (2009)

Optimal Mass Transport for Activities Location Problem Mamadou Koné, Babacar Mbaye Ndiaye, and Diaraf Seck

Abstract In this article, the major issue is the usefulness of models to design land use problem through the concepts of the Monge–Kantorovich, also known in the literature as the “Mass Optimal Transportation”. We present two main contributions for this land use problem. First, we propose a new continuous formulation of the activities location problems. Second, in order to solve the continuous problems, we discretize and compute numerically the optimal transportation by considering finite sums of weighted Dirac masses. In this specific case, the optimal transportation is a multi-valued map between the Dirac locations. In contrast to linear assignment problem, the Quadratic Assignment Problem (QAP) can handle the case where two or more land use decisions are interdependent, i.e., if there is a so called flow interaction between activities. The technic is globalized by the computation of Gromov–Wasserstein distance to solve the quadratic optimization problem with continuous variables. Keywords Activities location · Optimal transportation · Gromov–Wasserstein distance · Land use · Quadratic assignment

1 Introduction The activities location problem is often formulated as a Quadratic Assignment Problem (QAP) by Koopmans and Beckmann [1], which assigns n activities to n locations while minimizing the total cost location. The QAP is known to be NP-

This work was completed with the support of the NLAGA project M. Koné · B. M. Ndiaye () · D. Seck Laboratory of Mathematics of Decision and Numerical Analysis, University of Cheikh Anta Diop, Dakar, Senegal e-mail: [email protected]; [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_3

67

68

M. Koné et al.

complete [2]. The question would be as follows: what is the optimal way to locate activities in the transportation network? How are the locations of clinics within a hospital decided? How to locate optimally administrative services? To address these challenges, we are dealing with the most challenging combinatorial optimization problem. The main idea is the follows: Locations k and l are separated by a distances of dkl . On the other hand, entities i and j must exchange quantities of a given product fij . The cost of assigning i to k is cik but an assignment also induces a product routing cost which it is assumed to be proportional to the quantities of product to be exchanged and to the distance that separates the entities, i.e., fij dkl . The mathematical formulation of the Activities Location Problem (ALP ) is given as follows. Let: • X = {1, . . . , n} the set of activities and Y = {1, . . . , n} the set of potential sites for new activities, • F = (fij )n×n the matrix of flows from activity i to activity j , • D = (dkl )n×n the matrix of distances from site k to site l, • C = (cik )n×n the cost of assigning activity i to site k, independent of other locations, • Pn the set of all permutations of {1, . . . , n} in {1, . . . , n}, • πik the assignment of activity i to site k. The activity location problem can be modeled as a QAP, which is to find the minimum cost assignment (location) of n activities to n locations. For the ALP , the quadratic assignment formulation is shown in Eqs. (1.1)–(1.4) (see [3]): (ALP ) :

n  n 

Minimize π∈Pn

cik πik +

i=1 k=1

n  n  n  n 

ij kl πik πj l

(1.1)

i=1 j =1 k=1 l=1

s.t. n 

πik = 1,

∀ k ∈ {1, . . . , n}

(1.2)

πik = 1,

∀ i ∈ {1, . . . , n}

(1.3)

∀ i, k ∈ {1, . . . , n}.

(1.4)

i=1 n  k=1

πik ∈ {0, 1}, where  πik =

1,

if activity i is located at zone k,

0,

otherwise.

(1.5)

ikj l = fij dkl is the cost of locating activity i at location k and activity j at location l. ikj l in Eq. (1.1) is a cost variable representing the combination of quantitative

Optimal Mass Transport for Activities Location Problem

69

Fig. 1 (Left) five municipalities and transport links. (Center) the activities. (Right) location of potential activities [4]

and qualitative measures in ALP models. Equation (1.2) ensures that each location is assigned to only one activity. Equation (1.3) ensures that each activity is assigned to only one physical location. The following Fig. 1 gives an example of activity location. The ALP formulation in matrix form can be defined as: (T QAP ) :

min F, XDXT  + C, X X

s.t.

(1.6)

X ∈ Xn .

where Xn is the set of the n × n permutation matrices. For more details or a review of the literature, we refer the reader to Baldé and Ndiaye [5], Duranton and Puga [6], and Gueye [7] for both network design problems and activities location problems. Many algorithms and softwares have been proposed in the literature to solve this type of non linear programming problem, but present some limitations for large number of variables. Recently, in [8], we proposed finite linearization and an algorithm for the land use problem, based on a dichotomic translation of a hyperplane and can give better results compared to those obtained by other methods of solvers. In this article, we propose a new continuous reformulation for activities location problem using the optimal mass transportation. The article is organized as follows. In Sect. 2, we present some auxiliary results of optimal mass transport. In Sect. 3, we propose a new formulation of the ALP

70

M. Koné et al.

model, followed by the discrete optimal transport formulation for ALP in Sect. 4. The Sect. 5 presents simulations and comments of the results. Finally, in Sect. 6, we present conclusions and perspectives. We briefly provide an introduction to standard formulation of the Monge– Kantorovich problem. We refer the reader to [9] for an excellent account of the theory of optimal transportation.

2 The Optimal Mass Transport Problem In this section, we first review the two classical formulations of the optimal transport problem (i.e. Monge’s and Kantorovich’s formulations). In optimal transportation we wish to transport one mass distribution to another one such that the total cost is minimized. The problem was first studied by Monge[10], and a major breakthrough was made by Kantorovich. See [9] for an excellent account of the theory of optimal transportation. Problem 2.1 (Monge’s Continuous Formulation) The original formulation of optimal transport by Gaspard Monge considers two probability measures μX and μY over metric spaces X and Y ; and a measurable cost function c : X × X −→ R, which represents the cost of transporting a unit of mass from x ∈ X to y ∈ Y . The problem is to find a transport map T : X −→ Y that realizes 

M(T ) =

min

T# μX =μY

c(x, T (x)) dμX (x),

(2.1)

X

where T# denotes the push-forward of μX by T . The solution to (2.1) might not exist. However, a convex relaxation of the problem due to Kantorovich is guaranteed to have a solution. Problem 2.2 (Kantorovich Continuous Formulation) In 1942, Kantorovich introduced the following problem of optimal mass transport: 

K(π) :=

inf

π∈!(μX ,μY ) X×Y

c(x, y) dπ(x, y),

x ∈ X, y ∈ Y

(2.2)

where c(x, y) : X × Y → R is a measurable real cost function and !(μX , μY ) is the set of transportation plans, i.e., the set of all probability measure on X × Y with marginals μX and μY : ⎧ ⎫ ⎪ ⎪ ⎪ π ∈ P (X × Y ) : π(A × Y ) = μX (A) ⎪ ⎨ ⎬ !(μX , μY ) := and π(X × B) = μY (B) ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ∀A ⊆ X, B ⊆ Y measurable

(2.3)

Optimal Mass Transport for Activities Location Problem

71

Problem 2.3 (Kantorovich Duality) The Kantorovich problem is a linear minimizing problem with convex constraints, so it also admits a dual formulation: 

 inf

π∈!(μX ,μY ) X×Y

(ϕ,ψ)





c(x, y) dπ(x, y) = sup

ϕ dμX + X

ψ dμY

(2.4)

Y

for (ϕ, ψ) varying over the set of pairs of functions ϕ : X → [−∞, ∞) and ψ : Y → [−∞, ∞) which are integrable, i.e. ϕ ∈ L1 (μX ), ψ ∈ L1 (μY ), and satisfy ϕ(x) + ψ(y)  c(x, y), x ∈ X, y ∈ Y. Remark 2.4 Another interesting topic is that of regularized optimal transport, i.e. transport problems where the unknown transport map T (in the Monge formulation) is also penalized via a cost depending on its gradient ∇T or more generally on its incremental ratios (T (x) − T (x ))/|x − x | (for more details, see [11]). In this case the transport function, that is to say that the cost depends on  three variables C(x, T (x), ∇T(x)); this typically involves adding the integral of |∇T (x)|2 dx to the functional X c(x, T (x)) dμX (x) in order to obtain a Sobolev penelty. We give here some notation and background concepts for the optimal mass transport problem. • Let P (X) be the set all (Borel) probability measures on a space X and B (A) the set of all Borel sets of σ -algebra A. We note # the push forward-operator such that for B ∈ B (A), T# μ(B) = μ(T −1 (B)). • A measure μ on a set X is said to be fully supported if supp(μ) = X, where supp(μ) is the minimal closed subset A ⊂ X such that μ(X \ A) = 0. • For two probability measures μX ∈ P (X) and μY ∈ P (Y ), we denoted by !(μX , μY ) the set of all couplings or matching measures of μX and μY . • Recall that a map ϕ : X → Y between metric spaces (X, dX ) and (Y, dY ) is an isometric embedding if dY (ϕ(x), ϕ(x )) = dX (x, x ) for x, x ∈ X and ϕ#μX = μY . • A mapping T : X −→ Y Borel measurable, if T −1 (A) ∈ B (Y ) for all A ∈ B (X). The pushforward or image measure of T is defined to be the measure T# μ(A) ∈ B (X) given by written T# μ(A) = μ(T −1 (A)), for all A ∈ B (X). • For each 1  p  ∞, denote by Lp (μX ) the μX -measurable function f : X → R such that |f |p is μX -integrable. • A metric measure space (mm-space for short) is a triplet (X, dX , μX ) where (X, dX ) is a compact metric space and μX is Borel probability measure with full support supp(μ) = X. • By a measure network, we mean a triplet (X, dX , μX ). The naming convention arises from the case when X is finite; in such a case, we can view the pair (X, dX ) as a complete directed graph. Accordingly, the points of X are called nodes, pairs of nodes are called edges, and dX is called the edge weight function of X. • Let (X, dX , μX ), (Y, dY , μY ) be two measure network. A coupling between these two networks is a probability measure π ∈ X × Y with marginals μX and μY ,

72

M. Koné et al.

respectively. Stated differently, couplings satisfy the following property: {π ∈ P (X×Y ) | ∀(A, B) ∈ B (X)× B (Y ), π(A×Y ) = μX (A), π(X×B) = μY (B)}. • A probability measure on the measurable space (X, B (X)) is any measure μ on X such that μ(X) = 1. • A relation R on X and Y is called a correspondence: for any x ∈ X there exists y ∈ Y such that x and y are in relation R and, conversely, for any y ∈ Y there exists x ∈ X such that x and y are in relation R. • Consider sequences of probability measures on (X, B (X)), where (X, d) is a metric space and B (X) is the Borel σ -field. Definition 2.5 (Weak Convergence) Let {μk }k∈N be sequence of probability measures on (X, B (X)). We say that μn converges weakly to a probability measure μ on (X, B (X)), denoted by μk −−−→ μ, if n→∞



 f dμk → X

f dμ, k → ∞ for all f ∈ Cb (X), X

where f ∈ Cb (X) denotes the set of all continuous and bounded function f : X → R. Definition 2.6 (Lower Semicontinuous) A function f : X → R ∪ {±∞} is (sequentially) lower semicontinuous (l.s.c for short) at x ∈ X, if for every sequence xk converging to x we have f (x)  lim inf f (xk ), k→∞

or in other words ! f (x) = min lim inf f (xk ) : xk → x . k→∞

Remark 2.7 The following conditions are equivalent: (i) f is lower semicontinuous; (ii) we have f (x) = lim infy→x f (y) ∀x ∈ X, since the constant sequence xk = x, ∀k ∈ N is included in the infimum; (iii) for all λ ∈ R the sublevel set {f  λ} is closed. Let (X, d) a metric space with Borel σ -fiels B (X), and P (X) the set of all probability measures on (X, B (X)). Definition 2.8 M ⊆ P (X) is called tight if of every ε > 0, there exists a compact set K ⊂ X with μ(K c )  ε for all μ ∈ M

Optimal Mass Transport for Activities Location Problem

73

Theorem 2.9 (Prokhorov) Consider M ⊆ P (X). 1. If M is tight, then M is relatively sequentially compact. 2. Suppose that X is complete and separable. If M is relatively sequentially compact, then M is also tight. Definition 2.10 (p-Wasserstein Distance) The Wasserstein distance of parameter p ∈ [1, ∞) between two Borel probability measure μX and μY over the same metric spaces (X, d) is defined as: 

p

Wp (μX , μY ) =

inf

π∈!(μX ,μY ) X×Y

d p (x, y)dπ(x, y).

(2.5)

Definition 2.11 (p-GWD, [12, 13]) The p-(Gromov) Wasserstein distance between two mm-spaces (X, dX , μX ) and (Y, dY , μY ) with parameter p ∈ [1, ∞) p denoted Dp−GW is defined as: p

Dp−GW ((dX , μX ), (dY , μY )) :=

inf

π∈!(μX ,μY )

X (π),

(2.6)

where

X (π) =

1 2



 X×Y

X,Y (x, y, x , y )dπ(x, y)dπ(x , y ),

(2.7)

X×Y

with X,Y (x, y, x , y ) := |dX (x, x ) − dY (y, y )|p . 2 Definition 2.12 (The 2-GWD) The 2-Gromov Wasserstein (D2− GW ) distance between (μX , dX ) and (μY , dY ) measures the minimal distortion induced by a measure coupling between the two domains: 2 D2− GW ((μX , dX ), (μY , dY )) :=

inf

π∈!(μX ,μY )

G (π),

(2.8)

where

G (π) :=

1 2



 X×Y

|dX (x, x ) − dY (y, y )|2 dπ(x, y)dπ(x , y ).

(2.9)

X×Y

3 Continuous Optimal Transport Models of ALP In this section, given two metric measure spaces of equal mass, a mm-space for short, to be a triplet (X, dX , μX ) and (Y, dY , μY ), where (X, dX ) and (Y, dY ) are complete separable metric spaces, and μX and μY are Borel probability measures on X and Y respectively i.e. normalized as μX (x) = μY (y) = 1. We denote by

74

M. Koné et al.

!(μX , μY ) the set of all couplings of μX and μY . The new continuous optimal transport formulation of ALP is given by the following problem (P): ⎧ inf Q(π), ⎪ ⎪ π ∈!(μX ,μY ) ⎪ ⎪ ⎪ ⎪    ⎪ ⎪ 1 ⎪ ⎪ Q(π) = cX,Y (x, y) dπ(x, y) + X,Y (x, y, x , y ) dπ(x, y) dπ(x , y ) ⎪ ⎪ ⎪ 2 X×Y X×Y X×Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ subject to :  ⎪ ⎪ ⎪ dπ(x, y) = dμX (x), (C1 ) ⎪ ⎪ ⎪ Y ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ dπ(x, y) = dμY (y), (C2 ) ⎪ ⎪ ⎪ X ⎪ ⎪ ⎪ ⎩ π(x, y)  0.

(3.1) where cX,Y : X × Y −→ R+ represents the transportation cost per unit mass from location x ∈ X to location y ∈ Y and X,Y : X × Y × X × Y −→ R+ stands for the transportation cost per unit mass from of each location pair (x ↔ x ) to (y ↔ y ) for all x, x ∈ X, y, y ∈ Y . To ensure that all mass in μX is transported to μY , in other words we impose the constraints (C1 ) and (C2 ). π is called a transference plan from μX to μY with marginals dμX (x) and dμY (y). (C1 ) and (C2 ) are referred to as the assignment constraints (1.2), (1.3) and the constraints of the assignment constraints referred to as the assignement matrix. Remark 3.1 There are different interesting cases for the cost function X,Y . We given an example of the cost function X,Y (x, y, x , y ) := |dX (x, x )−dY (y, y )|p , 1  p < ∞, for all x, x ∈ X, y, y ∈ Y . It is common to take the quadratic cost X,Y (x, y, x , y ) := |dX (x, x ) − dY (y, y )|2 .

3.1 The Direct Method Fondamentally the existence theorem of the problem (P) will be focus on the concept (simple yet powerful) of direct method in the calculus of variation, which deals with Q : V → R ∪ {+∞}, where V is some function space. The main interest of the subject is to find minimizers for such functional, that is, functions v ∈ V such that: Q(v)  Q(u), ∀u ∈ V . The functional Q must be bounded inferiorly to have a minimizer. This means inf{Q(u) : u ∈ X} > −∞. The direct method consists of the following steps: (1) To construct a minimizer we take a minimizing sequence {un } ⊂ V such that lim Q(un ) → α := inf{Q(u) : u ∈ V } < +∞.

n→∞

(3.2)

Optimal Mass Transport for Activities Location Problem

75

(2) Prove that {un } admits a subsequence {unk } converging to some u0 ∈ V with respect to some (weak) topology τ in V . When Q has an integral representation of the from (3.1), this is usually a consequence of a prior cœrcivity conditions on the integral functions cost. (3) Establish the sequential lower semicontinuity of Q with respect to τ , i.e., α  Q(u∗ )  lim inf(un ) = α,

(3.3)

n→∞

whenever the sequence {un } ⊂ V converges weakly to v ∈ V with respect to τ . Thus, Q(u∗ ) = α and u∗ is the sought minimizer. (4) Conclude that u∗ minimizes Q. Indeed, inf Q(u) = lim Q(un ) = lim Q(unk )  Q(u∗ )  inf Q(u).

u∈V

n→∞

k→∞

u∈V

Remark 3.2 The Goal is to minimize Q in the set all couplings !(μX , μY ). Lemma 3.3 Let μX ∈ P (X) and μY ∈ P (Y ) two measures. Then !(μX , μY ) is compact in the weak topology. Proof 3.4 Let us start with the tightness of !(μX , μY ). Because μX and μY are both tight, for every ε there existes two compacts sets K1 ⊂ X and K2 ⊂ Y such that μX (X \ K1 )
−∞, in this case, for all ε > 0, there exists n0 such that for all nk  n0 , Q(πnk )  λ+ε, i.e. πnk ∈ Aλ+ε for all nk  n0 . As Aλ+ε is weakly closed, so it contains the weak limit

78

M. Koné et al.

of the sequence (πnk )nk n0 , namely π. Therefore, for all ε > 0, Q(π)  λ + ε, i.e.,

Q(π)  lim inf Q(πn ). n→+∞

By passing to the limit when n → +∞, which shows that Q is weakly lower semicontinuous. The second case is where λ = −∞. In this case, for all η < 0, there exists n0 such that for all nk  n0 , Q(πnk ) → η, i.e. πnk ∈ Aη for all nk  n0 . As Aη is weakly closed, it contains the weak limit of the sequence (πnk )nk n0 , namely π. We have therefore shown that π ∈ ∩η= 0; ================= ARRAY PARAMETERS param dX{xi in 1..n, xj in 1..n: xixj}; # dX[xi,xj] = flow from xi to xj param dY{yk in 1..n, yl in 1..n: ykyl}; # dY[yk,yl] = distance between yk and yl; ======================== VARIABLES var pi{1..n,1..n} binary; # 1, if xi is assigned to yk; ======================== OBJECTIVE minimize obj_function: sum{xi in 1..n, yk in 1..n} c[xi,yk]*pi[xi,yk] + 0.5*sum{xi in 1..n, xj in 1..n, yk in 1..n, yl in 1..n: xixj and ykyl and xiyl and ykxj} abs(dX[xi,xj]- dY[yk,yl])^2*pi[xi,yk]*pi[xj,yl]; ====================== CONSTRAINTS subject to constr2{xi in 1..n}: sum{yk in 1..n} pi[xi,yk] = muX[xi]; subject to constr1{yk in 1..n}: sum{xi in 1..n} pi[xi,yk] = muY[yk]; subject to bound1{xi in 1..n, yk in 1..n}: 0 0, c < 1 %⇒ lim d(c tn ) = M ,

(3.20)

c > 1 %⇒ lim d(c tn ) = 0 .

(3.21)

n→∞ n→∞

104

B. D. B. Diatta and P. Ngom

where (tn )n≥1 is a sequence of positive real numbers, d(t) the distance between the distribution of X(n) at time t and its stationary law ν (n) , and M is either a positive real or +∞. The characterization of the phenomenon is given by both Eqs. (3.20) and (3.21), where the left limit, i.e. the limit before the cut-off time, is given by (3.20), and the right limit, i.e. the limit after the cut-off time, given by (3.21). Limit M > 0 can be finite or infinite while the right limit is always 0. We give now a new characterization of the cut-off phenomenon which can be seen as an extended version of Definition 3.1. It takes into account the case where the left limit does not exist, but the sequence (d(ctn )) is bounded below by a positive constant N from a certain rank n0 . We call it extended cut-off. We have the following definition. Definition 3.2 (Extended Cut-off) A sequence of processes (X(n) )n≥1 is said to have an extended cut-off phenomenon at sequence of times (tn )n in the sense of distance d if for c > 0, ∃ (n0 ≥ 1, N > 0), ∀n ≥ n0 , d(c tn ) ≥ N, ∀ c < 1,

(3.22)

lim d(c tn ) = 0, ∀ c > 1.

(3.23)

n→∞

Unlike Eq. (3.20) in Definition 3.1, Eq. (3.22) takes into account all cases: case where the limit exists and is finite, case where the limit is infinite and case where the limit does not exist. Recall here the notion of exponential (ergodic) convergence of a process. Several definitions have been given by different authors (Down, Meyn and Tweedie [12], Roberts and Tweedie [21], Chen [5], Chen [6], Barrera et al. [3], Champagnat and Villemonais [4]). Let us consider then the characterization of the exponential ergodic convergence of a process (Chen [5]): let d be a distance between two probability distributions, ∃ R > 0, ρ > 0, such that ∀ t > 0, d(t) ≤ Re−ρt

(3.24)

The case of equality in (3.24) implies following relation, lim d(t)eρt = R.

t →∞

(3.25)

For the results we will give, Eqs. (3.24) and (3.25) are important assumptions. These results will be describe in the two following theorems. The first extends an important previous result (Barrera et al. [3], Theorem 3), and establishes the existence of the cut-off for a sequence of n-tuples of independent processes with the α-divergence measures.

Cut-off Phenomenon

105

Let us consider the sequence (τn ) of positive real numbers  τn = max

 log i , i = 1, . . . , n ρ(i,n)

(3.26)

where ρ(1,n) , . . . , ρ(n,n) are the values of ρ1 , . . . , ρn ranked in increasing order. Theorem 3.1 Let d be a α-divergence measure, with α ∈ R∗+ \ {1}. Let (Xi )i∈N∗ be a sequence of independent processes each exponentially converging in distribution with rate ρi . Denote by d(t) the distance to equilibrium of the n-tuple process (n) (n) X(n) = (X1 , . . . , Xn ) at time t ∈ R+ , i.e. d(t) = Div(Ft  ν (n) ), where Ft and ν (n) are respectively the distribution at time t and the stationary distribution of the process X(n) , and Div(.  .) a α-divergence measure. Assume that the following hypotheses in Lemma 2 (Barrera et al. [3]) are verified: (1)

There exists a positive function g, decreasing and tending to 0 as t tends to +∞, and a positive real t0 such that all t ≥ t0 and all i ≥ 1,    log di (t)    + ρ i  ≤ g(t).  t

(2) lim ρ(1,n) τn = +∞.

n→∞

(3) For any positive real c, lim

n→∞

g(cτn ) = 0. ρ(1,n)

Then we observe a cut-off phenomenon for the sequence of processes (X(n) ) in the sense of distance d at any sequence of times equivalent to (τn ). To understand the contribution of our article, let us first recall the result of Theorem 3 in Barrera et al. [3] which establishes the existence of the cut-off and locates it at any sequence of times equivalent to τkn , with k a positive integer. Dividing τn by k was a technical condition which was made to use the result of their Lemma 2. However, the disadvantage of this handling is that it simultaneously introduces the power k on distances of component measures (k = 2 for specific divergences: total variation, Hellinger, chi-square and Kullback-Leibler in Proposition 6 of Barrera et al. [3], page 1440). This leads to non-classical formulations for the probability metrics considered (square root of classical distances in Definition 4 of Barrera et al. [3], page 1439). The originality of our work through Theorem 3.1 is to establish the existence result of the cut-off phenomenon, with various distances (α-divergences families), where the cut-off instant does not depend on parameter k. This is made possible by

106

B. D. B. Diatta and P. Ngom

the good boundary properties of the α-divergences that we propose. We can see it in Lemma 3.1 where in the inequality relations concerning the distances between the product probability measures and the distances between component measures does not appear any power for the distances. The advantage of our results is to offer the possibility of directly using the simple distances (k = 1), instead of the square or any power distances (with k > 2), to characterize the cut-off phenomenon. The proof of Theorem 3.1 is given in the Appendix section. Our second theorem is an extended form of Theorem 9 in Barrera et al. [3]. It gives explicit expressions of distances around the cut-off time τn , as n tends to infinity, in case where components Xi,t of n-tuple Xt(n) are i.i.d. and where the processes Xi,t , i = 1, . . . , n converge exponentially with same rate ρ to its common stationary distribution ν.

 Theorem 3.2 Let (X(n) ) = (X1 , X2 , . . . , Xn ) be a sequence of stochastic processes such that all Xi have the same exponential convergence rate ρ. Let d be a α-divergence measure, with α ∈ R∗+ \ {1}. Assume both d and di verify (3.25). Then the sequence of cut-off time is logρ n , and we have the following results: If d is the Rényi divergence Rα (.  .) then,  lim d

n→∞

 log n + u = RRα e−ρu ρ

(3.27)

If d is the divergence Dα (.  .) then,  lim d

n→∞

 

1 log n +u = 1 − exp − α(1 − α)RDα e−ρu ρ α(1 − α)

(3.28)

If d is the Tsallis divergence Tα (.  .) then,  lim d

n→∞

  

1 log n +u = exp (α − 1)RTα e−ρu − 1 ρ α−1

(3.29)

In Theorem 3.2 we give a very important result concerning the left limit M. The novelty here is that M depends on the parameter α, and is explicitly known: M = 1 1 α(1−α) for the α-divergence Dα (..), and M = 1−α for the Tsallis α-divergence Tα (..). Note that the ergodic constants RRα , RDα and RTα depend on the distance from which they are calculated. Another remark is that the cut-off instant here is log n log n ρ instead of 2ρ (Barrera et al. [3]) when all components of the n-tuple have the same convergence rate ρ. The proof of Theorem 3.2 is given in Appendix section. The calculations are simple and use the equality relations in Lemma 3.1.

Cut-off Phenomenon

107

4 Simulations Results We give in this section some numerical and graphical aspects illustrating results that we established in the previous section. The Theorem 3.2 allows us to calculate the asymptotic distance around the cut-off instant. Note that this distance is a function of the variable u, and u = 0 corresponds to the cut-off instant logρ n . We observe the dynamic of the limit distance in a more or less symmetrical interval around u = 0. This makes it possible to illustrate the results of Theorem 3.1 in the particular case where the components Xi , i = 1, . . . , n of the n-tuple are i.i.d.  For our simulations we consider a fictitious sequence of n-tuples (X1 , . . . , Xn ) of processes where components Xi are i.i.d and have the same exponential convergence rate ρ. The results that will follow will be observed on the sequence as n tends to infinity (in practice we will take n enough large). For appreciating the interest of using α-divergence measures, we evaluate the impact of the parameter α in measuring the phenomenon. However only two of the three families of α-divergences allow us to do this. It’s about family distances Dα (.  .) and Tα (.  .), since for Rényi divergence Rα (.  .) we can’t do it yet because we work on fictitious processes. So for seeing the influence of parameter α we observe two cases: α ∈]0, 1[ and α ∈]1, +∞[. Each of the two cases corresponds to one aspect of the phenomenon with the limit M finite (left level) on first case and infinite on second case. Then several values for the parameter α will be fixed for each of the cases. This will allow us to appreciate the role of the parameter α in measuring the cut-off. Finally we will give some results on the influence of the convergence rate ρ and the ergodic constant R.

4.1 Influence of the Parameter α The parameter α is, very important in measuring the cut-off phenomenon with M finite in the case 0 < α < 1 and infinite in the case α > 1. Although there are some differences, the two distances Dα (.  .) and Tα (.  .) react globally in the same way under the effect of the parameter α. For illustration, two tables of numerical values of the asymptotic distance around the cut-off instant are given. Also graphical representations of the decrease curves of the asymptotic distances are made. Results with the α-Divergence Dα (.  .) The role of the parameter α is examined, as we have said, by observing two cases: 0 < α < 1 and α > 1. However numerical results (Table 1) are given only for the first case, while the graphical representations (Fig. 1) are established and analysed for the two cases. Regarding the first case (0 < α < 1), our calculations showed a symmetry of the effects of the parameter α. Thus the values at equal distance from the central value 0.5 of the interval ]0, 1[ have the same influence on the measurement of the phenomenon. This is why it is enough to examine the effects of α by reducing the interval ]0, 1[ to one of its parts, ]0, 0.5] or [0.5, 1[. Here we take for the parameter α several values between 0.1 and 0.5, the

108

B. D. B. Diatta and P. Ngom

Table 1 Distance Dα (..) around de cut-off instant logρ n with various values of the parameter α ∈ ]0, 1[, and ρ = 0.5 and R = 1 fixed. In bold characters, distance at cut-off instant ρ = 0.5 11.111 6.250 4.762 4.167 4.000 −13

α 0.1 0.2 0.3 0.4 0.5 u

R=1

11.111 6.250 4.762 4.167 4.000 −10

10.547 6.219 4.757 4.165 3.999 −7

5.397 4.334 3.753 3.459 3.369 −4

0.956 0.924 0.902 0.889 0.885 0

0.135 0.134 0.133 0.133 0.133 4

parameter α dependence

0.030 0.030 0.030 0.030 0.030 7

0.007 0.007 0.007 0.007 0.007 10

0.0015 0.0015 0.0015 0.0015 0.0015 13

10

50

parameter α dependence

30

d(u)

40

2 2.1 2.2 2.3 2.4

20

6 0

0

2

10

4

d(u)

8

0.1 0.2 0.3 0.4 0.5

−10

−5

0

u

5

10

−1

0

1

2

3

u

Fig. 1 Distance Dα (..) around the cut-off instant logρ n with various values of the parameter α ∈ ]0, 1[ (left) and values of the parameter α ∈]1, +∞[ (right), in both cases ρ = 0.5 and R = 1 are fixed

convergence rate ρ is fixed to 0.5 and constant R fixed to 1 (Table 1). To observe the phenomenon we take some values u such that logρ n + u describe the surroundings of the cut-off instant and, for each value of the parameter α, note a ceiling which shows that the left limit M is finite if 0 < α < 1 (Table 1, two first columns) and a floor at level 0 (Table 1, last column). This illustration of the phenomenon, with the influence of the parameter α, in the sense of divergence Dα (.  .) is supplemented by graphic representations where the two cases 0 < α < 1 and α > 1 are studied. Thus the left graphic (Fig. 1) visualize the phenomenon observed in the Table 1 where each upper level has as ordinate 1 M = α(1−α) . We can easily notice that the further we get from the central value 0.5 of the interval ]0, 1[ the higher the ceiling rises. The right graphic (Fig. 1) describes the second aspect of the phenomenon which appears when α > 1, with M = +∞. We can notice it with the absence of a ceiling and curves that decrease from infinity to 0. Added to this, the more the values of parameter increase, the more the decay rate of curves decreases.

Cut-off Phenomenon

109

Results with the α-Divergence Tα (.  .) For this distance the numerical values of the asymptotic distances are calculated with case α > 1. The observed values confirm well the absence of ceiling (left limit M = +∞) with values that increase exponentially as one moves away from u = 0 to the left (Table 2). As for the first distance, Dα (..), the measurement of the cut-off by the Tsallis divergence Tα (..) highlights the two aspects of the phenomenon, i.e. presence or absence of a ceiling (respectively finite or infinite left limit M). However, for this 1 distance, although the left limit M also depends on the parameter α, with M = 1−α , we note the loss of the symmetry of the effects of the parameter α ∈ ]0, 1[ (Fig. 2, left graphic). For the values of α > 1 another difference with the first distance is that the Tsallis divergence is less sensitive to the variations of the parameter (Fig. 2, right graphic). Table 2 Distance Tα (..) around de cut-off instant logρ n with various values of the parameter α ∈ ]1, +∞[, and ρ = 0.5 and R = 1 fixed. In bold characters, distance at cut-off instant α 2 2.1 2.2 2.3 2.4 u

ρ = 0.5 195,338 600,448 1,861,086 5,808,760 18,238,031 −5

R=1

87.38 124.87 179.66 260.05 378.42 −3

4.2 4.67 5.19 5.79 6.47 −1

1.72 1.82 1.93 2.05 2.18 0

0.83 0.86 0.89 0.92 0.95 1

0.08 0.08 0.09 0.09 0.09 5

0.03 0.03 0.03 0.03 0.03 7

0.01 0.01 0.01 0.01 0.01 9

parameter α dependence 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

0

d(u)

6 4 0

2

2

4

6

8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

8

10

10

parameter α dependence

d(u)

0.25 0.25 0.25 0.26 0.26 3

−5

0

u

5

0

2

4

6

u

Fig. 2 Distance Tα (..) around de cut-off instant logρ n with various values of the parameter α ∈]0, 1[ (left) and values of the parameter α ∈]1, +∞[ (right), in both cases ρ = 0.5 and R = 1 are fixed

110

B. D. B. Diatta and P. Ngom convergence rate ρ dependence 10

10

ergodic contant R dependence

6

8

0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

0

2

2

4

d(u)

6 4

d(u)

8

0.5 0.7 0.9 1.1 1.3 1.5

−10

−5

0

u

5

10

−10

−5

0

5

10

u

Fig. 3 Influence of the convergence rate ρ (left) and the ergodic constant R (right) in the cut-off phenomenon measured by the Dα (..) distance with α = 0.1 fixed

4.2 Influence of the Convergence Rate ρ and the Ergodic Constant R Like the parameter α, two other constants impact on the convergence of the process (X1 , X2 , . . . , Xn ), with n enough large, to its stationary distribution. These are the convergence rate ρ and the ergodic constant R. The influence of these two parameters on the phenomenon is measured by the distance Dα (.  .), where α is fixed to 0.1. So for seeing the effect of the convergence rate ρ, the value of R is fixed at 1 and ρ taking different values between 0.5 and 1.5. So when the values of ρ increase the convergence is more and more abrupt (Fig. 3 left graphic). The influence of the constant R is looked at with ρ = 0.5 and R taking various values between 0.5 and 3.5. Contrary to the effect of the convergence rate ρ, when the values of R increase, the convergence of the process is slower and slower (Fig. 3, right graphic). Additional calculations performed with the case α > 1 or also with the Tsallis divergence have shown the same effects of ρ and R on the process convergence.

5 Conclusion We have shown the existence of the cut-off phenomenon on a n-tuple of stochastic processes whose components are independent and have exponentially ergodic convergence, in the sense of parametric divergence measures depending on a parameter α. We also have given explicit expressions of asymptotic distances around the cut-off time for each α-divergence, when the components of the n-tuple are i.i.d. and all having the same convergence rate. The major contribution of our article is to have established the existence result of the cut-off with three families of probabilistic and parametric distances, generalizing

Cut-off Phenomenon

111

previous studies with particular distances (total variation, Hellinger, Chi-square, Kullback-Leibler). The characterization of the phenomenon was made with a simple expressions of our distances used and not with no power distances like the squares in previous works. A consequence of our approach was to locate the same (or same equivalent) cut-off instant for all our distances, but different from that of other previous studies in the literature. Another new feature that we have brought is that the left limit M which appears in one of the cut-off characterization equations is no longer equal to 1, as in previous studies of other authors, but depends on the parameter α of distances. This new result could offer interesting perspectives in the field. So depending on the value of parameter α, the choice of the correct distance should correspond to the interpretation applicable to a real given problem. Finally to study the influence of the parameter α we worked on a sequence of fictitious n-tuple of processes, assuming known the exponential convergence rate ρ and the ergodic constant R. We have noted that the parameter α plays a determining role in the evaluation of the cut-off phenomenon. The perspectives for future work would be to study the phenomenon on real processes like a sequence of independent binary Markov jump processes, a sequence of birth-death process, or a sequence of Ornstein–Uhlenbeck diffusion. Acknowledgments This research was supported, in part, by grants from Cheikh Anta Diop University and NLAGA Project. We are grateful to many seminar participants and to anonymous referees for comments.

Appendix In this section we give the proofs of our main results expressed in the Lemma 3.1, Theorem 3.1 and Theorem 3.2. Proof (of Lemma 3.1) We give now the proofs of the established relations. The demonstrations of equalities are straightforward, so we focus on the proofs of inequalities. For the divergence measure Dα , we establish the proof of inequalities. • Case 0 < α < 1 The demonstration of (3.11) uses the relation (3.4). So it is easy to check that 0 ≤ α(1 − α)Dα (μ1,i  μ2,i ) < 1,

112

B. D. B. Diatta and P. Ngom

then if we apply (3.4) with α(1 − α)Dα (μ1,i , μ2,i ) replacing xi we have n n /  (1 − α(1 − α)Dα (μ1,i  μ2,i )) ≥ 1 − α(1 − α)Dα (μ1,i  μ2,i ) i=1

i=1

1−

implying

n n /  (1 − α(1 − α)Dα (μ1,i  μ2,i )) ≤ α(1 − α)Dα (μ1,i  μ2,i ) , i=1

i=1

1 Multiplying the two members by α(1−α) leads us to the result. For the proof of (3.12) let us remark that n n



 /   1−α(1−α)Dα (μ1,i  μ2,i ) = exp exp log log 1−α(1−α)Dα (μ1,i  μ2,i ) , i=1

i=1

thus applying relation (3.5) to right member give us the following inequality n n

  /  −α(1 − α)Dα (μ1,i  μ2,i ) 1 − α(1 − α)Dα (μ1,i  μ2,i ) ≤ exp i=1

i=1

that leads us simply to the result. • Case α > 1 Equation (3.13) is obtained by applying the result (3.4) with xi ≤ 0 because α(1 − α) ≤ 0. For getting the relation (3.14) we have also used (3.5). For the Tsallis divergence measure we also establish the proof of inequalities. • Case 0 < α < 1 For inequality (3.16), it is easy to see that −1 < (α − 1)Tα (μ1,i  μ2,i ) ≤ 0 , such that we can use the relation (3.6) with −1 < xi ≤ 0. Then we have n /

n   (α − 1)Tα (μ1,i  μ2,i ) 1 + (α − 1)Tα (μ1,i  μ2,i ) ≥ 1 +

i=1

1 α−1

n

/

1 + (α − 1)Tα (μ1,i

i=1 n     μ2,i ) − 1 ≤ Tα (μ1,i  μ2,i ).

i=1

i=1

For having the relation (3.17) we first consider the following equality n n 

 /   1 + (α − 1)Tα (μ1,i  μ2,i ) = exp log 1 + (α − 1)Tα (μ1,i  μ2,i ) i=1

i=1

Cut-off Phenomenon

113

Then we have the inequality n n 

 /  1 + (α − 1)Tα (μ1,i  μ2,i ) ≤ exp (α − 1)Tα (μ1,i  μ2,i ) i=1

i=1

1 the two by using the relation (3.7), afterwards adding −1 and multiplying by α−1 members of the inequality gives us the result. • Case α > 1 We obtain the inequalities (3.18) and (3.19) by using respectively relations (3.6) and (3.7). 

Proof (of Theorem 3.1) Assuming the hypotheses of Lemma 2 verified, let us start by giving its results for k = 1, for any positive real c > 0. Define the distance di (t) = Div(Fi,t  νi ), where Fi,t is the distribution of the process Xi at time t and νi its stationary law. We have n 

c < 1 %⇒ lim

n→∞

n 

c > 1 %⇒ lim

n→∞

di (cτn ) = +∞,

(A.1)

di (cτn ) = 0.

(A.2)

i=1

i=1

Then di (t) is looked at as the version of d(t) at the component level Xi , i = (n) 1, . . . , n, with Ft = F1,t × · · · × Fn,t due to the independence assumptions of Xi , i = 1, . . . , n and ν (n) = ν1 × · · · × νn . Case 1: Rényi divergence measure. (n) (n) (n) In Lemma 3.1, if we note μ1 = Ft , μ2 = ν (n) , μ1,i = Fi,t and μ2,i = νi (n)

in Rényi divergence we have Rα Ft Then Eq. (3.9) can be rewritten by

d(t) =

 ν (n) and Rα (Fi,t  νi ), i = 1, . . . , n.

n 

di (t)

(A.3)

i=1

Replacing t by cτn in (A.3) then doing the limit for n tending to +∞, Eqs. (A.1) and (A.2) give respectively lim d(cτn ) = +∞, if c < 1 and lim d(cτn ) = 0, if c > 1,

n→∞

n→∞

which characterizes the cut-off, expressed by Eqs. (3.20) and (3.21), for the sequence (X(n) ) of processes in terms of Rényi divergence measure. Proofs for the distances Dα (.  .) and Tα (.  .) are analogous, so we only give that concerning the distance Dα (.  .)

114

B. D. B. Diatta and P. Ngom

α-divergence measure Dα (.  .)

 (n) Note by Dα Ft  ν (n) = d(t) and Dα (Fi,t  νi ) = di (t).

Case 2:

For α ∈ ]0, 1[, Eqs. (3.11) and (3.12) give the following left and right bounding for d(t)

    1 1 − exp − α(1 − α) di (t) ≤ d(t) ≤ di (t). α(1 − α) n

n

i=1

i=1

(A.4)

Replacing t with cτn in (A.4), considering limit as n → +∞ and applying the result (A.1) to left inequality, two cases may arise: case where limit M exists and case where d(ctn ) does not admit a limit. If limit M (finite or infinite) exists we have 1 ≤ lim d(cτn ) = M , ∀ 0 < c < 1. α(1 − α) n→∞ If d(cτn ) has no limit, there exists a rank n0 such that 1 ≤ d(cτn ) , ∀ 0 < c < 1. α(1 − α) 1 . For getting (3.23), In both cases we can always get Eq. (3.22), with N = α(1−α) we calculate limn d(ctn ) considering right inequality in (A.4) and applying the result (A.2). Thus we have the characterization of the cut-off for the sequence (X(n) ) in the sense of distance Dα (.  .) with α ∈ ]0, 1[. For α ∈]1, +∞[, Eqs. (3.13) and (3.14) can be rewritten like

d(t) ≥

n  i=1

   1 1 − exp − α(1 − α) di (t) . α(1 − α) n

di (t)

and d(t) ≤

i=1

(A.5) Note by t = cτn and applying respectively results (A.1) and (A.2) to equations of (A.5) we get expression of cut-off associated to Eqs. (3.20) and (3.21).  Proof (of Theorem 3.2) Assume that d(t) and di (t) verify the relation (3.25), we have d(t) ∼+∞ R e−ρt

and di (t) ∼+∞ R e−ρt ,

• For the Rényi divergence Rα (.  .) let us rewrite Eq. (3.9) by d(t) =

n  i=1

di (t)

Cut-off Phenomenon

115

We have now  lim d

n→∞

 n  log n log n + u = lim RRα e−ρ( ρ +u) n→∞ ρ i=1

= RRα e−ρu • For the α-divergence Dα (.  .), Eq. (3.10) can be rewritten by

 / 1 1− 1 − α(1 − α)di (t) , α(1 − α) n

d(t) =

i=1

then we have  lim d

n→∞

"  # n  / log n log n 1 + u = lim 1− 1 − α(1 − α)RDα e−ρ( ρ +u) n→∞ α(1 − α) ρ i=1

= lim

n→∞

1 α(1 − α)

 

 log n 1 − exp n log 1 − α(1 − α)RDα e−ρ( ρ +u)

using Taylor expansion on the neighbourhood of 0, at first order, of log(1 − x) we obtain  lim d

n→∞

    1 log n −ρ( logρ n +u) + u = lim 1 − exp −nα(1 − α)RDα e n→∞ α(1 − α) ρ 

1 1 − exp − α(1 − α)RDα e−ρu . = α(1 − α)

• For the Tsallis α-divergence Tα (.  .), Eq. (3.15) can be rewritten by  1 / 1 + (α − 1)di (t) − 1 α−1 n

d(t) =

i=1

then we have,  lim d

n→∞

    1 log n −ρ( logρ n +u) n 1 + (α − 1)RTα e −1 + u = lim n→∞ α − 1 ρ    

1 −ρ( logρ n +u) −1 exp n log 1 + (α − 1)RTα e = lim n→∞ α − 1

116

B. D. B. Diatta and P. Ngom

using Taylor expansion on the neighbourhood of 0, at first order, of log(1 + x) we obtain,       log n 1 log n + u = lim exp n(α − 1)RTα e−ρ( ρ +u) − 1 lim d n→∞ n→∞ α − 1 ρ =

 1  exp (α − 1)RTα e−ρu − 1 α−1



References 1. P. Aldous, P. Diaconis, Shuffling cards and stopping times. Am. Math. Mon. 93(5), 333–348 (1986) 2. D. Aldous, P. Diaconis, Strong uniform times and finite random walks. Adv. Appl. Math. 8, 69–97 (1987) 3. J. Barrera, B. Lachaud, B. Ycart, Cut-off for n-tuples of exponentially converging processes. Stoch. Process. Appl. 116, 1433–1446 (2006) 4. N. Champagnat, D. Villemonais, Exponential convergence to quasi-stationary distribution and Q-process. Probab. Theory Relat. Fields 164, 243–283 (2016) 5. M.F. Chen, Equivalence of exponential ergodicity and l 2 -exponential convergence for Markov chains. Stoch. Process. Appl. 87, 281–297 (2000) 6. M.F. Chen, From Markov Chains to Non-equilibrium Particle Systems (World Scientific Publishing Co. Inc., Singapore, 2004) 7. G.-Y. Chen, L. Saloff-Coste, The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13(3), 26–78 (2008) 8. A. Cichocki, S. Amari, Families of alpha-beta- and gamma- divergences: flexible and robust measure of similarities. Entropy 12(6), 1532–1568 (2010) 9. B. Cloez, M. Hairer, Exponential ergodicity for Markov processes with random switching. Bernoulli 21(1), 505–536 (2015) 10. P. Diaconis, The cutoff phenomenon in finite Markov chains. Proc. Nat. acad. Sci. U.S.A 93(4), 1659–1664 (1996) 11. A. Diédhiou, P. Ngom, Cutoff time based on generalized divergence measure. Stat. Prob. Lett. 79(10), 1343–1350 (2009) 12. D. Down, S.P. Meyn, R.L. Tweedie, Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23(4), 1671–1691 (1995) 13. A. Freslon, Cut-off phenomenon for random walks on free orthogonal quantum groups. Probab. Theory Relat. Fields 174, 731–760 (2019) 14. R.E. Funderlic, C.D. Meyer Jr., Sensitivity of the stationary distribution vector for an ergodic Markov chain. Linear Algebra Appl. 76, 1–17 (1986) 15. Y.-D. Kim A. Cichocki, H. Lee, S. Choi, Non-negative matrix factorization with α-divergence. Pattern Recogn. Lett. 29, 1433–1440 (2008) 16. B. Lachaud, Cut-off and hitting times for a sample of Ornstein-Uhlenbeck processes and its average. J. Appl. Probab. 42(4), 1069–1080 (2005) 17. S. Martinez, B. Ycart, Decay rates and cut-off for convergence and hitting times of Markov chains with countably infinite state space. Adv. Appl. Probab. 33(1), 188–205 (2001) 18. P. Ngom, B.D.B. Diatta, Model selection of stochastic simulation algorithm based on generalized divergence measures. Br. J. Math. Comput. Sci. 4(24), 3387–3402 (2014) 19. B. Póczos, J. Schneider, On the Estimation of α-divergences. J. Mach. Learn. Res. 15, 609–617 (2011)

Cut-off Phenomenon

117

20. R.D. Reiss, Approximate distributions of Order Statistics. Springer Series in Statistics (Springer, Berlin, 1989) 21. G.O. Roberts, R.L. Tweedie, Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 4, 341–363 (1996) 22. J.S. Rosenthal, On generalizing the cut-off phenomenon for random walks on groups. Adv. Appl. Math 16(3), 306–320 (1995) 23. L. Saloff-Coste, Random walks on finite groups. Encyclopaedia Math. Sci. 110, 263–346 (2004)

Stochastic Optimization in Population Dynamics: The Case of Multi-site Fisheries Sidy Ly and Diaraf Seck

Abstract In this paper we build and study stochastic models governing fisheries activities. The evolution of population is described by stochastic differential equations derived from the proposed model. By using tools of dynamic programming we also derive some Hamilton Jacobi Bellman equations that we will study in the theoretical and numerical point of view. Keywords Stochastic differential equations · Stochastic optimization · Dynamic programming principle · Hamilton-Jacobi-Bellman equations · Numerical simulations Mathematics Subject Classification (2010) 49L20, 49L20, 65C30, 35R60, 65N06

1 Introduction A stochastic differential equation appears as a generalization of differential equation with a term of white noise. Stochastic differential equations are used to model random trajectories, such as stock prices or movements of particles or species such as fish subject to diffusion phenomena. They also treat theoretically and numerically problems arising from partial differential equations. Their fields of application are notably vast in physics, biology, financial mathematics but in dynamics of populations. It is in this respect that we will make an application. The aim of this S. Ly () University Cheikh Anta Diop of Dakar, FASEG LMDAN, Dakar, Senegal e-mail: [email protected] D. Seck University Cheikh Anta Diop of Dakar, FASEG LMDAN, Dakar, Senegal IRD, UMMISCO, Dakar, Senegal e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_5

119

120

S. Ly and D. Seck

paper is to solve stochastic optimization problems related to fishery in the case of a single site and also in the case of several sites. What we call here FADS is the sites (Fish Aggregating Devices) that is to say very sophisticated objects that have the power to attract fishes. This paper is organized in three major parts consisting of modeling, theoretical and numerical resolution of stochastic optimization problems. Modeling is inspired by Allen [1] on the basis of two cases. In the first case, we consider the sea as one site. As a result, we neglect certain internal movements. Based up on other hypotheses, we model the fishery in two stochastic differential equations; one governing the density of species and the other the fishing effort. In the second case, we put L sites in the sea and assume that the boats can move from one site to another as well as the fish. Modeling in this case leads us to a 2L + 1 system of stochastic differential equations which L + 1 represents those of fish and L those of boats. For stochastic optimization problems, for each of the two cases we maximize a functional representing the profits under the constraints of stochastic differential equations resulting from the modeling. Using dynamic principle programming leads us to Hamilton-Jacobi-Bellman equation whose solution is given by Ladyzhenskaya [2]. In numerical simulations, we use finite differences methods to solve partial differential equation given by dynamic principle programming.

2 Modeling 2.1 Single Site Case In this section, we want to model the fishery by considering two sites one for fishes the other for fishermen (boats) see Fig. 1. Let x(t) and E(t) represent respectively the size of fishes and fishing effort in the system at time t. It is assumed that in a small time interval t, x can change into −1, 0 or 1 and E into −1, 0 or 1. Let’s consider X = [x; E]T the change in a small time interval t. In this model, catching by other predators is not taken account and we don’t look at growth of fishermen’s population. We denote by b the per capita birth, d the per capita death, c cost per unit of fishing effort, q the catchability in the site and p the price. Under these assumptions, as illustrated in Fig. 1, there are four possible changes for the two states in the time interval t not including the case where there is no change in the time interval and neglecting multiple births, deaths or transformations in time t which have probabilities of order (t)2 . The possible changes and there probabilities are given in Fig. 2. Now we are interested in finding the mean change E(x) and the covariance matrix E(x(x)T ) for the time interval t. Neglecting

Stochastic Optimization in Population Dynamics

121

Fig. 1 Modeling with two sites Change T X1 = [1; 0] T 2 X = [− 1; 0] T 3 X = [0; 1] T X4 = [0; − 1] X5 = [0; 0]

T

Probability p1 = bx t p2 = dx t + qxE t p3 = pqxE t p4 = cE t 4

p5 = 1 −

pi i=1

Fig. 2 Possible changes in the population of the fishes with corresponding probabilities

terms of order (t)2 , we have

E(X) =

5 

⎡ pi Xi = ⎣

bx − dx − qxE

⎤ ⎦ t

pqxE − cE

i=1

and E(X(X)T ) =

5 

⎡ pi (Xi )(Xi )T = ⎣

bx + dx + qxE 0

i=1

0

⎤ ⎦ t.

pqxE + cE

We now define the expectation vector f and the 2 × 2 symmetric positive definite covariance matrix G. f(t, X) = E(X)/t

and

G(t, X) = E(X(X)T )/t.

122

S. Ly and D. Seck

Noticing that t is small and E((X)2 ) = O((t)2 ), the covariance matrix is set equal to E(x(x)2)/t. Referring to [1] that leads us to the stochastic differential equation system: ⎧ √ dx(t) = (bx − dx − qxE)dt + bx + dx + qxEdW1 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎨ dE(t) = (pqxE − cE)dt + pqxE + cEdW2 (t) (2.1)

⎪ ⎪ ⎪ x(0) = x0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ E(0) = E0

The stochastic differential equation model for the dynamics of there interacting populations (2.1) can be rewritten as follow: dX(t) = f(t, X)dt + G1/2(t, X)dW(t) ⎡ with X(0) = X0 , G(t, X) = ⎣

bx + dx + qxE

0

(2.2) ⎤ ⎦ and W(t) =

0 pqxE + cE [W1 (t); W2 (t)] is the two-dimensional Brownian motions. T

2.2 3 Sites Case We put on the sea three sites. We suppose that fishes move randomly from the sea to a given site i and vice versa. They can also move from a site i to a site j = i and vice versa. We denote respectively by msi and mij the movements rate from the sea to the site i and from site i to the site j . We designate by xi (t) the size of the population at time t in the site i. We study the variation of the population x during an interval of time t. The time interval t is assumed to be sufficiently small that it can not have several births and several deaths at the same time in different sites. In this model, natural death and capture by other predators are not taken into account. The only way for fish to die is fishing. We denote by bi the per capita birth and qi the catchability in the site i (see Fig. 3). Under these assumptions, there are 21 possibilities for a population change x if we neglect multiple births, deaths or transformations in time t which have probabilities of order (t)2 . These possibilities are listed in the Fig. 4 along with their corresponding probabilities. The first component of the matrix x i represents the transformation of the population of fishes into the sea (off-site). The second, third and forth components represent respectively the transformation made into the site 1, 2 and 3. For example

Stochastic Optimization in Population Dynamics

Fig. 3 Evolution of fishery in the case of 3

Fig. 4 Possible changes in the population of the fishes with the corresponding probabilities

123

124

S. Ly and D. Seck

x 20 = [−1, 1, 0, 0]T represents the movement of one individual from the population in off-site xs to the population x1 during time interval t and the probability of this event is proportional to the size of the population xs and the time interval t that is, p20 = ms1 xs t. As a second example, x 10 = [0, 1, 0, 0]T represents a birth in the population x1 with as probability p10 = b1 x1 t. As third example, x 1 = [0, 0, 0, −1]T represents a death or the catching of one individual in site 3 and the probability of this event is proportional to the size of the population x3 , the fishing effort E3 and the time interval t that is, p1 = q3 x3 E3 t. It is 21  assumed that t > 0 is sufficiently small so that p21 > 0. Noticing that pi = 1. i=1

Now we are interested in finding the mean change E(x) =

21 

pi x i and the

i=1

covariance matrix E(x(x)T ) for the time interval t. Neglecting terms of order (t)2 , we have ⎡

bs xs − ds xs + m1s x1 + m2s x2 + m3s x3 − (ms1 + ms2 + ms3 )xs

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ b1 x1 − d1 x1 − q1 x1 E1 + ms1 xs + m21 x2 + m31 x3 − (m1s + m12 + m13 )x1 ⎥ ⎢ ⎥ ⎢ ⎥ E(x) = ⎢ ⎥ t ⎢ ⎥ ⎢ b2 x2 − d2 x2 − q2 x2 E2 + ms2 xs + m12 x1 + m32 x3 − (m2s + m21 + m23 )x2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ b3 x3 − d3 x3 − q3 x3 E3 + ms3 xs + m13 x1 + m23 x2 − (m3s + m31 + m32 )x3

and ⎡

δs ⎢a ⎢ 1s 21 ⎢  ⎢ E(x(x)T ) = pi x i (x i )T = ⎢ ⎢ a2s i=1 ⎢ ⎣ a3s

⎤ as1 as2 as3 δ1 a12 a13 ⎥ ⎥ ⎥ ⎥ ⎥ t. a21 δ2 a23 ⎥ ⎥ ⎦ a31 a32 δ3

E(x(x)T ) is a symmetric positive definite matrix with: δs = bs xs + ds xs + m1s x1 + m2s x2 + m3s x3 + (ms1 + ms2 + ms3 )xs , δ1 = b1 x1 + d1 x1 + q1 x1 E1 + ms1 xs + m21 x2 + m31 x3 + (m1s + m12 + m13 )x1 , δ2 = b2 x2 + d2 x2 + q2 x1 E2 + ms2 xs + m12 x1 + m32 x3 + (m2s + m21 + m23 )x2 , δ3 = b3 x3 + d3 x3 + q3 x3 E3 + ms3 xs + m13 x1 + m23 x2 + (m3s + m31 + m32 )x3 , as1 = a1s = −(ms1xs + m1s x1 ); as2 = a2s = −(ms2xs + m2s x2 ); as3 = a3s = −(ms3xs + m3s x3 ); a12 = a21 = −(m12 x1 + m21 x2 ); a13 = a31 = −(m13 x1 + m31 x3 ) and a23 = a32 = −(m23 x2 + m32 x3 ).

Stochastic Optimization in Population Dynamics

125

We now define the expectation vector f and the 4 × 4 symmetric positive definite covariance matrix g, as f(t, x1 , x2 , x3 ) = E(X)/t

and

g(t, x1 , x2 , x3 ) = E(X(X)T )/t.

Noticing that as t is small and E(X(X)T ) = O((t)2 ), the covariance matrix is set equal to E(X(X)T )/t. Refering to [1] (section 5 page 135) that leads us to the stochastic differential equation system: ⎧ ⎨ dX(t) = fdt + g 1/2 dW ⎩

(2.3) X(0) = X0

where W(t) = [Ws (t), W1 (t), W2 (t), W3 (t)]T . Now we are going to do the same thing for the boats. As for the fish, we start from a particular case of 3 sites and then generalize it to L sites. We assume that these 3 sites are deposited in the sea in order to attract the fish to be caught. To determine our model, we assume that we have only one boat that can move from one site i to another j with symmetric movements rates βij that is to βij = βj i for all i = j . Here we assume that we do not fish offsites and that on each site i there is a cost per unit of fishing effort ci and a catchability qi . In our model, we assume that we do not perform several costs at different sites at the same time and that we do not capture at different sites at the same time. The cost to leave a site i to go to another j is proportional to the distance between the two sites dij that means that ci = φij dij = φj i dj i where φj i > 0. Under these assumptions, there are also 13 changes for the variation of fishing effort E during a sufficiently small time interval t. These possibilities are listed in the Fig. 5 along with their corresponding probabilities. The first component of the matrix E i represents the evolution of the fishery in site 1. The second and third components represent respectively the evolution of fishery made into the site 2 and 3. For example E 12 = [−1, 1, 0]T represents the movement of the boat from site 1 to site 2 during time interval t and the probability of this event is proportional to the fishing effort E1 and the time interval t that is to say, p12 = β12 E1 t. As a second example, E 6 = [0, 1, 0]T represents a capture in site 2 with the probability proportional to the size of the population x2 , fishing effort E2 and time interval that is p6 = pq2 x2 E2 t where p represents the price of species. As a third example, E 4 = [0, 0, −1]T represents a cost per unit of fishing in site 3 and the probability of this event is proportional to the fishing effort E3 and the time interval t that is to say, p4 = c3 E3 t. It is assumed that t > 0 21  is sufficiently small so that p13 > 0. Noticing that pi = 1. i=1

126

S. Ly and D. Seck

Fig. 5 Possible changes in the population of the fishes with corresponding probabilities

Then neglecting terms of order (t)2 , the mean change E(E) and the covariance matrix E(E(E)T ) for the time interval t are given by: ⎡

β21 E2 + β31 E3 − (β12 + β13 )E1 + pq1 x1 E1 − c1 E1

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ E(E) = pi E = ⎢ β12 E1 + β32 E3 − (β21 + β23 )E2 + pq2 x2 E2 − c2 E2 ⎥ t ⎢ ⎥ i=1 ⎣ ⎦ 13 

i

β13 E1 + β23 E2 − (β31 + β32 )E3 + pq3 x3 E3 − c3 E3

and ⎡

γ1 b12 b13

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ E(E(E) ) = pi E (E ) = ⎢ b21 γ2 b23 ⎥ t. ⎥ ⎢ i=1 ⎦ ⎣ b31 b32 γ3 T

13 

i

i T

E(y(y)T ) is a symmetric positive definite matrix with: γ1 = β21 E2 + β31 E3 + (β12 + β13 )E1 + pq1 x1 E1 + c1 E1 , γ2 = β12 E1 + β32 E3 + (β21 + β23 )E2 + pq2 x2 E2 + c2 E2 , γ3 = β13 E1 + β23 E2 + (β31 + β32 )E3 + pq3 x3 E3 + c3 E3 ,

Stochastic Optimization in Population Dynamics

127

b12 = b21 = −(β12 E1 + β21 E2 ); b32 = b23 = −(β32E3 + β23 E2 ); b13 = b31 = −(β13 E1 + β31 E3 ); We now define the expectation vector f1 and the 3 ×3 symmetric positive definite covariance matrix g1 , as follow f1 (t, E1 , E2 , E3 ) = E(E)/t

and

g1 (t, E1 , E2 , E3 ) = E(E(E)T )/t.

Noticing that as t is small and E(E(E)T ) = O((t)2 ), the covariance matrix is set equal to E(E(E)T )/t. Referring to [1] (section 5 page 135) that leads us to the stochastic differential equation system: ⎧ 1/2 ⎪ ⎨ dY(t) = f1 dt + g1 dW1 ⎪ ⎩ Y(0) = Y 0

(2.4)

) *T Where W1 (t) = W11 (t), W12 (t), W13 (t) .

2.3 L Sites Case In this section, we want to model the fishery by considering the evolution of the resource (fish) and boat movement between different sites. For this, we put on the sea L sites where L is a positive integer (≥ 3). Here we call sites F.A.D (Fish aggregating devices). They are some objects that have power to attract fish. We suppose that fish move randomly from the sea to a given site i and vice versa. They can also move from a site i to a site j = i and vice versa. In this model, capture by other predators is not taken into account. We denote by bi the per capita birth, di per capita death in the site i and qi the catchability in the site i. We denote respectively by msi and mij movements rate from the sea to the site i and movements rate from site i to the site j . We designate by xi (t) the size of the population at time t in the site i and Ei (t) the fishing effort. To determine our model, we assume that we have only one boat that can move from one site i to another j with symmetric movements rates βij that is to say βij = βj i for all i = j . Here we assume that we do not fish outside the sites and that on each site i there is a cost per unit of fishing effort ci and a catchability qi . In our model, we assume that we do not charge several costs at different sites at the same time and that we do not capture at different sites at the same time (see Fig. 6). Under These assumptions, we have (L + 1)(L + 2) + 1 possibilities (L ≥ 2) corresponding to the evolution of fishery. With the same arguments as case of single site, we have stochastic model for a boat defined as follow:

128

S. Ly and D. Seck

Fig. 6 Pattern for L sites

⎧ dX(t) = fdt + g 1/2 dW ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1/2 ⎪ ⎨ dY(t) = f1 dt + g1 dW1 (2.5)

⎪ ⎪ ⎪ X(0) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Y(0) =

X0 Y0

where vectors X, Y, W, W1 , f and f1 are defined by: ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ 1⎞ ws xs E1 w1 ⎜ w1 ⎟ ⎜ x1 ⎟ ⎜ E2 ⎟ ⎜ w2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 1⎟ Xt = ⎜ . ⎟; Yt = ⎜ . ⎟; W = ⎜ . ⎟; W1 = ⎜ . ⎟; ⎝ .. ⎠ ⎝ .. ⎠ ⎝ .. ⎠ ⎝ .. ⎠ xL ⎡

EL

w1L

wL L 

⎤

L 

bs xs − ds xs + ⎢ ⎥ mis xi − msi xs ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ i=1 i=1 fs ⎢ ⎥ ⎢ ⎥ ⎢ ⎥   ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ b1 x1 − d1 x1 − q1 n1 E1 + ms1 xs − m1s x1 + mi1 xi − m1i x1 ⎥ ⎢ ⎥ ⎢ f1 ⎥ ⎢ ⎥ ⎢ ⎥ i = 1 i = 1 f=⎢ ⎥=⎢ . ⎥ ⎢ ⎥ ⎢ .. ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ fL L−1 L−1 ⎢ ⎥ ⎣b x − d x − q n E + m x − m x +  m x −  m x ⎦ L L L L L L L sL s Ls L iL i Li L i=1

i=1

Stochastic Optimization in Population Dynamics

129

and ⎡

L 

⎤

L 

βi1 Ei − β1i E1 + pq1 x1 E1 − c1 E1 ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ f11 ⎥ ⎢ i=2 i=2 ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢  ⎥  ⎥ ⎢ 2⎥ ⎢ ⎢ βi2 Ei − β2i E2 + pq2 x2 E2 − c2 E2 ⎥ ⎢ f1 ⎥ ⎥=⎢ . ⎥ f1 = ⎢ ⎥ ⎢ . ⎥ ⎢ i=2 i=2 ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎢ ⎥ ⎥ ⎣ ⎢ ⎦ . ⎥ ⎢ L−1 L L−1 ⎥ ⎢  f1 ⎦ ⎣ βiL Ei − βLi EL + pqL xL EL − cL EL i=1

i=1

The symmetric positive definite matrices g and g1 are given by: ⎤ δs as1 . . . asL ⎢ a1s δ1 . . . a1L ⎥ ⎥ ⎢ g=⎢ . . . . ⎥ ⎣ .. .. . . .. ⎦ aLs aL1 . . . δL ⎡

where δs = b s x s + d s x s +

L 

mis xi +

i=1

δi = bi xi −d1i xi +qi ni Ei +msi xs +mis xi +



mj i xj +

j =i

aij = −(mij xi + mj i xj )

L 

msi xs

i=1



mij xi

for i, j = 1, . . . , L

j =i

∀i = j

and ⎤ γ1 b12 . . . b1L ⎢ b21 γ2 . . . b2L ⎥ ⎥ ⎢ g1 = ⎢ . . . . ⎥ ⎣ .. .. . . .. ⎦ bL1 bL1 . . . γL ⎡

where γi =

 j =i

βj i Ej +

 j =i

βij Ei + pqi xi Ei + ci Ei

for i, j = 1, . . . , L.

130

S. Ly and D. Seck

and bij = −(βij Ei + βj i Ej )

∀i = j

3 Stochastic Optimization 3.1 Position of the Problem In this section, we want to maximize the profits defined as the difference between the total capture in the different sites and the costs related to the fishing activity. The functional governing these profits is defined as the average value of all benefits over a time interval [0; T ] T > 0 and it is given by ⎡ J [Z(t), E (.)] = E ⎣

 0

T

⎛ ⎞ ⎤ L L    ⎝ pqi xi Ei − cj Ei ⎠ dt ⎦ . i=1

i=1 j =i

where vector Z(t) = Zt = (Xt ; Et ), (X(t); E(t)) = (Xt ; Et ) and cj = φij dij represents the cost to leave a site i to go to another j and it is proportional to the distance between the two sites dij . In the development of this functional, we do not take into account the costs incurred by the boats to leave the beach and go to the first fishing site as well as the return costs of the boats to the beach after fishing. Then the stochastic optimization problem is to maximize J [Z(t), E (.)] under the constraints: ⎧ dXt = fdt + g 1/2 dW ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1/2 ⎪ ⎨ dEt = f1 dt + g1 dW1 (3.1) ⎪ ⎪ ⎪ X(0) = X0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ E(0) = E0 with Z(t) = Zt = (Xt ; Et ), (3.1) can be rewittren as follow: ⎧ ⎨ dZt = Fdt + Gdζ ⎩  where: F =

Z(0) = (X0 , E0 )

  1    g2 0 W f . ;ζ = and G is matrix G = 1 W1 f1 0 g12

Stochastic Optimization in Population Dynamics

131

This leads us to the following stochastic optimization problem that we want maximize the functional: ⎡ ⎛ ⎞ ⎤  T  L L   ⎝ pqi xi Ei − cj Ei ⎠ dt ⎦ (3.2) J [Zt , E (.)] = E ⎣ 0

i=1 j =i

i=1

under the counstraints: ⎧ ⎨ dZt = Fdt + Gdζ ⎩

(3.3) Z(0) = (X0 , E0 )

3.2 Dynamic Programming Principle To solve the problem defined by (3.2) and (3.3), let U (Z, t), known as the value function, be the excepted value of the objective function (3.2) from t to T when an optimal policy is followed from t to T , given Zt = z: ⎡ U (z, t) =

max

p(.)∈[pmin ;pmax ]

E⎣



⎛ ⎞ ⎤ L L    ⎝ pqi xi Ei − cj Ei ⎠ ds ⎦ .

T

t

(3.4)

i=1 j =i

i=1

This functional gives the remaining optimal cost by assuming that we arrive at Zt in time t < T . The final condition imposed by function U is: U (z, T ) = 0

(3.5)

Then, by the principle of optimality, U (z, t) =

max

p(.)∈[pmin ;pmax ]

⎡  E⎣

t+dt

⎛ ⎞ ⎤ L L    ⎝ pqi xi Ei − cj Ei ⎠ ds + U (z + dZt , t + dt)⎦ .

t

i=1

i=1 j =i

(3.6) By Taylor’s expansion, we have: ∂U ∂U ∂U dXt + dEt + dt U (z + dZt , t + dt) = U (z, t) + ∂X ∂E ∂t 

1 Uxx (dXt )2 + UEE (dEt )2 + Ut t (dt)2 + o(dt)2 + 2 (3.7)

132

S. Ly and D. Seck

We have: ⎛ dZt = ⎝

dXt

⎞

⎛

⎠=⎜ ⎝

dEt

fdt + g 1/2 dW f1 dt

1/2 + g1 dW1

⎞ ⎟ ⎠

and (dXt )2 = f2 (dt)2 + 2fg 1/2dWdt + g(dW)2 1/2

(dEt )2 = f1 2 (dt)2 + 2f1 g1 dW1 )dt + g1 (dW1 )2 By using the Brownian motion properties: E(dWt ) = 0 and E(dWt )2 = dt then we have: ⎛ E(dZt ) = ⎝

fdt

⎞ ⎠

(3.8)

f1 dt and E(dXt )2 = f2 (dt)2 + gdt

E(dEt )2 = f1 2 (dt)2 + g1 dt.

(3.9)

Computing the expectation and Substituting (4.9) and (3.9) into (3.7), we have: E(U (z + dZt , t + dt)) = U (z, t) + (Uz .F)dt +

1 ∂U dt + T rUzz .G2 dt. ∂t 2

(3.10)

Substitute (3.10) into (3.4), we obtain: ⎛ ⎞ ⎤ L L    ⎝ ⎢ pqi xi Ei − cj Ei ⎠ ds + U (z, t) ⎥ ⎥ ⎢ t ⎥ ⎢ i=1 i=1 j =i ⎦ ⎣ 1 ∂U 2 dt + T rUzz .G dt +(Uz .F)dt + ∂t 2 ⎡ U (z, t) =

max

p(.)∈[pmin ;pmax ]

t +dt

We assume that dt is sufficiently small that we have:

U (z, t) =

max

p(.)∈[pmin ;pmax ]

⎞ ⎡⎛ ⎤ L L    ⎢⎝ ⎥ pqi xi Ei − cj Ei ⎠ dt ⎢ ⎥ ⎢ i=1 ⎥ i=1 j=i ⎣ ⎦ ∂U 1 2 2 dt + T rUzz .G dt + o(dt) +U (z, t) + (Uz .F)dt + ∂t 2

(3.11) Note that we have suppressed the arguments of the functions involved in (3.11).

Stochastic Optimization in Population Dynamics

133

Cancelling the term U on both sides of (3.11), dividing the remainder by dt, and letting dt −→ 0, we obtain the Hamilton-Jacobi-Bellman (HJB) equation: ⎡ ⎤ L L    ∂U 1 ⎣ pqi xi Ei − cj Ei + Uz .F + T rUzz .G2 ⎦ = 0 + max ∂t p(.)∈[pmin ;pmax ] 2 i=1 j =i

i=1

(3.12) for the value function U (z, t) with the boundary condition U (z, T ) = 0

(3.13)

⎡ ⎤ ⎧ L L  ⎪   ∂U 1 ⎪ 2 ⎪ ⎪ pqi xi Ei − cj Ei + Uz .F + T rUzz .G ⎦ = 0 ⎨ ∂t + p(.)∈[pmax;p ] ⎣ 2 min max i=1 i=1 j =i ⎪ ⎪ ⎪ ⎪ ⎩ U (z, T ) = 0

(3.14) Noticing that terms in square brackets are continuous in p so the maximum always exists in [pmin ; pmax ]. We denote this maximum by p. So that our problem becomes: ⎧    L  L ⎪  ∂U 1 ⎪ 2 ⎪ ⎪ + U T rU = .F + .G c E − pqi xi Ei z zz j i ⎨ ∂t 2 i=1 j =i i=1 (3.15) ⎪ ⎪ ⎪ ⎪ ⎩ U (z, T ) = 0 For simplify calculations, we put index s = 0, δs = a00 and for i = 1 . . . L, δi = aii and γi = bii then Uz .F =

L  i=0

∂U  i ∂U + f1 ∂xi ∂Ei L

fi

i=0

and T rUzz .G2 =

L  i,j =0

L  ∂ 2U ∂ 2U aij + bij . ∂xi ∂xj ∂Ei ∂Ej i,j =1

134

S. Ly and D. Seck

So Eq. (3.15) becomes: ⎧ ⎞ ⎛ L L L L ⎪ 2U 2U     ⎪ ∂U ∂U ∂U ∂ ∂ 1 ⎪ ⎪ + fi + f1i + ⎝ aij + bij ⎠ ⎪ ⎪ ⎪ ∂t ∂xi ∂Ei 2 ∂xi ∂xj ∂Ei ∂Ej ⎪ ⎪ i=0 i=0 i,j =0 i,j =1 ⎪ ⎨ L  L   = cj E i − pqi xi Ei ⎪ ⎪ ⎪ ⎪ i=1 j =i i=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ U (z, T ) = 0 (3.16) By using variables changing s = T − t, the system (3.16) can be rewrite as follow ⎧ ⎛ ⎞ L L L L ⎪ 2V 2V     ⎪ ∂V ∂V ∂V ∂ ∂ 1 ⎪ ⎪ − + fi + f1i + ⎝ aij + bij ⎠ ⎪ ⎪ ⎪ ∂t ∂xi ∂Ei 2 ∂xi ∂xj ∂Ei ∂Ej ⎪ ⎪ i=0 i=0 i,j =0 i,j =1 ⎪ ⎨ L  L   cj E i − pqi xi Ei = ⎪ ⎪ ⎪ ⎪ i=1 j =i i=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V (z, 0) = 0 (3.17) where V (z, s) = U (z, T − s). Before to give the solution of problem (3.17), let us recall for a nonintegral positive number l that C l;l/2 () is Banach space of functions u(x; t) that are continuous in  with all derivatives of the form Dtr Dxs for 2r + s < l and have finite norm: (l)

(l)

|u| =< u > +

|l| 

(j )

< u >

j =0

where (j )

(0)

< u > = max|u|; < u > = 



(0)

|Dtr Dxs (u)|

2r+s=j

(l) (l) < u >(l)  =< u >x, + < u >t, (l)

< u >x, =



(l=|l|)

< Dtr Dxs (u) >x,

2r+s=|l| (l/2)

< u >x, =

 0 0 and 0 < l. All coefficients belong to the class C l;l/2() then problem (3.17) has a unique solution from the class C 2+l;1+l/2(). It satisfies the inequality:

|V |(l+2) 

(l)    L L      < k cj E i − pqi xi Ei    i=1 j =i i=1

(3.18)



where k is a constant not depending on h. Proof Coefficients are C ∞ () and holderian exponent 1. So that by using theorem (5.1) page(320) in [2], we show that problem(3.17) has a unique solution from the class C 2+l;1+l/2(). It satisfies the inequality (3.18). 

4 Stochastic Optimization and Numerical Simulations in the Case of Single Site This section is devoted to stochastic optimization and numerical simulations in the case of single site whose modeling is done in Sect. 1.

4.1 Stochastic Optimization Before solving stochastic optimization problem, let us recall stochastic differential equations system obtained in Sect. 1 ⎧ √ dx(t) = (bx − dx − qxE)dt + bx + dx + qxEdW1 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎨ dE(t) = (pqxE − cE)dt + pqxE + cEdW2 (t) (4.1)

⎪ ⎪ ⎪ x(0) = x0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ E(0) = E0

The stochastic differential equation model for the dynamics of there interacting populations (4.1) can be rewritten as form: dX(t) = b(t, X)dt + G(t, X)dW(t)

(4.2)

136

S. Ly and D. Seck

⎡ with X(0) = X0 , G(t, X) = ⎣

bx + dx + qxE

0

⎤ ⎦ and W(t) =

0 pqxE + cE [W1 (t); W2 (t)]T is the two-dimensional Brownian motions. Now we show that this system (4.1) has a unique solution. For this, let  = [xmin ; xmax ]×[Emin ; Emax ] and A = [pmin ; pmax ] where xmin , xmax , Emin , Emax , pmin and pmax are positive real numbers such that xmin < xmax , Emin < Emax and pmin < pmax . Theorem 4.1 Assume that X0 is independent of the future of the Brownian motion beyond time t = 0 and for any t ∈ [0; +∞), p ∈ A and for X ∈  a progressively measurable process such that, for any T > 0 

T

E

 |Xs |2 ds < +∞

t

then system (4.1) has a unique solution.     x x1 in  and p ∈ A. To prove the theorem, we Proof Let X = ,Y = E1 E show that: |b(t, X, p) − b(t, Y, p)| + |G(t, X, p) − G(t, Y, p)| ≤ K |X − Y| for t ∈ [0; +∞). We have |b(t, X, p) − b(t, Y, p)| = max (|(b − d)(x − x1 ) − q(xE − x1 E1 )| ; |pq(xE − x1 E1 ) − c(E − E1 )|) ≤ K1 |X − Y| where K1 = max(|b − d| + qEmax+ qxmax ; pqEmax + cxmax )  √bx + dx + qxE − √bx1 + dx1 + px1 E1  ; |G(t, X, p) − G(t, Y, p)| = max √  √  pqxE + cE − pqx1 E+ cE1  ≤ K2 |X − Y| .  b + d + q(xmax + Emax ) pq(xmax + Emax + c) where K2 = max √ ; √ . 2 (b + d + qEmax )xmax 2 (pqxmax + c)Emax When we choose K = max(K1 ; K2 ) that end the proof. 



In Fig. 7, we plot solution of system (4.1). blue and red curves respectively represent biomass and fishing in the deterministic and stochastic cases with initial conditions x(0) = 10 and E(0) = 5. We notice that when we increase fishing effort to a maximal value, biomass considerably decreases that carries away extinction of species. So when species become rare, fishing effort also decreases and goes to zero.

Stochastic Optimization in Population Dynamics

137

18 16 14 12

E(t)

10 8 6 4 2

x(t)

0 0

10

20

30

40

50

60

70

80

90

100

Fig. 7 Representation of the solution of system (4.1). We plot the solution for T = 100, b = 0.05, d = 0.02, q = 0.01, p = 1, and c = 0.02

4.1.1 Position of the Problem In this section, we want to maximize the profits defined as the difference between the total capture in the different sites and the costs related to the fishing activity. The functional governing these profits is defined as the average value of all benefits over a time interval [0; T ] T > 0 and it is given by 

T

J [xt , E (.)] = E

 (pqxE − cE) dt .

(4.3)

0

where positive constants p and c are respectively price and cost per unit of fishing effort. Then the stochastic optimization problem is to maximize J [xt , E (.)] under the constraints ⎧ √ ⎨ dx(t) = f (t, x)dt + g(t, x)dW (4.4) ⎩ x(0) = x0 where f (t, x) = bx(t)−dx(t)−qx(t)E(t), g(t, x) = and W is Brownian motion.

√ bx(t) + dx(t) + qx(t)E(t)

4.1.2 Dynamic Programming Principle To solve the problem defined by (4.3) and (4.4), let U (x, t), known as the value function, be the excepted value of the objective function (4.3) from t to T  U (x, t) =

max

E(.)∈[Emin ;Emax ]

T

E t

 (pxE − cE) ds .

(4.5)

138

S. Ly and D. Seck

This functional gives the remaining optimal cost by assuming that we arrive at xt in time t < T . The final condition imposed by function U is: U (x, T ) = 0

(4.6)

Then, by the principle of optimality,  U (x, t) =

max

E(.)∈[Emin ;Emax ]

t +dt

E

 , t + dt) . − cE) ds + U (x + dx (pqxE t

t

(4.7) By Taylor’s expansion, we have: U (x + dxt , t + dt) = U (x, t) +

 1 ∂U ∂U dxt + dt + Uxx (dxt )2 + Utt (dt)2 + o(dt)2 ∂x ∂t 2

(4.8) We have: dxt = f dt + g 1/2 dW

(dxt )2 = f 2 (dt)2 + 2fg 1/2 dW dt + g(dW )2 .

and

By using the Brownian motion properties: E(dWt ) = 0 and E(dWt )2 = dt then we have: E(dxt ) = f dt

(4.9)

E(dxt )2 = f 2 (dt)2 + gdt.

(4.10)

and

Computing the expectation and Substituting (4.9) and (4.10) into (4.8), we have: E(U (x + dxt , t + dt)) = U (x, t) + (Ux f )dt +

1 ∂U dt + Uxx gdt + terms in (dt)2 . ∂t 2

(4.11) Substitute (4.11) into (4.5), we obtain: ⎡ U (x, t) =

max

E(.)∈[Emin ;Emax ]

⎢ ⎣

⎤ (pqxE − cE) ds + U (x, t) + (Ux f )dt + ⎥ ⎦ ∂U 1 dt + Uxx .gdt + terms in (dt)2 ∂t 2

t +dt t

Stochastic Optimization in Population Dynamics

139

We assume that dt is sufficiently small that we have: ⎡ U (x, t) =

max

E(.)∈[Emin ;Emax ]

⎤ (pqxE − cE) dt + U (x, t) + (Ux f )dt + ⎣ ⎦ 1 ∂U dt + Uxx gdt + o(dt)2 ∂t 2 (4.12)

Note that we have suppressed the arguments of the functions involved in (4.12). Cancelling the term U on both sides of (4.12), dividing the remainder by dt, and letting dt −→ 0, we obtain the Hamilton-Jacobi-Bellman (HJB) equation:   ∂U 1 + max pqxE − cE + Ux f + Uxx .g = 0 ∂t 2 E(.)∈[Emin ;Emax ]

(4.13)

for the value function U (x, t) with the boundary condition U (x, T ) = 0.

(4.14)

Substituting f and g by their value in (4.13), we have: ∂U + max ∂t E(.)∈[Emin ;Emax ]

  1 ∂2U ∂U + (bx + dx + qxE) 2 = 0 pqxE − cE + (bx − dx − qxE) ∂x 2 ∂x

(4.15) Noticing that terms in square brackets are continuous in E so the maximum always exists in [Emin ; Emax ] where Emin and Emax are positive real values that verify Emin < Emax . We notice this maximum by E. So that our problem becomes ⎧ 2 ⎪ ⎪ ∂U + 1 (bx + dx + qxE) ∂ U + (bx − dx − qxE) ∂U = cE − pqxE ⎨ ∂t 2 ∂x 2 ∂x ⎪ ⎪ ⎩ U (x, T ) = 0 (4.16) 4.1.3 Existence and Uniqueness By using variables changing s = T −t, the system (4.16) can be rewritten as follow: ⎧ ∂V 1 ∂ 2V ∂V ⎪ ⎪ (x; s) − (bx + dx + qxE) 2 (x; s) − (bx − dx − qxE) (x; s) = pqxE − cE ⎨ ∂s 2 ∂x ∂x ⎪ ⎪ ⎩ V (x, 0) = 0

(4.17) where V (x; s) = U (x; T − s).

140

S. Ly and D. Seck

1 0.8 0.6 0.4 0.2 0 -0.2 5 4.5

4 3.5

3 2.5

2 1.5

1 0.5

0

0

1

2

3

4

5

6

7

8

9

10

(a) 1.2 1 0.8 0.6 0.4 0.2 0 5 4.5

4 3.5

3 2.5

2 1.5

1 0 0.5 5

0

0

1

2

3

4

5

6

7

8

9

10

(b) Fig. 8 Representation of solution of (4.16) for b = 0.01, d = 0.02, q = 0.1, p = 0.08, c = 0.5, T = 10, a = 5, E = 0.2 (case (a)) and E = 0.4 (case (b))

Let functions f ; g and h defined on [a; +∞) × [0; T ] by: f (x; t) = bx − dx − qxE; g(x; t) = bx + dx + qxE and h(x; t) = pqxE − cE. System (4.17) has a unique solution because it is a particular case of the system (3.17) and the solution is assured by theorem 3.1 and represented in Figs. 8 and 9.

4.2 Numerical Simulations In this section, we do numerical simulations of the solution (4.17) by using finite difference methods and assuming that b ≥ d + qE. For this, we must solve our partial differential equation in [0; a] × [0; T ] where a > 0 and T > 0. We respectively subdivide interval [0; a] and [0; T ] in N and in K equal intervals with T a and s = . respective steps x and s where x = N +1 K +1

Stochastic Optimization in Population Dynamics

141

30 20 10 0 -10 -20 5 4.5

4 3.5

3 2.5

2 1.5

1 0.5

0

1

0

8

9

10

5

7

2

4

6

3

8

9

10

5

7

2

4

6

3

(c)

1500 1000 500 0 -500 -1000 5 4.5

4 3.5

3 2.5

2 1.5

1 0.5

0

1

0

(d) Fig. 9 Representation of solution of (4.16) for b = 0.01, d = 0.02, q = 0.1, p = 0.08, c = 0.5, T = 10, a = 5, E = 0.7 (case (c)) and E = 1 (case (d))

So for i ∈ {0, . . . , N + 1} and j ∈ {0, . . . , K + 1}, we have xi = ix and j sj = j s. We designate V (xi ; sj ) by Vi for all i ∈ {0, . . . , N + 1} and j ∈ {0, . . . , K + 1}. With initial condition at t = 0 of the problem, we have V (xi , 0) = Vi0 = 0 for i ∈ {0, . . . , N + 1}. So we have (N + 2)(K + 2) equations for (N + 2)(K + 2) unknown. By discretizing, we have for all j ∈ {0, . . . , K}: for i = 0 j

j

j

V − 2V1 + V0 ∂ 2V (x; s) ≈ 2 2 ∂x (x)2 j

j

V − V0 ∂V (x; s) ≈ 1 ∂x (x) j +1

(4.18)

(4.19) j

V − V0 ∂V (x; s) ≈ 0 . ∂s (s)

(4.20)

142

S. Ly and D. Seck

for i ∈ {1, . . . , N} j

j

j

− 2Vi + Vi−1 V ∂ 2V (x; s) ≈ i+1 2 ∂x (x)2 j

(4.21)

j

V − Vi ∂V (x; s) ≈ i+1 ∂x (x) j +1

(4.22)

j

V − Vi ∂V (x; s) ≈ i . ∂s (s)

(4.23)

for i = N + 1 j

j

j

− 2VN + VN+1 V ∂ 2V (x; s) ≈ N−1 2 ∂x (x)2 j

(4.24)

j

V − VN+1 ∂V (x; s) ≈ N ∂x (x) j +1

(4.25)

j

V − VN+1 ∂V (x; s) ≈ N+1 . ∂s (s)

(4.26)

Substituting equations from (4.18) to (4.26) in (4.17), we have for all j ∈ {0, . . . , K}: for i = 0 " j " j # # j +1 j j j j V0 − V0 V2 − 2V1 + V0 V1 − V0 + A(x0 ) + B(x0 ) = C(x0 ) (s) (x)2 (x) (4.27) for i ∈ {1, . . . , N} " j " j j j # j# j Vi+1 − 2Vi + Vi−1 Vi+1 − Vi − Vi + B(xi ) + A(xi ) = C(xi ) (s) (x)2 (x) (4.28)

j +1

Vi

for i = N + 1 j +1

(s)

"

j

VN+1 − VN+1

+ A(xN+1 )

j

j

j

VN−1 − 2VN + VN+1 (x)2

#

" + B(xN+1 )

j

j

VN − VN+1 (x)

# = C(xN+1 )

(4.29)

Stochastic Optimization in Population Dynamics

143

where for i ∈ {0, . . . , N + 1} 1 A(xi ) = − (bxi + dxi + qExi ) 2 B(xi ) = −(bxi − dxi − qExi ) C(xi ) = pqExi − cE. With simple calculations, Eqs. (4.27), (4.28) and (4.29) respectively become: j +1

V0 j +1

Vi j +1

(4.30)

j

− Vi j j j + αi Vi+1 + βi Vi + γi Vi−1 = C(xi ) (s)

(4.31)

j

VN+1 − VN (s)

j

− V0 j j j + α0 V2 + β0 V1 + γ0 V0 = C(x0 ) (s)

j

j

j

+ αN+1 VN−1 + βN+1 VN+1 + γN+1 VN = C(xN+1 )

(4.32)

where α0 =

A(x0 ) −2A(x0) B(x0 ) A(x0 ) B(x0 ) ; β0 = + − ; γ0 = x x (x)2 (x)2 (x)2

  A(xi ) B(xi ) 2A(xi ) B(xi ) A(xi ) αi = ; βi = − ; γi = + + x x (x)2 (x)2 (x)2 and αN +1 =

A(xN +1 ) A(xN +1 ) B(xN +1 ) −2A(xN +1 ) B(xN +1 ) ; γN +1 = ; βN +1 = − + x x (x)2 (x)2 (x)2

When we introduce vectorial notations for all j ∈ {0, . . . , K + 1}: ⎞ j V0 ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ = ⎜ . ⎟. ⎜ . ⎟ ⎝ . ⎠ j VN+1 ⎛

V (j )

144

S. Ly and D. Seck

scheme (4.30), (4.31) and (4.32) can be under the following form: ⎧ j +1 = (I − sM) × V j + sC ⎨V ⎩

(4.33) j ∈ {0, . . . , K + 1}

where I is identity matrix ⎡

γ0 ⎢ γ1 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ . ⎢ . ⎢ . M=⎢ . ⎢ . ⎢ . ⎢ . ⎢ . ⎢ . ⎢ ⎢0 ⎢ ⎣0 0

β0 α0 0 . . . . . . . . . β1 α1 0 0 . . . . . . γ2 β2 α2 0 0 . . . .. . 0 γ β α 0 3

3

3

.. .. . . .. .

.. .. .. .. . . . . .. .. .. .. . . . . .. .. .. .. . . . . . . . . . . . . . 0 0 γN−1 ... ... ... ... 0 0 ... ... ... ... ... 0

... ... ...

... ... ...

0 0 0

... .. . .. . .. . βN−1 γN αN+1

...

0 .. . .. .

..

.

..

.

αN−1 βN γN+1

0 0 αN βN+1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, ⎡

C(x0 )

⎤

⎢ ⎥ ⎢ ⎥ ⎢ C(x ) ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ C=⎢ ⎥. ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ C(xN ) ⎥ ⎢ ⎥ ⎣ ⎦ C(xN+1 ) and Proposition 4.2 If norm of I −tM is less than 1 then scheme (4.33) is convergent for the norm .∞ . Where A∞ = max

1≤i≤n

with matrix A = (aij )1≤i,j ≤n

n  j =1

|aij |.

Stochastic Optimization in Population Dynamics

145

Proof Assume that norm of I − tM is less than 1, scheme (4.33) is equivalent to: for i = 0, j +1 j j j V0 = (1 − sγ0 )V0 − sβ0 V1 − sα0 V2 + sC(x0 ) . When we pass at norm, we have: j +1 j j j |V0 | ≤ |(1 − sγ0 )||V0 | + | − sβ0 ||V1 | + | − sα0 ||V2 | + s|C(x0 )| . With our assumption, we obtain: j +1

j

|V0

j

j

| ≤ |(1 − sγ0 )||V0 | + | − sβ0 ||V1 | + | − sα0 ||V2 | + s|C(x0 )| j ≤ (|1 − sγ0 | + |sβ0 | + |sα0 |)V0 ∞ + sC∞ j ≤ V ∞ + sC∞ (4.34) With same arguments, we show that for i = {1, . . . , N} j +1

|Vi

| ≤ V j ∞ + sC∞

(4.35)

and for i = N + 1, we have: j +1

|VN+1 | ≤ V j ∞ + sC∞ .

(4.36)

Equations (4.34), (4.35) and (4.36) show that for i = {0, . . . , N + 1} and j = {0, . . . , M} V j +1 ∞ ≤ V j ∞ + sC∞ .

(4.37)

An obvious recurrence gives us: V j ∞ ≤ V 0 ∞ + j sC∞ ≤ V 0 ∞ + T C∞

(4.38)

for all j = {0, . . . , M + 1} which shows that the scheme is stable and ends the proof. 

References 1. E. Allen, Modeling with Ito Stochastic Differential Equations (Springer, Dordrecht, 2007) 2. O.A. Ladyzhenskaya, V.A. Solonikov, N.N. Uralceva, Linear and Quasi linear Equations of Parabolic Type (AMS, Providence, 1968)

A Hurwitz Like Characterization of GUAS Planar Switched Systems Daouda Niang Diatta, Moussa Balde, and Aminata D. T. Keita

Abstract This paper deals with the stability problem for the planar linear switched system defined by the equations x(t) ˙ = u(t)A1 x(t) + (1 − u(t))A2 x(t), x = (x1 , x2 ) ∈ R2 .

(0.1)

where the real matrices A1 , A2 ∈ R2 are Hurwitz and u(.) : [0, +∞) −→ {0, 1} is a measurable function. We give a Hurwitz like characterization of globally uniformly asymptotically stable planar switched systems. Another contribution of this paper is a new version of the main result in Shorten and Narendra (Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable second order linear time-invariant systems, in Proceedings of the 1999 American Control Conference (1999), pp. 1410–1414) using real algebraic geometry tools. This new approach gives a Hurwitz like characterization of switched systems which share a same strict or large common quadratic Lyapunov function and improves the main result in Balde et al. (Int J Control 82(10):1882–1888, 2009). Keywords Planar switched systems · Asymptotic stability · Quadratic Lyapunov functions

This work was completed with the support of the NLAGA Simons Project. D. N. Diatta () Université Assane Seck de Ziguinchor, Ziguinchor, Senegal e-mail: [email protected] M. Balde Université Cheikh Anta Diop de Dakar, Dakar, Senegal e-mail: [email protected] A. D. T. Keita African Institute for Mathematical Sciences, Mbour, Senegal e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_6

147

148

D. N. Diatta et al.

1 Introduction Let A1 and A2 be two 2 × 2 real Hurwitz matrices. We focus on the problem of finding necessary and sufficient conditions on A1 and A2 under which the switched system (S): x(t) ˙ = u(t)A1 x(t) + (1 − u(t))A2 x(t), x = (x1 , x2 ) ∈ R2 is globally uniformly asymptotically stable (GUAS), with respect to a measurable switching functions u(.) : [0, ∞) −→ {0, 1}. This problem has been studied in [4] when both A1 and A2 are diagonalizable in C and in [1] when at least one among A1 and A2 is not diagonalizable. In both cases the stability conditions are given in terms of coordinate-invariant parameters. Unfortunately the parameters used in the diagonalizable case become singular in the nondiagonalizable one and therefore the two cases were studied separately. More recently, in [2], the authors unify the previous studies by reformulating them in terms of new invariants that permit to treat all cases at the same time. Hence, they reduce the cases to be studied from 24 to 6 cases. In this paper we reduce such cases to be studied from 6 to 1, therefore give a Hurwitz like characterization of globally uniformly asymptotically stable planar switched systems. Another contribution of our paper is a new version of the main result in [7] using real algebraic geometry tools. This new approach give a Hurwitz like characterization of switched systems which share a same strict or large common quadratic Lyapunov function and simplify the main result given in [2]. In Sect. 2, we remind some classical notions concerning stability of switched systems and introduce the Sturm-Habitcht theorem which will be very useful to set our stability results. In Sect. 3, we state our main result and present its proof thereafter.

2 Mathematical Preliminaries 2.1 Stability Notions Let Bδ ⊂ R2 be the ball of radius δ, centered at the origin. We denote by U the set of measurable functions defined on [0, +∞) with values in {0, 1}. Given x0 ∈ R2 , we denote by γx0 ,u(.) (.) the trajectory of (S) starting at x0 and corresponding to the control u(.). Definition 2.1 (Unboundness) The switched system (S) is unbounded at the origin if there exist a initial position x0 and a control u(.) such that the trajectory γx0 ,u(.) (t) goes to infinity as t goes to infinity.

A Hurwitz Like Characterization of GUAS Planar Switched Systems

149

Definition 2.2 (Uniform Stability) The switched system (S) is uniformly stable at the origin if and only if for any ball Bε one can find a ball Bδ such that for any control u(.) ∈ U and any initial position x0 ∈ Bδ , the trajectory γx0 ,u(.) (.) stay in the ball Bε . Definition 2.3 (Global Uniform Asymptotical Stability) The switched system (S) is globally uniformly asymptotically stable at the origin (GUAS for short) if it is uniformly stable at the origin and globally uniformly attractive, i.e, for any balls Bε and Bδ , for any initial position x0 ∈ Bδ , for any u(.) ∈ U, there exist a time T > 0 such that γx0 ,u(.) (t) stay in the ball Bε for every t ≥ T . According to the following proposition, the stability properties of the switched system (S) do not change if we allow measurable switching functions taking values in [0, 1] instead of {0, 1}. We will name convexified switched system, the switched system (S) with u(.) taking values in [0, 1]. Proposition 2.4 ([6]) The switched system (S) and its convexified has the same stability behavior. More precisely, the switched system (S) is GUAS (resp. uniformly stable or unbounded) if and only if its convexified is GUAS (resp. uniformly stable or unbounded). Definition 2.5 (Common Lyapunov Function) V : R2 −→ R+ is a common Lyapunov function (CLF for short) for the switched system (S) if V is continuous, positive definite and strictly decreasing along non-constant trajectories of (S). If V is a positive definite, continuous and non-increasing along non-constant trajectories of (S) function then V is called a non-strict common Lyapunov function. Definition 2.6 (Common Quadratic Lyapunov Function) V : R2 −→ R+ is a common quadratic Lyapunov function (CQLF for short) for the switched system (S), if it exists a 2 × 2 positive definite symmetric matrix P such that: • V (x) = x T P x; • AT1 P + P A1 and AT2 P + P A2 are negative definite. We recall that, for the switched system (S), the existence of a common Lyapunov function is equivalent to GUAS (see for instance [5]). Moreover the existence of a non-strict Lyapunov function guarantees the uniform stability of (S).

2.2 A Useful Real Algebraic Geometry Tool Hereafter we recall some classical results from real algebraic geometry namely the Sturm-Habitcht theorem which allows to determine precisely the number of real roots, in any interval [a, b] ⊂ R of a given polynomial P ∈ R[x]. This real algebraic classical tool will be very useful to characterize switched systems which share a common quadratic Lyapunov function.

150

D. N. Diatta et al.

Definition 2.7 (Sturm Sequence) Let P ∈ R[x], (fi (x)), i ∈ {1, .., s} a sequence of elements of R[x] and [a, b] an interval of R. By definition,(fi (x)), i ∈ {1, .., s} is a Sturm sequence associated to P on [a, b] if the following conditions are satisfied: • • • •

f1 = P ; fs does not vanish on the interval [a, b]; fi (α)fi+2 (α) < 0, for any i ∈ {1, .., s − 2}, α ∈ [a, b] such that fi+1 (α) = 0; for any α ∈ [a, b] such that f1 (α) = 0, there exist ε > 0 such that f1 (α + ε)f2 (α + ε) < 0 and f1 (α − ε)f2 (α − ε) < 0.

Definition 2.8 (Sign Variations) Let A = (a0 , . . . , an ) ∈ Rn+1 . We call variation of the sequence A, denoted by V (A) or V (a0 , . . . , an ), the number of pairs (i, i + k) with k > 0 such that ai ai+k < 0 and when k > 1, ai+r = 0 for r ∈ {1, .., k − 1}. Example Consider the sequence A = (1, −1, 0, 5, 3, 3, −7), its variation corresponds to the number of its sign variations hence V (A) = 3. Theorem 2.9 (Real Roots Counter [3]) Let P ∈ R[x] and (fi (x)), i ∈ {1, .., s} a Sturm sequence associated to P on [a, b] and consider the function w from R to N defined by: w(y) = V (f1 (y), . . . , fs (y)). The number of distinct real roots of P in (a, b] is equal to w(a) − w(b). For the effective computation of a Sturm sequence associated to P , the following theorem provides an algorithm based on the Euclidean division algorithm. Theorem 2.10 (Sturm-Habicht [3]) Let [a, b] be an interval of R, P ∈ R[x] a square-free polynomial, (f1 (x), . . . , fs (x)) the sequence of element of R[x] defined by: f1 = P ;

(2.1)

f2 = P ;

(2.2)

fi−2 = fi−1 qi − fi ,

(2.3)

for i > 2,

with deg(fi ) < deg(fi−1 ). Then the sequence (fi (x))i∈[1,s] is a Sturm sequence associated to P on [a, b]. Remark 2.11 Remark that for i > 2, −fi is the remainder of the Euclidean division of fi−2 by fi−1 and s is the smallest integer i such that fi does not vanish in [a, b].

A Hurwitz Like Characterization of GUAS Planar Switched Systems

151

3 Stability Behavior of a Switched Planar System In this section we remind the notations and objects that was used in [2] to state their stability result. We will use the same invariants to state our stability results which extend the ones given in [2]. In the following the word invariant will indicate any object which is invariant with respect to coordinate transformations. As usual, we denote by det(X) and tr(X) the determinant and the trace of a matrix X. If X ∈ R2 × R2 the discriminant is defined as: δX = tr(X)2 − 4 det(X)

(3.1)

Given a pair of matrices X and Y , we define the following invariant: (X, Y ) =

1 (tr(X) tr(Y ) − tr(XY )) 2

(3.2)

Since A1 and A2 are suppose to be two real Hurwitz matrices, so we have: det(A1 ), det(A2 ) ∈ R∗+ and tr(A1 ), tr(A2 ) ∈ R∗−

(3.3)

By means of these invariants the following invariants associated to (S) are defined: ⎧ tr(Ai ) ⎪ ⎪ √|δA | , if δA1 δA2 = 0 ⎪ ⎨ i tr(A )  0 τi := 6|δ i | , if δA1 δA2 = 0, but δAj = Aj ⎪ ⎪ ⎪ ⎩ tr(Ai ) 2 , if δA1 = δA2 = 0 k =

τ1 τ2 tr(A1 ) tr(A2 ) (2 tr(A1 tr(A2 ))

− tr(A1 ) tr(A2 ))

(3.4)

(3.5)

 = 4((A1 , A2 )2 − (A1 , A1 )(A2 , A2 ))

(3.6)

√ 2(A1 , A2 ) +  τ1 t1 +τ2 t2 R = √ e 2 det(A1 ) det(A2 )

(3.7)

where for i = 1, 2 ⎧ 

tr(A1 ) tr(A2 )(kτi +τ3−i ) π ⎪ √ , if δAi < 0 − arctan ⎪ ⎪ 2 ⎪

 √2  ⎨ 2τ1 τ2  , if δAi > 0 ti := arctanh tr(A √1 ) tr(A2 )(kτi −τ3−i ) ⎪ ⎪ 2  ⎪ ⎪ ⎩ tr(A1 A2 )− 1 tr(A1 ) tr(A2 )τi , if δAi = 0 2

We remind now the main stability result in [2] of Balde, Boscain and Mason.

(3.8)

152

D. N. Diatta et al.

Theorem 3.1 (Balde–Boscain–Mason) Consider the switched system (S). Then: √ √ 1. if (A1 , A2 ) > − det(A1 ) det(A2 ) and tr(A1 A2 ) > −2 det(A1 ) det(A2 ) then the switched √ system (S) admits a CQLF. √ If − det(A1 )√det(A2 ) < (A1 , A2 ) ≤ det(A1 ) det(A2 ) then the condition tr(A1 A2 ) > −2 det(A1 ) det(A2 ) is automatically satisfied. As a consequence the system admits √ a CQLF. 2. if (A1 , A2 ) < −√ det(A1 ) det(A2 ) then the switched system (S) is unbounded. 3. (A1 , A2 ) = − det(A1 ) det(A2 ) then the switched system (S) is uniformly stable but not GUAS. √ √ 4. if (A1 , A2 ) > det(A1 ) det(A2 ) and tr(A1 A2 ) ≤ −2 det(A1 ) det(A2 ) then the switched system (S) is GUAS uniformly stable (but not GUAS) or unbounded respectively if R < 1, R = 1, R > 1.

3.1 Statement of Our Main Result In this sub-section we state our mains results which characterizes completely GUAS two-dimensional bilinear switched systems like (S). Note that the Theorem 3.3 show that the first condition of Theorem 3.1 is not only sufficient but also necessary. Theorem 3.2 (Characterization of GUAS Switched Systems) The switched system (S) is globally uniformly asymptotically stable (GUAS) if and only if:

7  7 (A1 , A2 ) ∈ − det(A1 ) det(A2 ), det(A1 ) det(A2 ) cosh(τ1 t1 + τ2 t2 ) . Theorem 3.3 (Characterization of Switched Systems with a CQLF) The switched system (S) admits a common quadratic Lyapunov function if and only if  7  7 1 (A1 , A2 ) ∈ − det(A1 ) det(A2 ), tr(A1 ) tr(A2 ) + det(A1 ) det(A2 ) 2 Along the next two sub-sections, we will give, among other, the complete proof of our main result.

A Hurwitz Like Characterization of GUAS Planar Switched Systems

153

3.2 Strict or Large Common Quadratic Lyapunov Function For shortness we will denote by: d1 := det(A1 ), d2 := det(A2 ) T1 := tr(A1 ), T2 = tr(A2 ) n = tr(A1 A2 ) m = (A1 , A2 ) Since we have suppose A1 and A2 to be two real Hurwitz matrices, so we have: d1 , d2 ∈ R∗+ and T1 , T2 ∈ R∗−

(3.9)

By means of these invariants we can define the following invariants associated to (S): ⎧ ⎪ √Ti , if δA1 δA2 = 0 ⎪ ⎪ ⎨ |δAi | T τi := 6|δi | , if δA1 δA2 = 0, but δAj = 0 (3.10) Aj ⎪ ⎪ ⎪ ⎩ Ti , if δ = δ = 0 A1 A2 2 k =

τ1 τ2 T1 T2 (n − 2m)

R =

m+

and λ = t1 τ1 + t2 τ2

(3.11)

7

m2 − d 1 d 2 λ √ e d1 d2

(3.12)

where for i = 1, 2 ⎧   ⎪ T1 T2 (kτi +τ3−i ) π ⎪ √ , if δAi < 0 ⎪ 2 − arctan ⎪ 4 m2 −d1 d ⎪ 2 ⎨  √ ti := arctanh 4τ1 τ2 m2 −d1 d2 , if δA > 0 i T1 T2 (kτi −τ3−i ) ⎪ ⎪ ⎪ √ ⎪ ⎪ 4 m2 −d1 d2 ⎩ (n−2m)τi , if δAi = 0

(3.13)

3.2.1 Systems with a Strict Common Quadratic Lyapunov Function The following theorem of Narenda and Shorten, from [7], provides a necessary and sufficient condition for the existence of common quadratic Lyapunov function for the switched system (S). For any u ∈ [0, 1], let D(u) and N(u) be the following expressions: D(u) = det(uA1 + (1 − u)A2 ) and N(u) = det(uA1 + (1 − u)A−1 2 ) × d2

154

D. N. Diatta et al.

Theorem 3.4 (Narenda and Shorten) The switched system (S) admits a common quadratic Lyapunov function if and only if for any u ∈ [0, 1], D(u) > 0 and N(u) > 0. Lemma 3.5 ([2]) For any u ∈ [0, 1], we have: D(u) = (d1 + d2 − 2m)u2 + 2(m − d2 )u + d2 N(u) = (d1 d2 + 1 − n)u2 + (n − 2)u + 1 Thanks to the Theorem 3.4, we would like to express the necessary and sufficient condition for the existence of a common quadratic Lyapunov function “for any u ∈ [0, 1], D(u) > 0 and N(u) > 0” into necessary and sufficient condition with respect to our chosen invariants. Since from Lemma 3.5 it appears that D(u) and N(u) are polynomials, so one can use the Sturm-Habitcht Theorem 2.10 to study the sign variations of D(u) and N(u) on [0, 1]. Study of the Sign Variations of D(u), u ∈ [0, 1] It is clear that D(0) = d2 and D(1) = d1 . Since A1 and A2 are real Hurwitz matrices, it appears that D(0), D(1) ∈ R∗+ , so one can deduce directly the sign of D on [0, 1] by studying its number of real roots in [0, 1]. The main rule is the following: “D(u) > 0 on [0, 1] if and only if D has no root in [0, 1]”. And we use Sturm Habicht theorem to count the number of real roots of D(u) = (d1 + d2 − 2m)u2 + 2(m − d2)u + d2, u ∈ [0, 1]. Before we build a Sturm sequence associated to D(u) on [0, 1], let us point out some important remarks. 2 Remark 3.6 If d1 + d2 − 2m = 0 i.e m = d1 +d 2 , then D(u) = 2(m − d2 )u + d2 . As ∗ D(0), D(1) ∈ R+ and D is affine, therefore we have D(u) > 0 for all u ∈ [0, 1].

Suppose now that d1 + d2 − 2m = 0 and let f1 (u) = D(u), f2 (u) = D (u) = 2(d1 + d2 − 2m)u + 2(m − d2 ). Using the Euclidean division algorithm we obtain, 2 −d d 1 2 according to the Sturm-Habicht Theorem 2.10, f3 (u) = dm as the opposite 1 +d2 −2m of the remainder of the Euclidean division of f1 (u) by f2 (u). Since f3 (u) does not depend on u, so in our case s = 3. Remark 3.7 The sign variations of D(u) in [0, 1] depends on the number of roots of D in [0, 1] which depends, according to Theorem 2.9, on w(0) − w(1) where w(y) = V (f1 (y), f2 (y), f3 (y)). • f1 (0) = d2 , f2 (0) = 2(m − d2 ), f3 (0) = • f1 (1) = d1 , f2 (1) = 2(d1 − m), f3 (1) =

m2 −d1 d2 d1 +d2 −2m m2 −d1 d2 d1 +d2 −2m

We know that d1 , d2 ∈ R∗+ . Let us construct a sign table that will give us the sign of f2 (0), f2 (1) and f3 (0) = f3 (1) in terms of m. The particular values of m that should appear in the sign table are: d1 , d2 ,

7 d1 + d2 7 , − d1 d2 , d1 d2 2

A Hurwitz Like Characterization of GUAS Planar Switched Systems

155

It will be necessary to order these values before giving the sign table. That is the purpose of the next lemma. Lemma 3.8

√ √ 2 1. If d1 < d2 then − d1 d2 < d1 < d1 d2 < d1 +d < d2 . 2 √ √ d1 +d2 2. If d2 < d1 then − d1 d2 < d2 < d1 d2 < 2 < d1 . √ √ 2 = d1 . 3. if d1 = d2 then − d1 d2 < d2 = d1 d2 = d1 +d 2 √ √ 2 √ Proof Suppose d1 < d2 . Then ( d1 − d2 ) = d1 + d2 − 2 d1 d2 > 0, hence √ 2 2 d1 d2 < d1 +d 2 . Since√0 < d1 < d2 , we have d1 + d2 < 2d2 and d1 < d1 d2 , hence d1 +d2 < d2 and d1 < d1 d2 . The same arguments prove the second case. The third 2 point is obvious.  For the sign table of D, we will consider the previous three cases for making sure that we will not neglect any case. From the sign tables given in Figs. 1, 2 and 3, we obtain the following proposition. Proposition 3.9

√ 1. For all u ∈ [0, 1], D(u) > 0 if and only if m > − d1 d2 . 2. For all u ∈ [0, 1], √ D(u) ≥ 0 and there exists u0 ∈ (0, 1) such that D(u0 ) = 0 if and only if m = − d1 d2 . √ 3. D changes sign twice in [0, 1] if and only if m < − d1 d2 .

m

√ - d1d2

−∞

d1 + d2 − 2m

+

m2 − d1d2

+

√ d1d2

d1 +

+

0

−

−

0

0

−

−

0

−

f3

+

f2(0)

−

−

f2(1)

+

+

f1

+

+

0

d1 + d2 2

+

−

−

+

+

+

+

−

−

−

−

0

−

− +

+∞

d2

0

−

+

+ − +

+

w(0)

2

1

1

1

1

1

2

1

1

1

1

w(1)

0

0

1

1

1

1

2

1

1

1

1

w(0) − w(1)

2

1

0

0

0

0

0

0

0

0

0

Fig. 1 The sign table of D when d1 < d2

156

D. N. Diatta et al.

√ - d1d2

−∞

m d1 + d2 − 2m

+

m2 − d1d2

+

√ d1d2

d2 +

+

0

−

−

0

0

−

−

0

+

f3

+

f2(1)

+

+

f2(0)

−

−

f1

+

+

0

d1 + d2 2

+

−

−

+

+

+

+

−

−

+

+

0

+

+ +

+∞

d1

0

+

+

+

−

+

+

w(0)

2

1

1

1

1

0

0

1

1

1

1

w(1)

0

0

1

1

1

0

0

1

1

1

1

w(0) − w(1)

2

1

0

0

0

0

0

0

0

0

0

Fig. 2 The sign table of D when d2 < d1

m

− d1

−∞

d1 +

0

+∞

2(d1 − m)

+

m2 − d21

+

0

f3

+

0

f2(0)

−

−

0

+

f2(1)

+

+

0

−

f1

+

+

−

0

−

− + −

+

2

1

1

1

1

w(1)

0

0

1

1

1

w(0) − w(1)

2

1

0

0

0

w(0)

Fig. 3 The sign table of D when d1 = d2

A Hurwitz Like Characterization of GUAS Planar Switched Systems

157

Proof From the sign tables in Figs. 1, 2 and 3 it appears √ that D has respectively, √ in [0, 1]√zero, one and two distinct roots when m > − d1 d2 , m = − d1 d2 and m < − d1 d2 . Since D(0), D(1) ∈ R∗+ , it comes the announced results.  Study of the Sign Variations of N(u), u ∈ [0, 1] In this paragraph, we achieve the sign variations study of N(u) = (d1 d2 + 1 − n)u2 + (n − 2)u + 1 in [0, 1], by using the same strategy as in the previous sub-section. Before we build the Sturm sequence associated to N(u) on [0, 1], let us point out some important remarks. Remark 3.10 If d1 d2 − n + 1 = 0 i.e n = d1 d2 + 1, then N(u) = (d1 d2 − 1)u + 1. As N(0) = 1 > 0, N(1) = d1 d2 > 0 and N is affine, therefore we have N(u) > 0 for all u ∈ [0, 1]. Suppose now d1 d2 − n + 1 = 0 and let g1 (u) = N(u), g2 (u) = N (u) = 2(d1 d2 − n + 1)u + (n − 2). Using the Euclidean division algorithm we obtain, 2 1 d2 according to the Sturm-Habicht Theorem 2.10, g3 (u) = 4(dn 1−4d d2 −n+1) as the opposite of the remainder of the Euclidean division of g1 (u) by g2 (u). Since g3 (u) does not depend on u, so in our case s = 3. Remark 3.11 The sign variations of N(u) in [0, 1] depends on the number of roots of N in [0, 1] which depends, according to Theorem 2.9, on w(0) − w(1) where w(y) = V (g1 (y), g2 (y), g3 (y)). n2 −4d1 d2 4(d1 d2 −n+1) 2 1 d2 g3 (1) = 4(dn 1−4d d2 −n+1)

• g1 (0) = 1, g2 (0) = (n − 2), g3 (0) = • g1 (1) = d1 d2 , g2 (1) = 2d1d2 − n,

We know that d1 , d2 ∈ R∗+ . Let us construct a sign table that will give us the sign of g2 (0), g2 (1) and g3 (0) = g3 (1) in terms of n. The particular values of n that should appear in the sign table are: 7 7 2, 2d1 d2 , d1 d2 + 1, −2 d1 d2 , 2 d1 d2 It will be necessary to order these values before giving the sign table. That is the purpose of the next lemma. Lemma 3.12

√ √ 1. If d1 d2 < 1 then −2√d1 d2 < 2d1 d2√ < 2 d1 d2 < d1 d2 + 1 < 2. 2. If d1 d2 > 1 then −2 d√ 1 d2 < 2 < 2 d1 d2 < d1 d2 + 1 < 2d1 d2 . 3. If d1 d2 = 1 then 2 = 2 d1 d2 = d1 d2 + 1 = 2d1 d2 . Proof then (d1 d2 )2 < d1 d2 , hence 2d1 d2 < 1 d2 < 1. Since d1 d2 > 0√ √ Suppose d√ 2 2√d1 d2 . Since ( d1 d2 − 1) = d1 d2 − 2 d1 d2 + 1 > 0 and d1 d2 < 1, hence 2 d1 d2 < d1 d2 + 1 < 2. The same arguments prove the second case. The third point is obvious.  For the sign table of N, we will consider the previous three cases for making sure that we will not neglect any case. From the three previous sign tables we obtain the following proposition.

158

D. N. Diatta et al.

n

√

−∞

-2 d1d2

d1d2 − n + 1

+

n2 − 4d1d2

+

g3

+

g2(0)

√

2d1d2

2 d1d2

+

+

0

−

−

0

0

−

−

0

−

−

−

g2(1)

+

+

g1

+

+

0

d1d2 + 1 +

−

−

+

+

+

+

−

−

−

−

0

−

− +

+∞

2

0

−

−

+

+

+

+

w(0)

2

1

1

1

1

1

2

0

1

1

1

w(1)

0

0

1

1

1

1

2

0

1

1

1

w(0) − w(1)

2

1

0

0

0

0

0

0

0

0

0

Fig. 4 The sign table of N when d1 d2 < 1 n

√

−∞

-2 d1d2

d1d2 − n + 1

+

n2 − 4d1d2

+

g3

+

g2(1)

√

2

2 d1d2

+

+

0

−

−

0

0

−

−

0

+

+

+

g2(0)

−

−

g1

+

+

0

d1d2 + 1 +

−

−

+

+

+

+

−

−

+

+

0

+

+ +

+∞

2d1d2

0

+

+

+

−

+

+

w(0)

2

1

1

1

1

0

0

0

1

1

1

w(1)

0

0

1

1

1

0

0

0

1

1

1

w(0) − w(1)

2

1

0

0

0

0

0

0

0

0

0

Fig. 5 The sign table of N when d1 d2 > 1

Proposition 3.13

√ 1. For all u ∈ [0, 1], N(u) > 0 if and only if n > −2 d1 d2 . 2. For all u ∈ [0, 1], N(u) √ ≥ 0 and there exists u1 ∈ (0, 1) such that N(u1 ) = 0 if and only if n = −2 d1 d2 . √ 3. N changes sign twice in [0, 1] if and only if n < −2 d1 d2 .

A Hurwitz Like Characterization of GUAS Planar Switched Systems

n

−2

−∞

159

2 +

+∞

0

2− n

+

n2 − 4

+

0

g3

+

0

g2(0)

−

−

0

+

g2(1)

+

+

0

−

g1

+

+

−

0

−

− + −

+

2

1

1

0

1

w(1)

0

0

1

0

1

w(0) − w(1)

2

1

0

0

0

w(0)

Fig. 6 The sign table of D when d1 d2 = 1

Proof From the sign tables in Figs. 4, 5 and 6, it appears √ that N has respectively, √ in [0, 1]√zero, one and two distinct roots when n > −2 d1 d2 , n = −2 d1 d2 and n < −2 d1 d2 . Since N(0), N(1) ∈ R∗+ , it comes the announced results.  We remind in the next proposition the expression of the Theorem 3.3 and give its proof. Proposition 3.14 (Characterization of Switched Systems with a CQLF) The switched system (S) admits a common quadratic Lyapunov function if and only if  7  7 1 m ∈ − d 1 d 2 , T1 T2 + d 1 d 2 2 Proof From Theorem 3.4, the switched system (S) admits a common quadratic Lyapunov function if and only if for all u ∈ [0, 1], D(u) > 0 and N(u) > 0. From Proposition 3.9 and Proposition 3.13: 7 for all u ∈ [0, 1], D(u) > 0 if and only if m > − d1 d2 7 for all u ∈ [0, 1], N(u) > 0 if and only if n > −2 d1 d2 Since m = 12 (T1 T2 − n), it comes the announced result.



160

D. N. Diatta et al.

3.2.2 Systems with a Large Common Quadratic Lyapunov Function We will need the following lemma, from [7], to prove the Proposition 3.16 which characterize switched system (S) with a non-strict common quadratic Lyapunov function. Lemma 3.15 ([7]) Let A ∈ R2×2 be a Hurwitz matrices and consider the linear time invariant systems: SA : x˙ = Ax SA−1 : x˙ = A−1 x Then, any Lyapunov function for SA of the form V (x) = x T P x, is also a Lyapunov function for SA−1 . Proposition 3.16 (Switched Systems with a NCQLF) The switched system (S) admits a non-strict common quadratic Lyapunov function if 7 7 1 m = − d1 d2 or m = T1 T2 + d1 d2 2 √ Moreover, the switched system (S) is GUAS when m = 12 T1 T2 + d1 d2 and not √ GUAS when m = − d1 d2 . √ Proof Suppose m = − d1 d2 . According to Proposition 3.9, we have: for all u ∈ [0, 1], D(u) ≥ 0 and there exists an unique u0 ∈ (0, 1) such that D(u0 ) = 0. Therefore the system (S) cannot be GUAS. Because D(u0 ) = det(u0 A1 + (1 − u0 )A2 ) = 0 implies one of the eigenvalues of B = u0 A1 + (1 − u0 )A2 is equal to zero and the other one is equal to tr(B) = tr(u0 A1 + (1 − u0 )A2 ) = u0 tr(A1 ) + (1 − u0 ) tr(A2 ) < 0 since A1 and A2 are real Hurwitz matrices. Hence, if we consider the control function u(t) = u0 , for all t ∈ R+ , it appears that the convexified of the switched system (S) is not GUAS, so according to Proposition√ 2.4 the switched system (S) is not GUAS. It remains now to prove that, for m = − d1 d2 , the switched system (S) admits a non-strict common quadratic Lyapunov function. For this task we refer the reader to [2] where such a non-strict common quadratic Lyapunov function is given. √ Suppose now m = 12 T1 T2 + d1 d2 . The trick here is to consider the following switched system (S ∗ ): 2 x(t) ˙ = u(t)A1 x(t) + (1 − u(t))A−1 2 x(t), x = (x1 , x2 ) ∈ R

A Hurwitz Like Characterization of GUAS Planar Switched Systems

161

If we plane to analyze (S ∗ ) as we did with (S) in the previous sub-section, we ∗ shall study the sign variations of D ∗ (u) = det(uA1 + (1 − u)A−1 2 ) and N (u) = −1 det(uA1 + (1 − u)A2 ) × d2 . The key point is that for all u ∈ [0, 1]: D ∗ (u) = d2−1 × N(u) N ∗ (u) = d2 × D(u) √ Since m = 12 T1 T2 + d1 d2 corresponds, according to Proposition 3.13, to the √ existence of an unique u1 ∈ (0, 1) such that N(u1 ) = 0, so for m = 12 T1 T2 + d1 d2 we have D ∗ (u1 ) = 0. So by the same analyze as in the first part of the current √ proof we obtain that, when m = 12 T1 T2 + d1 d2 , the switched system (S ∗ ) is not GUAS but admits a non-strict common quadratic Lyapunov function (the one given in [2] adapted to the invariants of (S ∗ )). Let us call V such a non-strict common quadratic Lyapunov function. It comes that V is a non-strict quadratic Lyapunov function for SA1 and for SA−1 . Hence, according to Lemma 3.15, V is non-strict 2 quadratic Lyapunov function for SA2 . In conclusion if V is a non-strict quadratic √ Lyapunov function for SA1 and for SA2 , so when m = 12 T1 T2 + d1 d2 , V is a nonstrict common quadratic Lyapunov function for the switched system (S). It remains √ to prove that when m = 12 T1 T2 + d1 d2 the switched system (S) is GUAS to finish the proof. This task will be achieve in the next sub-section (more precisely in Remark 3.20). 

3.3 Global Uniform Asymptotic Stable Switched Systems The object of this sub-section is to establish the results announced in Theorem 3.2 We recall partially Theorem 3.1 from [2] in the next proposition. Proposition 3.17 (Balde–Boscain–Mason) Consider the switched system (S). Then: √ 1. if m < −√d1 d2 , then the switched √ system (S) is unbounded, 2. if m > d1 d2 and n ≤ −2 d1 d2 then the switched system (S) is GUAS, uniformly stable (but not GUAS) or unbounded respectively if R < 1, R = 1, R > 1.

162

D. N. Diatta et al.

√ We recall that m = 12 (T1 T2 − n). So n ≤ −2 d1 d2 if and only if m ≥ 12 T1 T2 + √ d1 d2 . Since T1 , T2 ∈ R∗− (because A1 and A2 are real Hurwitz matrices), the second point of the previous proposition can be reformulate simply as follow: √ if m ≥ 12 T1 T2 + d1 d2 then the switched system (S) is GUAS, uniformly stable (but not GUAS) or unbounded respectively if R < 1, R = 1, R > 1. √

m+

m2 −d d

1 2 λ √ We recall that R = e . Hence d1 d2 by squaring this last equality we obtain:

7 √ m2 − d1 d2 = e−λ R d1 d2 − m and

7 m2 − d1 d2 = d1 d2 e−2λ R2 − 2e−λ Rm d1 d2 + m2 Therefore m=

7

 d1 d2

e−λ R2 + eλ 2R

 (3.14)

In the followings, we study the variations of m with respect to the parameter R. Before we start this study, let us proof the following lemma which will be very useful. √ Lemma 3.18 For m ≥ 12 t1 t2 + d1 d2 we have the following relations: 1. λ = τ1 t1 + τ2 t2 ∈ R− . 2. R ≥ eλ . Proof From the relations in (3.4), it appears τ1 , τ2 ∈ R− . Let us now prove that t1 ≥ 0 and t2 ≥ 0. We recall that: ⎧   ⎪ T1 T2 (kτi +τ3−i ) π ⎪ √ , if δAi < 0 ⎪ 2 − arctan ⎪ 4 m2 −d1 d ⎪ 2 ⎨  √ ti := arctanh 4τ1 τ2 m2 −d1 d2 , if δA > 0 i T1 T2 (kτi −τ3−i ) ⎪ ⎪ ⎪ ⎪ √ 2 ⎪ ⎩ 4 m −d1 d2 , if δ = 0 Ai (n−2m)τi k =

τ1 τ2 T1 T2 (n −

2m)

 Since for any x ∈ R, arctan(x) ∈ − π2 , π2 , it comes ti ≥ 0 if δAi < 0. As m ≥ √ √ 1 ≤ n−T1 T2 −2 d1 d2 . We have m = 12 (T1 T2 −n), hence 2 T1 T2 + d1 d2, so n−2m √ n − 2m ≤ −2 m + d1 d2 < 0. Since τi ≤ 0, it comes that ti ≥ 0 if δAi = 0. Since n − 2m < 0, it appears that k < 0, hence (kτi − τ3−i ) > 0. Knowing that arctanh(x) ≥ 0 when x ≥ 0, it comes ti √ ≥ 0 if δAi > 0. This finish the proof of λ = τ1 t1 +τ2 t2 ∈ R− when m ≥ 12 T1 T2 + d1 d2 . Now let us prove by contradiction

A Hurwitz Like Characterization of GUAS Planar Switched Systems

163

√ m+ m2 −d1 d2 λ √ that R ≥ eλ . If we had R < eλ , we would have 0 < e < eλ . d1 d2

2 7 7 Thus m + m2 − d1 d2 < d1 d2 . Therefore (m2 − d1 d2 ) < −m m2 − d1 d2 , so 7 √ m2 − d1 d2 < −m that is impossible because m ≥ 12 T1 T2 + d1 d2 > 0.  Now let us study the sign variations of m with √ respect to R. As R is use in [2] to inspect the behavior when m ≥ 12 T1 T2 + d1 d2 > 0, then we can consider deservedly that R ≥ eλ , according to Lemma 3.18. Therefore the sign variations of m with respect to R will be studied in [eλ , +∞). If we differentiate m with respect to R we obtain:  −λ 2  dm 7 e R − eλ = d1 d2 (3.15) dR 2R2 λ λ Hence ddm R = 0 if and only if R = −e or R = e . According to Lemma 3.18 λ ∈ R− , so we obtain the following table: √ It comes out from this table that for m ≥ 12 T1 T2 + d1 d2 :

R1⇔m>

7 7 7

d1 d2 cosh(λ) d1 d2 cosh(λ) d1 d2 cosh(λ)

So we can reformulate the second point of the Proposition 3.17 into the following proposition (Fig. 7).

R

eλ

dm dR

0

1

+∞

+

+∞ √ d1d2 cosh(λ)

m √ d1d2

Fig. 7 Variations of m with respect to R

164

D. N. Diatta et al.

Proposition 3.19 Consider the switched system (S). Then: 8  √ √ 1. If m ∈ 12 T1 T2 + d1 d2 , d1 d2 cosh(λ) then the switched system (S) is GUAS. √ 2. If m = d1 d2 cosh(λ) then the switched system (S) is uniformly stable (but not GUAS).√ 3. If m > d1 d2 cosh(λ) then the switched system (S) is unbounded. Remark 3.20 The first √ point of the previous proposition, in the particular case when m = 12 T1 T2 + d1 d2 , complete the remaining part of the proof of the Proposition 3.16. If we gather Theorem 3.3, Theorem 3.4 and Proposition 3.19 we obtain our main result and its corollaries: Theorem 3.21 (Characterization of GUAS Switched Systems) The switched system (S) is globally uniformly asymptotically stable (GUAS) if and only if:

7  7 (A1 , A2 ) ∈ − det(A1 ) det(A2 ), det(A1 ) det(A2 ) cosh(τ1 t1 + τ2 t2 ) . Corollary 3.22 The switched system (S) is uniformly stable not GUAS if and only if: √ √ det(A1 ) det(A2 ) cosh (A1 , A2 ) = − det(A1 ) det(A2 ) or (A1 , A2 ) = (τ1 t1 + τ2 t2 ). Corollary 3.23 The √ switched system (S) is unbounded if and √ only if: det(A1 ) det(A2 ) cosh (A1 , A2 ) < − det(A1 ) det(A2 ) or (A1 , A2 ) > (τ1 t1 + τ2 t2 ) Acknowledgments Many thanks for the reviewers for their relevant comments and suggestions.

References 1. M. Balde, U. Boscain, Stability of planar switched systems: the nondiagonalizable case. Commun. Pure Appl. Anal. 7(1), 1–21 (2008) 2. M. Balde, U. Boscain, P. Mason, A note on stability conditions for planar switched systems. Int. J. Control 82(10), 1882–1888 (2009) 3. S. Basu, R. Pollack, M.F. Roy, Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10, 2nd edn. (Springer, Berlin, 2006) 4. U. Boscain, Stability of planar switched systems: the linear single input case. SIAM J. Control Optim. 41(1), 89–112 (2002) 5. W.P. Dayawansa, C.F. Martin, A converse Lyapunov theorem for a class of dynamical systems which undergo switching. IEEE Trans. Automat. Control 44(4), 751–760 (1999) 6. P. Mason, U. Boscain, Y. Chitour, Common polynomial Lyapunov functions for linear switched systems. SIAM J. Control Optim. 45(1), 226–245 (2006) 7. R. Shorten, K. Narendra, Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable second order linear time-invariant systems, in Proceedings of the 1999 American Control Conference (1999), pp. 1410–1414

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak Coura Baldé, Mountaga Lam, and Samuel Bowong

Abstract The aim of this study is to understand oral poliovirus vaccine (OPV) evolution and identify various biological aspects that substantially influence the spread of OPV virus leading to circulating vaccine derived poliovirus (cVDPV) outbreak. We also investigate the significance of vaccination transmission called contact vaccination. We develop a model of ordinary differential equation describing the dynamic of polio disease taking into account the evolution of OPV and pathogenic virus circulating in the environment (wild poliovirus WPV and cVDPV). Our model shows there exists a persistence threshold of OPV virus and a critical vaccination threshold !crit which is explicitly determined. Our model is governed by two states according the vaccination rate ! and the OPV persistence parameter τ . This means no epidemic whenever τ < 1 and ! ≥ !crit and epidemic if τ ≥ 1 for all vaccination rate !. Furthermore, we show that there existes a disease free equilibrium which is globally asymptotically stable (GAS) whenever τ < 1 and ! ≥ !crit while if τ < 1 and ! < !crit there exists a unique endemic equilibrium which is GAS. Moreover when τ > 1 there exists two endemic equilibria. Our analysis shows also that contact vaccination can decrease the critical vaccination threshold. However, OPV virus can spread extensively and then persist in the environment before the critical vaccination threshold is achieved. We also show at the endemic level OPV virus could be dominant in the environment compared to pathogenic virus. Our findings assess the usefulness of OPV vaccine but inherit the main problem of OPV mutation. Keywords Live attenuated virus · OPV virus evolution · Environment · Global stability

C. Baldé () · M. Lam Department of Mathematics and Computer Science, Faculty of Science and Technic, Cheikh Anta Diop University, Dakar, Senegal e-mail: [email protected] S. Bowong Faculty of Science, University of Douala, Douala, Cameroon © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_7

165

166

C. Baldé et al.

Mathematics Subject Classification (2010) 92D25, 92D30, 92B05

1 Introduction Epidemics of Polio disease strongly decreased since the launch of Global Polio Eradication Initiative in 1988. The disease was endemic in more than 125 countries with an estimated 350,000 cases annually [9]. To date, with the intensive use of live attenuated oral poliovirus vaccine (OPV), only three countries remain endemic Afghanistan, Pakistan and Nigeria. However the disease remains in concern. Between 2000–2006 Polio disease outbreaks were reported in Haiti and the Dominican Republic, Philippines, Madagascar, China, Cambodia, and Indonesia [8, 9]. A recombinant strain derived from OPV vaccine, named circulating vaccine derived poliovirus (cVDPV), was at the origin of these outbreaks. Indeed many studies have shown that, live attenuated strain contained in OPV vaccination evolve to pathogenic virus cVDPV through a process of reversion of the attenuating mutations or by recombination with other viruses of the same type [3, 6, 8, 13, 17]. Generally, those who receive OPV vaccine excrete the vaccine viruses during 3 to 4 weeks. Then OPV viruses start circulating in the community. However, OPV viruses may die out relatively quickly in a community with high immunization coverage rate because they cannot find sufficient numbers of susceptible individuals to continue transmission [5]. Although in the presence of sufficient numbers of fully susceptible in a community, OPV viruses can continue to circulate and evolve toward increased neurovirulence over time, which can lead to outbreaks of circulating vaccinederived polioviruses (cVDPVs) [5]. To date the way of how OPV viruses evolve to cVDPV is still unknown. No data track the genetic changes that occur in OPV sequences as these viruses transmit largely asymptomatically through real populations with suboptimal immunity [5, 17]. As the main strategy against Polio disease, vaccination with live attenuated OPV vaccine continues in many countries in routine immunizations or supplementary immunization activities (SIAs). OPV vaccine is popular in part because it can induce immunity to close contacts of the vaccine recipient known as contact immunity. Indeed the live viruses multiply briefly in the vaccine recipient, may be shed in body fluids or excrement, and can be contracted by another person. Producing little or no illness, this contact may benefit an additional person, and further increases the number of vaccinated individuals in the population. Although contact immunity is an advantage of OPV vaccine, the genetic plasticity of poliovaccine strains threatens the benefit of vaccination campaigns. Little research focuses on modelling the evolution of OPV vaccine. A study by Radboud J. Duintjer Tebbens et al. [6] reports such concerns. Attempts have been made to understand the dynamic spread of OPV viruses leading to cVDPV outbreak [5, 9, 19, 20]. One author investigate the epidemiological impact of reversion, and the creation of cVDPVs assuming the present polio vaccination strategy in developing countries (continuous OPV vaccination) [20]. And then investigate the

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

167

significance of the role of contact vaccination in decreasing the required vaccination coverage rate to control pathogenic wild-type virus spread [19]. Others study the transmission dynamic of OPV virus leading to cVDPV outbreak by considering an heterogenous networks of contacts between individuals [9]. All these studies disregard the dynamic of OPV and cVDPV viruses in the environment. Yet the presence of these viruses in sewage is a proven indicator of their evolution in the environment [12, 22]. Our goal in this paper is to give an understanding of how OPV virus evolve over time once they start circulating in the environment and to identify various biological aspects that substantially influence the spread of OPV viruses leading to cVDPVs outbreak. We mainly use a dynamical differential system and show that there exists a persistence threshold for OPV virus and a critical vaccination threshold. We also investigate the impact of the role of contact vaccination to control the spread of polio disease. The paper is structured as follow: in Sect. 2 we give the formulation of the model. In Sect. 3 we analyse the model. In Sect. 4 we present numerical simulation for different scenarios to identify various biological aspects that favor OPV mutation to cVDPV leading to polio outbreak.

2 Model Framework 2.1 Model Formulation The considered model classifies the human population according to their disease ˜ vaccinated individuals V˜ , infected indistatus, namely: susceptible individuals S, ˜ Thus, the total human population at time t, viduals I˜ and recovered individuals R. ˜ N(t) is ˜ + V˜ (t) + I˜(t) + R(t). ˜ N˜ (t) = S(t)

(2.1)

We denote by G2 (t) the population of viruses at time t that contains pathogenic virus (circulating vaccine derived polioviruses (cVDPV) and the wild poliovirus (WPV)) circulating in the environment. Susceptible individuals may become infected by contact with infected individuals or by ingestion of free pathogenic viruses content in the drinking water. So we denote by λ˜ = βh

G2 I˜ + βG , G2 + K N˜

(2.2)

the force of infection. This has been motivated in our previous study [1]. We assume that natural mortality at rate μ occurs in each compartment and the birth rate ν is

168

C. Baldé et al.

proportional to the total population. We also assume that the vaccine confers life long immunity. In what follows we derive the dynamic of the human population. The population of susceptibles increases by newborn children who missed vaccination at a rate (1 − !). This population decreases by infection of susceptibles through contact with infected individuals or by ingestion of pathogenic viruses (cVDPV and WPV). It also decreases by secondary immunization or contact vaccination of susceptibles. There is a couple of options in the literature regarding the transmission of OPV virus leading to secondary immunization [19, 20]. Since secondary immunity occurs through direct contact of vaccinated individuals with V˜ susceptible individuals, we choose the following standard incidence βV , where N˜ βV is the vaccinated-susceptible transmission rate per unit of time. V˜ ˜ S˙˜ = (1 − !)ν N˜ − βV S˜ − (λ˜ + μ)S. N˜ The population of vaccinated individuals is generated following the vaccination at rate ! of newborn children and susceptibles individuals which become vaccinated through secondary immunization. This population decreases when vaccinated individuals go to the recovered compartment at rate γ . V˜ V˙˜ = !ν N˜ + βV S˜ − (μ + γ )V˜ . N˜ The population of infected individuals contains those which can transmit the disease. As 1 in 200 infections leads to irreversible paralysis (usually in the legs) and among those paralyzed only 5 to 10% die when their breathing muscles become immobilized, we assume that disease-induced mortality is negligible. Then the population of infected individuals is generated by the infection of susceptibles individuals through contact with infected individuals or by ingestion of pathogenic viruses (cVDPV and WPV). The population decreases when infected individuals go to the recovered compartment at rate α. I˙˜ = λ˜ S˜ − (α + μ)I˜. Because it is more convenient to work with the model in terms of proportions of the population, we apply the variable transformations S=

V˜ I˜ R˜ S˜ ,V = ,I = ,R = . N˜ N˜ N˜ N˜

Noting that X˜ ˜˙ 1 N, X˙ = X˜˙ − ˜ N N˜ 2

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

169

˜ V˜ , I˜, R. ˜ Then we obtain the dynamic of human population in terms where X˜ is S, of proportions: ⎧ S˙ = (1 − !)ν − βV V S − (λ + ν)S, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V˙ = !ν + βV V S − (ν + γ )V , ⎪ ⎪ ⎪ I˙ = λS − (α + ν)I, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ R = γ V + αI − νR,

(2.3)

G2 . In the following we compute the dynamic of OPV G2 + K virus and pathogenic viruses (WPV and cVDPV). We denote by G1 the live attenuated viruses contained in OPV vaccine and circulating in the environment. Modelling OPV virus evolution might be challenging since their dynamic depend on several factors as mentioned in [5]. However, our interest here is to describe the variation of OPV virus once they start circulating in the environment after routine immunization. Thus we assume that OPV virus strains replicate in the vaccine recipient at rate ξ1 and then these viruses are shed at rate κ in the environment. Since vaccine recipient shed OPV virus during 3 to 4 weeks and there is an amount of viruses shed in the environment per day, we assume there is a threshold concentration of OPV vaccine in the environment r. Thus we have the following logistic equation for the growth of OPV virus.

G1  . f (G1 ) = κξ1 G1 1 − r where λ = βh I + βG

Furthermore, vaccine viruses can mutate to cVDPV or recover the same properties of transmissibility as WPV at a rate δ. OPV viruses decreases at rate η1 due to climatic changes or lack of susceptible individuals to pursue transmission. Thus we have the following differential equation for OPV viruses.

 ˙ 1 = κξ1 G1 1 − G1 − η1 G1 − δG1 . G r We denote by G2 (t) the population of pathogenic viruses (cVDPV, WPV). This population increases when a proportion of infectious individual shed a rate ξ2 of pathogenic virus in the environment. Moreover, these viruses increase when OPV viruses revert back and decrease at a rate η2 . ˙ 2 = ξ2 I + δG1 − η2 G2 . G

170

C. Baldé et al.

Fig. 1 Diagram flow of the transmission dynamics of Polio disease

Thus, the corresponding system of differential equations governing the dynamics of polio disease under consideration is ⎧ S˙ = (1 − !)ν − βV V S − (λ + ν)S, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V˙ = !ν + βV V S − (ν + γ )V , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I˙ = λS − (α + ν)I, ⎪ ⎨ ⎪ ⎪ R˙ = γ V + αI − νR, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ G1  ⎪ ⎪ ˙ ⎪ 1 − − η1 G1 − δG1 , = κξ G G 1 1 1 ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ G2 = ξ2 I + δG1 − η2 G2 ,

(2.4)

G2 (a flow diagram of the model is depicted in Fig. 1, G2 + K the associated state variables and parameters are described in Table 1). The fourth where λ = βh I + βG

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

171

Table 1 Variables and parameters with units for model system (2.5) Symbols S V I R G1 G2 ν ! κ βh βG βV δ γ α ξ1 ξ2 η1 η2 μ K r

Description Susceptible individuals Vaccinated individuals Infected individuals Recovered individuals Vaccine viruses (OPV) Pathogenic viruses (WPV, cVDPV) Birth rate Fraction of newborns vaccinated with OPV OPV virus shedding rate Transmission rate from infected-susceptibles contact Transmission rate from G2 Transmission rate from vaccinated-susceptibles contact Rate of reversion OPV to cVDPV or WPV Recovery rate from OPV vaccine Recovery rate from WPV (cVDPV) infections Growth rate of G1 in the vaccinee Contribution of infected individuals on G2 Rate of decreasing viruses in G1 Rate of decreasing viruses in G2 Natural mortality rate Virus 50% infectious dose, sufficient to cause infection Threshold concentration of OPV virus in the environment

Unit Unitless

cells.ml−1 cells.ml−1 day−1 Unitless Unitless day−1 day−1 day−1 day−1 day−1 day−1 day−1 day−1 day−1 day−1 day−1 cells.ml−1 cells.ml−1

equation of (2.4) is redundant so we drop it on the system, and we obtain the following system (2.5). The model below will be rigorously analyzed ⎧ ⎪ S˙ = (1 − !)ν − βV V S − (λ + ν)S, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎪ V = !ν + βV V S − (ν + γ )V , ⎪ ⎪ ⎪ ⎪ ⎨ I˙ = λS − (α + ν)I, ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ⎪ ˙ 1 = κξ1 G1 1 − G1 − η1 G1 − δG1 , G ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ G2 = ξ2 I + δG1 − η2 G2 .

(2.5)

172

C. Baldé et al.

3 Mathematical Analysis 3.1 Basic Properties We define $ ξ2 + δr % .  = (S, V , I, G1 , G2 ) ∈ R5+ , S + V + I ≤ 1, G1 (t) ≤ r, G2 (t) ≤ η2 It is clear that for any positive initial condition trajectories remains positives. Summing all the equations of system (2.5) yieds to S˙ + V˙ + I˙ = ν − ν(S + V + I ) − αI − γ V ≤ ν − ν(S + V + I ).

(3.1)

Applying the Gronwall inequality to the above equation yields S + V + I ≤ 1 for all t ≥ 0 if S(0) + V (0) + R(0) ≤ 1. Now we only have to show that the domain is invariant for G1 and G2 . The fourth equation of model system (2.5) is easy to solve. Thus we have G1 (t) = a

Ke(δ+η1)(τ −1)t  , 1 + Ke(δ+η1 )(τ −1)t

(3.2)

κξ1 r(τ − 1) aG1 (0) and a = . ,K = δ + η1 (1 − aG1 (0)) τ Note that if τ < 1, G1 (t) → ∞ as t → ∞ while G1 (t) > 0 if τ ≥ 1. That is OPV virus dies out if τ < 1 as t → ∞ while OPV virus persist otherwise. Let’s define τ as the persistence parameter of OPV virus in the environment. We have for all t positive, where τ =

a

Ke(δ+η1 )(τ −1)t  ≤ a ≤ r, 1 + Ke(δ+η1 )(τ −1)t

(3.3)

Thus G1 (t) ≤ r. Finally, using the fact that I (t) ≤ 1, G1 (t) ≤ r, and the Gronwall inequality, we have G2 (t) ≤

ξ2 + δr , η2

ξ2 + δr . This implies that  is a positively-invariant set under η2 the flow described by (2.5) so that no solution path leaves through any boundary of . Hence it is sufficient to consider the dynamics of the model (2.5) in . In this region, the model can be considered as being epidemiologically and mathematically well-posed. whenever G2 (0) ≤

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

173

The right-hand side of system (2.5) is a continuously differentiable map (C1 ). Then by the Cauchy-Lipschitz theorem for any given solution X0 ∈ R5+ , there exist a unique maximal solution, φ(t, X0 ), to the Cauchy problem of the differential equation (2.5).

3.2 The Disease Free Equilibrium of Model System (2.5) and Its Stability First, we set ! = 0 in system (2.5). Then we compute the basic reproductive numbers (Rv ) of vaccinated individuals (V ) and the basic reproductive numbers (R0 ) of infected individuals (I ) through cVDPV and WPV, defined to be the number of secondary transmissions of a single infectious individual in an otherwise fully susceptible population. From the definition, if considering vaccinated individuals we fix pathogenic viruses (G2 ), OPV viruses (G1 ) and infectious individuals (I ) to zero, and vice versa. Applying the next generation method (Van den Driessche and Watmough [18]) we have for V = 0, the only equilibrium point is X = (S0 , 0, 0, 0, 0), where S0 = 1. Then F = βV S0 , V = ν + γ .

(3.4)

βV . γ +ν

(3.5)

Thus Rv = FV−1 =

One has to think of Rv as the number of secondary immunization caused by a single vaccinated individual in a fully susceptible population. Applying the same method for I = 0, G1 = 0, G2 = 0 the only equilibrium point is X = (S0 , 0, 0, 0, 0) then ⎛ ⎜ βh S0 0 ⎜ ⎜ F=⎜ ⎜ 0 0 ⎜ ⎝ 0 0

⎞ ⎛ ⎞ βG S0 α+ν 0 0 K ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎟ ⎟ and V = ⎜ ⎜ 0 −κξ ⎟. + δ + η 0 1 1 0 ⎟ ⎜ ⎟ ⎟ ⎝ ⎠ ⎠ ξ2 −δ η2 0

Thus, to the resulting system yields the reproduction numbers R0 =

βG ξ2 βh + = Rh0 + Renv 0 , α+ν Kη2 (α + ν)

(3.6)

174

C. Baldé et al.

where Rh0 is the number of secondary infection caused by an individual initially infected and Renv 0 is the expected number of infections caused through exposure to the virus circulating in the environment. As in [20] we set the condition Rv < Rh0 . Since the vaccine virus is attenuated, substantially reducing both transmissibility as well as virulence.

3.2.1 The Disease Free Equilibrium Now we introduce the vaccination coverage rate (! = 0) and compute the disease free equilibrium (DFE) of system (2.5). By definition, the DFE has I¯ = 0 this ¯ 2 = 0 and G ¯ 1 = 0 and we must have implies, at equilibrium, G ν − ν S¯ . V¯ = γ +ν

(3.7)

We solve the second equation in (2.5) and obtain V¯ =

!ν . γ + ν − βV S¯

(3.8)

Then we equate (3.7) and (3.8) !ν ν − ν S¯ . = γ +ν γ + ν − βV S¯

(3.9)

This yields the quadratique equation Rv S¯ 2 − (Rv + 1)S¯ + (1 − !) = 0.

(3.10)

We compute the unique positive solution of (3.10) yielding the DFE since the other solution is not in the domain  (see Appendix A). 7 (Rv − 1)2 + 4!Rv S¯ = , 2Rv 7 (R − 1) + (Rv − 1)2 + 4!Rv ν v . V¯ = γ +ν 2Rv (Rv + 1) −

(3.11) (3.12)

Proposition 3.1 Model system (2.5) exhibits a critical vaccination coverage rate !crit . The DFE is locally asymptotically stable (LAS) if ! > !crit and τ < 1. !crit is explicitly determined.

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

175

Proof The Jacobian of model system (2.5) at the DFE is ⎛

βG S¯ −βV S¯ −βh S¯ 0 − ⎜ −βV V¯ − ν K ⎜ ¯ − (γ + ν) ⎜ βV V¯ β S 0 0 0 V ⎜ ⎜ βG S¯ ⎜ 0 0 βh S¯ − α − ν 0 ⎜ K ⎜ ⎝ 0 0 0 (δ + η1 )(τ − 1) 0 0 0 ξ2 δ −η2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

(3.13)

Thus the Jacobian is split into the eigenvalue (δ + η1 )(τ − 1),

(3.14)

and a bloc square matrix ⎛ ⎝

AB

⎞ ⎠,

(3.15)

0 C where  A=

⎞ ⎛  ¯ S β G ¯ ¯ ¯ −βV S −βV V − ν , B = ⎝ −βh S − K ⎠ , βV V¯ βV S¯ − (γ + ν) 0 0

(3.16)

⎛

⎞ βG S¯ ¯ β S − α − ν C=⎝ h K ⎠. ξ2 −η2

(3.17)

T r(A) = βV S¯ − βV V¯ − ν − (γ + ν),

(3.18)

We have

By replacing expression (3.7) into (3.18) !ν < 0, T r(A) = −βV V¯ − ν − V¯

(3.19)

det(A) = βV V¯ (γ + ν) > 0.

(3.20)

and

176

C. Baldé et al.

Thus the eigenvalues of matrix A are negatives. Then for the DFE to be locally stable the followings conditions are needed T r(C) = βh S¯ − α − ν − η2 < 0,

(3.21)

¯ 2 βG Sξ > 0. det(C) = −η2 (βh S¯ − α − ν) − K

(3.22)

Straightforward calculation gives the condition 1 S¯ < . R0

(3.23)

We will analyse two cases. 1 1 If R0 < 1, then S¯ < 1 < . Thus the stability of the DFE depends only of the R0 eigenvalue (δ + η1 )(τ − 1). We conclude that the DFE is locally asymptotically stable if τ < 1. 2 If R0 > 1, Note that the only information about S we have is contained in (3.9). We write that equation in the form ¯ = ψ(S)

!ν ν − ν S¯ = 0. − γ +ν γ + ν − βV S¯

1 Notice that ψ(S) is an increasing function. Hence the inequality S¯ < is R0

1  equivalent to ψ > 0, which is R0 !ν γ + ν − βV

1 R0

−

ν 1  > 0. 1− γ +ν R0

(3.24)

One finds by elementary manipulations that this inequality is equivalent to

1  Rv  ! > 1− 1− = !crit . R0 R0

(3.25)

Hence the stability of the DFE depend not only of the persistence parameter τ but also of the vaccination rate !. We conclude that the DFE is locally asymptotically

1  Rv  stable if ! > !crit = 1 − 1− and τ < 1.  R0 R0

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

177

Remark 3.2 We have two cases (1) If R0 < 1 and τ > 1, the DFE become unstable which means that there is no level of vaccination to avoid polio outbreak. This highlights the fact that as long as OPV viruses are circulating in the population polio outbreak is possible unless anybody into the population is vaccinated. (2) If R0 > 1 and τ < 1, there exists a vaccination threshold. This suggests that for the disease to die out from the population, the vaccination coverage rate must be above the critical vaccination rate !crit .

3.3 Global Stability of the DFE Theorem 3.3 The DFE of model system (2.5) is globally asymptotically stable (GAS) if ! ≥ !crit and τ < 1. Global asymptotic stability can be established by applying theory of asymptotically autonomous system [2]. Consider the following systems x˙ = f (x, t), x ∈ Rn ,

(3.26)

x˙ = g(x), x ∈ R ,

(3.27)

n

where f is the asymptotically autonomous system and g the limit system of (3.26) (see [2]). Theorem 2.3 in [2] stated that if (3.27) has a locally asymptotically stable equilibrium then every trajectory of (3.26) which intersect its bassin of attraction tends to the equilibrium. By finding a Lyapunov function for system (3.27) one can show that the equilibrium is globally asymptotically stable and thus all trajectories of (3.26) tend to the equilibrium. Solving the fourth equation of model system (2.5) leads us to the following asymptotically autonomous system ⎧ ⎪ S˙ = (1 − !)ν − βV V S − (λ + ν)S, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎪ V = !ν + βV V S − (ν + γ )V , ⎨ ⎪ I˙ = λS − (α + ν)I, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

Ke(δ+η1 )(τ −1)t  ⎪ ⎪ ˙ ⎪ ⎩ G2 = ξ2 I + aδ − η2 G2 . 1 + Ke(δ+η1)(τ −1)t

(3.28)

178

C. Baldé et al.

Remind that if τ < 1, G1 (t) → 0 as t → ∞. This leads us to the following limit system of system (3.28) ⎧ ˙ ⎪ ⎪ S = (1 − !)ν − βV V S − (λ + ν)S, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V˙ = !ν + βV V S − (ν + γ )V , ⎪ ⎪ ⎪ I˙ = λS − (α + ν)I, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ G2 = ξ2 I − η2 G2 .

(3.29)

System (3.29) is well posed in $ ξ2 % ,  = (S, V , I, G2 ) ∈ R4+ , S + V + I ≤ 1, G2 (t) ≤ η2 ¯ V¯ , 0, 0) where S¯ and V¯ the restriction of . And has a disease free equilibrium (S, are defined in (3.11). Proof We set the following Lyapunov function for model (3.29). S V F = S − S¯ − S¯ ln + V − V¯ − V¯ ln + I + wG2 . ¯ S V¯

(3.30)

This function is defined and continuous in  . It is straightforward to show that the ¯ V¯ , 0, 0) and positive for all the other positive function F is zero at the DFE (S, value in  and thus the DFE is the global minimum of F . The time derivative of F along trajectories is

V¯  ˙ S¯  ˙ 2. + V˙ 1 − + I + wG F˙ = S˙ 1 − S V

(3.31)

We have

 ¯ ¯ ˙ − S ) = (1 − !)ν − βV V S − βh SI − βG G2 S − νS 1 − S . S(1 S G2 + K S

(3.32)

It can be shown from model system (3.29) at the DFE ¯ (1 − !)ν = βV V¯ S¯ + ν S.

(3.33)

Substituting (3.33) into (3.32) gives 

S¯  S¯  ¯ ¯ = βV V S +ν S¯ −βV V S −βh SI −βG f (G2 )S −νS 1− , S˙ 1− S S

(3.34)

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

179

where f (G2 ) =

G2 . G2 + K

(3.35)

Thus the relation (3.31) becomes 

S¯  F˙ = βV V¯ S¯ + ν S¯ − βV V S − βh SI − βG f (G2 )S − νS 1 − S

¯ V ˙ 2. + V˙ 1 − ) + I˙ + wG (3.36) V We split F into F1 and F2 such that F = F1 + F2 , where

V¯  S¯  + V˙ 1 − , F1 = (βV V¯ S¯ + ν S¯ − βV V S − νS) 1 − S V

(3.37)

S¯  ˙ ˙ 2. F2 = (−βh SI − βG f (G2 )S) 1 − + I + wG S

(3.38)

and

Computation of F1 gives

S S¯ 2 S¯ + βV V¯S¯ − βV V¯ + βV V S¯ F1 = ν S¯ 2 − − S S S¯ V¯ + 2!ν − (γ + ν)V − !ν − βV S V¯ + (γ + ν)V¯ . V

(3.39)

The second equation of system (3.29) gives at the disease free equilibrium the following relation ¯ (γ + ν)V¯ = !ν + βV V¯ S.

(3.40)

Replacing (3.40) into (3.39) gives

S S¯ S¯ + βV V¯ S¯ 2 − − F1 = ν S¯ 2 − − S S S¯ V V + !ν − (γ + ν)V + βV V S¯ ≤ !ν ¯ V V¯

S V V¯ + !ν 2 − − V S¯ V¯ − (γ + ν)V + βV V S¯ ≤ 0.

180

C. Baldé et al.

The first inequality follow from the fact that the arithmetic mean is always greater

S V¯ S¯ ≤ 0 and 2 − − than or equal to the geometric mean. Therefore 2 − − S V S¯ V ≤ 0 with equality only if S = S¯ and V = V¯ . In the final inequality we use the V¯ !ν following equality γ + ν = + βV S¯ which derives from (3.40). V¯ Computation of F2 gives ¯ ¯ ¯ + βG f (G2 )S¯ − βG G2 S + βG G2 S − (α + ν)I + w(ξ2 I − η2 G2 ) F2 = βh SI K K ¯ ¯ h − 1)I + βG G2 S + w(ξ2 I − η2 G2 ). (3.41) ≤ (α + ν)(SR 0 K G2 S¯ ≤ 0 since The last inequality follow from the fact that βG f (G2 )S¯ − βG K G2 S¯ βG f (G2 )S¯ ≤ βG . Moreover since Rh0 = R0 −Renv 0 inequality (3.41) becomes K ¯ ¯ 0 −1)I −(α+ν)SR ¯ env I +βG G2 Sη2 +w(ξ2 I −η2 G2 ). F2 ≤ (α+ν)(SR 0 Kη2 Replacing the following equality Renv = 0 following inequality ¯ 0 − 1)I + βG F2 ≤ (α + ν)(SR

with w = βG

(3.42)

βG ξ2 into (3.42) we have the Kη2 (α + ν)

S¯ (−ξ2 I + η2 G2 ) + w(ξ2 I − η2 G2 ), Kη2 (3.43)

S¯ , we have Kη2 ¯ 0 − 1)I. F2 ≤ (α + ν)(SR

(3.44)

Combining (3.41) and (3.44) yields ¯ 0 − 1)I. F˙ = F1 + F2 ≤ (α + ν)(SR Using the fact that S¯ ≤

(3.45)

1 ⇔ ! ≥ !crit see proof 3.2.1, we obtain the result R0 ! ≥ !crit ⇒ F ≤ 0.

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

181

¯ V¯ , 0, 0), Thus, since the set in  where F˙ = 0 is reduced to the DFE, (S, and is clearly invariant, by Lasalle invariance principle [10] the DFE is globally asymptotically stable in  . Then using theorem 2.3 in [2], the trajectories ¯ V¯ , 0, 0 respectively. Thus all S(t), V (t), I (t), G2 (t) of system (3.28) tend to S, ¯ V¯ , 0, 0, 0) if τ < 1 and ! ≥ !crit . Thus the trajectories of system (3.28) tend to (S, ¯ V¯ , 0, 0, 0), is globally asymptotically disease free equilibrium of system (2.5), (S, stable in  if τ < 1 and ! ≥ !crit . 

3.4 Uniform Persistence Let the stable manifold of the DFE of model (2.5) be $ % 0 = (S, V , I, G1 , G2 ) ∈ , I = G1 = G2 = 0 .

(3.46)

Theorem 3.4 The susceptibles S and vaccinated V compartments are always uniformly persistent. The OPV virus compartment G1 is uniformly persistent if τ ≥ 1. Proof We have

S˙ = (1 − !)ν − βV V S − βh I + βG Using the fact that I (t) < 1, V (t) < 1 and inequality

 G2 + ν S. G2 + K

(3.47)

G2 < 1, we have the following G2 + K

S˙ > (1 − !)ν − (βV + βh + βG + ν)S.

(3.48)

Using the fluctuation lemma [15, 16] we have S∞ >

(1 − !)ν . βV + βh + βG + ν

(3.49)

We recall the equation of V(t) V˙ = !ν + βV V S − (ν + γ )V > !ν − (ν + γ )V .

(3.50)

Using the fluctuation lemma again we have V∞ >

!ν . ν +γ

(3.51)

Using the result in 3.2, if τ ≥ 1, G1 (t) > 0. Thus G1 (t) is uniformly persistent if τ ≥ 1. 

182

C. Baldé et al.

Theorem 3.5 The infected compartments I and G2 are uniformly persistent if ! < !crit . Proof We apply Theorem 8.17 from [16] to prove uniform weak-persistence. For the sake of simplicity, we use the notation X = (S, V , I, G1 , G2 ) ∈  for the state of the system. Let us choose ρ(X) = I + G2 . The invariant extinction space of the infected compartments is 0 = {(S, V , I, G1 , G2 ) ∈ , I = G1 = G2 = 0}. We use the usual notation w(X) for the ω-limit set of a point X in . We examine ˜ := ∪X∈0 w(X). It is clear that all solutions starting from the extinction the set  ¯ thus  ˜ = {X}. ¯ Let M = {X}, ¯ space 0 converge to the disease-free equilibrium X, ˜ ⊂ M, where M is isolated (due to instability of the disease-free then we have  equilibrium [4]), compact, invariant and acyclic. It remains us to show that M is weakly ρ-repelling, then by [theorem 8.17, Chapter 8 in [16]] the weak persistence follows. Suppose that X¯ is not ρ repelling which means that there is a solutions of (2.5) which converges to the disease free equilibrium. Then for any  > 0, for t sufficiently large we have S(t) > S¯ − . Furthermore one can choose m > 0 satisfying 1 G2

− m. G2 + K K Thus we get the following differential inequality ⎧ 1 ⎪ ⎪ ⎨ I˙ > βh (S¯ − )I + βG ( − m)G2 (S¯ − ) − (α + ν)I, K ⎪ ⎪ ⎩G ˙ 2 > ξ2 I − η2 G2 .

(3.52)

˜ System (3.52) reduces to u(t) ˙ > Cu(t) where u(t) = (I, G2 ) and

 ⎞ ¯ −  − α − ν ( 1 − m)βG S¯ −  β S h ⎠. C˜ = ⎝ K −η2 ξ2 ⎛

(3.53)

We remind the matrix C defined in (3.17) ⎞ βG S¯ ¯ C = ⎝ βh S − α − ν K ⎠ . ξ2 −η2 ⎛

(3.54)

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

183

We have if det(C) ≤ 0 then s(C) > 0. Where s(C) is the stability modulus of the matrix C which is defined as the maximum real part of the eigenvalues of the matrix C. We have ¯ 0−1>0 det(C) ≤ 0 ⇔ SR

(3.55)

This condition is equivalent to ! < !crit . Thus one can choose  small enough such that (S¯ − )R0 − 1 > 0. Now let’s find condition under which the stability modulus ˜ > 0 of the matrix C˜ is positive. One condition is det(C) ˜ ≤ 0. Straightforward s(C) calculation gives

 ˜ = βG mξ2 (S¯ − ) − η2 (α + ν) (S¯ − )R0 − 1 det(C)

(3.56)

Relation (3.56) is negative if m
0. As C˜ is irreductible and quasipositive then by corollary A27 in that s(C) ˜ [16] there exists some ζ > 0 such that |u(t)| ≥ ζ |u(0)|es(C)t , t > 0. Thus u(t) → ∞, t → ∞ which contradicts our supposition above. Thus M is weaklyrepelling and all the conditions of theorem 8.17, Chapter 8 [16] are satisfied then the weakly uniform persistence hold. To show that uniform persistence also holds, we use Theorem 4.5 from [16]. Our flow is continuous and the subspaces 0 and  are clearly invariant. The existence of a compact attractor follows from the boundedness of the solutions. Thus, all conditions of Theorem 4.5 in [16] hold, from which we obtain the uniform persistence of I +G2 . The uniform persistence of I +G2 implies that at least one of the two infected compartments I (t), G2 (t) is uniformly weakly persistent. Let us assume that I (t) is uniformly weakly persistent, the other case can be handled in a similar way. Using again Theorem 4.5 from [16], we obtain that from the uniform weak persistence of I (t), also the uniform persistence of I (t) follows. This means that there exists some I > 0 such that lim inft →∞ I (t) > I . We know from theorem 2 below that all of the three susceptible compartments S(t), is always uniformly persistent. We will use the fluctuation lemma to show that G2 (t) is also uniformly persistent. Rearranging the equation for G2 , we obtain Thus one can choose

˙ 2 + η2 G2 = ξ2 I + δG1 > ξ2 lim inf I > ξ2 I . G t →∞

(3.58)

184

C. Baldé et al.

According to the fluctuation lemma in [16] we obtain lim inf G2 > t →∞

ξ2 I . η2

(3.59) 

i.e. G2 (t) is uniformly persistent.

3.5 Existence of the Endemic Equilibrium and Its Stability We denote the endemic equilibrium by (S ∗ , V ∗ , I ∗ , G∗1 , G∗2 ) obtained by setting the right hand side of system (2.5) equals to zero. Thus we have the following theorem. Theorem 3.6 Model system (2.5) has at least one endemic equilibrium if ! < !crit and τ < 1 and at least two endemic equilibria if τ > 1. Proof For the existence of the endemic equilibrium see Appendix B.



Theorem 3.7 If τ < 1, the endemic equilibrium, (S ∗ , V ∗ , I ∗ , 0, G∗2 ), of system (2.5) is globally asymptotically stable (GAS) in  \ 0 whenever ! < !crit . The proof of global stability of the endemic equilibrium (S ∗ , V ∗ , I ∗ , 0, G∗2 ) derives from the framework of Zhisheng Shuai and P. Van Den Driessche [14] and theorem Theorem 2.3 in [2]. Proof Consider the limit system defined in (3.29) and the invariant domain $ ξ2 % .  = (S, V , I, G2 ) ∈ R4+ , S + V + I ≤ 1, G2 (t) ≤ η2 We define  0 = {(S, V , I, G2 ) ∈  , I = G2 = 0}. Following the same notation in Zhisheng Shuai and P. Van Den Driessche [14]. We set D1 = S − S ∗ − S ∗ ln

S V I + V − V ∗ − V ∗ ln ∗ + I − I ∗ − I ∗ ln ∗ , S∗ V I

D2 = G2 − G∗2 − G∗2 ln

G2 . G∗2

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

185

The derivative of D1 along trajectories is

V ∗  ˙ I∗  S∗  + V˙ 1 − +I 1− . D˙1 = S˙ 1 − S V I

(3.60)

To make the calculations easier to follow we calculate the derivatives of all terms on the right-hand side of the equation for (3.60) separately. Thus we have

S∗  S∗  G2 = ((1−!)ν −βV V S −βh SI −βG S −νS) 1− . S˙ 1− S G2 + K S

(3.61)

The first equation of system (2.5) gives at the endemic equilibrium the following equality (1 − !)ν = βV V ∗ S ∗ + νS ∗ + βh S ∗ I ∗ + βG

G∗2 S∗. G∗2 + K

(3.62)

Substituting (3.62) into (3.61), and using the same notation as in (3.35) we have

S∗  = (βV V ∗ S ∗ − βV V S + νS ∗ − νS − βh SI − βG f (G2 )S S˙ 1 − S

S¯∗  + βh S ∗ I ∗ + βG f (G∗2 )S ∗ ) 1 − . (3.63) S We have

V∗ V∗ V˙ 1− = !ν +βV V S −(ν +γ )V −!ν −βV V ∗ S +(ν +γ )V ∗ . V V

(3.64)

The second equation of system (2.5) gives at the endemic equilibrium the following relation (γ + ν) =

!ν + βV S ∗ . V∗

(3.65)

Replacing (3.65) into (3.64) gives

V∗ V V∗ + βV V S − βV S ∗ V = 2!ν − !ν ∗ − !ν V˙ 1 − V V V − βV V ∗ S + βV V ∗ S ∗ .

(3.66)

We have

I∗  = βh SI + βG f (G2 )S − (α + ν)I − βh SI ∗ I˙ 1 − I I∗ + βG f (G2 )S + (α + ν)I ∗ . I

(3.67)

186

C. Baldé et al.

The third equation of system (2.5) gives at the endemic equilibrium the following relation (α + ν) = βh S ∗ + βG f (G∗2 )

S∗ . I∗

(3.68)

Replacing (3.68) into (3.67) gives

S∗ I∗  = βh SI + βG f (G2 )S − (βh S ∗ + βG f (G∗2 ) ∗ )I − βh SI ∗ I˙ 1 − I I ∗ I + βG f (G2 )S + βh S ∗ I ∗ + βG f (G∗2 )S ∗ . (3.69) I Replacing (3.63), (3.66) and (3.69) into (3.60), straightforward calculation gives

S∗ S  S∗ S − ∗ + νS ∗ 2 − − ∗ D˙ 1 ≤ βV V ∗ S ∗ 2 − S S S S



∗ ∗ S V S V  − ∗ + !ν 2 − − ∗ + βh S ∗ I ∗ 2 − S S V V

∗ S f (G I ) Sf (G2 )I ∗  2 + − . − + βG f (G∗2 )S ∗ 2 − S f (G∗2 ) I ∗ S ∗ f (G∗2 )I

(3.70)

Since the arithmetic mean is greater or equal than the geometric mean. We have the following inequality

S∗ f (G2 ) I Sf (G2 )I ∗  + − ∗− ∗ D˙1 ≤ βG f (G∗2 )S ∗ 2 − ∗ S f (G2 ) I S f (G∗2 )I

S∗ S∗ S∗ f (G2 ) I Sf (G2 )I ∗ ≤ βG f (G∗2 )S ∗ 2 − − ln + ln + − ∗− ∗ ∗ S S S f (G2 ) I S f (G∗2 )I G2 Sf (G2 )I ∗  G2 I I Sf (G2 )I ∗ − ln ∗ + ln ∗ + ∗ − ∗ + ln ∗ − ln ∗ . (3.71) ∗ G2 G2 I I S f (G2 )I S f (G∗2 )I Using the inequality 1 − x + ln x ≤ 0 for x > 0 with equality holding if and only if x = 1, we have

f (G ) f (G∗ )G 2 2 2 + D˙1 ≤ βG f (G∗2 )S ∗ f (G∗2 ) f (G2 )G∗2 G2 G2 G2 I I  + ∗ − ∗ − ln ∗ − ∗ + ln ∗ . G2 G2 G2 I I

(3.72)

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

187

Straightforward calculation gives the following equality 

f (G ) f (G∗2 )G2  f (G2 ) f (G∗2 )G2 G2 2 + − 1 1 − . + = f (G∗2 ) f (G2 )G∗2 G∗2 f (G∗2 ) f (G2 )G∗2

(3.73)

This leads us to the following inequality  $ f (G ) f (G∗2 )G2  2 − 1 1 − f (G∗2 ) f (G2 )G∗2

G G2 I I % 2 + ∗ − ln ∗ − ∗ + ln ∗ . G2 G2 I I

D˙1 ≤ βG f (G∗2 )S ∗

(3.74)

f (G2 ) 1 G2 is monotone nonincreasing, and f (G2 ) = is = G2 G2 + K G2 + K monotone nondecreasing. We have the following inequality Using

G G2 I I  2 D˙1 ≤ βG f (G∗2 )S ∗ − ln ∗ − ∗ + ln ∗ =: a12 G12 . ∗ G2 G2 I I

(3.75)

Similarly

G∗  ˙2 1− 2 . D˙2 = G G2

(3.76)

At the endemic equilibrium, the fifth equation of system (2.5) gives the following relation, η2 =

ξ2 I ∗ . G∗2

Thus D2 we have the following inequality,

ξ2 I ∗  D˙2 ≤ ξ2 I − ∗ G2 1 − G2

I G2 ≤ ξ2 I ∗ 1 + ∗ − ∗ − I G2

G∗2  G2 I G∗2  I ∗ G2

I G∗ I G2 G2 G2 I I  ≤ ξ2 I ∗ 1 + ∗ − ∗ − ∗ 2 + ln ∗ − ln ∗ + ln ∗ − ln ∗ I G2 I G2 G2 G2 I I

G  G2 I I 2 ≤ ξ2 I ∗ ln ∗ − ∗ + ∗ − ln ∗ := a21 G21 . G2 G2 I I

188

C. Baldé et al.

The fourth inequality follows from the fact that 1 − x + ln x ≤ 0 for x > 0. Then following the framework in [14], a weighted digraph with two vertices and two arcs can be constructed such that the weights are a12 and a21. We have along the only cycle, G12 + G21 = 0. By theorem 3.5 in [14] there exists c1 , c2 such that D = c1 D1 + c2 D2 is a Lyapunov function for system (3.29). By Theorem 3.3 a21 a21 . Therefore the function D = D1 + in [14], c1 a12 = c2 a21 thus c1 = c2 a12 a12 D2 is a Lyapunov function for system (3.29). Thus by Lyapunov theorem [21], the endemic equilibrium is stable. Furthermore the largest invariant set in  where D˙ = 0 is reduced to the endemic equilibrium S = S ∗ , V = V ∗ , I = I ∗ and G2 = G∗2 . Therefore by Lasalle invariance principle the endemic equilibrium of system (3.29) is globally asymptotically stable in  \  0 . Hence all the trajectories S(t), V (t), I (t), G2 (t) of system (3.28) tend to S = S ∗ , V = V ∗ , I = I ∗ and G2 = G∗2 from theorem 2.3 in [2]. Furthermore if τ < 1, G1 (t) → 0 as t → ∞. From this the global asymptotic stability of the endemic equilibrium of system (2.5) follows. 

4 Numerical Simulation Here we perform numerical simulation to support our theoretical analysis.

4.1 Sensitivity Analysis Herein, we carried out the sensitivity analysis to identify parameters that are most influential in determining disease dynamics [11]. Thus a Latin Hypercube Sampling (LHS) scheme is performed for each input parameter using a uniform distribution over the range of biologically realistic values, with descriptions and references given in Table 2. For each parameter we sample 1000 values. This gives a matrix called LHS matrix with 1000 rows and columns depending to the number of parameters. Using system (2.5) and a time period of 400 months, solutions are then simulated, using each row of the LHS matrix. Four outcome measures are calculated for each run: susceptible individuals, vaccinated individuals, infected individuals, the OPV and cVDPV concentration over the model’s time span. Then Partial Rank Correlation Coefficients (PRCC) and corresponding p-values are computed. An output is assumed sensitive to an input if the corresponding PRCC is less than −0.50 or greater than 0.50, and the corresponding p-value is less than 5%. Table 3 summarize the four parameters with most significant PRCCs index for each output variable over the time period of 4, 15 and 60 days. Table 3 shows that there is a positive correlation between the OPV virus shedding rate κ, the replication rate ξ1 and the evolution of OPV virus (G1 ). It also shows a positive correlation between the mutation rate of OPV virus δ and the evolution of

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak Table 2 Numerical values for the parameters of model system (2.5)

Parameters ! βh βV βG γ α δ ξ1 ξ2 η1 η2 κ K r

Range 0.01−1 0.01−1 0.01−1 0.01−1 10−5 −1 [1/16, 1/35] 1/60−1/730 0.001−1 0.001−1 0.001−1 0.001−1 0.001−1 1−2 0.1−2

189 Nominal value Variable Variable Variable Variable 1/7 1/35 0.42 0.21 0.5 Variable 0.05 Variable 1.5 1

Reference Assumed Assumed Assumed Assumed [5] [5] [9] Assumed Assumed Assumed Assumed Assumed Assumed Assumed

Table 3 PRCCs results for system (2.5) Sensitivity index Parameters with significant sensitivity index PRCC Day 4 Day 15 (A) Sensitivity analysis of susceptible individuals S as output of interest !(−), ν, βG (−), K !(−), ν, βG (−), K (B) Sensitivity analysis of vaccinated individuals V as output of interest γ (−), !, ν, γ (−), ν, !, η2 (C) Sensitivity analysis of infected individuals I as output of interest !(−), βG , K(−), βh !(−), βG , K(−), βV (−) (D) Sensitivity analysis of OPV viruses G1 as output of interest κ, ξ1 , r, δ(−) ξ1 , κ, r, δ(−) (E) Sensitivity analysis of cVDPV viruses G2 as output of interest η2 (−), ξ2 , δ, ξ1 η2 (−), ξ2 , !(−), ξ1

Day 60 !(−), ν, βG (−), η2 ν, !, γ (−), η2 !(−), ν, γ, βG ξ1 , κ, r, δ(−) η2 (−), !(−), ξ2 , δ

pathogenic virus cVDPV and WPV (G2 ). This suggests that a better knowledge of these parameters could help to better understand how OPV viruses evolve towards cVDPV in the environment.

4.2 Numerical Analysis Here we simulate various scenarios. Since the sensitivity analysis has shown that the OPV shedding rate κ is a sensitive parameter, for all the simulations we make a variation of the persistence parameter τ by making variation of the OPV shedding rate κ. Note that the persistence parameter τ is an increasing function of the OPV shedding rate κ. Figure 2 shows that system (2.5) could be in one of these two

190

C. Baldé et al.

Fig. 2 State behavior of system (2.5) relatively to the vaccination rate ! and the persistence parameter τ . Here !crit = 0.814, βG = 0.015, βh = 0.15, βV = 0.05. All the others parameters are as in Table 2

regimes, epidemic or no epidemic, depending on the vaccination rate ! and the persistence parameter τ . No epidemic occurs if τ < 1 and ! ≥ !crit . This means there is no OPV viruses circulating in the environment and the vaccination coverage rate ! is above the critical vaccination treshold !crit . However if τ ≥ 1, epidemic occur for all vaccination rate !. This highlights that if OPV virus persist circulation in the environment there is no level of vaccination to stop polio transmission. Hence the entire population should be vaccinated to avoid cVDPV outbreak. Figure 3 shows the evolution of OPV viruses and pathogenic viruses (WPV, cVDPV) in the environment relatively to the vaccination rate ! and the persistence parameter τ . Figure 3a shows that if ! < !crit and there is no OPV virus circulating in the environment, the amount of pathogenic viruses decreases as the vaccination rate ! increases. Once the critical vaccination rate !crit = 0.156 is achieved, pathogenic viruses stop circulation in the environment as long as τ < 1. However if τ ≥ 1, as shown in Fig. 3b, pathogenic viruses reappear. This highlights the risk of pathogenic viruses creation as long as OPV viruses persist circulation in the environment. Figure 4 shows how infected individuals evolve relatively to the vaccination rate ! and the persistence parameter τ . Figure 4a shows that if ! ≥ !crit and τ < 1, as shown in Fig. 4b, there is no infected individuals into the population. However some individuals become infected once τ ≥ 1. This is due to the persistence of OPV viruses in the environment leading to the creation of pathogenic

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

191

Fig. 3 (a) OPV virus (G1 ) and pathogenic virus (G2 ) relatively to the vaccination rate ! at the endemic level, (b) OPV virus (G1 ) and pathogenic virus (G2 ) relatively to the OPV persistence parameter τ at the endemic level. Here !crit = 0.154, βG = 0.005, βh = 0.75, βV = 0.05, and all the others parameters are as in Table 2

192

C. Baldé et al.

Fig. 4 Infected individuals relatively to the vaccinatiion rate ! and the OPV persistence parameter τ . Here !crit = 0.148, βG = 0.005, βh = 0.75, βV = 0.05. All the other parameters are as in Table 2

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

193

viruses. In this situation, a silent transmission of pathogenic viruses to unvaccinated individuals through contact with the contaminated environment may occurs. This leads later to a transmission between individuals in the population. Yet increasing the vaccination rate decreases the prevalence of the epidemic. Figure 5 shows how infected individuals, OPV virus and pathogenic viruses (cVDPV and WPV) evolve relatively to the vaccination rate !. Figure 5a shows that maintaining the vaccination coverage rate relatively high decreases considerably the number of infected individuals. However Fig. 5b suggests that OPV virus could spread extensively and persist in the community before the critical vaccination threshold is achieved. Thus pathogenic viruses (cVDPV, WPV) never interupt circulation in the environment. This highlights the significance of vaccination time. The significance of the role of contact vaccination in decreasing the vaccination coverage rate has been proved in [19] where a model for LAVV (Live Attenuated Vaccine Viruses) like OPV vaccine has been analyzed. However the model does not take into account the indirect transmission of the disease through the environment. Note that if considering only direct transmission between individuals which means that Renv 0 = 0, then

Rv Rv  1  1  1− . !crit = 1 − h 1 − h ≤ 1 − R0 R0 R0 R0 This means that if individuals are getting infected through the environment more individuals should be vaccinated than if we consider only direct transmission. This is consistent with findings in [1] where the relevance of the indirect transmission of polio disease has been analyzed. Figure 6 is a representation of the critical vaccination rate as a function of the secondary immunization rate and Renv 0 . Figure 6a shows that increasing the number of secondary immunization rate decrease the critical vaccination rate even if the secondary immunization rate is below one which means that transmission of OPV virus between individuals no longer progress into the population. However increasing Renv increases the critical vaccination rate. Figure 6b shows that when 0 the number of individuals getting infected through the environment Renv 0 increases, increasing the secondary immunization rate decrease the critical vaccination rate even if the secondary immunization rate is below one. Furthermore when the secondary immunization rate is over one it highly decreases the critical vaccination threshold. This assess the usefullness of secondary immunization as a way to increase herd immunity. Figure 7 depicts how OPV virus and pathogenic virus evolve in the environment. It compares the amount of OPV virus and pathogenic virus (WPV, cVDPV) at the endemic level relatively to the vaccination rate ! and the persistence parameter τ . Figure 7a shows the set of value for the vaccination rate and the persistence parameter where epidemic occurs or not. Figure 7b shows the environment state by comparing the amount of circulating OPV and Pathogenic viruses (WPV, cVDPV). If τ < 1 and ! ≥ !crit there is no viruses circulating in the environment. If τ < 1

194

C. Baldé et al.

Fig. 5 (a) Infected individuals relatively to the vaccination rate !. (b) OPV viruses and pathogenics viruses relatively to the vaccination rate !. Here !crit = 0.714, ! = 0.4, βG = 0.7, βh = 0.91, βV = 0.5 and all the others parameters are as in Table 2

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

195

(a) 0.7

1.4 0.6

1.2

R env 0

1

0.5

0.8 0.4

0.6 0.3

0.4 0.2

0.2

0.2

0.4

0.6

0.8

Rv

1

1.2

(b)

Fig. 6 Critical vaccination threshold as a function of the expected number of infection caused through exposure to the virus circulating in the environment Renv 0 and the secondary immunization number Rv

and ! < !crit only pathogenic viruses are circulating in the environment. Once τ ≥ 1, OPV viruses persist circulating in the environment. Then for all vaccination rate !, OPV viruses could be dominant.

196

C. Baldé et al.

Fig. 7 (a) Behavior of model system (2.5) relatively to the vaccination rate and the OPV persistence parameter τ . Here !crit = 0.199. (b) Comparition of OPV virus (G1 ) and pathogenic virus (G2 ) in the environment at the endemic level relatively to the vaccination rate ! and the OPV persistence parameter τ

5 Conclusion In this work we develop a dynamical differential system to analyze OPV evolution leading to the creation of cVDPV. Our study shows that there exists a persistence threshold of OPV virus and a critical vaccination threshold !crit which is explicitly determined. Moreover we show that there is no epidemic whenever τ < 1 and ! ≥ !crit while an epidemic occurs if τ ≥ 1 for all vaccination rate !. This means that, if OPV virus stop circulating in the environment, maintaining vaccination coverage rate above the critical vaccination rate ! ≥ !crit is sufficient to avoid Polio outbreak. However as long as OPV virus are persistent, the entire population should be vaccinated to prevent Polio outbreak. Furthermore we compute equilibrium and show that the disease free equilibrium is globally asymptotically stable (GAS) whenever τ < 1 and ! ≥ !crit while if τ < 1 and ! < !crit the unique endemic equilibrium is GAS. We also show there exists two endemic equilibria if τ > 1. In our study we also investigate contact vaccination and show that it can strongly decrease the critical vaccination threshold. The model also highlight the significance of vaccination time to overcome the risk of pathogenic viruses persistence in the environment. However our study has many limitations. Indeed we simplify modelling OPV evolution. In reality reversion will likely occur through accumulated mutations. And parameters such as the OPV shedding rate κ and the replication rate ξ1 might likely depend on past immune state of the infected individuals. Despite these limitations our study points out realistic phenomenon consistent with the Global Poliomyelite Eradication Initiative (GPEI) plan. Indeed our study shows the usefullness of environmental surveillance (ES) to asses stop circulation of pathogenic viruses and the need to maintain high level of vaccination after certified end of polioviruses circulation.

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

197

Appendix A We show here that Eq. (3.10) has a unique solution in $ ξ2 + δr % .  = (S, V , I, G1 , G2 ) ∈ R5+ , S + V + I ≤ 1, G1 (t) ≤ r, G2 (t) ≤ η2 Solving Eq. (3.10) gives two solutions: S¯ =

(Rv + 1) −

(Rv + 1) + S¯ =

7 7

(Rv − 1)2 + 4!Rv , 2Rv

(A.1)

(Rv − 1)2 + 4!Rv . 2Rv

(A.2)

We have to show that S¯ is not in the domain . Suppose that S¯ ≤ 1 thus we have (Rv + 1) + 7

7

(Rv − 1)2 + 4!Rv ≤ 1, 2Rv

(Rv − 1)2 + 4!Rv ≤ Rv − 1.

(A.3) (A.4)

Elevate to the square the last equation gives (Rv − 1)2 + 4!Rv ≤ (Rv − 1)2 .

(A.5)

Thus we have the following inequality 4!Rv ≤ 0.

(A.6)

Which contradicts the hypothesis of positivity of the parameters. It can be shown in a similar manner that S¯ lie in the domain .

Appendix B We will use persistence theory to show existence of the endemic equilibrium. The existence of an endemic equilibrium follows from the uniform persistence and the uniform boundedness of the solutions (see e.g. [7], Theorem D.3). For convenience to the reader we recall the theorem: Theorem B.1 Let φ be a semidynamical system defined on a subset E, the closure of an open set, in a locally compact metric space X. Let M be a compact isolated invariant set for the dynamical system φ. Suppose that ∂E, the boundary of E, is

198

C. Baldé et al.

invariant under φ. Assume that φ is dissipative and that the boundary flow φ∂ is isolated and acyclic with acyclic covering M. Let W + (A) be the attracting set of a compact invariant set A. Then φ is uniformly persistent if and only if ˚ = ∅, W + (Mi ) ∩ E

for each Mi ∈ M

(B.1)

Theorem B.2 Let the hypotheses of Theorem B.1 hold for a dynamical system φ, with X = Rn+ (thus the invariant boundary ∂E is composed of the coordinate faces), and let (B.1) hold. Then there is a rest point in the interior of E. System (2.5) is uncoupled thus we can consider the following system ⎧ ⎪ S˙ = (1 − !)ν − βV V S − (λ + ν)S, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V˙ = !ν + βV V S − (ν + γ )V , ⎪ ⎪ ⎨ ⎪ I˙ = λS − (α + ν)I, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

Ke(δ+η1 )(τ −1)t  ⎪ ⎪ ˙ ⎪ ⎩ G2 = ξ2 I + aδ − η2 G2 . 1 + Ke(δ+η1)(τ −1)t

(B.2)

Solving the fourth equation of system (2.5) equals to zero gives two equilibria, G∗1 = r(τ − 1) ∗ ∗ . Note that G¯1 exists only if τ ≥ 1. However if τ = 1, 0 and G¯1 = τ ∗ G∗1 = G¯1 = 0. From the proof of theorem 3.5 we have shown that all hypothesis of theorem B.1 hold. We have also shown the uniform persistence of S, V , I and G2. Thus by theorem B.1 the hypothesis (B.1) hold. It follows there exists a rest point (S ∗ , V ∗ , I ∗ , 0, G∗2 ) in  \ 0 if τ < 1 and two rest points (S ∗ , V ∗ , I ∗ , 0, G∗2 ) and (S ∗ , V ∗ , I ∗ , G∗1 , G∗2 ) in  \ 0 if τ ≥ 1. 

References 1. C. Balde, M. Lam, A. Bah, S. Bowong, J.J. Tewa, Theoretical assessment of the impact of environmental contamination on the dynamical transmission of polio. Int. J. Biomath. 12(2) (2019) 2. C. Castillo-Chavez, H.R. Thieme, Asymptotically Autonomous Epidemic Models. Mathematical Population Dynamics: Analysis of Heterogeneity (2003), pp. 33–50 3. F. Delpeyroux , F. Colbère-Garapin, R. Razafindratsimandresy, S. Sadeuh-Mba, M.L. Joffret, D. Rousset, B. Blondel, Eradication of poliomyelitis and emergence of pathogenic vaccinederived polioviruses: from Madagascar to Cameroon. Med. Sci. 29(11) (2013) 4. A. Dénes, L. Székely, Global dynamics of a mathematical model for the possible re-emergence of polio. Math. Biosci. (2017). https://doi.org/10.1016/j.mbs.2017.08.010

OPV Virus Evolution: Assessing the Risk of cVDPV Outbreak

199

5. R.J. Duintjer Tebbens et al., Review and assessment of poliovirus immunity and transmission: synthesis of knowledge gaps and identification of research needs. Risk Anal. 33(4), 606–646 (2013) 6. R.J. Duintjer Tebbens et al., Oral poliovirus vaccine evolution and insights relevant to modeling the risks of circulating vaccine derived polioviruses (cVDPVs). Risk Anal. 33(4), 680–702 (2013) 7. M. Famulare, S. Chang, J. Iber, K. Zhao, J.A. Adeniji, D. Bukbuk, M. Baba, M. Behrend, C.C. Burns, M.S. Obersteb, Sabin vaccine reversion in the field: a comprehensive analysis of Sabin-like poliovirus isolates in Nigeria. J. Virol. 90(1), 317–331 (2015) 8. Genetic Evolution of Vaccine-Derived Poliovirus (VDPV) in an immunodeficient patient. Epidemiol. Bull. 20(11), 249–271 (2004) 9. J.-H. Kim, S.-H. Rho, Transmission dynamics of oral polio vaccine viruses and vaccine-derived polioviruses on networks. J. Theor. Biol. 364, 266–274 (2015) 10. J.P. LaSalle, S. Lefschetz, Stability by Liapunov’s Direct Method (Academic Press, Cambridge, 1961) 11. S. Marino, I.B. Hogue, C.J. Ray, D.E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 178–196 (2008) 12. A.K. Ndiaye, P.A.M. Diop, O.M. Diop, Environmental surveillance of poliovirus and non-polio enterovirus in urban sewage in Dakar, Senegal (2007–2013). Pan Afr. Med. J. 19, 243 (2014). https://doi.org/10.11604/pamj.2014.19.243.3538 13. OMS, Organisation Panaméricaine de la Santé, Guide pratique: introduction du vaccin antipoliomyélitique inactivé (IPV) 14. Z. Shuai, P. Van Den Driessche, Global stability of infectious disease model using Lyapunov functions. SIAM J. Appl. Math. 73(4), 1513–1532 (2013) 15. H. Smith, An Introduction to Delay Differential Equations with Applications to the life sciences (Springer, Berlin, 2010) 16. H.L. Smith, H.R. Thieme, Dynamical Systems and Population Persistence, vol. 118. Graduate Studies in Mathematics (American Mathematical Society, Providence, 2011) 17. H.L. Smith, P. Waltman, The Theory of the Chemostat, vol. 13. Cambridge Studies in Mathematical Biology (Cambridge University Press, Cambridge, 1995) 18. P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002) 19. B.G. Wagner, D.J.D. Earn, Circulating vaccine derived polio viruses and their impact on global polio eradication. Bull. Math. Biol. 70, 253–280 (2008) 20. B.G. Wagner, D.J.D. Earn, Population dynamics of live-attenuated virus vaccines. Theor. Popul. Biol. 77, 79–94 (2010) 21. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Text in Applied Mathematics, 2nd edn. (2003) 22. B. Wijayanto, N. Susanti, V. Setiawaty, Characterization and identification of poliovirus from the environment in Indonesia 2015. Bali Med J 7(3), 539–543 (2018)

A Scalable Engineering Combination Therapies for Evolutionary Dynamic of Macrophages Moctar Kande, Raphaël M. Jungers, Diaraf Seck, and Moussa Balde

Abstract The study of human immunodeficiency virus (HIV) has been the subject of massive scientific research in recent years. Some works have focused on the evolution of the virus mutations developed against the immune system or the combination design therapy. But most of these models are too complex to analyze in details and to design an optimal combination therapy. The main objective of this paper is to analyse the stability and optimal control of a HIV combination therapy nonlinear model, which takes into account the mutations and latent cells. We use a L1 controller that stabilize the evolutionary dynamics of HIV disease. Because of the positive nature of the system, this problem can be solved with a scalable iterative algorithm that finds the best medication.Therefore, following recent work of V. Jonsson and R. Murray we introduce a similar algorithm to solve the combination therapy design problem. We obtain efficient results for this nonlinear model and thereby show that optimal control theory can be applied on more complex and realistic models. Keywords HIV infection · Stability · Combination of therapies · Optimal control

M. Kande () · M. Balde Faculté des Sciences et Techniques (FST), Département de Mathématiques et Informatique (DMI), Université Cheikh Anta Diop de Dakar (UCAD), Dakar, Senegal e-mail: [email protected]; [email protected] R. M. Jungers UCL/ICTEAM, Louvain-la-Neuve, Belgium e-mail: [email protected] D. Seck Faculté des Sciences et Techniques (FST), Faculté des Sciences Economiques et de Gestions (FASEG), Laboratoire mathématique de la décision et d’analyse numérique(LMDAN), Université Cheikh Anta Diop de Dakar (UCAD), Dakar, Senegal e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_8

201

202

M. Kande et al.

1 Introduction The Human Immunodeficiency virus (HIV) is still one of the main public health problem across the world, with more than 1 million deaths in 2018, and 379 million people living with HIV. It has been progressively understood that mutations and latent cells are the main factors that complicate the treatment of people infected with HIV. Recently, several mathematical models have been proposed, some describe the interactions between viruses and the immune system or antiretroviral therapy, in order to understand the evolution of mutations during the infection of the body by the HIV virus [1–5]. HIV mutations make ineffective the anti-HIV monotherapy, this phenomenon has received considerable attention in the biology and biomedical communities. The resistance of HIV to a monotherapy has led to use a combination of therapies to combat the progression of the disease, especially against mutations. An effective combination of therapies can stop the evolution of mutations, but it can not cure the disease. In addition, a combination of therapies can cause very harmful side effects (neurological disorders, acute diarrhea, metabolic disorders) if the dose is exceeded, therefore it is necessary to optimize the combination therapy to find the best dose possible. In this context, optimal control techniques have allowed a better understanding on the application of the therapy [6–9]. It is also necessary for people infected with HIV to take antiretroviral therapies for life, because if they stop the treatment the disease may be reactivated because of the existence of latent cells, such as macrophages. Latently infected cells are classically defined as cells that contain integrated HIV DNA and are transcriptionally silent, but upon activation are capable of producing infectious virus. Macrophages play a crucial role in innate and adaptative immunity in response to microorganisms. They are an important cellular target during HIV-1 infection and the infected cells are considered as a long-term active reservoir, which play an important role in the final stages of the infection, the AIDS phase. In recent years, enormous progress has been made in the mathematical modeling of the evolution of HIV and the application of combination therapies [1, 6–8, 10– 12]. However, interactions between viruses and the immune system are poorly understood, as they are very complicated to be modeled. The models of mutations are complicated to analyze, both theoretically and numerically because of the number of equations and parameters they present [1, 4]. In addition, most existing models are not able to reproduce the three stages of infection (initial acute infection, a long asymptomatic period and AIDS phase) through numerical simulation, but they just reproduce the first two. Recently, in 2007 Hadjiandreou and al have proposed the first HIV ODE model time invariant parameters that can represent the clinical course of HIV. The model proposed by Hadjiandreou and al was very sensitive to the variation of parameters, it is in this sense that Esteban A. Hernandez-Vargas and Richard H. Middleton [13] have studied a simplified model of Hadjiandreou and al. Their model besides being robust to the variation of the parameters can also reproduce the three stages of the evolution of HIV.

Viral Dynamic Modelling

203

The main objective of this paper is the study of the macrophages model proposed by Esteban A. Hernandez-Vargas and Richard H. Middleton and more precisely the stability analysis of the proposed model. Then we add the mutations and the combination therapies, which are modeled as a Hill function. Finally we apply the optimal control techniques to our nonlinear model, which take in account the mutations, CD4+ T cell, macrophages and the combination therapies. We use a L1 controllers that stabilize the evolutionary dynamics of HIV disease. Because of the positive nature of the system, this problem can be solved with a scalable iterative algorithm that finds the best medication.Therefore, following recent work of V. Jonsson and R. Murray [10–12] we introduce a similar algorithm to solve the combination therapy design problem. The model proposed by V. Jonsson and R. Murray besides being simple, is linear and just takes into account the evolution of mutations. Despite the complexity of our nonlinear model, with the theory of optimal control we obtain efficient and realistic results.

2 Mathematical Models In this part we present the mathematical model proposed in [14] and study the behavior of the model around its equilibrium points.This model has five state variables: uninfected CD4 + T cells (T ), infected CD4 + T cells (T ∗ ), uninfected macrophages (M), infected macrophages (M ∗ ) and viral load (V ). In this model, we assumed that the CD4+T cells and macrophages, that are the main targets of the virus are produced respectively by the thymus at a constant rate ST and SM and both can die respectively at the rate δT and δM . CD4 + T cells become CD4 + T cells infected at a speed kT T V and the macrophages at a speed kM MV . Infected CD4+T cells and infected macrophages, respectively, have the following mortality rates δT ∗ and δM ∗ and the viral load is produced at the rates PT T ∗ and PM M ∗ by the infected CD4+ T cells and infected macrophages and the rate of degradation of the virus is δV . So we get the following system of differential equations: ⎧ ⎪ ⎪ T˙ ⎪ ⎪ ⎪ ⎨ T˙∗ M˙ ⎪ ⎪ ⎪ M˙ ∗ ⎪ ⎪ ⎩ V˙

= = = = =

sT − kT T V − δT T , k T T V − δT ∗ T ∗ , sM − kM MV − δM M, kM MV − δM ∗ M ∗ , pT T ∗ + pM M ∗ − δV V .

(2.1)

The values δ1T , δ 1∗ , δ1M , δ 1 ∗ and δ1V represent, respectively, the lifespan of T M uninfected CD4+T cells, infected CD4+T cells, uninfected macrophages, infected macrophages and the virus.

204

M. Kande et al.

2.1 Stability Analysis of the System The model (2.1) was proposed and studied by Esteban A. Hernandez-Vargas in [14]. In the analysis of the stability of the model we will use a different approach, which will be based on the basic reproductive ratio of the epidemic noted R0 (see Definition 2.1). We will carry out a detailed study on the stability of the system around its equilibrium points.

2.1.1 Existence of the Equilibrium Points To determine the equilibrium points we must solve the following equation: ⎧ ⎧ ⎪ ⎪ =0 ⎪ T˙ = 0 ⎪ sT − kT T V − δT T ⎪ ⎪ ⎪ ⎪ ∗ ∗ = 0 ˙ ⎪ ⎪ ∗ T T V − δ T =0 k ⎨ ⎨ T T ˙ ⇐⇒ M =0 sM − kM MV − δM M = 0 after calculation, we obtain the ⎪ ⎪ ⎪ ⎪ ∗ ˙ ⎪M = 0 ⎪ kT MV − δM ∗ M ∗ =0 ⎪ ⎪ ⎪ ⎪ ⎩ V˙ = 0 ⎩ p T ∗ + p M∗ − δ V = 0 T M V following results: sT , k T V + δT

T∗ =

V kT sT , δT ∗ k T V + δT

sM , k M V + δT

M∗ =

kM sM V . ∗ δM k M V + δM

T = M=

where V is the solution of the polynomial pM kM sM pT kT sT V V + − δV V = 0, δT ∗ k T V + δT δM ∗ k M V + δM this equation is in the form aV 3 + bV 2 + cV = 0 and admits three solutions V0 = 0,

V1 =

−b +

√

b2 − 4ac , 2a

V2 =

−b −

√ b2 − 4ac , 2a

where a = δT ∗ δM ∗ δV k T k M , b = δT ∗ δM ∗ δV (kM δT + kT δM ) − pT kT sT kM δM ∗ − pM kM sM kT δT ∗ , c = δT ∗ δM ∗ δV δT δM − pT kT sT δM δM ∗ − pM kM sM δT ∗ δT .

Viral Dynamic Modelling

205

In conclusion the system (2.1) admits three equilibrium points that we will note X0 = (T0 , T0∗ , M0 , M0∗ , V0 ), X1 = (T1 , T1∗ , M1 , M1∗ , V1 ) and X2 = (T2 , T2∗ , M2 , M2∗ , V2 ). Remark X0 represents the equilibrium point without infection and it exists when V0 = 0, in other words, when the viral load is zero. We have T0 (t) =

sT + λT exp (−δT t), δT

M0 (t) =

T0∗ = 0,

sM + λM exp (−δM t), δM

M0∗ = 0.

So when V0 = 0, lim T0 (t) = δsTT and lim M0 (t) = δsM . M t →∞ t →∞ The equilibrium points X1 and X2 exist if and only if we have respectively V1 > 0 and V2 > 0, because we can not have a negative viral load for biological reasons. Definition 2.1 The basic reproductive ratio of the epidemic noted R0 is the total amount of healthy cells that an infected CD4 + T cell and an infected macrophage cell can infect during HIV evolution. In our model an infected CD4 + T cell that has a life expectancy kT pT virions, which have a life expectancy cells, which have a lifetime of pT kT sT δT ∗ δT δV pM kM sM δM ∗ δM δV

1 δT

1 δV

1 δT ∗

produces

will infect some of the sT healthy

. So, an infected CD4 + T cell can infect R1 =

healthy cells and an infected macrophage, by analogy can infect R2 = . Finally, the basic reproductive ratio of the epidemic, R0 = R1 + R2 =

pT kT sT pM kM sM + . δT ∗ δT δV δM ∗ δM δV

Proposition 2.2 If there is an infection (V > 0), the system (2.1) has a single equilibrium point and this point exists if and only if c < 0 so, if and only if R0 =

pT kT sT pM kM sM + > 1. δT ∗ δT δV δM ∗ δM δV

Proof We assume there is infection (V > 0) and that c < 0, it is clear that −b + √ √ b2 − 4ac > 0 and −b − b2 − 4ac < 0 whatever the value taken by b, because a > 0. So only the point X1 exists (V1 > 0). If c > 0 we have, R0 =

pT kT sT pM kM sM + < 1. δT ∗ δT δV δM ∗ δM δV

If b takes positive values, then we get V1 < 0 and V2 < 0, so neither X1 nor X2 exists.

206

M. Kande et al.

If b < 0, from the expression of b noted above the following condition is obtained kM pT kT sT δT ∗ δT δV k M + k T

+

δM δT

kT pM kM sM > 1. δM ∗ δM δV kT + kM δδT M

This result is in contradiction with the fact c > 0, because we have R0 < 1. Now let’s suppose that c = 0, it means that R0 = 1, so we have δpT∗kδTT sδTV < 1 and pM kM sM δM ∗ δM δV

T

< 1. We obtain from the equation aV 3 + bV 2 + cV = 0, V =

V =−

−b a ,

pT kT sT kM δM ∗ b pM kM sM kT δT ∗ δT ∗ δM ∗ δV (kM δT + kT δM ) = + − a δT ∗ δM ∗ δV k M k T δT ∗ δM ∗ δV k M k T δT ∗ δM ∗ δV k M k T =

pT kT sT δT pM kM sM δM δT δM × + × − − δT ∗ δT δV kT δM ∗ δM δV kM kT kM

≤

δT δM δT δM + − − kT kM kT kM

≤ 0. Remark If R0 = δpT∗kδTT sδTV + δpM∗kδMMsδMV ≤ 1, it means that c ≥ 0, b is necessarily T M nonnegative (b ≥ 0) and there is not infection. Finally, if √ there is an infection (V > 0) only the equilibrium point X1 define by −b+ b2 −4ac , such as c < 0, that exists.  V1 = 2a 2.1.2 Stability of Equilibrium Points We will study the stability of the uninfected equilibrium point X0 = (T0 , T0∗ , M0 , M0∗ , V0 ) and the infected equilibrium point X1 = (T1 , T1∗ , M1 , M1∗ , V1 ), where T0 =

sT , δT

T0∗ = 0,

T1 =

sT , kT1 V1 + δT

M1 =

sM , k M V1 + δT

M0 =

sM , δM

T1∗ = M1∗ =

M0∗ = 0,

V0 = 0,

V1 kT sT , ∗ δT k T V1 + δT V1 kM sM . δM ∗ k M V1 + δM

We start with the first equilibrium point X0 . Proposition 2.3 The compact set  = {(T , T ∗ , M, M ∗ , V ) ∈ R5 : 0 ≤ T + T ∗ ≤ δsTT , , V ≤ K, K > 0} is a positive invariant set for the system (2.1). 0 ≤ M + M ∗ ≤ δsM M

Viral Dynamic Modelling

207

Proof Let t be the flow of the system (2.1), the compact set  is a positive invariant set for the system (2.1) if ∀ Y = (y1 , y2 , y3 , y4 , y5 ) ∈ , we have t (Y ) ∈  for ∀ t > 0. So we have, ⎧ ⎪ y˙1 = sT − kT y1 y5 − δT y1 , ⎪ ⎪ ⎪ ⎪ ⎨ y˙2 = kT y1 y5 − δT ∗ y2 , y˙3 = sM − kM y3 y5 − δM y3 , ⎪ ⎪ ⎪ y˙4 = kM y3 y5 − δM ∗ y4 , ⎪ ⎪ ⎩ y˙ = p y + p y − δ y . 4 T 2 M 4 V 5 y˙1 + y˙2 = sT − δT y1 − δT ∗ y2 ≤ sT − δT (y1 + y2 ), because the mortality rate of infected cells is higher than the mortality rate of uninfected cells (δT ∗ > δT ), so if we pose Y1 = y1 + y2 we get Y˙1 ≤ sT − δT Y1 , Finally we find Y1 ≤ δsTT + (Y1 (0) − sT δT )exp(−δT t), we can assume that when t = 0, the number of infected CD4 + T cells is negligible and the concentration of uninfected CD4 + T cells is less than sT /δT . So we get, Y1 (0) < δsTT and Y1 = y1 + y2 ≤ δsTT . By analogy, we also have , since the viral load is bounded, so there exists a real K > 0 Y2 = y3 + y4 ≤ δsM M such that y5 ≤ K. We conclude that t (Y ) ∈ , hence  is a positive invariant set for the system (2.1).  Theorem 2.4 The uninfected equilibrium point X0 is globally asymptotically stable in  if R0 ≤ 1 and is unstable if R0 > 1. Proof Before proving the theorem (2.4) let us recall the Lyapunov–Lasalle theorem. Theorem 2.5 Let  ⊂ D ⊂ Rn be a compact positively invariant set with respect to the system dynamics (1). Let V : D −→ Rn be a continuously differentiable function such that V˙ (x(t)) ≤ 0 in . Let E ⊂  be the set of all points in  where V˙ (x) = 0. Let H ⊂ E be the largest invariant set in E. Then every solution starting in  approaches M as t −→ ∞. Let’s put F (T , T ∗ , M, M ∗ , V ) =

pT ∗ pM ∗ T + ∗ M + V, δT∗ δM

F (T , T ∗ , M, M ∗ , V ) is a Lyapunov function for the nonlinear system (2.1) and we have F˙ = [ pδT ∗kTδVT + pδM k∗MδVM − 1]δV V , since T ≤ δsTT and M ≤ δsM , then we get M T M pT kT sT pM kM sM ˙ + − 1]δV V . It is obvious that if R0 ≤ 1 we obtain F˙ ≤ 0. F ≤[ δT ∗ δT δV

δM ∗ δM δV

Now we suppose that H is the set of solutions of the system where F˙ = 0, we can see that all paths in  approach the largest positively invariant subset of the

208

M. Kande et al.

set H thanks to the Lyapunov–Lasalle Theorem. So, when V = 0 we have on the boundary of , T (t) =

sT + λT exp (−δT t), δT lim T (t) =

t →∞

sT , δT

T ∗ = 0,

M(t) =

sM + λM exp (−δM t), δM

lim M(t) =

t →∞

sM . δM

M ∗ = 0.

Hence all solution paths in  converge asymptotically towards the uninfected equilibrium point X0 . To demonstrate the instability of the uninfected equilibrium point, we must calculate the Jacobian matrix of the system (2.1). This Jacobian Matrix evaluated to the point X0 is defined as follows ⎞ 0 0 −kT T0 −δT 0 ⎜ 0 −δ ∗ 0 0 k T T0 ⎟ ⎟ ⎜ T ⎟ ⎜ J (X0 ) = ⎜ 0 0 −δM 0 −kM M0 ⎟ . ⎜ ∗ k M ⎟ ⎝ 0 0 0 −δM M 0 ⎠ 0 pT 0 pM −δV ⎛

Let PJ (X0 ) (x) = det (J (X0 )−xI5 ) be the characteristic polynomial of J (X0 ). After calculation we get PJ (X0 ) (x) = −(δT + x)(δM + x)A(x) where A(x) is in the following form A(x) = x 3 + αx 2 + βx + γ , ∗ +δ , α = δT∗ + δM V ∗ ∗ ∗ δ − β = δT δM + δT∗ δV + δM V ∗ pT kT sT δM δT

pT kT sT − pMδkMM sM , δT ∗ p k s δ − M δMM M T . equilibrium point X0 is asymptotically

∗ δ − γ = δT∗ δM V Finally, the uninfected stable if and only if all the roots of the polynomial PJ (X0 ) (x) have a real negative part and that is possible if the Routh–Hurwitz stability criterion is satisfied for the polynomial A(x). The roots of A(x) have a negative real part if and only if we have α > 0, γ > 0 and αβ − γ > 0, we can remark that γ > 0 ⇐⇒ R0 < 1. Hence if R0 > 1 the uninfected equilibrium point X0 is unstable. 

Theorem 2.6 The infected equilibrium point X1 exists if R0 > 1 and when it exists, it is unstable.

Viral Dynamic Modelling

209

Proof The characteristic polynomial PJ (X1 ) (x) = det (J (X1 ) − xI5 ) of the Jacobian matrix J (X) evaluated to the infected steady state X1 is in the following forms, PJ (X1 ) (x) = H0 (x) + H1 (x) + H2 (x), PJ (X1 ) (x) = x 5 + Ax 4 + Bx 3 + Cx 2 + Dx + E, H0 (x) = −(δT∗ + x)(kM V1 + δM + x)(kT V1 + δT + x) × ∗ + x)(δ + x), (δM V H1 (x) = pM kM M1 (δT∗ + x)(δM + x)(kT V1 + δT + x), ∗ + x)(δ + x)(k V + δ + x). H2 (x) = pT kT T1 (δM T M 1 M A necessary condition of the Routh–Hurwitz stability criterion is satisfied if A > 0, B > 0, C > 0, D > 0 and E > 0. From the expression of H0 (x) we can easily calculate ∗ A = −(δT∗ + δM + δV + δT + δM + kT V1 + kM V1 ),

A is negative because all the parameters values are assumed positive and also V1 > 0.This proves the instability of the infected steady state X1 , when R0 > 1.  Numerical analysis of this model has been well studied by A. Hernandez-Vargas and Richard H. Middleton in [13]. They have shown the ability of this model to reproduce the clinical data of HIV evolution and the impact of macrophages on exploding viral load in the last phase of infection. They also prove that this model is not sensitive to a variation of the parameters. However, this model does not significantly reproduce the last phase for viral load. So, to have the three stage of the evolution of HIV, they add the proliferation of CD4 + T cells and macrophage cells in model (2.1), modeled as a kinetic function of Michaelis–Menten.

2.2 Mutations and Therapy In this part, we will add the mutations and the combination of therapies to the model (2.1). Despite the fact that the model (2.1) describes the evolution of HIV fairly well, it remains too simplified. Because it does not take into account the mutations of the virus. These mutations are due either to the reaction of the immune system or to reaction against therapy.

2.2.1 Model of Mutations Uninfected CD4+T cells (T ) and uninfected macrophages (M) are still considered and we add the cell proliferation terms using Michaelis–Menten kinetics in the

210

M. Kande et al.

V V following form CρTT+V and CρMM+V respectively for uninfected CD4 + T cells and uninfected macrophages. We introduce the mutations as follows: Uninfected CD4+ T cells and uninfected macrophages can be infected by all different viral strains. It is assumed that a strain i is obtained during the replication of this same strain with a probability qii , or it is obtained from the mutation of the strain j with a probability qij . Thus, a viral strain i can infect CD4 + T and macrophages respectively at rates kT and kM . The bilinear terms inthe equations of infected  CD4 + T cells and infected macrophages become kT T nj=1 qij Vj and kM M nj=1 qij Vj . We will note a CD4 + T cell infected by the strain i (Ti∗ ), macrophages infected by i (Mi∗ ) and Vi represents the density of the viral strain i. We obtain the following modified systems, no cells proliferation model with mutations

⎧ T˙ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎨ T˙i M˙ ⎪ ⎪ ⎪ M˙ ∗ ⎪ ⎪ ⎩ ˙i Vi

= sT − kT T V − δT T ,  = kT T nj=1 qj i Vj − δT ∗ Ti∗ , = sM − kM MV − δM M,  = kM M nj=1 qj i Vj − δM ∗ Mi∗ , = pT Ti∗ + pM Mi∗ − δV Vi .

(2.2)

and cells proliferation model with mutations ⎧ T˙ ⎪ ⎪ ⎪ ⎪ T˙∗ ⎪ ⎨ i M˙ ⎪ ⎪ ⎪ M˙ ∗ ⎪ ⎪ ⎩ i V˙i

V = sT + CρTT+V T − k T T V − δT T , n = kT T j =1 qj i Vj − δT ∗ Ti∗ , V M − kM MV − δM M, = sM + CρMM+V n = kM M j =1 qj i Vj − δM ∗ Mi∗ , = pT Ti∗ + pM Mi∗ − δV Vi .

(2.3)

These systems  contain (3n + 2)  equations, where n is the number of strains. But if we pose T ∗ = ni=1 Ti∗ , M ∗ = ni=1 Ti∗ and V = ni=1 Vi in the system (2.2) we go back to the system (2.1). A theoretical analysis of the evolution of the mutation system (2.3) is very complex. However the numerical analysis will also allow us to understand the evolution of this system, in particular the evolution of the total viral load and CD4+ T cells.

2.2.2 Numerical Analysis of the Mutation Model We will use the values of the parameters used in [14] for the numerical analysis of the model of mutations. For the initial conditions, it will be assumed that all strains are born with the same concentration Vi (0) = 10−3 copies/ml, CD4 + T cells in the initial state are equal to 1000 cells/mm3 , uninfected macrophages are equal to 150 cells/mm3 and infected CD4 + T cells and infected macrophages are assumed equal to zero.

Viral Dynamic Modelling

211

2500

800

2000

Virus Load

uninfected CD4+ T cells

Evolution of uninfected CD4+

1000

600 400 200

Evolution of Virus Load

1500 1000 500

0

0 0

1

2

3

4

5

6

7

8

9 10

0

1

2

Time (years)

3

4

5

6

7

8

9 10

Time (years)

infected CD4+

100

50

0 0

1

2

3

4

5

6

7

Time (years)

8

9 10

uninfected macrophages

Evolution of infected CD4+ Evolution of uninfected macrophages

150

15000

10000

5000

0 0

1

2

3

4

5

6

7

8

9 10

Time (years)

Fig. 1 Evolution of uninfected (T ) and total infected CD4 + T cells T ∗ , uninfected macrophages M and the total viral load V of the nonlinear system (2.3)

The uninfected CD4 + T cells, in the presence of strains, tend more rapidly to 200 cells/mm3 in the mutation model than the simplified model presented in [13]. Mutations accelerate the appearance of the last phase of HIV infection, which is AIDS. Figure 1 shows the three phases of HIV evolution, initial acute infection, a long asymptomatic period and AIDS phase. We can observe the long life of macrophages in the evolution of uninfected macrophages. Modeling the proliferation of CD4 + cells and macrophage cells as a form Michaelis–Menten kinetics allows the system (2.3) to have a certain level of robustness in relation to the variation of the parameters sT , sM , kT , kM , δT , δM , δT ∗ , δM ∗ and δV [14]. An increase in parameter values ρT and ρM leads to a rapid decrease in CD4 + cells below 200 cells/mm3 and enables to reach more quickly the last phase of HIV infection. We can notice in Fig. 4 the system (2.3) is more sensitive to the variation of the parameter ρM than to the parameter ρT (Figs. 2 and 3). In the Fig. 4, we checked the changes that the viral load suffered if we increase the values of the parameters ρT and ρM from 1 to 4%. The mutation system is not sensitive to the variation of the parameter ρT all the time, on the other hand a slight variation of the value of ρM leads to an explosion of viral load more quickly. But it can be noted that the system (2.3) is robust to the variation of ρM for the first 6 years as can be seen in Fig. 4.

212

M. Kande et al.

Evolution of uninfected CD4+

10000

800

8000

600 rho =0.003 M

400 200 rho =0.0057

Evolution of virus load

rho =0.0057

6000

M

4000 2000

rho M =0.003

M

0

0 0

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

Time ( years)

Evolution of infected CD4

Evolution of uninfected macrophage 4

100

M

50

0 1

2

3

4

5

6

7

Time ( years)

10

4

rho M =0.003

3 rho =0.0057

rho M =0.0057 rho =0.003

0

0

Time (years) 150

infected CD4

virus load

1000

uninfected macrophage

uninfected CD4+ T cells

Fig. 2 Evolution of uninfected (T ) and infected CD4 + T cells T ∗ , uninfected macrophages M and the total viral load V for ρT = 0.01, 0.02, . . . 0.1

8

9

10

M

2 1 0 0

1

2

3

4

5

6

7

8

9

10

Time ( years)

Fig. 3 Evolution of uninfected (T ) and infected CD4 + T cells T ∗ , uninfected macrophages M and the total viral load V for a variation of ρM of 10, 20, 30, 40, 50, 60, 70, 80 and 90%

Viral Dynamic Modelling

213

1500

1500 normal values 2% increase rho T

normal values 1% increase rho T

1000

1000

1% increase rho M

500

2% increase rho M

500

0

0 0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

5

6

7

1500

1500

normal values 4% increase rho T

normal values 3% increase rho T

1000

1000

3% increase rho M

4% increase rho M

500

500

0

0 0

1

2

3

4

5

6

7

0

1

2

3

4

Fig. 4 The variation of the parameters ρT and ρM for the verification of the robustness of the system (2.2) in the viral load

In the following, we assume the uninfected CD4 + T cells and the uninfected macrophages are constants. Indeed, for a short period of time, the number of healthy T cells and macrophages remain mainly constant so they are the same as the initial condition because uninfected CD4 + T cells and uninfected macrophages decrease really slowly during the latency period [13, 14]. So, the system (2.2) and (2.3) become the following linear system ⎧ n ∗ ∗ ⎪ ⎨ T˙i = kT T j =1 qj i Vj − δT ∗ Ti , n ∗ M˙i = kM M j =1 qj i Vj − δM ∗ Mi∗ , ⎪ ⎩ V˙ = p T ∗ + p M ∗ − δ V . i T i M i V i

(2.4)

2.2.3 Combination of Therapies In this part, we add the therapy to the model (2.4). We assumed that the combination of therapies is modeled as a Hill function in the following form [10, 11], ψik (%k ) = f (Kik ) =

%nk n %nk + Kik

214

M. Kande et al.

f (Kik ) is a function of Kik , which is the association constant for the virus/antibody binding reaction %j + Vi −→ %j Vi , for each neutralization reaction representing the rate at which a neutralizing macromolecule %k (assumed to remain at constant concentrations) neutralizes mutant i. To obtain a linear pharmacodynamics it will be assumed that n = 1 and Kik >> %k , so we have ψik (%k ) = f (Kik ) =

%k . Kik

The Hill equation has been used in pharmacology to model nonlinear drug doseresponses, such as the concentration dependent effects of drugs on cell viability or virus neutralization. We notice that the Hill function is a biological analog to actuator saturation, in that there is a law of diminishing returns in terms of the effect of ever increasing drug concentrations on the system [15]. We obtain the model of the following combination of therapies, ⎧ n ∗ ˙∗ ⎪ ⎨ Ti = kT T j =1 qj i Vj − δT ∗ Ti , n M˙i∗ = kM M j =1 qj i Vj − δM ∗ Mi∗ , ⎪ ⎩ V˙ = p T ∗ + p M ∗ − δ V − m i T i M i V i k=1

(2.5) %k Kik Vi .

Therapy is added to the strain equation, because it acts on the different viruses present in the system and m represents the different therapies possible to combine. This linear system (2.5) can be rewritten as following ⎞ −δT ∗ In 0n×n kT T Q X˙ = ⎝ 0n×n −δM ∗ In kM MQ ⎠ X, pT In pM In −δV In − ψL ⎛

(2.6)

where X = (T1∗ , T2∗ , . . . Tn∗ , M1∗ , M2∗ . . . Mn∗ , V1 , V2 , . . . Vn ), Q = [qj i ]1≤i,j ≤n , 0n×n isthe null matrix of order n × n, In is the identity matrix of size n and ψL = lk diag( m k=1 Kik )n×n .

3 Optimal Control In what follows we will apply the optimal control techniques used by V. Jonsson and R. Murray [10–12], using the theory of positive systems [16, 17], on our model of the combination of therapies, especially on the system (2.6). V. Jonsson and R. Murray, in their work to design treatment protocols to prevent mutations escape from monotherapy. They worked on three articles, in the first article they developed a suboptimal algorithm for the systematic design of feedback strategies to stabilize the evolutionary dynamics of a generic disease model, such as HIV and cancer, using an H∞ approach. The lack of scalability of their H∞ algorithm led them to publish a

Viral Dynamic Modelling

215

second article, in which they propose a scalable solution to the combination therapy problem by reformulating it as a second order cone program (SOCP). In the first and second article they supposed that the effect of drugs is linear, in the third they modeled the effect of drugs as a piecewise linear function and they developed a new algorithm to find a suboptimal controller. However, the model they used is too simple and does not represent well the evolution of HIV infection, because their model only takes into account the concentration of mutations. Thus, Eloise Ganhy and Cassandra de Cock de Rameyen in their master thesis under the supervision of Raphael M. Jungers [18], introduced a new model of mutations that takes into account CD4 + T cells and they applied the control techniques used in [10, 11]. Their model represents better HIV infection than the model proposed by V. Jonsson and R. Murray and they obtained good results except on the gain of closed-loop system, which takes very large values, despite a low dosage of drugs. In addition, their model represents only the first two phases of HIV infection and does not present the last phase, the AIDS phase. Their model converges towards the latent phase, despite the presence of mutations and CD4 + T cells. In this part we will recall all the theory that will be needed in the design of algorithms and the resolution of our optimal control problems. The set of nonnegative real numbers is denoted by R+ . The inequality X > 0, (X ≥ 0) means that all elements of the matrix or vector X are positive (nonnegative). X ) 0 means that X is a symmetric and positive definite matrix. The matrix X is H urwitz if all eigenvalues have negative real part and the matrix X is Metzler if all off-diagonal elements are nonnegative. We define 1n as being the vector of all ones of dimension n. Definition 3.1 ([17]) 1. Given matrices A ∈ Rn×n , B ∈ Rn×nu , C ∈ Rnz ×n and D ∈ Rnz ×nu , a Linear Time Invariant dynamical system (LT I ) can be described as 

x(t) ˙ = Ax(t) + Bu(t) z(t) = Cx(t) + Du(t)

(3.1)

where x(t) is the state, u(t) is the input and z(t) is the output. The transfer matrix of system (3.1) is given by: G(s) = C(sI − A)−1 B + D, G(s) is defined in the resolvent of the matrix A and the corresponding impulse response is g(t) = CeAt Bθ (t)+Dδ(t), where θ (t) is the Heaviside step function and δ(t) the Dirac delta function. 2. The maximum norm of vector x ∈ Rn is given by x∞ = max (|x1 |, |x2 |, . . . , |xn |) .

216

M. Kande et al.

3. The induced matrix norm for M ∈ Rr×m is given by Mp−ind =

sup

w∈Rm \{0}

|Mw|p |w|p

where |w|p = (|w1 |p + |w2 |p + . . . + |wm |p ). If M has nonnegative entries, we have M∞−ind < γ if and only if M1n < γ 1n . 4. The p norm of an element w of some Lebesgue space Lp [0, ∞) is given by wp =

"  k

∞

#1

p

|wk | dt p

.

0

5. For a r × m matrix transfer function G(s) = C(sI − A)−1 B + D, the induced norm of the corresponding impulse response g(t) = CeAt Bθ (t) + Dδ(t) is gp−ind = sup w

g ∗ wp wp

p

for w ∈ Lm [0, ∞) and g ∗ w ∈ Lrp [0, ∞) the convolution of g and w. Theorem 3.2 ([16]) The LTI system defined in (3.1) is internally positive if and only if (1) A is Metzler (2) B, C, D ≥ 0. Theorem 3.3 ([16, 17]) Let g(t) = CeAt Bθ (t) + Dδ(t) where CeAt B ≥ 0 for t ≥ 0 and D ≥ 0 while A is H urwitz. Then gp−ind = G(0)p−ind for p = 1, 2 or ∞. Theorem 3.4 ([16, 17]) Let g(t) = CeAt Bθ (t) + Dδ(t), then the following statements are equivalent: (1) A H urwitz and g∞−ind < γ . (2) ∃ξ ∈ Rn+ \ {0} such that 

AB CD



ξ 1n




0, a controller is said to be a L1 controller if the following conditions hold: (1) (A + ELF ) is asymptotically stable. z∞ < γ. (2) Tzw ∞−ind = sup0 0 such that the solution to the convex program: min

%∞

s.t.

(A + EL1 F )d + I ≺ 0

%∈Rm +

L1 = (I ⊗ %)

(3.7)

Viral Dynamic Modelling

219

is a stabilizing controller for system (12), where (A + EL1 F )d is a diagonal matrix comprised of the diagonal elements of A + EL1 F . Proof Let A + EL1 F = (A + EL1 F )d + (A + EL1 F )ij , where (A + EL1 F )d is the diagonal elements of the matrix A + EL1 F and (A + EL1 F )ij are the elements off-diagonal of the matrix A + EL1 F , so the matrix M = (A + EL1 F )ij ∈ R3n×3n , with (A + EL1 F )ij > 0 if i = j and (A + EL1 F )ij = 0 if i = j . By the Perron Frobenius such that the spectral radius ρ(M) = r ≤  theorem, there exists r > 0 maxi (A + EL1 F )ij . Let  = maxi (A + EL1 F )ij and M = I − (I − M). We note that −(I − M) ≺ 0. So we have the closed loop dynamics A + EL1 F = (A + EL1 F )d + I − (I − M) ≺ (A + EL1 F )d + I ≺ 0, yielding the desired stability.  3.1.2 A Suboptimal L1 Controller The system (3.6) must satisfy the conditions of Theorem 3.6 to keep its stability and have a certain level of robustness γ . The matrix A + EL1 F is Metzler, because we have ⎛

⎞ −δT ∗ In 0n kT T Q A + EL1 F = ⎝ 0n −δM ∗ In kM MQ ⎠ , pT In pM In −δV In − ψL L1 ∈ D is a positive diagonal matrix and B, C and F ≥ 0. Stability and robustness 3n are achieved for our model if there exist ξ ∈ R3n + , μ ∈ R+ such as Aξ + B1n + Eμ < 0 Cξ + D1n + Gμ < γ 1n F ξ + H 1n ≥ μ which can be rewritten as (A + EL1 F )ξ + 13n < 0 Cξ < γ 1mn+1 Fξ ≥ 0 because we assume that D ∈ R+ , therefore the condition F ξ +H 1n ≥ μ is replaced by F ξ + H 1n ≥ 0. We have μ = L(F ξ + H 1n ), since D = 0, G = 0 and H = 0, we have μ = LF ξ and F ξ ≥ 0. Even more, minimizing the term Cξ is simply minimizing the term CV where V is the virus population, this is justified by the fact that if ξ satisfies Theorem 3.4, then −ξ < x(t) < ξ for all solutions to the equation x˙ = Ax + Bw with x(0) = 0 and w∞ ≤ 1. In conclusion, for any given value of the regularizers λ1 ∈ R+ \ {0} and λ2 ∈ R+ \ {0}, we can find by solving the following non-convex program in order to

220

M. Kande et al.

compute the antibody: min

m ξ ∈R3n + ,%∈R+

Cξ ∞ + λ1 %1 + λ2 %2 (A + EL1 F )ξ + 13n < 0

s.t.

Cξ ∞ < γ L1 = (I ⊗ %) ξ ≥ 0. The antibody concentrations % that yield an optimal ∞−induced closed loop norm. This problem is bilinear because of the product L1 x and there are no known convex reformulations of this problem. As is standard for bilinear problems, we propose to search for a solution by alternatively considering one variable constant and optimizing on the other. This allows us to state the following iterative algorithm, based on the two following convex programs: Program 1 min

ξ ∈R3n + ,γ ∈R+

s.t.

γ (A + EL1 F )ξ + 13n ≤ 0 Cξ ∞ ≤ γ L1 = (I ⊗ %) ξ ≥ 0.

The output of this program is a value of ξ ∈ R3n + and γ , when the value of % is fixed. It will be noted P 1% when % is fixed to % . Program 2 min

γ ∈R+ ,%∈Rm +

s.t.

Cξ ∞ + λ1 %1 + λ2 %2 (A + EL1 F )ξ + 13n ≤ 0 Cξ ∞ ≤ γ L1 = (I ⊗ %) ξ ≥ 0.

The output of this program is a value of % ∈ Rm + , and γ when the value of ξ is fixed. It will be noted P 2ξ ,λ1 ,λ2 when ξ is fixed to ξ .

Viral Dynamic Modelling

221

Algorithm 1 L1 Algorithm 1) Set  > 0. Initialisation 2) Solve for initial stabilizing controller % : Solve (3.7) to obtain a controller %0 . Set (ξ , γ ) = P 1% . Set (% , γ ) = P 2ξ ,0,0 . 3) Find (λ 1 , λ 2 , %) that minimize γ : while γ − γ >  do i) Set (ξ , γ ) = P 1% . ii) Set(% , γ ) = P 2ξ ,λ1 ,λ2 . iii) Set γ = γ .

3.1.3 Implementation and Results The initial problem and Program 1 are linear and can be solved by the function (linprog.m) in matlab using an interior-point method, we choose to use CVX modeling system for a performance in time and the Program 2 is a second order cone program. we have implemented our programs with the CVX toolbox combined with the SDPT3 solver. CVX is a software package that runs in Matlab. It transforms Matlab into a modeling language to solve a disciplined convex programs (DCP). A disciplined convex programs describe objective and constraints using expressions formed from a set of basic conventions (linear, affine, convex, concave functions), a restricted set of operations or rules (that preserve convexity). CVX supports a number of standard problem types, including linear programs (LP) and quadratic programs (QP), second-order cone programs (SOCP), and semidefinite programs (SDP) [19]. The algorithm implemented in SDPT3 is a primal-dual path-following algorithm. At each iteration, we first compute a predictor search direction aimed at decreasing the duality gap as much as possible. After that, the algorithm generates a Mehrotratype corrector step with the intention of keeping the iterates close to the central path. However, we do not impose any neighborhood restrictions on our iterates. Initial iterates need not be feasible the algorithm tries to achieve feasibility and optimality of its iterates simultaneously. It should be noted that in our implementation, the user has the option to use a primal-dual path-following algorithm that does not use corrector steps [20]. We applied the algorithm on the linearized system and then we will compare the behavior of the nonlinear systems of mutations (2.2) and (2.3), and the linearized system of mutations (2.4). Our linearized system of mutations is stable, if and only if the matrix (A+EL1 F ) is Hurwitz, in other words, if and only if all real part of the eigenvalues of the matrix (A + EL1 F ) are negative. By construction, the matrix (A + EL1 F ) is Metzler, all elements in the off-diagonal of this matrix are nonnegative and we have for some  > 0, A+EL1F ≺ (A+EL1 F )d +I . By Proposition (3.8), (A+EL1F )d +I ≺

222

M. Kande et al.

0, then we have necessarily A + EL1 F ≺ 0 that conclude the stability of the matrix (A + EL1 F ).

3.1.4 Parameters for the Implementation in Matlab A main goal of our experiments is to compare the results obtained from the combination of therapies applied to our model to the results obtained by V. Jonsson and R. Murray [10, 11]. Thus, we take the same efficiency of the association constant 3I C50 Kij = 3ri +ln(2)−IijC50ij to calculate the matrix , where (IC50) is the half maximal inhibitory antibody concentration and these values are shown in [11]. The Q matrix is used to compute the matrix A, and q = n1a u(1 − u)k = 1.443.10−6 (where u = 3 × 10−5 mutations/nucleotide base pair/replication cycle is the mutation rate for HIV reverse transcriptase, k , 3000 is the size of the genome in residues and na = 19 is the number of amino acids that can be mutated to), is the probability that one specific amino acid mutates in another one [10]. Units of concentrations are respectively copies/ml, cells/mm3 , μg/ml for the virus, cells and the drug dose and time is measured in days. Since the first line represents the mutation of the wild type, only one mutation is needed and α = 1 − (n − 1)q. For the rest of the mutants, two mutations in a row are required hence the q 2 and β = 1 − q − (n − 1)q 2 , ⎛

α ⎜q ⎜ ⎜ .. Q=⎜ ⎜. ⎜. ⎝ .. q

⎞ q ··· ··· q β q2 · · · q2 ⎟ ⎟ . . ⎟ q 2 β . . .. ⎟ ⎟. .. . . . . 2 ⎟ . . .q ⎠ q2 · · · q2 β

For initial conditions, it is assumed that an HIV-infected person starts taking therapy at the beginning of the latency phase, because HIV is detected in people infected after the first phase of infection, which is described by a strong immune system reaction against the virus. So, we will assume that the Initial conditions for our model coincide with the infected equilibrium point X2 (see Sect. 2). In other words, we assume that all infected CD4 + T cells, infected macrophages and viral strains have respectively the same concentration at the equilibrium point X2 and their sum is equal to the infectious equilibrium. The endemic equilibrium point depends on the parameter values. Thus, the values of the parameters are taken as we reach a concentration of CD4 + T cells which is approximately equal to 200 cells/mm3 . In the application of the therapy, we will take into account five drugs and eighteen mutations and we will observe the concentration of drugs obtained according to the application of regularizers.

Viral Dynamic Modelling

223

3.1.5 Experimental Results After implementation in matlab we have the following results: 1. Solving the initial problem with  = 0.1, we obtain a concentration of drugs % = (0.1811.10−8, 0.1808.10−8, 0.1748.10−8, 0.1817.10−8, 0.1810.10−8) µg/ml, which can stabilize our linearized and non-linearized system (2.3) for eighteen mutations (It can be noted that this quantity is very small compared to the quantity which stabilizes the model proposed in [10, 11]). More precisely this quantity of drug makes the viral load go to zero in a few weeks for our linearized model and stabilize the nonlinear system (2.2) to a small amount of concentration of the viral load and cell concentration remain at the initial conditions. We can see that in the Fig. 5 for the linearized system and Fig. 6 for the nonlinear system (2.2). These drugs can only control the system (2.3) for some period of time before the virus load explode and CD4+T cells go progressively to zeros, as one can observe in Fig. 7 2. We solve our algorithm combined with program1 and 2 without regularizations, λ1 = λ2 = 0, and no condition on the sum of the drugs % for eighteen mutations. We get % = (7.4481, 7.4481, 7.4287, 7.4481, 7.4481)µg/ml, this concentration of drug is very high and can stop the evolution of mutations more quickly compared to the initial concentration. The drug concentration obtained without a regulator can eliminate the mutations of the linearized and non-linearized system

900 Total virus concentration CD4+ T cells concentration

Viral and CD4+T cells Concentration

800 700 600 500 400 300 200 100 0 0

1

2

3

4

5

6

7

8

9

10

Times (Months)

Fig. 5 Concentration of the total viral load and CD4+ T Cells for the linearized system for the concentration of drugs % = (0.1811.10−8 , 0.1808.10−8 , 0.1748.10−8 , 0.1817.10−8 , 0.1810.10−8 ) µg/ml, without noise

224

M. Kande et al. 3000

Viral and CD4+T cells concentration

Total virus concentration CD4+ T cells concentration

2500

2000

1500

1000

500

0 0

1

2

3

4

5

6

7

8

9

10

Times (Months)

Fig. 6 Concentration of the total viral load and CD4+ T Cells for the nonlinear system (2) for the concentration of drugs % = (0.1811.10−8 , 0.1808.10−8 , 0.1748.10−8 , 0.1817.10−8 , 0.1810.10−8 ) 3000 Total virus concentration CD4+ T cells concentration

CD4+T cells viral load

2500 2000 1500 1000 500 0 0

500

1000

1500

Time (Days) Fig. 7 Concentration of the total viral load and CD4+ T Cells for the nonlinear system (3) for the concentration of drugs % = (0.1811.10−8 , 0.1808.10−8 , 0.1748.10−8 , 0.1817.10−8 , 0.1810.10−8 )

in a few days. However, in terms of the application of combination therapies such a concentration of drugs is not applicable in an HIV-infected person, because the side effects can be fatal. For this concentration, we found a some level of robustness γ = 73.5431 of the linearized system.

Viral Dynamic Modelling

225

3. L1 and L2 regularizations: λ1 = λ2 = 0.5, our suboptimal algorithm gives the following results: % = (0, 0, 4.2309, 3.7649, 0) and γ = 155.3237. For the regularizers λ1 = λ2 = 0.5, the concentration of combination of drugs obtained is sparse and is small compared to the drugs obtained with no regularizations, but it can still eliminate the mutations of the linearized and both nonlinear systems. With this concentration, the linearized system is more sensitive to noise than without regularizers. 4. L1 regularization: λ1 = 0.5 and λ2 = 0, we obtain the following drugs concentration % = (0, 0, 4.2309, 3.7649, 0) and γ = 155.3237. These results are the same results obtained with the combined L1 and L2 regularizations. 5. L2 regularization: λ1 = 0 and λ2 = 0.5, we obtain the following results: % = (15.9292, 15.9292, 14.4098, 15.9292, 15.9292) and γ = 73.5601. The drugs concentration obtained with the L2 regulator is greater than the concentration obtained with the combined L1 and L2 regularizations, on the other hand, they have almost the same value of γ . This means that the linearized system has the same level of robustness for the concentration of drugs obtained in both cases, with the regularization L2 and no regularizations (Fig. 8). The concentration of combination of drugs obtained in each case is large, to get small drug concentrations, we add an additional constraint on the sum of drugs in  the algorithm ( j %j ≤ 1) to force it to find another local minimum (Fig. 9).

Fig. 8 Evolution of each strain viral for the linearized system for the concentration of drugs % = (0.1811.10−8 , 0.1808.10−8 , 0.1748.10−8 , 0.1817.10−8 , 0.1810.10−8 ) with a disturbance w, whose values are taken in a uniform distribution between 0 and 1. For this concentration of drugs, the linear system is sensitive to the noise

226

M. Kande et al.

CD4+T cells and viral load concentration

1500 Total virus concentration CD4+ T cells concentration

1000

500

0 0

50

100

150

200

250

300

350

400

450

500

Time (days) Fig. 9 Evolution of the concentration of the total virus (blue) and (red) the concentration of CD4+T cells the concentration of drugs % = (0, 0, 4.2309, 3.7649, 0). With this concentration of drugs in the nonlinear system (2.3) the concentration of the virus go the zero and the CD4+T cells come back at the initial condition

1. No regularizations, λ1 = λ2 = 0 we obtain % = (0, 0, 0.8180, 0.1820, 0) and γ = 1662. 2. L1 and L2 regularizations: λ1 = λ2 = 0.5, % = (0, 0, 0.6894, 0.1669, 0) and γ = 1862. 3. L1 regularization: λ1 = 0.5, λ2 = 0, % = (0, 0, 0.6894, 0.1669, 0) and γ = 1862. 4. L2 regularization: λ2 = 0.5, λ1 = 0, % = (0.0517, 0.0401, 0.6832, 0.1240, 0.0517) and γ = 1905. We observe that we have a smaller quantity of drugs but on the other hand we have a bigger γ . For these amounts of drugs the linearized system is more sensitive to the noise and it can be noted that in all the previous cases the concentration of the drugs can stabilize the evolution of the mutations for all the linearized and non-linearized systems (Fig. 10).

Viral Dynamic Modelling

227

CD4+ T cells et viral concentration

1400 Total virus concentration CD4+ T cells concentration

1200

1000

800

600

400

200

0 0

50

100

150

200

250

300

350

400

450

500

Times (days)

Fig. 10 Evolution of the concentration of the total virus (blue) and (red) the concentration of CD4+T cells the concentration of drugs % = (0, 0, 0.6894, 0.1669, 0). We can remark that with this concentration of drugs in the nonlinear system (2.2) the concentration of the virus go the zero and the CD4+T cells come back at the initial condition

4 Conclusion and Perspectives In this work, we have pushed further the system-theoretic analysis of the HIV evolution in the human body. We started with the model presented by Esteban A. Hernandez-Vargas, which is one of the models that better represents the evolution of HIV, we studied stability of the system according to the values of the basic reproduction rate R0 . We then added mutations and therapy to the above model, and we used optimal control techniques for positive systems in order to propose a slighty different scalable iterative algorithm that finds the best combination of drugs and that can minimize the induced L1 norm. We obtained satisfying results but our solution was more sensitive to noise than results obtained by Vanessa Jonsson and al. We observe that, with a small value of drug doses, the linearized version and system (2.2) and (2.3) behave differently. The nonlinear system converges to a nonzero equilibrium while the linearized reaches zero. With no regularization and no condition on the sum of drugs we obtained high drug concentrations and strong robustness to disturbances compared to the results obtained by applying regularizations. However, the concentration of drugs obtained in both conditions can eliminate the viral load, and in the nonlinear system CD4+T cells come back to the initial condition. We believe that our new model is interesting because control techniques give efficient results and this model considers a more realistic HIV dynamics (Fig. 11).

228

M. Kande et al.

CD4+T cells and viral load

1400 Total virus concentration CD4+ T cells concentration

1200 1000 800 600 400 200 0 0

200

400

600

800

1000

Times (days) Fig. 11 Evolution of the concentration of the total virus (blue) and (red) the concentration of CD4+T cells the concentration of drugs % = (0, 0, 0.6894, 0.1669, 0). We can remark that with this concentration of drugs in the nonlinear system (2.3) the concentration of the virus go a nonzero equilibrium and the CD4+T cells come back at the initial condition

This work was an initial attempt at making use of nonlinear optimal control techniques in an involved model taking into account mutations and therapy. Yet, our models do not take into account the response of the immune system, it would be interesting to add the response of the immune system in our models of mutations. In addition, the doses of drugs are assumed to be constant but in reality this is not possible, this assumption could be adjusted by considering a drug dose that varies over time.

References 1. M.A. Nowak, R.M. May, Mathematical biology of hiv infections: antigenic variation and diversity threshold. Math. Biosci. 106, 1–21 (1991) 2. A.S. Perelson, Modelling Viral and Immune System Dynamics. Macmillan Magazines Ltd, London, 2002 3. S. Bonhoeffer, M.A. Nowak, Mutation and the evolution of virulence. Roy. Soc. 258(1352), 133–140 (1994) 4. D. Wodarz, M.A. Nowak, The effect of different immune responses on the evolution of virulent cxcr4-tropic HIV. Proc Biol Sci. 265(1411), 2149–2158 (1998) 5. R.F. Stengel, Mutation and control of the human immunodeficiency virus, in Presented at the 13th Yale Workshop on Adaptive and Learning Systems, Yale University, New Haven, CT May 2005 (2005) 6. N.K. Dhingra, M. Colombino, M.R. Jovanovi´c, On the convexity of a class of structured optimal control problems for positive systems, in 2016 European Control Conference, June 29–July 1 2016 (2016)

Viral Dynamic Modelling

229

7. M. Colombino, N.K. Dhingra, M.R. Jovanovi´c, R.S. Smith, Convex reformulation of a robust optimal control problem for a class of positive systems, in 2016 IEEE 55th Conference on Decision and Control (CDC) ARIA Resort & Casino (2016), pp. 5263–5268 8. N.K. Dhingra, M. Colombino, M.R. Jovanovi´c, Structured decentralized control of positive systems with applications to combination drug therapy and leader selection in directed networks. National Science Foundation Under Awards ECCS-1739210 and CNS-1544887 (2018) 9. M. Colombino, N.K. Dhingra, M.R. Jovanovi, A. Rantzer, R.S. Smith, On the optimal control problem for a class of monotone bilinear systems, in 22nd International Symposium on Mathematical Theory of Networks and Systems, July 11–15 2016 (2016), pp. 411–413 10. V. Jonsson, N. Matni, R.M. Murray, Reverse engineering combination therapies for evolutionary dynamics of disease: an H∞ approach, in 52nd IEEE Conference on Decision and Control (2013), pp. 2060–2065 11. V. Jonsson, A. Rantzer, R.M. Murray, A scalable formulation for engineering combination therapies for evolutionary dynamics of disease, in American Control Conference (ACC) (2014), pp. 2771–2778 12. V. Jonsson, N. Matni, R.M. Murray, Synthesizing combination therapies for evolutionary dynamics of disease for nonlinear pharmacodynamics, in 53rd IEEE Conference on Decision and Control (2014), pp. 2352–2358 13. E.A. Hernandez-Vargas, R.H. Middleton, Modeling the three stages in hiv infection. J. Theor. Biol. 320, 33–40 (2013) 14. E.A. Hernandez Vargas, A Control Theoretic Approach to Mitigate Viral Escape in HIV (Hamilton Institute National University of Ireland, Maynooth, 2011) 15. V. Jonsson, Robust control of evolutionary dynamics. Ph.D Thesis, California Institute of Technology (2016) 16. A. Rantzer, Scalable control of positive systems. Eur. J. Control 24, 72–80 (2015) 17. A. Rantzer, Distributed control of positive systems. Automatic Control LTH, Lund University (2014) 18. E. Ganhy, C. de Cock de Rameyen, Optimal control techniques applied to hiv combination therapy. Master’s Thesis, Université Catholique de Louvain (2014–2015) 19. M.C. Grant, S.P. Boyd, The CVX Users’ Guide (2018) 20. R.H. Tütüncü, K.C. Toh, M.J. Todd, a matlab software package for semidefinite-quadraticlinear programming, version 3.0. Pittsburgh, PA 15213, USA (2001)

Exact Steady Solutions for a Fifteen Velocity Model of Gas Amah d’Almeida

Abstract Existence and boundedness of the solutions of the boundary value problem for a fifteen velocity tridimensional discrete model of gas is proved for bounded boundary conditions and exact analytic solutions are built. An application to the determination of the accommodation coefficients on the boundaries of a flow in a rectangular box is performed. Keywords Discrete velocity models · Boundary value problem · Exact solutions · Accommodation coefficient Mathematics Subject Classification (2010) Primary 76P05; Secondary 35L60, 35Q20

1 Introduction The complexity of the Boltzmann equation, the basic equation of gas dynamics, leads to the development of models equations having its main features but easier to handle mathematically or numerically. Discrete velocity models of gas are media composed of particles whose velocities belong to a given finite set of vectors. The nonlinear integro-differential Boltzmann equation is replaced by a system of coupled semi-linear partial differential equations which describe the evolution of the number densities associated with each of the selected velocities. The mathematical aspect of the study of discrete velocity models concern the proof of the global existence and the uniqueness of the solutions of the initial boundary value problem associated to the kinetic equations and interesting results have been obtained in the one-dimensional case [1, 4, 5, 9–12]. The situation is quite different for multi-

A. d’Almeida () Laboratoire d’Analyse, de Modélisation Mathématiques et Applications Faculté des Sciences, Université de Lomé, Lomé, Togo e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_9

231

232

A. d’Almeida

dimensional problems even in the steady case. In [14], using techniques based on the fractional steps method, the problem of existence and uniqueness of the solution of the initial boundary value problem is solved for the two velocity Carleman model. In the steady case, the boundary value problem for the two-dimensional four velocity Broadwell model is investigated in [3] and the existence of a solution is proved. These models are both one speed discrete velocity models and can not describe energetic phenomena. In this work, we prove the existence and the boundedness of the solutions of the boundary value problem for a fifteen velocity three speed discrete model of gas and build exact analytic solutions which show that the solutions are not unique in general. However, the equilibium solutions are unique and we use them to compute accommodation coefficients on the boundaries of a flow in a rectangular box. The paper is organized as follows. In Sect. 2 we briefly describe the model, state the boundary value problem and present the main result of the paper which is proved in Sect. 4. In Sect. 3 we establish the positivity of the solution of the boundary value problem. The exact analytic solutions are presented in Sect. 5 and an application to the determination of accommodation coefficients is performed for a gas flow in a rectangular box in Sect. 6.

2 Statement of the Problem The steady flow of a gas in an infinite channel with rectangular cross section is a problem of gas dynamics the modelling of which can lead to the boundary value problem in consideration here. We choose the origin O of the orthonormal reference (O, e-1 , e-2 , e-3 ) of R3 associated to the cartesian coordinates (x , y , z ) so that the edges of the cross section are located on the lines x = 0, x = a, y = 0 and y = b, 0 < b ≤ a. The channel is infinite in the direction of e-3 .

2.1 The Discrete Velocity Model The components of the velocities of the fifteen discrete velocity model we shall use to solve the problem are in the basis (-e1 , e-2 , e-3 ) : u- 1 = c(0, 0, 0), u-2 = c(1, 0, 0), u-3 = c(0, 1, 0),u4 = c(0, 0, 1), u-i+3 = −ui , i = 2, 3, 4, u- 8 = c(1, 0, 1), u-9 = c(1, 0, −1), u- 12 = c(0, 1, 1), u-13 = c(0, 1, −1),uj +2 = −uj , j = 8, 9, 12, 13. We denote by Ni (t , x , y , z ) the number density of particles of velocity u-i in point M(t , x , y , z ) at time t . The Ni are continuous functions of t , x , y and z . The kinetic equations of the model are given in the Appendix. The fact that the channel is infinite in the z direction suggests to seek for the steady problem solutions independent of the space variable z . This assumption retained together with the steadiness of the flow implies the vanishing of the partial derivatives with respect to t and z . The kinetic equations reduce to:

Exact Steady Solutions

233

√ 0 = 2cs [(N2 + N3 + N5 + N6 )(N4 + N7 ) − (N8 + N9 + N10 + N11 + N12 + N13 )N1 ] √ − 2cs [(N14 + N15 )N1 ] √ 2 3cs [(N3 + N6 )(N8 + N9 ) − N2 (N12 + N13 + N14 + N15 )] c ∂N = ∂x √ + 5cs [N3 (N14 + N15 ) + N5 (N8 + N9 ) + N6 (N12 + N13 ) − 3N2 (N10 + N11 )] √ + 2cs [N1 (N8 + N9 ) − N2 (N4 + N7 )] + 2cs(N3 N6 + N4 N7 − 2N2 N5 ) √ 3 c ∂N ∂y = 3cs [(N3 + N6 )(N12 + N13 ) − N3 (N8 + N9 + N10 + N11 )] √ + 5cs [N2 (N10 + N11 ) + N5 (N8 + N9 ) + N6 (N12 + N13 ) − 3N3 (N14 + N15 )] √ + 2cs [N1 (N12 + N13 ) − N3 (N4 + N7 )] + 2cs(N2 N5 + N4 N7 − N3 N6 ) √ 0 = 2cs [N1 (N8 + N11 + N12 + N15 ) − N4 (N2 + N3 + N5 + N6 )] + 2cs(N2 N5 + N3 N6 √ −2N4 N7 ) + 5cs [N7 (N8 + N11 + N12 + N15 ) − N4 (N9 + N10 + N13 + N14 )] √ 5 −c ∂N 3cs [(N3 + N6 )(N10 + N11 ) − N5 (N12 + N13 + N14 + N15 )] = √∂x + 5cs [N2 (N10 + N11 ) + N3 (N14 + N15 ) + N6 (N12 + N13 ) − 3N5 (N8 + N9 )] √ + 2cs [N1 (N10 + N11 ) − N5 (N4 + N7 )] + 2cs(N3 N6 + N4 N7 − 2N2 N5 ) √ 6 −c ∂N 3cs [(N2 + N5 )(N14 + N15 ) − N6 (N8 + N9 + N10 + N11 )] = √∂y + 5cs [N2 (N10 + N11 ) + N3 (N14 + N15 ) + N5 (N8 + N9 ) − 3N6 (N12 + N13 )] √ + 2cs [N1 (N14 + N15 ) − N6 (N4 + N7 )] + 2cs(N2 N5 + N4 N7 − 2N3 N6 ) √ 0 = 2cs [N1 (N9 + N10 + N13 + N14 ) − N7 (N2 + N3 + N5 + N6 )] + 2cs(N2 N5 √ +N3 N6 − 2N4 N7 ) + 5cs [N4 (N9 + N10 + N13 + N14 ) − N7 (N8 + N11 + N12 + N15 )] √ 8 c ∂N 12 N15 − N8 N11 ) + 5cs(N2 N11 + N3 N15 + N4 N9 + N6 N12 − 3N5 N8 ∂x = 2cs(N √ √ −N7 N8 ) + 3cs [N2 (N12 + N15 ) − N8 (N3 + N6 )] + 6cs [N9 (N12 + N15 ) − N8 (N13 + N14 )] √ √ +2 2cs(N9 N11 + N12 N14 + N13 N15 − 3N8 N10 ) + 2cs(N2 N4 − N1 N8 ) √ 9 c ∂N 3cs [N2 (N13 + N14 ) − N9 (N3 + N6 )] + 2cs(N13 N14 − N9 N10 ) = ∂x √ √ + 2cs(N2 N4 − N1 N9 ) + 5cs(N2 N10 + N3 N14 + N6 N13 + N7 N8 − N4 N9 − 3N5 N9 ) √ √ + 6cs [N8 (N13 + N14 ) − N9 (N12 + N15 )] + 2 2cs(N8 N10 + N12 N14 + N13 N15 − 3N9 N11 ) √ 10 −c ∂N 3cs [N5 (N13 + N14 ) − N10 (N3 + N6 )] + 2cs(N13 N14 − N9 N10 ) = √ ∂x √ + 2cs(N5 N7 − N1 N10 ) + 5cs(N3 N14 + N5 N9 + N6 N13 + N7 N11 − N4 N10 − 3N2 N10 ) √ √ + 6cs [N11 (N13 + N14 ) − N10 (N12 + N15 )] + 2 2cs(N9 N11 + N12 N14 + N13 N15 − 3N8 N10 ) √ 11 −c ∂N 3cs [N5 (N12 + N15 ) − N11 (N3 + N6 )] + 2cs(N12 N15 − N8 N11 ) = √ ∂x √ + 2cs(N4 N5 − N1 N11 ) + 5cs(N3 N13 + N5 N8 + N6 N12 + N4 N10 − 3N2 N11 − N7 N11 ) √ √ + 6cs [N10 (N12 + N15 ) − N11 (N13 + N14 )] + 2 2cs(N8 N10 + N12 N14 + N13 N15 − 3N9 N11 ) √ 12 c ∂N ∂y = 3cs [N3 (N8 + N11 ) − N12 (N2 + N5 )] + 2cs(N8 N11 − N12 N15 ) √ √ + 2cs(N3 N4 − N1 N12 ) + 5cs(N2 N11 + N3 N15 + N4 N13 + N5 N8 − N7 N12 − 3N6 N12 ) √ √ + 6cs [N13 (N8 + N11 ) − N12 (N9 + N10 )] + 2 2cs(N8 N10 + N9 N11 + N13 N15 − 3N12 N14 ) √ 13 c ∂N ∂y = 3cs [N3 (N9 + N10 ) − N13 (N2 + N5 )] + 2cs(N9 N10 − N13 N14 ) √ √ + 2cs(N3 N7 − N1 N13 ) + 5cs(N2 N10 + N3 N14 + N5 N9 + N7 N12 − N4 N13 − 3N6 N13 ) √ √ + 6cs [N12 (N9 + N10 ) − N13 (N8 + N11 )] + 2 2cs(N8 N10 + N9 N11 + N12 N14 − 3N13 N15 ) √ −c ∂N14 = 3cs [N6 (N9 + N10 ) − N14 (N2 + N5 )] + 2cs(N9 N10 − N13 N14 )+ √ ∂y √ 2cs(N6 N7 − N1 N14 ) + 5cs(N2 N10 + N5 N9 + N6 N13 + N7 N15 − N4 N14 − 3N3 N14 ) √ √ + 6cs [N15 (N9 + N10 ) − N14 (N8 + N11 )] + 2 2cs(N8 N10 + N9 N11 + N13 N15 − 3N12 N14 ) √ −c ∂N15 = 3cs [N6 (N8 + N11 ) − N15 (N2 + N5 )] + 2cs(N8 N11 − N12 N15 )+ √ √ ∂y 2cs(N4 N6 − N1 N15 ) + 5cs(N2 N11 + N5 N8 + N6 N12 + N4 N14 − N7 N15 − 3N3 N15 ) √ √ + 6cs [N14 (N8 + N11 ) − N15 (N9 + N10 )] + 2 2cs(N8 N10 + N9 N11 + N12 N14 − 3N13 N15 ).

(2.1)

234

A. d’Almeida

2.2 The Boundary Conditions Let M(t , x , y , z ) be a point of a wall in contact with the gas flow at the time t , n-(t , x , y , z ) and u- w (t , x , y , z ) respectively, the inward-pointing (into the gas) unit vector normal to the wall and the velocity of the wall at M. The velocities of the discrete model can be arranged into three groups corresponding to emerging, grazing and impinging particles. This leads to the partition of the set of the velocity numbers into three sets: E = {i, (ui − u- w ).n > 0} , G = {i, (ui − u-w ).n = 0} , I = {i, (ui − u-w ).n < 0} . In discrete kinetic theory, the boundary conditions are prescribed when the microscopic densities of the emerging particles are given at the wall [6, 8]. For the problem in consideration, the particles of microscopic densities N1 , N4 and N7 are grazing particles for all the boundaries. Hence the boundary conditions have the form: ⎧ ⎪ N2 (0, y ) = ⎪ ⎪ ⎪ ⎪ N8 (0, y ) = ⎪ ⎪ ⎪ ⎪ N9 (0, y ) = ⎪ ⎪ ⎪ ⎪ ⎪ N5 (a, y ) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ N10 (a, y ) = ⎨ N11 (a, y ) = ⎪ N3 (x , 0) = ⎪ ⎪ ⎪ ⎪ ⎪ N12 (x , 0) = ⎪ ⎪ ⎪ ⎪ N13 (x , 0) = ⎪ ⎪ ⎪ ⎪ N6 (x , b) = ⎪ ⎪ ⎪ ⎪ N14 (x , b) = ⎪ ⎪ ⎩ N15 (x , b) =

ψ2 (y ) ψ8 (y ) ψ9 (y ) ψ5 (y ) (y ) ψ10 (y ) ψ11 ψ3 (x ) (x ) ψ12 (x ) ψ13 ψ6 (x ) (x ) ψ14 (x ) ψ15

(2.2)

The functions ψk , k ∈ B = {2, 3, 5, 6, 8, . . . , 15} are continuous, differentiable and positive functions. The boundary value problem is the system (2.1) with the boundary conditions (2.2). We prove in the sequel the following result: Theorem 2.1 (Main Theorem) The problem (2.1) and (2.2) has positive and bounded solution N = (N1 , . . . , N15 ) if the boundary data ψk , k ∈ B and their first derivatives are bounded.

3 Positivity of the Microscopic Densities Nk The expressions of the microscopic densities N1 , N4 and N7 in terms of the other microscopic densities Nk , k ∈ B are obtained by the resolution of the algebraic

Exact Steady Solutions

235

relations given by their kinetic equations in the form: 9 ⎧ ⎪ (N2 N5 + N3 N6 )(N2 + N3 + N5 + N6 )2 ⎪ ⎪ ⎪ N1 = ⎪ ⎪ 2(N8 + N11 + N12 + N15 )(N9 + N10 + N13 + N14 ) ⎨ N1 (N8 + N11 + N12 + N15 ) N4 = ⎪ ⎪ (N2 + N3 + N5 + N6 ) ⎪ ⎪ ⎪ N (N9 + N10 + N13 + N14 ) 1 ⎪ ⎩ N7 = (N2 + N3 + N5 + N6 )

(3.1)

Hence they are positive if Nk , k ∈ B are positive. We can take without loss of generality s = 1. We denote by cQk the second member of the differential equation for Nk , k ∈ B and we consider the boundary value problem: ⎧ ∂Nk ⎪ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ∂Nk ⎪ ⎪ ⎪ − ⎪ ⎪ ∂x ⎪ ⎪ ∂Nk ⎪ ⎪ ⎪ ⎪ ∂y ⎨ ∂Nk − ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ Nk (0, y ) ⎪ ⎪ ⎪ ⎪ Nk (a, y ) ⎪ ⎪ ⎪ ⎪ Nk (x , 0) ⎪ ⎪ ⎩ Nk (x , b)

= Qk (N), k = 2, 8, 9 = Qk (N), k = 5, 10, 11 = Qk (N), k = 3, 12, 13 = Qk (N), k = 6, 14, 15 = = = =

ψk (y ), ψk (y ), ψk (x ), ψk (x ),

k k k k

(3.2)

= 2, 8, 9 = 5, 10, 11 = 3, 12, 13 = 6, 14, 15

whose solutions are the number densities Nk , k ∈ B. Proposition 3.1 The solution (N2 , . . . , N15 ) of problem (3.2) when it exists, belongs to C+12 . We give two proofs for this result.

3.1 The First Proof The first proof is based on the fixed point method techniques that we use in the sequel to prove the existence of the solution of the problem.

236

A. d’Almeida

 0σ We set ρ(N) = 15 k=1 Nk and for σ > 0, Qk (N) = Qk (N)+σ Nk ρ(N), k ∈ B then the system (3.2) is equivalent to the system: ⎧ ∂Nk ⎪ ⎪ + σ Nk ρ(N) ⎪ ⎪ ∂x ⎪ ⎪ ∂N ⎪ k ⎪ ⎪ − + σ Nk ρ(N) ⎪ ⎪ ∂x ⎪ ⎪ ∂Nk ⎪ ⎪ ⎪ ⎪ ∂y + σ Nk ρ(N) ⎨ ∂Nk − + σ Nk ρ(N) ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ Nk (0, y ) ⎪ ⎪ ⎪ ⎪ Nk (a, y ) ⎪ ⎪ ⎪ ⎪ Nk (x , 0) ⎪ ⎪ ⎩ Nk (x , b)

= Q0σ k (N), k = 2, 8, 9 = Q0σ k (N), k = 5, 10, 11 = Q0σ k (N), k = 3, 12, 13 = Q0σ k (N), k = 6, 14, 15 = = = =

ψk (y ), ψk (y ), ψk (x ), ψk (x ),

k k k k

(3.3)

= 2, 8, 9 = 5, 10, 11 = 3, 12, 13 = 6, 14, 15

The solutions of (3.2) are positive if and only if those of (3.3) are positive. Let J = [0, a] × [0, b]. We denote by C the set of continuous functions defined p on J , and by C+ its subset of positive functions. C p and  C+ respectively denote p their cartesian products for p ≥ 2. For M ∈ C , |M| = |M1 |, . . . , |Mp | . We introduce the following norms: If z = (x, y) ∈ J and M = (M1 , . . . , Mp ) ∈ C p then z = |x| + |y|, Mi 0 = supz∈J |Mi (z)| and M1 = supi∈ Mi 0 , with  = {1, . . . , p}. Proposition 3.2 The solution (N2 , . . . , N15 ) of problem (3.3) when it exists, belongs to C+12 for sufficiently large σ . Proof For M ∈ C 12 we consider the problem ⎧ ∂Nk ⎪ ⎪ + σ Nk ρ(|M|) = ⎪ ⎪ ∂x ⎪ ⎪ ∂Nk ⎪ ⎪ ⎪ − + σ Nk ρ(|M|) = ⎪ ⎪ ∂x ⎪ ⎪ ∂Nk ⎪ ⎪ ⎪ + σ Nk ρ(|M|) = ⎪ ⎨ ∂y ∂Nk − + σ Nk ρ(|M|) = ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ N = k (0, y ) ⎪ ⎪ ⎪ ) ⎪ (a, y = N ⎪ ⎪ k ⎪ ⎪ = ⎪ ⎪ Nk (x , 0) ⎩ = Nk (x , b)

Q0σ k (|M|), k = 2, 8, 9 Q0σ k (|M|), k = 5, 10, 11 Q0σ k (|M|), k = 3, 12, 13 Q0σ k (|M|), k = 6, 14, 15 ψk (y ), ψk (y ), ψk (x ), ψk (x ),

k k k k

= 2, 8, 9 = 5, 10, 11 = 3, 12, 13 = 6, 14, 15

(3.4)

Exact Steady Solutions

237

which is a linear boundary value problem associated to (3.3). It is solved for given M ∈ C 12 by splitting it into the following boundary value problems: ⎧ ∂Nk ⎪ ⎪ ⎪ ∂x + σ Nk ρ(|M|) = ⎪ ⎪ ⎨ ∂Nk − + σ Nk ρ(|M|) = ∂x ⎪ ⎪ ⎪ Nk (0, y ) = ⎪ ⎪ ⎩ Nk (a, y ) = ⎧ ∂Nk ⎪ ⎪ + σ Nk ρ(|M|) = ⎪ ⎪ ⎪ ⎪ ∂y ⎨ ∂Nk − + σ Nk ρ(|M|) = ∂y ⎪ ⎪ ⎪ ⎪ N (x , 0) = k ⎪ ⎪ ⎩ = Nk (x , b)

Q0σ k (|M|), k = 2, 8, 9 Q0σ k (|M|), k = 5, 10, 11

(3.5)

ψk (y ), k = 2, 8, 9 ψk (y ), k = 5, 10, 11 Q0σ k (|M|), k = 3, 12, 13 Q0σ k (|M|), k = 6, 14, 15

(3.6)

ψk (x ), k = 3, 12, 13 ψk (x ), k = 6, 14, 15

The unique solution of (3.4) is given by: Nk

(x , y )

=

Nk

=

x

 + y g (x , y ) +

ψk

 + y f (x − a, y ) +

k = 2, 8, 9, (x , y )



ψk

k = 5, 10, 11,

0

 Nk (x , y ) = ψk x g − (x , y ) + k = 3, 12, 13,



y 0

 Nk (x , y ) = ψk x f − (x , y − b) + k = 6, 14, 15.

+ Q0σ k (|M|)(s, y )f (x − s, y )ds,



a x

+ Q0σ k (|M|)(s, y )f (x − s, y )ds,

− Q0σ k (|M|)(x , s)f (x , y − s)ds,



b y

− Q0σ k (|M|)(x , s)f (x , y − s)ds,

(3.7) with:

  x g + (x , y ) = exp −σ 0 ρ(|M|)(s, y )ds ,

  y g − (x , y ) = exp −σ 0 ρ(|M|)(x , s)ds g + (x , y ) g − (x , y ) , f − (x , y − a) = − . f + (x − a, y ) = + g (a, y ) g (x , a)

(3.8)

Thus the operator T˜ defined on C 12 by T˜ (M) = N where N is the unique solution of (3.4) is well defined and a solution of (3.3) is a fixed point of T˜ . For sufficiently large σ , Q0σ k is positive ∀k ∈ B hence as ψk is positive ∀k ∈ B Nk (x , y ) ≥ 0, ∀k ∈ B , ∀(x , y ) ∈ J . 

238

A. d’Almeida

k Remark 3.3 The kinetic equations for Nk , k ∈ B have the form ± ∂N ∂α = Fk (N) − Nk Gk (N) where α ∈ {x, y}, Fk is a positive quadratic function and Gk a positive linear function of Ni , i = k whose coefficients are the transition The √ probabilities. √ sufficient condition on σ for the positivity of Q0σ is σ ≥ 4 2 as 4 2cs is the k highest value of the transition probabilities of the model.

3.2 The Second Proof The second proof is based on the particular form of the system of the kinetic equations of Nk , k ∈ B. Proof Let:

N˜ i (x , y ) = N˜ i (x , y ) = N˜ i (x , y ) =



x

0 x exp a

 y exp 0

 y exp b

N˜ i (x , y ) = exp

 ρ(N)(s, y )ds Ni (x , y ),  ρ(N)(s, y )ds Ni (x , y ),  ρ(N)(x , s)ds Ni (x , y ),  ρ(N)(x , s)ds Ni (x , y ),

i = 2, 8, 9 i = 5, 10, 11 i = 3, 12, 13

(3.9)

i = 6, 14, 15

Then for i ∈ {2, 8, 9}

  ∂N ∂ N˜ i i x = ρ(N)N˜ i + exp 0 ρ(N)(s, y )ds , ∂x

  ∂x x = ρ(N)N˜ i + exp 0 ρ(N)(s, y )ds [Fi (N) − Ni Gi ] 

 x = exp 0 ρ(N)(s, y )ds Fi (N) + N˜ i [ρ(N) − Gi (N)] .

(3.10)

Similarly,

  ∂ N˜ i x )ds F (N) + N ˜ i [ρ(N) + Gi (N)] , i = 5, 10, 11, = − exp ρ(N)(s, y i a ∂x

  ∂ N˜ i y , s)ds F (N) + N ˜ i [ρ(N) − Gi (N)] , i = 3, 12, 13, = exp ρ(N)(x i 0 ∂y

  ∂ N˜ i y , s)ds F (N) + N ˜ i [ρ(N) + Gi (N)] , i = 6, 14, 15. = − exp ρ(N)(x i b ∂y (3.11)

Exact Steady Solutions

239

We thus have: $ %  x  s N˜ i (x , y ) = ψi (y ) + 0 exp 0 Gi (N)(a, y )da Fi (N)(s, y )ds

  x exp 0 [ρ(N) − Gi (N)] (s, y )ds , i = 2, 8, 9 ! a  a N˜ i (x , y ) = ψi (y ) + x exp s G i (N)(t, y )dt Fi (N)(s, y )ds a (s, y )ds , i = 5, 10, 11 exp − x [ρ(N) $ + Gi (N)] %   s y N˜ i (x , y ) = ψi (x ) + 0 exp 0 Gi (N)(x , a)da Fi (N)(x , s)ds

  y exp 0 [ρ(N) − Gi (N)] (x , s)ds , i = 3, 12, 13 $

  % b b N˜ i (x , y ) = ψi (x ) + y exp s Gi (N)(x , t)dt Fi (N)(x , s)ds

  b exp − y [ρ(N) + Gi (N)] (x , s)ds , i = 6, 14, 15

(3.12)

As Fk (N) and ψk are positive functions ∀k ∈ B, N˜ k , k ∈ B are positive and so are Nk , k ∈ B. 

4 Existence and Boundedness of the Solution Let: D0 = N1 + N4 + N7 , D1 = N2 + N8 + N9 , D2 = N5 + N10 + N11 , D3 = N3 + N12 + N13 , D4 = N6 + N14 + N15 , Q0 √ = 2 [N12 N15 + N13 N14 − (N8 N11 + N9 N10 )] +4√2(N4 N12 + N13 N15 − N8 N10 − N9 N11 ) +2 5 [N3 (N14 + N15 ) + N6 (N12 + N13 ) − N5 (N8 + N9 ) − N2 (N10 + N11 )] (4.1) The densities Di , i = 0, . . . , 4 are the sums of the microscopic densities Nk whose velocities have the same projection in the linear space spanned by (-e1 , e-2 ). From the system (2.1) we deduce by addition the system: ⎧ 0 ⎪ ⎪ ⎪ ∂D1 ⎪ ⎪ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ⎨ ∂D2 ∂x ⎪ ∂D3 ⎪ ⎪ ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ∂D 4 ⎪ ⎩ ∂y

= N3 N6 + N2 N5 − 2N4 N7 = 2(N3 N6 + N4 N7 − 2N2 N5 ) + Q0 (N) = Q1 (N) = −Q1 (N) = 2(N2 N5 + N4 N7 − 2N3 N6 ) − Q0 (N) = Q2 (N) = −Q2 (N)

(4.2)

240

A. d’Almeida

Hence the boundary value problem for the determination of Di ,i = 1, 2, 3, 4 is ⎧ ∂D1 ⎪ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ∂D2 ⎪ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ∂D3 ⎪ ⎪ ⎪ ⎪ ⎨ ∂y ∂D4 ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ D1 (0, y ) ⎪ ⎪ ⎪ ⎪ D2 (a, y ) ⎪ ⎪ ⎪ ⎪ D3 (x , 0) ⎪ ⎪ ⎩ D4 (x , b)

= 3(N3 N6 − N2 N5 ) + Q0 (N) = Q1 (N) = −Q1 (N) = 3(N2 N5 − N3 N6 ) − Q0 (N) = −Q1 (N) (4.3)

= Q1 (N) = = = =

(ψ2 + ψ8 + ψ9 )(y ) = φ1 (y ) + ψ )(y ) = φ (y ) (ψ5 + ψ10 11 2 + ψ )(x ) = φ (x ) (ψ3 + ψ12 13 3 + ψ )(x ) = φ (x ) (ψ6 + ψ14 15 4

We put: D = (D1 , D2 , D3 , D4 ), ρ + (D) = D1 + D2 and ρ − (D) = D3 + D4 and consider for σ > 0 the following problem ⎧ ∂D1 ⎪ ⎪ + σ D1 ρ + (D) ⎪ ⎪ ∂x ⎪ ⎪ ∂D2 ⎪ ⎪ ⎪ + σ D2 ρ + (D) ⎪ ⎪ ∂x ⎪ ⎪ ∂D3 ⎪ ⎪ ⎪ + σ D3 ρ − (D) ⎪ ⎨ ∂y ∂D4 + σ D4 ρ − (D) ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ D1 (0, y ) ⎪ ⎪ ⎪ ⎪ D2 (a, y ) ⎪ ⎪ ⎪ ⎪ D3 (x , 0) ⎪ ⎪ ⎩ D4 (x , b)

= Q1 (N) + σ D1 ρ + (D)

= Qσ1 (N)

= −Q1 (N) + σ D2 ρ + (D) = Qσ2 (N) = −Q1 (N) + σ D3 ρ − (D) = Qσ3 (N) = Q1 (N) + σ D4 ρ − (D) = = = =

= Qσ4 (N)

(4.4)

φ1 (y ) φ2 (y ) φ3 (x ) φ4 (x )

The system (4.4) is obtained from system (4.3) by adding σ Di ρ ± (D) to the two members of the kinetic equation for Di so the two systems of equations are equivalent.

4.1 Existence of Solutions of (4.4) Proposition 4.1 The problem (4.4) has a solution which belongs to C+4 for sufficiently large σ .

Exact Steady Solutions

241

Proof Consider for M ∈ C 4 , the following boundary value problem: ⎧ ∂D1 ⎪ ⎪ + σ D1 ρ + (|M|) = Q1 (|M|) + σ |M1 |ρ + (|M|) = Qσ1 (|M|) ⎪ ⎪ ∂x ⎪ ⎪ ∂D2 ⎪ ⎪ ⎪ + σ D2 ρ + (|M|) = −Q1 (|M|) + σ |M2 |ρ + (|M|) = Qσ2 (|M|) ⎪ ⎪ ∂x ⎪ ⎪ ∂D3 ⎪ ⎪ − − σ ⎪ ⎪ ∂y + σ D3 ρ (|M|) = −Q1 (|M|) + σ |M3 |ρ (|M|) = Q3 (|M|) ⎨ ∂D4 + σ D4 ρ − (|M|) = Q1 (|M|) + σ |M4 |ρ − (|M|) = Qσ4 (|M|) ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ D1 (0, y ) = φ1 (y ) ⎪ ⎪ ⎪ ⎪ = φ2 (y ) D2 (a, y ) ⎪ ⎪ ⎪ ⎪ D3 (x , 0) = φ3 (x ) ⎪ ⎪ ⎩ = φ4 (x ) D4 (x , b)

(4.5)

 Lemma 4.2 The problem (4.5) has for given M ∈ C+4 an unique solution which belongs to C+4 for sufficiently large σ . Proof The problem (4.5) is a linear problem associated with (4.4) and it is solved for given M ∈ C+4 by splitting it into the two following boundary value problems: ⎧ ∂D1 σ ⎪ + ⎪ ⎪ ∂x + σ D1 ρ (M) = Q1 (M) ⎪ ⎪ ⎨ ∂D2 + σ D2 ρ + (M) = Qσ2 (M) (4.6) ∂x ⎪ ⎪ ⎪ = φ1 (y ) D1 (0, y ) ⎪ ⎪ ⎩ = φ2 (y ) D2 (a, y ) and

⎧ ∂D3 ⎪ ⎪ + σ D3 ρ − (M) ⎪ ⎪ ⎪ ∂y ⎪ ⎨ ∂D4 + σ D4 ρ − (M) ∂y ⎪ ⎪ ⎪ ⎪ ⎪ D3 (x , 0) ⎪ ⎩ D4 (x , b)

= Qσ3 (M) = Qσ4 (M) = =

(4.7)

φ3 (x ) φ4 (x )

the unique solution of (4.5) is given by:  + y g (x , y ) +



x

Qσ1 (M)(s, y )f + (x − s, y )ds,  a  D2 (x , y ) = φ2 y f + (x − a, y ) − Qσ2 (M)(s, y )f + (x − s, y )ds,  y x  − D3 (x , y ) = φ3 x g (x , y ) + Qσ3 (M)(x , s)f − (x , y − s)ds, 0  b  Qσ4 (M)(x , s)f − (x , y − s)ds. D4 (x , y ) = φ4 x f − (x , y − b) −

D1

(x , y )

=

φ1

0

y

(4.8)

242

A. d’Almeida

with:

  x g + (x , y ) = exp −σ 0 ρ + (M)(s, y )ds ,

  y g − (x , y ) = exp −σ 0 ρ − (M)(x , s)ds − g + (x , y ) − (x , y − a) = g (x , y ) . f + (x − a, y ) = + , f g (a, y ) g − (x , a)

(4.9)

We compute the Qσk and get: √ Qσ1 √ (M) = 2(M12 M15 + M13 M14 ) + 3M3 M6 + 4 2(M12 M14 + M13 M15 ) +2 5 [M3 (M14 + M15 ) + M6 (M 12 + M13 )] + σ (M2 + M8 + M9 )2 √  + (σ − 2) (M8 M11 + M9 M10 ) + σ − 4 2 (M8 M10 + M9 M11 )

√  + (σ − 3) M2 M5 + σ − 2 5 [M2 (M10 + M11 ) + M5 (M8 + M9 )] √ Qσ2 √ (M) = −2(M12 M15 + M13 M14 ) − 3M3 M6 − 4 2(M12 M14 + M13 M15 ) −2 5 [M3 (M14 + M15 ) + M6 (M 12 + M13 )] + σ (M5 + M10 + M11 )2 √  + (σ + 2) (M8 M11 + M9 M10 ) + σ + 4 2 (M8 M10 + M9 M11 )

√  + (σ + 3) M2 M5 + σ + 2 5 [M2 (M10 + M11 ) + M5 (M8 + M9 )]

√  Qσ3 (M) = (σ − 2) (M12 M15 + M13 M14 ) + σ − 4 2 (M12 M14 + M13 M15 )

√  + σ − 2 5 [M3 (M14 + M15 ) + M6 (M12 + M13 )] + σ (M3 + M12 + M13 )2 √ √ +2 5 [M2 (M10 + M11 ) + M5 (M8 + M9 )] + 4 2 (M8 M10 + M9 M11 ) +3M2 M5 + 2 (M8 M11 + M9 M10 ) + (σ − 3) M3 M6  √ Qσ4 (M) = (σ + 2) (M12 M15 + M13 M14 ) + σ + 4 2 (M12 M14 + M13 M15 )

√  + σ + 2 5 [M3 (M14 + M15 ) + M6 (M12 + M13 )] + (σ + 3) M3 M6 +σ√ (M6 + M14 + M15 )2 − 2 (M8 M11 + M9 M10√ ) − 3M2 M5 −2 5 [M2 (M10 + M11 ) + M5 (M8 + M9 )] − 4 2 (M8 M10 + M9 M11 ) √ Obviously Qσ1 and Qσ3 are positive when σ ≥ 4 2. Moreover ∀σ : Qσ2 (M) + Qσ4 (M) = σ [D2 (D1 + D2 ) + D4 (D3 + D4 )] Qσ2 (M).Qσ4 (M) = −Q21 + σ [D2 (D1 + D2 ) + D4 (D3 + D4 )] Q1 +σ 2 [D2 (D1 + D2 ) − D4 (D3 + D4 )]

(4.10)

(4.11)

As the sum is strictly positive and the product has not constant sign we have two cases: either Qσ2 (M) and Qσ4 (M) are positive or one of them is positive and the other negative. When one of the Qσk , k = 2, 4 is negative, the corresponding Dk is positive. √ In the case where the Qσk , k = 2, 4 are positive, for σ ≥ 4 2, Qσi is positive ∀i ∈  hence as φi is positive ∀i ∈  Di (x , y ) > 0, i = 1, 3, ∀(x , y ) ∈ J and

Exact Steady Solutions

243

Di (x , y ) > 0, i = 2, 4 if and only if: a σ +  Q (M)(s, y )f (x − s, y )ds φ2 y > x 2 f + (x − a, y )  a

Qσ2 (M)(s, y )f + (a − s, y )ds, b σ −  y Q4 (M)(x , s)f (x , y − s)ds φ4 x > f − (x , y − b)  b Qσ4 (M)(x , s)f − (x , b − s)ds. = =

x

(4.12)

y

As 0 < f + (a − s, y ) < 1, ∀(s, y ) ∈ J , 0 < f − (x , b − s) < 1, ∀(x , s) ∈ J , we have a σ Q (M)(s, y )f + (a − s, y )ds ≤ a sup(x ,y )∈J Qσ2 (M) xb 2σ (4.13) − σ y Q4 (M)(x , s)f (x , b − s)ds ≤ b sup(x ,y )∈J Q4 (M) and it sufficient that φ2 > aQσ2 (M)0 and φ4 > bQσ4 (M)0 to have D ∈ C+4 . Naturally one of these conditions applies when one of the Qσk , k = 2, 4 is positive. − Otherwise, we have Q1 = Q+ 1 − Q1 with √ (M) = 2(M12 M15 + M13 M14 ) + 3M3 M6 + 4 2(M12 M14 + M13 M15 ) Q+ 1√ +2 5 [M3√(M14 + M15 ) + M6√(M12 + M13 )] √ √ = 2(1 + M13 ) 3 D4 + (3 − 2 5)M6 D3 + (4 √ 5 − 5 − 4 2)M6 (M12 √ √ + 2 2)D√ D4 − 2M14 (M12 + 2 2M13 ) − 2M15 (M13 + 2 2M12 ) +2( 5 − 1 − 2 2)M3√ Q− (M) = 3M2 M5 + 4 2 (M8 M10 + M9 M11 ) + 2 (M8 M11 + M9 M10 ) 1√ +2 5 [M2√(M10 + M11 ) + M5√(M8 + M9 )] √ √ = 2(1 (4 5 − 5 − 4 2)M5 (M√ 1 D2 + (3 − 2 5)M5 D1 + √ 8 + M9 ) √ + 2 2)D√ +2( 5 − 1 − 2 2)M2 D2 − 2M10 (M8 + 2 2M9 ) − 2M11 (M9 + 2 2M8 ) (4.14) Hence + − Q− Qσ2 (M) =√ 1 (M) − Q1 (M) + σ D2 (D1 + D2 ) ≤ Q1 (M) + σ D2 (D1 + D2 ) ≤ 2(1 + 2 2)D1 D2 + σ D2 (D1 + D2 ) − + Qσ4 (M) =√ Q+ 1 (M) − Q1 (M) + σ D4 (D3 + D4 ) ≤ Q1 (M) + σ D4 (D3 + D4 ) ≤ 2(1 + 2 2)D3 D4 + σ D4 (D3 + D4 ) (4.15) √ and for σ ≥ 2(1 + 2 2):

Qσ2 (M) ≤ σ D2 (2D1 + D2 ) ≤ σ ρ + (M) 2 Qσ4 (M) ≤ σ D4 (2D3 + D4 ) ≤ σ ρ − (M) 2

So Qσ2 (M)0 and Qσ4 (M)0 are bounded for bounded M ∈ C+4 .

(4.16) 

244

A. d’Almeida

Thus the operator T defined by T (M) = D where D is the unique solution of (4.5) is well defined and satisfies: Lemma 4.3 T is continuous and compact on bounded sets of C+4 . Proof We have T (M) = D if and only if D is given by the relations (4.8) from which we deduce:          D1 (x , y ) ≤ φ y  g + (x , y ) +  x Qσ (M)(s, y )f + (x − s, y )ds  , 1 1 0          D2 (x , y ) ≤ φ y  f + (x − 1, y ) +  x Qσ (M)(s, y )f + (x − s, y )ds  , 2 2 a          D3 (x , y ) ≤ φ x  g − (x , y ) +  y Qσ (M)(x , s)f − (x , y − s)ds  , 3 3 0          D4 (x , y ) ≤ φ x  f − (x , y − b) +  y Qσ (M)(x , s)f − (x , y − s)ds  . 4 4 b + − In one hand using the Generalized Mean ) Theorem, * Value * )as g and * g )are strictly positive * ) functions, we can find c1 ∈ 0, x , c2 ∈ x , a , c3 ∈ 0, y and c4 ∈ y , b such that:

 x σ + 0a Qσ1 (M)(s, y )f+ (x − s, y )ds Q (M)(s, y )f (x − s, y )ds xy 2σ Q (M)(x , s)f − (x , y − s)ds 0b σ3 − y Q4 (M)(x , s)f (x , y − s)ds

= = = =

 x Qσ1 (M)(c1 , y ) 0 f + (x − s, y )ds, a Qσ2 (M)(c2 , y ) x f + (x − s, y )ds,  y Qσ3 (M)(x , c3 ) 0 f − (x , y − s)ds, b Qσ4 (M)(x , c4 ) y f − (x , y − s)ds.

In in * the ) other hand * ) accordance * * )Value Theorem we can find d1 ∈ ) with the Mean 0, x , d2 ∈ x , a , d3 ∈ 0, y and d4 ∈ y , b such that:  x + 0a f+ (x − s, y )ds f (x − s, y )ds xy − f (x , y − s)ds 0b − y f (x , y − s)ds

= = = =

x f + (x − d1 , y ), (a − x )f + (x − d2 , y ), yf − (x , y − d3 ), (b − y )f − (x , y − d4 ).

Hence letting  a A+ (y ) = exp σ 0 ρ + (M)(s − a, y )ds  b A− (x ) = exp σ 0 ρ − (M)(x , s − b)ds ,

(4.17)

we get:     D1 (x , y ) D2 (x , y )     D3 (x , y ) D4 (x , y )

      ≤ φ1 y  + Qσ1 (M)(c 1, y ) ,     ≤ φ2 y  A+ y  + Qσ2 (M)(c 2, y ) ,     , c ) , ≤ φ3 x  + Qσ3 (M)(x 3    ≤ φ4 x  A− x  + Qσ4 (M)(x , c4 ) .

  since g ± (x , y ) < 1 .

(4.18)

Exact Steady Solutions

245

From which we infer  T (M)1 ≤ max φ1 0 , φ2 0 A+ 0 , φ3 0 , φ4 0 A− 0 + Qσ (M)1 (4.19) Thus T is continuous since A± , φi and Qσi , i ∈  are continuous. Hence if M is bounded then D = T (M) is bounded. Otherwise if D is the solution of (4.5) then ∀ i ∈ , ⎧ ∂Di ⎪ ⎨ + σ Di ρ + (M) = Qσi (M), ∂x ∂Di ⎪ ⎩ + σ Di ρ − (M) = Qσi (M), ∂y

i = 1, 2 i = 3, 4.

Thus ⎧ ∂Di ⎪ ⎨ = Qσi (M) − σ Di ρ + (M), i = 1, 2 ∂x ∂Di ⎪ ⎩ = Qσi (M) − σ Di ρ + (M) i = 3, 4. ∂y And ⎧   ∂Di  ⎪  ⎪ ⎨   ≤ Qσi (M) + σ Di ρ + (M)  ∂x   ∂Di  ⎪ ⎪ ⎩   ≤ Qσi (M) + σ Di ρ + (M) ∂y

i = 1, 2 i = 3, 4.

Moreover the first three equations of the system of the conservation equations of the model, given in the Appendix, written in terms of the Di , i ∈  taking into account the independence in the variables t and z give the system: ⎧ ∂ (D1 − D2 ) ∂ (D3 − D4 ) ⎪ ⎪ + =0 ⎪ ⎪ ∂x ∂y ⎪ ⎨ ∂ (D1 + D2 ) =0 ⎪ ∂x ⎪ ⎪ + D ∂ (D ) ⎪ 3 4 ⎪ ⎩ =0 ∂y From which we deduce: ⎧ ∂D ∂D3 1 ⎪ + =0 ⎨ ∂x ∂y ∂D2 ∂D4 ⎪ ⎩ + =0 ∂x ∂y

(4.20)

246

A. d’Almeida

We differentiate the Eqs. (4.20) with respect to y and get as the Di are differentiable functions of x and y the system: ⎧ 2 ∂ D1 ∂ 2 D3 ⎪ ⎪ =0 ⎨ + ∂x ∂y ∂y 2 ∂ 2 D2 ∂ 2 D4 ⎪ ⎪ ⎩ + =0 ∂x ∂y ∂y 2

(4.21)

We integrate (4.21) with respect to x and get: ⎧  x ∂ 2 D3 ∂D1 ⎪ ⎪ = − (s, y )ds + 1 (y ) ⎨ 0 ∂y ∂y 2  a ∂ 2 D4 ∂D2 ⎪ ⎪ ⎩ = (s, y )ds + 2 (y ) x ∂y ∂y 2 Then we integrate with respect to y and we have: ⎧  y  x ∂ 2 D3  y ⎪ ⎪ (s, t)dsdt + ⎨ D1 (x , y ) = − 0 0 0 1 (t)dt + Γ1 (x ) ∂y 2   ∂ 2 D4  y ⎪ ⎪ ⎩ D2 (x , y ) = 0y xa (s, t)dsdt + 0 2 (t)dt + Γ2 (x ) 2 ∂y Using the boundary conditions we have: 

 y D1 (0, y ) = 0 1 (t)dt + Γ1 (x ) = φ1 (y )  y D2 (a, y ) = 0 2 (t)dt + Γ2 (x ) = φ2 (y )

Hence the functions Γi are constant and i =

(4.22)

dφi , i = 1, 2. We thus have: dy

⎧  x ∂ 2 D3 ∂D1 ⎪ )ds + dφ1 (y ) ⎪ (x , y ) = − (s, y ⎨ 0 ∂y dy ∂y 2 2 ⎪ ⎪ ∂D2 (x , y ) =  a ∂ D4 (s, y )ds + dφ2 (y ) ⎩ x ∂y dy ∂y 2

(4.23)

Similarly, by differentiating (4.20) with respect to x , we obtain: ⎧  y ∂ 2 D1 dφ3 ∂D3 ⎪ ⎪ ⎨ (x , y ) = − (x , s)ds + (x ) 0 ∂x dx ∂x 2 2  b ∂ D2 dφ ∂D4 ⎪ ⎪ ⎩ (x , y ) = y (x , s)ds + 4 (x ) 2 ∂x dx ∂x

(4.24)

Exact Steady Solutions

247

Using the expressions (4.8) of Di , we get: ⎧ 2 σ 2 + ∂ D1 ⎪ (y ) ∂ g (x , y ) + ∂Q1 (M) (x , y ) ⎪ (x , y ) = φ ⎪ 1 ⎪ ∂x ⎪ ∂x 2 ∂x 2 ⎪ σ ⎪ 2D 2f + ⎪ ∂ ∂ 2 ⎪ , y ) = φ (y ) − a, y ) + ∂Q2 (M) (x , y ) ⎪ (x (x ⎨ 2 ∂x ∂x 2 ∂x 2 σ (M) 2D 2g− ∂Q ∂ ∂ 3 ⎪ 3 ⎪ (x , y ) = φ3 (x ) (x , y ) + (x , y ) ⎪ ⎪ 2 2 ⎪ ∂y ∂y ∂y ⎪ ⎪ ⎪ ∂Qσ4 (M) ∂ 2f − ⎪ ∂ 2 D4 ⎪ ⎩ (x , y ) = φ4 (x ) (x , y − b) + (x , y ) 2 2 ∂y ∂y ∂y

(4.25)

As ∂ 2g+ ∂x 2 ∂ 2f + ∂x 2 ∂ 2g− ∂y 2 ∂ 2f − ∂y

2

(x , y ) = σ 2 ρ + (M)g + (x , y ) 2

(x − a, y ) =

∂ 2g+ 1 2 (x , y ) = σ 2 ρ + (M)f + (x − a, y ) g + (a, y ) ∂x 2

(x , y ) = σ 2 ρ − (M)g − (x , y ) 2

(x , y − b) =

1

∂ 2 g−

g − (x , b)

∂x 2

(x , y ) = σ 2 ρ − (M)f − (x , y − b) 2

we have:     2  ∂ D1     2  + 2    +    ∂Qσ1 (M)           2 (x , y ) ≤ φ1 y σ ρ (M) x , y  g x , y +  ∂x (x , y ) ,     ∂x  ∂ 2 D2     2  + 2    +    ∂Qσ2 (M)           2 (x , y ) ≤ φ2 y σ ρ (M) x , y  f x , y +  ∂x (x , y ) ,     ∂x  ∂ 2 D3     2  − 2    −    ∂Qσ3 (M)           ∂y 2 (x , y ) ≤ φ3 x σ ρ (M) x , y  g x , y +  ∂y (x , y ) ,  2     ∂ D4     2  − 2    −    ∂Qσ4 (M)           ∂y 2 (x , y ) ≤ φ4 x σ ρ (M) x , y  f x , y +  ∂y (x , y ) .

and it exists x0 ∈]0, x [, x1 ∈]x , a[, y0 ∈]0, y [ and y1 ∈]y , b[ such that    ∂D1     ∂y (x , y )    ∂D2     ∂y (x , y )    ∂D3     ∂x (x , y )    ∂D4    (x , y )   ∂x

 2     ∂ D3   dφ1     ≤ a (x0 , y ) +  (y ) , 2 ∂y  2   dy   ∂ D4   dφ2     ≤ a (x1 , y ) +  (y ) , 2   dy   ∂y  ∂ 2 D1   dφ3     (x , y0 ) +  (x ) , ≤ b 2 ∂x   dx   2  ∂ D2   dφ4  + . (x , y ) (x ) ≤ b  1    dx ∂x 2

(4.26)

248

A. d’Almeida

Thus if M is bounded, 1, 2 and

dφi ∂Di ∂Di and ,∀i ∈  are uniformly bounded if ,i = ∂x ∂y dy

dφi , i = 3, 4 are bounded and it exists α and β such that ∀i ∈  dx    ∂Di     ∂x  < α in [0, a]

and

   ∂Di     ∂y  < β in [0, b] .

Given z1 = (x1 , y1 ) ∈ J and z2 = (x2 , y2 ) ∈ J . We deduce from the Mean Value Theorem, that it exists z0 = (x0 , y0 ) ∈ [z1 , z2 ] ⊂ J such that Di (z1 ) − Di (z2 ) = dDi (z0 )(z1 − z2 ) with $ % [z1 , z2 ] = z ∈ R2 /z = t (z1 − z2 ) + z2 , t ∈ [0, 1] and dDi (z0 )(h) =

∂Di ∂Di (z0 )h1 + (z )h2 ∂x ∂y 0

∀h = (h1 , h2 ) ∈ R2 .

Hence     Di (z ) − Di (z ) = dDi (z )(z − z ) 1 2 0 1 2 ≤ dDi (z0 )0 z1 − z2  with dDi (z0 )0 = suph≤1 = suph≤1

|dDi (z0 )(h)| h    ∂Di  ∂Di    ∂x (z0 )h1 + ∂y (z0 )h2  . |h1 | + |h2 |

But     ∂Di ∂Di  ≤  ∂Di (z )0 |h1 | +  ∂Di (z )0 |h2 |  (z )h + (z )h 2 0   ∂x 0 1 0 ∂y ∂x  ∂y 0  ∂Di ∂Di ≤ max  (z0 ) ,  (z0 ) h. ∂x ∂y 0 0

Exact Steady Solutions

249

Thus    ∂Di  ∂Di      ∂x (z0 )h1 + ∂y (z0 )h2  ∂Di ∂Di ≤ max  (z0 ) ,  (z0 ) |h1 | + |h2 | ∂x ∂y 0 0 ≤ max(α, β).   That is dDi (z0 ) ≤ max(α, β). Then Di (z1 ) − Di (z2 ) ≤ max(α, β)z1 − z2 .    It is sufficient that z1 − z2  < to have Di (z1 ) − Di (z2 ) <  for all max(α, β) i ∈ . We prove that for all solution D of (4.5):   ∀ > 0, ∃ξ > 0, z1 − z2  < ξ ⇒ Di (z1 ) − Di (z2 ) < 

∀ z1 , z2 ∈ J.

The set of the solutions of (4.5) is thus equicontinuous so T is compact on every  bounded subset of C+4 . Lemma 4.4 Every solution of the equation D = λT (D), 0 < λ < 1, is bounded. Proof D is a solution of D = λT (D) if and only if ⎧ ∂D1 ⎪ ⎪ + σ D1 ρ + (D) ⎪ ⎪ ∂x ⎪ ⎪ ∂D2 ⎪ ⎪ ⎪ + σ D2 ρ + (D) ⎪ ⎪ ∂x ⎪ ⎪ ∂D3 ⎪ ⎪ ⎪ + σ D3 ρ − (D) ⎪ ⎨ ∂y ∂D4 + σ D4 ρ − (D) ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ D1 (0, y ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D2 (a, y ) ⎪ ⎪ ⎪ ⎪ D3 (x , 0) ⎩ D4 (x , b)

= λQσ1 (D) (4.27.1) = λQσ2 (D) (4.27.2) = λQσ3 (D) (4.27.3) = λQσ4 (D) (4.27.4) = λφ1 (y ) = λφ2 (y ) = λφ3 (x ) = λφ4 (x )

(4.27)

(4.27.5) (4.27.6) (4.27.7) (4.27.8)

Making the sums (4.27.1) + (4.27.2) and (4.27.3) + (4.27.4), we obtain for the determination of the partial macroscopic densities ρ + (D) and ρ − (D) the following system of partial differential equations: ∂

[ρ + (D)] + (1 − λ)σ )ρ + (D)*2 = 0 (4.28.1) ∂x *2 ) ∂ [ρ − (D)] + (1 − λ)σ ρ − (D) = 0 (4.28.2) ∂y

(4.28)

250

A. d’Almeida

The unique solution of the system (4.28) is obviously ⎧ 1 ⎪ ρ + (x , y ) = ⎪ ⎪ ⎨ (1 − λ)σ x + h+ (y ) ⎪ ⎪ ⎪ ⎩ ρ − (x , y ) =

(4.29) 1 (1 − λ)σy + h− (x )

The problem (4.27) is a two point boundary value problem and only a part of the data are given at each boundary namely D1 (0, y) on the line x = 0, D2 (a, y ) on the line x = a, D3 (x , 0) on the line y = 0 and D4 (x , b) on the line y = b. We thus introduce the positive functions of y , αk+ and the positive functions of x , αk− , k = 0, 1 such that  D2 0, y  D1 a, y  D4 x , 0 D3 x , b

= α0+ (y )D1 (0, y ) = α1+ (y )D2 (a, y ) = α0− (x )D3 (x , 0) = α1− (x )D4 (x , b).

(4.30)

We emphazise the fact that the relations (4.30) are by no means reflection laws and are obtained merely by comparing functions of the same variables at the boundaries of the rectangle J and consequently are general. In the particular case of impermeable boundaries, the vanishing of the normal velocity on each boundary yields the relations  D2 0, y =  D1 a, y =  D4 x , 0 = D3 x , b =

D1 (0, y ) D2 (a, y ) D3 (x , 0) D4 (x , b)

at at at at

x = 0 ∀y ∈ [0, b] , x = a ∀y ∈ [0, b] , y = 0 ∀x ∈ [0, a] , y = b ∀x ∈ [0, a]

(4.31)

which amount to take αk+ (y ) = 1 and αk− (x ) = 1 ∀(x , y ) ∈ J , k = 0, 1. With the relations (4.30) we can compute the values of ρ ± at the boundaries from which we get:  h+ y = )  h− x = )

1

* 1 + α0+ (y ) λφ1 (y )

= )

1

* 1 + α1+ (y ) λφ2 (y )

+ σ (λ − 1)

1 1 * * = ) + σ (λ − 1). 1 + α0− (x ) λφ3 (x ) 1 + α1− (x ) λφ4 (x ) (4.32)

Exact Steady Solutions

251

From the systems (4.29) and (4.32) we deduce  ρ+ x , y =  ρ− x , y =

1 (1 − λ)σ x + )

1

* 1 + α0+ (y ) λφ1 (y ) 1

1 (1 − λ)σy + ) − * 1 + α0 (x ) λφ3 (x )

(4.33) .

Thus for 0 < λ < 1 , ρ + and ρ − are continuous and bounded as φi , i = 1, 3 and αk± , k = 0, 1 and so are the number densities Di ,∀i ∈ .  Remark 4.5 We point out the fact that for λ = 1 the solutions ρ + and ρ − of (4.33) are not singular and moreover satisfy partial conservation equations. Accordingly they depend upon one variable. Remark 4.6 The proof done for M ∈ C+4 can be extended to M ∈ C 4 using |M|. Finally we conclude to the existence of a solution of problem (4.4) by using the fixed point theorem of Schaefer [13] Theorem 4.7 Let T be a continuous and compact mapping of a Banach space X into itself, such that the set {x ∈ X, x = λT (x)} is bounded ∀ λ, 0 < λ < 1. Then T has a fixed point. The number densities D1 = N2 + N8 + N9 , D2 = N5 + N10 + N11 , D3 = N3 + N12 + N13 , D4 = N6 + N14 + N15 exist, are continuous and bounded. Hence the Ni , i ∈ B which are positive functions of x and y exist, are continuous and bounded and so are N1 , N4 and N7 .

5 Exact Solutions We seek in this section exact solutions for the symmetrical model obtained by assuming that the number densities of particles having the same projection of their velocities in the plane z = 0 are equal. For this symmetrical model we have: N1 = N4 = N7 , N2 = N8 = N9 , N3 = N12 = N13 , N5 = N10 = N11 , N6 = N14 = N15

(5.1)

252

A. d’Almeida

The microscopic densities are now N1 , N2 , N3 , N5 , N6 and letting σ = : 8 unknown √ √ s 8( 2 + 5) + 7 the boundary value problem reduces to: 3 ⎧ ∂N2 ⎪ ⎪ = σ (N3 N6 − N2 N5 ) = Q1 (N) (5.2.1) ⎪ ⎪ ∂x ⎪ ⎪ ∂N ⎪ 5 ⎪ ⎪ = −Q1 (N) (5.2.2) ⎪ ⎪ ∂x ⎪ ⎪ ∂N ⎪ 3 ⎪ ⎪ = −Q1 (N) (5.2.3) ⎪ ∂y ⎪ ⎪ ⎪ ⎨ ∂N6 = Q1 (N) (5.2.4) (5.2) ∂y ⎪ 6 ⎪ ⎪ ⎪ 2 N5 ⎪ N1 (x , y ) = N3 N6 +N (5.2.5) ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ N2 (0, y ) = φ2 (y ) ⎪ ⎪ ⎪ ⎪ N5 (a, y ) = φ5 (y ) ⎪ ⎪ ⎪ ⎪ N3 (x , 0) = φ3 (x ) ⎪ ⎩ N6 (x , b) = φ6 (x ) We put ρ + (N) = N2 +N5 and ρ − (N) = N3 +N6 and infer from the sums ((5.2.1)+ (5.2.2)) and ((5.2.3)+(5.2.4)) that: ∂ρ + (N) ∂ρ − (N) = =0 ∂x ∂y

(5.3)

Hence ρ + (N) and ρ − (N) are functions of one variable and N5 (x , y ) = ρ + (N)(y ) − N2 (x , y ) N6 (x , y ) = ρ − (N)(x ) − N3 (x , y )

(5.4)

The system (5.2) becomes:  ⎧ ∂N3 ∂N2 ⎪ ⎪ N2 − = − = σ ⎪ ∂x ∂y ⎪ ⎪ ⎪ ⎨ N (0, y ) = φ (y ) 2

2

ρ+ 2

N2 (a, y ) = ρ + (y ) − φ5 (y ) ⎪ ⎪ ⎪ ⎪ ⎪ N (x , 0) = φ3 (x ) ⎪ ⎩ 3 N3 (x , b) = ρ − (x ) − φ6 (x )

2

− N3 −

ρ− 2

2

+

ρ − −ρ + 4 2

2



(5.5)

Exact Steady Solutions

253

ρ+ ρ− (y ) − N2 (x , y ), F3 (x , y ) = (x ) − N3 (x , y ) the Letting F2 (x , y ) = 2 2 system (5.5) takes the form: # " ⎧ −2 − ρ+2 ∂F ∂F ρ ⎪ 2 3 ⎪ ⎪ = − = −σ F22 − F32 + ⎪ ⎪ ∂x ∂y 4 ⎪ ⎪ ⎪ + ⎪ ρ ⎪ ⎪ ⎪ F (0, y ) = (y ) − φ2 (y ) ⎪ ⎨ 2 2 + ) = φ (y ) − ρ (y ) (a, y F ⎪ 2 5 ⎪ ⎪ 2 ⎪ − (x ) ⎪ ρ ⎪ ⎪ ⎪ F3 (x , 0) = − φ3 (x ) ⎪ ⎪ 2 ⎪ − ⎪ ⎪ ⎩ F (x , b) = φ (x ) − ρ (x ) . 3 6 2

(5.6)

The system (5.6) has a simpler form but its exact resolution is complicated. However it allows to find exact solutions of the problem (5.2) in particular cases.

5.1 Maxwellian Solutions The thermodynamic equilibrium densities of a discrete velocity model of gas are called maxwellian densities of the model. The unique maxwellian densities of the symmetrical model associated to the macroscopic density ρ, the mean velocity U- = (U, V , 0) and the total energy E are the positive functions NiM , i = 1, 2, 3, 5, 6 given by: N1M =

ρ(5−6e) 9

ρ [6e + 3(u + v) − 2] [6e + 3(u − v) − 2] 72(3e − 1) ρ [6e + 3(u + v) − 2] [6e − 3(u − v) − 2] N3M = 72(3e − 1) ρ [6e − 3(u + v) − 2] [6e − 3(u − v) − 2] N5M = 72(3e − 1) ρ [6e − 3(u + v) − 2] [6e + 3(u − v) − 2] N6M = 72(3e − 1) V E U u = , v = , e = 2. c c c N2M =

(5.7)

254

A. d’Almeida

They are solutions of the system [6]: ⎧ ⎪ ⎪ρ ⎪ ⎪ ⎪ ρU ⎪ ⎪ ⎨ ρV ⎪ 2ρE ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ 0

= = = = = =

3(N1M + N2M + N3M + N5M + N6M ) 3c(N2M − N5M ) 3c(N3M − N6M ) c2 [2N1M + 5(N2M + N3M + N5M + N6M )] N2M N5M − N3M N6M 2 . N2M N5M + N3M N6M − 2N1M

(5.8)

6 6 1 1 ρ + 2 (y ) − 4c1 , F3 (x , y ) = ρ − 2 (x ) − 4c1 The solution F2 (x , y ) = 2 2 for c1 ≥ 0 of (5.6) leads to the maxwellian solution: N1 (x , y ) = N2 (x , y ) = N5 (x , y ) = N3 (x , y ) = N6 (x , y ) =

√ c1 ρ + (y ) 2 ρ + (y ) 2 ρ − (x ) 2 ρ − (x ) 2

6 1 ρ + 2 (y ) − 4c1 26 1 − ρ + 2 (y ) − 4c1 26 1 + ρ − 2 (x ) − 4c1 26 1 − ρ − 2 (x ) − 4c1 . 2 +

(5.9)

Taking into account the boundary conditions, we get: ρ + (y ) = φ2 (y ) + φ5 (y ) ρ − (x ) = φ3 (x ) + φ6 (x ) c1 = φ2 (y )φ5 (y ) = φ3 (x )φ6 (x ).

(5.10)

The validity of the third relation (5.10) imposes the dependence of the boundary data in the form: φ5 (y ) = φ6 (x ) =

c1 φ2 (y ) c1 . φ3 (x )

(5.11)

We thus can express the Maxwellian solutions in the following form: √ N1 (x , y ) = c1 N2 (x , y ) = φ2 (y ) N5 (x , y ) = φ c(y1 ) (x , y )

N3 = N6 (x , y ) =

2

φ3 (x ) c1 . φ3 (x )

(5.12)

Exact Steady Solutions

255

The solutions (5.12) are associated to the macroscopic variables: ρ

=3

√ c1 + φ2 + φ3 +

ρU = 3c(φ2 −

c1 ) φ2 c1 ) φ3

c1 φ2

+

ρV = 3c(φ3 − 8 √

2ρE = c2 2 c1 + 5 φ2 + φ3 +

c1 φ3



(5.13) c1 φ2

 : + φc1 . 3

So they are merely particular expressions of the unique maxwellian solutions (5.7) of the model associated to the macroscopic variables ρ, U , V and E.

5.2 Non Maxwellian Solutions For ρ + , ρ − , k and l constant such that kρ − − lρ + = 0 a solution of (5.2) is given by: k M(x , y ) l N3 (x , y ) = M(x ,y )   x  − y k2 − l2 + − . M(x , y ) = c0 exp σ kρ − lρ + − k l kρ − lρ +

N2 (x , y ) =

(5.14)

This solution is non maxwellian whatever ρ + , ρ − , k and l when kρ − − lρ + = 0 as c0 is a non zero scaling parameter. Moreover when ρ + = ρ − = ρ constant we have a solution given by: ) *  ρ c2 c3 −c2 + c3 tanh c1 + c2 x + c3 y +  2 2 2 σ c2 − c3  * ) c2 c3 ρ c3 − c2 tanh c1 + c2 x + c3 y . N3 (x , y ) = +  2 2 2 σ c2 − c3 N2 (x , y ) =

(5.15)

The fact that we have several solutions of different kinds (pure maxwellian and pure non maxwellian) for constant ρ + and ρ − shows the non uniqueness of the solutions of the system (5.2) in general. Furthermore the solutions (5.14) for ρ − = ρ + = ρ constant and the solutions (5.15) are two different non maxwellian solutions of (5.2).

6 Steady Flow in Box We investigate in this section the flow of a discrete gas in a box in order to compute accommodation coefficients. In the statement of a flow problem, in contrast to the boundary value problem (4.3) in which they are assumed known, the boundary

256

A. d’Almeida

conditions φi depend upon the accommodation coefficients which describe the interactions between the particles of the gas and those of the boundaries of the flow domain. The accommodation coefficients are unknowns of the problem and classically one has to prescribe reflection laws to get additional relations for their determination which is achieved only when the mathematical problem is solved [7], [6]. √ We choose the reference quantities n0 = c1 and a respectively for the densities and the lenghts and introduce the following dimensionless variables and parameters:



x = xa , y = ya , ε = ab , Kn = (sn0 a)−1 , i = Ni , φi = φi , Q  = Q2 . N n0

n0

(6.1)

sn0

The problem (5.2) is put in the nondimensional form: ⎧ : 8 √ √ 2 ⎪ 8( 2+ 5)+7  ⎪ ∂N ⎪ 3 N N ) 2 N 6 − N 5 = Q( = N ⎪ 3K ε ⎪ n ⎪ ∂x ⎪ ⎪ 5 ⎪ ∂N ⎪  N)  ⎪ = −Q( ⎪ ⎪ ∂x ⎪ ⎪ ⎪ 3 ∂N ⎪ ⎪  N)  ⎪ = −εQ( ⎪ ⎪ ⎪ ⎨ ∂y 6 ∂N N ) = εQ( ⎪ ⎪ ∂y 6 ⎪ ⎪     ⎪ ⎪N 1 = N3 N6 +2 N2 N5 ⎪ ⎪ ⎪ ⎪N 2 (0, y) = φ2 (y) ⎪ ⎪ ⎪ ⎪ ⎪ 5 (1, y) = φ5 (y) N ⎪ ⎪ ⎪ ⎪ 3 (x, 0) = φ3 (x) ⎪ N ⎪ ⎪ ⎩N 6 (x, ε) = φ6 (x)

(6.2)

The microscopic densities of the discrete gas in Maxwellian equilibrium with a wall are the maxwellian densities associated with 1, the macroscopic velocity and the total energy of the wall ([7, 8]). Assume that the nondimensional macroscopic velocity and total energy of the box are U-w = (uw (x, y), vw (x, y), 0) and ew (x, y). The microscopic densities of the gas in maxwellian equilibrium with the box are: 1w = (5−6ew ) N ) 9 * 2 (6ew + 3uw − 2)2 − 9vw  N2w = 72(3ew − 1) ) * (6ew + 3vw − 2)2 − 9u2w 3w = N 72(3ew − 1) ) * 2 (6ew − 3uw − 2)2 − 9vw 5w = N 72(3ew − 1) ) * (6ew − 3vw − 2)2 − 9u2w  N6w = 72(3ew − 1)

(6.3)

Exact Steady Solutions

257

It is usually assumed, when the exchanges of mass or energy of a gas and its surrounding only result from the collisions of its particles with its boundaries, that only the microscopic densities of the reflected particles are known near the walls [2]. We can compare these densities to those of the fictitious gas in equilibrium with each wall and introduce the functions li (y), i = 2, 5 and lj (x), j = 3, 6 such that: ! 2 (0, y) l2 (y) [6ew (0, y) + 3uw (0, y) − 2]2 − 9vw 72[3ew (0, y) − 1] ! l5 (y) [6ew (1, y) + 3vw (1, y) − 2]2 − 9u2w (1, y) 5 (1, y) = N 72[3ew (1, y) − 1] ! 2 (x, 0) l3 (x) [6ew (x, 0) − 3uw (x, 0) − 2]2 − 9vw 3 (x, 0) = N 72[3ew (x, 0) − 1] ! l6 (x) [6ew (x, ε) − 3vw (x, ε) − 2]2 − 9u2w (x, ε) 6 (x, ε) = . N 72[3ew (x, ε) − 1] 2 (0, y) = N

(6.4)

Using the form (5.12) of the maxwellian solutions (5.6) we have: 2 (0, y) = φ2 (y) N 5 (1, y) = 1 N φ2 (y) 3 (x, 0) = φ3 (x) N 6 (x, ε) = 1 . N φ3 (x)

(6.5)

We can thus explicitly determine the functions lk = 2, 3, 5, 6 which are given by: 72φ2(y)[3ew (0, y) − 1] l2 (y) = ! 2 (0, y) [6ew (0, y) + 3uw (0, y) − 2]2 − 9vw 72[3ew (1, y) − 1] l5 (y) = ! φ2 (y) [6ew (1, y) + 3vw (1, y) − 2]2 − 9u2w (1, y) 72φ3(x)[3ew (x, 0) − 1] l3 (x) = ! 2 (x, 0) [6ew (x, 0) − 3uw (x, 0) − 2]2 − 9vw 72[3ew (x, ε) − 1] . l6 (x) = ! φ3 (x) [6ew (x, ε) − 3vw (x, ε) − 2]2 − 9u2w (x, ε)

(6.6)

We introduce now reflection laws. We prescribe that particles of opposite velocities are reflected with the same accommodation coefficients. That is: l2 (y) = l5 (y), ∀y ∈ [0, ε] l3 (x) = l6 (x), ∀x ∈ [0, 1] .

(6.7)

258

A. d’Almeida

We infer from these additional relations: φ2 (y) =

1y εx φ3 (x) =  0y , 0x , 72(3ew (0,y)−1) , l3 (x) 0y 1y

= 72(3ew0x(x,0)−1) , l2 (y) = εx 7 7 0y = [3ew (1, y) − 1] 0y , 1y = [3ew (0, y) − 1] 1y , √ √ 0x = [3ew (x, ε) − 1] 0x , εx = [3ew (x, 0) − 1] εx , 2 (1, y), 0y = [6ew (1, y) − 3uw (1, y) − 2]2 − 9vw 2 2 (1, y), 1y = [6ew (1, y) − 3uw (1, y) − 2] − 9vw 0x = [6ew (x, 0) + 3vw (x, 0) − 2]2 − 9u2w (x, 0), εx = [6ew (x, ε) − 3vw (x, ε) − 2]2 − 9u2w (x, ε).

(6.8)

The relations (6.8) give the boundary data φj in terms of the macroscopic variables of the box’s walls. In fact the walls do not move freely as we assume in our computations. For a solid rectangle (for instance a cross section of a solid channel) all the walls have the same velocity. In our two dimensional case the velocities of two normal walls are functions of independent variables and they are equal if and only if they are constant. Hence if we take into account the fact that the box is solid we have: ; (6e +3u −2)2 −9v 2 φ2 = (6ew +3v w−2)2 −9u2w w ; w w 2 (6e −3u −2) −9v 2 (6.9) φ3 = (6ew −3v w−2)2 −9u2w w

w

w

72(3ew −1) 2 ][(6e −3u −2)2 −9v 2 ] [(6ew +3uw −2)2 −9vw w w w

l2 = √

= l3 .

The accommodation coefficients are equal although the boundary conditions are different in this more realistic case. The accommodation coefficients on each wall are thus known in terms of the total energy and the components of the macroscopic velocity of the solid box.

7 Conclusion We show that the boundary value problem for the fifteen velocity discrete model has bounded solution. Only positivity and boundedness are assumed for the data. The solution is not unique in general. Some exact analytic maxwellian and non maxwellian solutions are built. An application to the determination of the accommodation coefficients on the boundaries of a gas flow in a box is performed. The fact that we have two completely analytic expressions of the maxwellian densities permits to compute exactly the accommodation coefficients.

Exact Steady Solutions

259

Appendix The Kinetic Equations of the Model ∂N1 ∂t√

=

√ 2cs [(N2 + N3 + N5 + N6 )(N4 + N7 ) − (N8 + N9 + N10 + N11 + N12 + N13 )N1 ]

− 2cs [(N14 + N15 )N1 ] √ ∂N2 ∂N2 3cs [(N3 + N6 )(N8 + N9 ) − N2 (N12 + N13 + N14 + N15 )] + c ∂x = ∂t√ + 5cs [N3 (N14 + N15 ) + N5 (N8 + N9 ) + N6 (N12 + N13 − 3N2 (N10 + N11 )] √ + 2cs [N1 (N8 + N9 ) − N2 (N4 + N7 )] + 2cs(N3 N6 + N4 N7 − 2N2 N5 ) √ ∂N3 ∂N3 ∂t + c ∂y = 3cs [(N3 + N6 )(N12 + N13 ) − N3 (N8 + N9 + N10 + N11 )] √ + 5cs [N2 (N10 + N11 ) + N5 (N8 + N9 ) + N6 (N12 + N13 ) − 3N3 (N14 + N15 )] √ + 2cs [N1 (N12 + N13 ) − N3 (N4 + N7 )] + 2cs(N2 N5 + N4 N7 − N3 N6 ) √ ∂N4 ∂N4 1 (N8 + N11 + N12 + N15 ) − N4 (N2 + N3 + N5 + N6 )] + 2cs(N2 N5 ∂t + c ∂z = 2cs [N √ +N3 N6 − 2N4 N7 ) + 5cs [N7 (N8 + N11 + N12 + N15 ) − N4 (N9 + N10 + N13 + N14 )] √ ∂N5 ∂N5 3cs [(N3 + N6 )(N10 + N11 ) − N5 (N12 + N13 + N14 + N15 )] − c ∂x = ∂t√ + 5cs [N2 (N10 + N11 ) + N3 (N14 + N15 ) + N6 (N12 + N13 ) − 3N5 (N8 + N9 )] √ + 2cs [N1 (N10 + N11 ) − N5 (N4 + N7 )] + 2cs(N3 N6 + N4 N7 − 2N2 N5 ) √ ∂N6 ∂N6 ∂t − c ∂y = 3cs [(N2 + N5 )(N14 + N15 ) − N6 (N8 + N9 + N10 + N11 )] √ + 5cs [N2 (N10 + N11 ) + N3 (N14 + N15 ) + N5 (N8 + N9 ) − 3N6 (N12 + N13 )] √ + 2cs [N1 (N14 + N15 ) − N6 (N4 + N7 )] + 2cs(N2 N5 + N4 N7 − 2N3 N6 ) √ ∂N7 ∂N7 1 (N9 + N10 + N13 + N14 ) − N7 (N2 + N3 + N5 + N6 )] + 2cs(N2 N5 ∂t − c ∂z = 2cs [N √ +N3 N6 − 2N4 N7 ) + 5cs [N4 (N9 + N10 + N13 + N14 ) − N7 (N8 + N11 + N12 + N15 )] √ ∂N8 ∂N8 ∂N8 2cs(N12 N15 − N8 N11 ) + 5cs(N2 N11 + N3 N15 + N4 N9 + N6 N12 ∂t + c ∂x + c ∂z = √ √ −3N5 N8 − N7 N8 ) + 3cs [N2 (N12 + N15 ) − N8 (N3 + N6 )] + 6cs [N9 (N12 + N15 ) − N8 N13 ] √ √ √ − 6csN8 N14 + 2 2cs(N9 N11 + N12 N14 + N13 N15 − N8 N10 ) + 2cs(N2 N4 − N1 N8 ) √ ∂N9 ∂N9 ∂N9 3cs [N2 (N13 + N14 ) − N9 (N3 + N6 )] + 2cs(N13 N14 − N9 N10 ) + c ∂x − c ∂z = ∂t√ √ + 2cs(N2 N4 − N1 N9 ) + 5cs(N2 N10 + N3 N14 + N6 N13 + N7 N8 − N4 N9 − 3N5 N9 ) √ √ + 6cs [N8 (N13 + N14 ) − N9 (N12 + N15 )] + 2 2csN8 N10 + N12 N14 + N13 N15 − 3N9 N11 ) √ ∂N10 ∂N10 ∂N10 3cs [N5 (N13 + N14 ) − N10 (N3 + N6 )] + 2cs(N13 N14 − N9 N10 ) − c ∂x − c ∂z = ∂t√ √ + 2cs(N5 N7 − N1 N10 ) + 5cs(N3 N14 + N5 N9 + N6 N13 + N7 N11 − N4 N10 − 3N2 N10 ) √ √ + 6cs [N11 (N13 + N14 ) − N10 (N12 + N15 )] + 2 2csN9 N11 + N12 N14 + N13 N15 − 3N8 N10 ) √ ∂N11 ∂N11 ∂N11 3cs [N5 (N12 + N15 ) − N11 (N3 + N6 )] + 2cs(N12 N15 − N8 N11 ) − c ∂x + c ∂z = ∂t√ √ + 2cs(N4 N5 − N1 N11 ) + 5cs(N3 N13 + N5 N8 + N6 N12 + N4 N10 − 3N2 N11 − N7 N11 ) √ √ + 6cs [N10 (N12 + N15 ) − N11 (N13 + N14 )] + 2 2csN8 N10 + N12 N14 + N13 N15 − 3N9 N11 ) √ ∂N12 ∂N12 ∂N12 ∂t + c ∂y + c ∂z = 3cs [N3 (N8 + N11 ) − N12 (N2 + N5 )] + 2csN8 N11 − N12 N15 ) √ √ + 2cs(N3 N4 − N1 N12 ) + 5cs(N2 N11 + N3 N15 + N4 N13 + N5 N8 − N7 N12 − 3N6 N12 ) √ √ + 6cs [N13 (N8 + N11 ) − N12 (N9 + N10 )] + 2 2csN8 N10 + N9 N11 + N13 N15 − 3N12 N14 ) √ ∂N13 ∂N13 ∂N13 ∂t + c ∂y − c ∂z = 3cs [N3 (N9 + N10 ) − N13 (N2 + N5 )] + 2csN9 N10 − N13 N14 ) √ √ + 2cs(N3 N7 − N1 N13 ) + 5cs(N2 N10 + N3 N14 + N5 N9 + N7 N12 − N4 N13 − 3N6 N13 ) √ √ + 6cs [N12 (N9 + N10 ) − N13 (N8 + N11 )] + 2 2csN8 N10 + N9 N11 + N12 N14 − 3N13 N15 )

260

A. d’Almeida

√ ∂N14 ∂N14 ∂N14 3cs [N6 (N9 + N10 ) − N14 (N2 + N5 )] + 2csN9 N10 − N13 N14 )+ − c ∂y − c ∂z = √∂t √ 2cs(N6 N7 − N1 N14 ) + 5cs(N2 N10 + N5 N9 + N6 N13 + N7 N15 − N4 N14 − 3N3 N14 ) √ √ + 6cs [N15 (N9 + N10 ) − N14 (N8 + N11 )] + 2 2csN8 N10 + N9 N11 + N13 N15 − 3N12 N14 ) √ ∂N15 ∂N15 ∂N15 3cs [N6 (N8 + N11 ) − N15 (N2 + N5 )] + 2csN8 N11 − N12 N15 )+ − c ∂y + c ∂z = √∂t √ 2cs(N4 N6 − N1 N15 ) + 5cs(N2 N11 + N5 N8 + N6 N12 + N4 N14 − N7 N15 − 3N3 N15 ) √ √ + 6cs [N14 (N8 + N11 ) − N15 (N9 + N10 )] + 2 2csN8 N10 + N9 N11 + N12 N14 − 3N13 N15 ).

The Conservation Equations of the Model The dimension of the summational invariant space of the model is five for the binary collisions. The macroscopic variables of the model are thus well defined. They are the macroscopic density N, the macroscopic velocity U- = (U, V , W ) and the total energy E given by: N = N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9 + N10 + N11 + N12 + N13 +N14 + N15 NU = c(N2 − N5 + N8 + N9 − N10 − N11 ) NV = c(N3 − N6 + N12 + N13 − N14 − N15 ) NW = c(N4 − N7 + N8 − N9 − N10 + N11 + N12 − N13 − N14 + N15 ) 2NE = c2 [N2 + N3 + N4 + N5 + N6 + N7 + 2(N8 + N9 + N10 + N11 )] +2c2 (N12 + N13 + N14 + N15 )

The conservation equations of the model are: ∂(NU ) ∂(NV ) ∂(NW ) ∂N =0 ∂t + ∂x + ∂y + ∂z ∂(NU ) ∂(N2 +N5 +N8 +N9 +N10 +N11 ) +N10 −N11 ) + c + c ∂(N8 −N9∂z = ∂t ∂x ∂(NV ) ∂(N3 +N6 +N12 +N13 +N14 +N15 ) ∂(N12 −N13 +N14 −N15 ) +c ∂t + c ∂y ∂z ∂(NW ) ∂(N8 −N9 +N10 −N11 ) ∂(N12 −N13 +N14 −N15 ) +c ∂t + c ∂x ∂y 11 +N12 +N13 +N14 +N15 ) +c ∂(N4 +N7 +N8 +N9 +N10 +N =0 ∂z ∂[N2 −N5 +2(N8 +N9 −N10 −N11 )] ∂(2NE) 2 +c ∂t ∂x 13 −N14 −N15 )] +c2 ∂[N3 −N6 +2(N12∂y+N 11 +N12 −N13 −N14 +N15 )] +c2 ∂[N4 −N7 +2(+N8 −N9 −N10∂z+N =0

0 =0

Exact Steady Solutions

261

References 1. J.M. Bony, Existence globale à données de Cauchy petites pour les modèles discrets de l’équation de Boltzmann. Commun. Partial Difffer. Equ. 16, 533–545 (1991) 2. C. Cercignani, Mathematical Methods in Kinetic Theory (Plenum Press, New York, 1969) 3. C. Cercignani, R. Illner, M. Shinbrot, A boundary value problem for the two dimensional broadwell model. Commun. Math. Phys. 114, 687–698 (1988) 4. A. d’Almeida, Exact solutions for discrete velocity models. Mech. Res. Commun. 34, 405–409 (2007) 5. A. d’Almeida, Transition of unsteady flows of evaporation to steady state. C. R. Mecanique 336, 612–615 (2008) 6. A. d’Almeida, R. Gatignol, Boundary Conditions for Discrete Models of Gases and Applications to Couette Flows. Computational Fluid Dynamics (Springer, Berlin, 1995), pp. 115–130 7. A. d’Almeida, R. Gatignol, The half space problem in discrete kinetic theory. Math. Models Methods Appl. Sci. 13, 99–119 (2003) 8. R. Gatignol, Kinetic theory boundary conditions for discrete velocity gases. Phys. Fluids. 20, 2022–2030 (1977) 9. R. Illner, M. Shinbrot, C. Cercignani, A boundary value problem for discrete velocity models. Duke Math. J. 55, 889–900 (1987). 10. I. Kolodner, On the Carleman’s model for the Boltzmann equation and its generalizations. Ann. Math. Pure Appl. 73, 11–32 (1963) 11. T. Natta, K.A. Agossémé, A.S. d’Almeida, Existence and uniqueness of solution of the ten discrete velocity model C1 . J. Adv. Math. Comput. Sci. 29, 1–12 (2018) 12. T. Platkowski, R. Illner, Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory. SIAM. Rev. 30, 213–255 (1988) 13. D.R. Smart, Fixed Point Theorems. (Cambridge University Press, Cambridge, 1974) 14. R. Temam, Sur la résolution exacte et approchée d’un problème hyperbolique non linéaire de T. Carleman. Arch. Rat. Mech. Anal. 35, 351–362 (1969)

Monotony and Comparison Principle in Non Autonomous Size Structured Models Mamadou Lamine Diagne, Mamadou Moustapha Mbaye, and Ousmane Seydi

Abstract Studying asymptotic properties for non autonomous partial differential equations is often challenging question due to the lack of general theory. We prove in this paper some monotony properties of a class of general non linear non autonomous size(age)-structured population dynamic models. Our results are applied to an example in order to show how one can prove some global asymptotic properties by using comparison principle. Keywords Size structure · Age structure · Periodic · Stability · Threshold Mathematics Subject Classification (2010) 47H30, 34G20, 37C75, 26A18

1 Introduction Over the past decades, nonlinear size(age)-structured population dynamics models have generated a great deal of interest among applied mathematicians in fields of population dynamics (biology, ecology, epidemiology . . . ). The first non linear age structured models where introduced by M. Gurtin and R. C. MacCamy and F. Hopensteadt in 1974 (see [32]). The book of Webb [32] on theory of nonlinear age-dependent population dynamics has laid the fondations for several developments in these areas. Since then many development have been made in the

M. L. Diagne Université de Thiès, Thiès, Senegal e-mail: [email protected] M. M. Mbaye Université Cheikh Anta Diop, Dakar, Senegal e-mail: [email protected] O. Seydi () Ecole Polytechnique de Thiès, Thiès, Senegal e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_10

263

264

M. L. Diagne et al.

study of size(age)-structured population models namely existence and uniqueness of solutions, positivity as well as asymptotic properties with a widely different approaches. We refer for example to the recent monographs [12, 18] for more completeness and references. In this work we are mainly interested in exploring the monotone properties of a system of non autonomous size(age)-structured population dynamics models and its application to study the asymptotic behavior of the solutions. More precisely the system considered is the following ⎧  u(t, ·))(a), a ∈ (0, m), t > t0 ⎪ ∂t u(t, a) + ∂a (ϑ(t, a)u(t, a)) = G(t, ⎪ ⎪ ⎪ ⎪ ⎨ ϑ(t, 0)u(t, 0) = F (t, u(t, ·)), t > t0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(t0 , ·) = u0 ∈ L1+ ([0, m], Rn ).

(1.1)

Note that the existence, uniqueness and positivity of the solutions of the system of partial differential equations (1.1) have been fully study in [14, 15]. Thus we only concentrate our efforts in the monotone properties. Let us mention that according to Smith [25] the early works in monotone dynamical systems are due to Müller [22] in (1926), Kamke [13] in 1932 and Krasnoselskii [16, 17] respectively in 1964 and 1968, Matano [20, 21] respectively in 1979 and 1984, and Hirsch [10] in 1982. Let us also mention the work of Smith [24, 25] and Smith and Thieme [26– 29] where very significant contributions have been made on monotone dynamical systems. More recently Magal, Seydi and Wang [19] provide a theory on monotone dynamical systems by using integrated semigroup approach and cover a wide range of partial differential equations arising in population dynamics. In the best of our knowledge, the monotone properties of the size(age)-structured model considered in this work is not yet treated by the theory developed the literature. In fact in the concern to give simple tools for the study of the asymptotic behavior of solutions, we are interested to prove results on monotony for the fairly general class of models (1.1). It is well known that the study of global (even local) asymptotic behavior for non autonomous differential equations is a difficult task. We show in this paper the utility of such monotone theory in proving the global stability of periodic equilibrium without using Lyapunov function. More precisely using our results combined together with the renewal theory developed by Thieme [30], we were able to introduce a threshold condition for an example of a size(age)-structured model which makes possible to study the extinction of the considered population without using Lyapunov function. The non autonomous system considered in this work is mainly motivated by the phenomenon of invasive species in ecological problem where the mode of reproduction as well as the environment is an important fact that changes over time. It is therefore natural to assume that the rates related to these factors in more realistic and interesting situations depend on time and size (or age). One of the objectives is to know how individual variability influences the dynamics of the

Monotony and Comparison Principle

265

whole population, conceived as an individual frequency distribution that evolves over time. For more details size-structured models we refer the reader to [2, 3, 6– 9, 11, 14, 15, 31]. Let us mention among other some works that are closely related to our paper. Tucker and Zimmerman [31] deal with global existence of solutions and local stability properties. Howard [11] obtained very interesting results on stability and chaotic behavior. Rong et al. [9] proposed a size-structured model coupled with a hybrid system of ordinary differential equations. Ackleh et al. [1] have developed a monotone approximation, based on the technique of superior and inferior solutions, to solve a non linear model of structured size with competition between individuals. The remaining part of the manuscript decomposed into two sections. Section 2 contains the main results of this paper and the proofs of the theorems. More precisely we prove the monotony of the evolution semiflow generated by (1.1) with respect to the initial distributions. We also give some comparison principles by introducing upper and lower solutions for (1.1). Some consequences of our results are also presented. In Sect. 3 we proposed a size structured model describing invasive species and apply our results to prove some extinction phenomenon. We also introduce a threshold quantity for our example by using the renewal theorem of Thieme [30].

2 Main Results In this section we will give the main results of this paper. The proof of the main theorems are postponed in the next section. To this aim, the following set of assumptions will be required C1. For k = 1, 2, F is continuous from R+ × L1 ([0, m], Rn ) into Rn and for each ξ > 0 and σ > 0 there exists LF := LF (ξ, σ ) > 0 such that F (t, ϕ1 ) − F (t, ϕ2 ) ≤ LF ϕ1 − ϕ2 L1 for all ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ), with ϕ1 L1 ≤ ξ , ϕ2 L1 ≤ ξ and t ∈ [t0 , t0 + σ ].  is continuous from R+ × L1 ([0, m], R) into L1 ([0, m], R) and C2. For k = 1, 2, G for each ξ > 0 and σ > 0 there exists LG  := LG  (ξ, σ ) > 0 such that  ϕ2 )L1 ≤ LG  ϕ1 ) − G(t, G(t,  ϕ1 − ϕ2 L1 for all ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ), with ϕ1 L1 ≤ ξ , ϕ2 L1 ≤ ξ and t ∈ [t0 , t0 + σ ]. C3. ϑ is defined from R+ × [0, m] into R+ with the following properties C3-(i) ϑ(t, a) > 0 for each (t, a) ∈ R+ × [0, m) and ϑ(t, m) = 0. C3-(ii) ϑ is differentiable with respect to a ∈ [0, m] and for each a ∈ [0, m], the map t → ϑ(t, a) is continuous from R+ into R+ .

266

M. L. Diagne et al.

C3-(iii) There exists a constant Lϑ > 0 such that for each t ∈ R+ and a1 , a2 × [0, m] we have 0 < ϑ(t, a) < Lϑ and |ϑ(t, a1 ) − ϑ(t, a2 )| ≤ Lϑ |a1 − a2 |.

(2.1)

C4. We assume that 0L1 ≤ ϕ ⇒ 0Rn ≤ F (t, ϕ), ∀ϕ ∈ L1+ ([0, m], Rn ). C5. We assume that for each ξ > 0 and σ > 0 there exists α = α(ξ, σ ) > 0 such that  ϕ) + αϕ 0L1 ≤ ϕ ⇒ 0L1 ≤ G(t, for all ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ), with ϕ1 L1 ≤ ξ , ϕ2 L1 ≤ ξ and t ∈ [t0 , t0 + σ ]. By Assumptions C1–C5 its has been proven in [14] that there exists a maximally defined non negative mild solution of (1.1) in the sense of Definition 2.4 below. Since we are interested on the monotony of the evolution semiflow generated by (1.1) we will make the following additional assumptions C4’ We assume that 0L1 ≤ ϕ1 ≤ ϕ2 ⇒ 0Rn ≤ F (t, ϕ1 ) ≤ F (t, ϕ2 ) for all ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ). C5’ We assume that for each ξ > 0 and σ > 0 there exists α = α(ξ, σ ) > 0 such that  ϕ1 ) + αϕ1 ≤ G(t,  ϕ2 ) + αϕ2 0L1 ≤ ϕ1 ≤ ϕ2 ⇒ 0L1 ≤ G(t, for all ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ), with ϕ1 L1 ≤ ξ , ϕ2 L1 ≤ ξ and t ∈ [t0 , t0 + σ ]. Remark 2.1 Let us note that Assumptions C4’–C5’ imply C4–C5. Therefore it is clear that the existence of maximally defined non negative mild solutions will follow from the results in [14].

Monotony and Comparison Principle

267

Next we will give a step by step derivation of the solutions of (1.1) along the characteristics. Note that with the above assumptions our system (1.1) can be rewritten as ⎧ ⎪ ∂t u(t, a) + ϑ(t, a)∂a u(t, a) = G(t, u(t, ·))(a), a ∈ (0, m), t > t0 ⎪ ⎪ ⎪ ⎪ ⎨ (2.2) ϑ(t, 0)u(t, 0) = F (t, u(t, ·)), t > t0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(t , ·) = u ∈ L1 ([0, m], Rn ) 0 0 + where we have set  ϕ), ∀t ≥ 0 and ϕ ∈ L1 ([0, m], R). G(t, ϕ) = −∂a ϑ(t, ·)ϕ + G(t, Using C3-(iii) it follows that |∂a ϑ(t, a)| ≤ Lϑ , ∀t ≥ 0, a ∈ [0, m]

(2.3)

and we have the following lemma. Lemma 2.2 Let Assumptions C1–C3 and C4’–C5’ be satisfied. Then the map G : R+ × L1 ([0, m], R) → L1 ([0, m], R) satisfies the following properties (i) G is continuous from R+ ×L1 ([0, m], R) into L1 ([0, m], R) and for each ξ > 0 and σ > 0 there exists LG := LG (ξ, σ ) > 0 such that G(t, ϕ1 ) − G(t, ϕ2 )L1 ≤ LG ϕ1 − ϕ2 L1 for all ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ), with ϕ1 L1 ≤ ξ , ϕ2 L1 ≤ ξ and t ∈ [t0 , t0 + σ ]. (ii) For each ξ > 0 and σ > 0 there exists α = α(ξ, σ ) > 0 such that 0L1 ≤ ϕ1 ≤ ϕ2 ⇒ 0L1 ≤ G(t, ϕ1 ) + αϕ1 ≤ G(t, ϕ2 ) + αϕ2 for all ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ), with ϕ1 L1 ≤ ξ , ϕ2 L1 ≤ ξ and t ∈ [t0 , t0 + σ ]. Let t1 ≥ t0 and a1 ∈ [0, m] be given. Then the characteristics starting at the point (t1 , a1 ) are given by the following system of differential equations ⎧ ⎧ ⎨ dγ1 (s) ⎨ dγ2 (s) = 1, s > t0 = ϑ(s, γ2 (s)), s > t1 and ds ds ⎩ γ (t ) = t , ⎩ γ (t ) = a . 1 0 1 2 1 1

(2.4)

268

M. L. Diagne et al.

Denote by t → θ (t, t1 ; a1 ) the solution of the γ2 -equation through a1 at time t1 . That is to say that γ2 (t) = θ (t, t1 ; a1), ∀t ≥ t1 . Note that θ : R2+ × [0, m) → R has the properties of an evolution flow, namely we have  θ (t, r; ·) ◦ θ (r, s; a) = θ (t, s; a), ∀t, s, r ∈ R+ , a ∈ [0, m] (2.5) θ (t, t; a) = a, ∀t ∈ R+ , ∀a ∈ [0, m]. Observe that since ϑ is differentiable with respect to a by using classical arguments on ordinary differential equations we have ∂θ (t, r; a) ∂θ (t, r; a) =− ϑ(r, a), for 0 ≤ r ≤ t, a ∈ [0, m] ∂r ∂a

(2.6)

and ∂θ (t, r; a) = exp ∂a



t r

 ∂ϑ(s, θ (s, r; a)) ds , for 0 ≤ r ≤ t, a ∈ [0, m]. ∂a (2.7)

Therefore (2.6) and (2.7) imply that ∂θ (t, r; a) = −ϑ(r, a) exp ∂r

 r

t

 ∂ϑ(s, θ (s, r; a)) ds , for 0 ≤ r ≤ t, a ∈ [0, m] ∂a (2.8)

Moreover since ϑ is positive, it follows that for each r ≥ 0 and a ∈ [0, m] the map t ∈ [r, +∞) → θ (t, r, a) is increasing. Furthermore by (2.7) one knows that for each t ≥ r ≥ 0 the map a ∈ [0, m] → θ (t, r, a) is also increasing. Since θ (r, t, ·) is the inverse of θ (t, r, ·) for t ≥ r it follows that a1 , a2 ∈ [0, m], a1 < a2 ⇔ θ (t, r; a1 ) < θ (t, r; a2 ) ⇔ θ (r, t; a1 ) < θ (r, t; a2 ), ∀t, r ∈ R+ .

(2.9)

Remark 2.3 For later reference let us observe that (2.8) with the fact that ϑ is positive imply that for each fixed a ∈ [0, m] and t > 0 the map r → θ (t, r, a) is decreasing in [0, t].

Monotony and Comparison Principle

269

Finding the solution along the characteristics means to evaluate u(γ1 (s), γ2 ◦ γ1 (s)), s ≥ 0. In fact if we seek a solution t → u(t, ·) of (1.1) (equivalently of (2.2)) locally defined in time along the characteristics, the following condition must be imposed t1 ∈ [t0 , t0 + δ] for some δ > 0 as well as γ1 (s) ∈ [t1 , t0 + δ], ∀s ∈ I and γ2 ◦ γ1 (s) ∈ [a1 , m], ∀s ∈ I where I ⊂ [t0 , +∞) is to be precise. In fact by setting I = [t0 , 2t0 − t1 + δ] we have γ1 (s) = t1 + s − t0 ∈ [t1 , t0 + δ], ∀s ∈ I and by (2.9) we have γ2 ◦ γ1 (s) = θ (t1 + s − t0 , t1 ; a1) so that a1 ≤ θ (t1 + s − t0 , t1 ; a1 ) ≤ θ (t0 + δ, t1 ; a1 ) ≤ θ (t0 + δ, t1 ; m) = m, ∀s ∈ I, ∀a1 ∈ [0, m]. Therefore for each t1 ≥ t0 , a1 ∈ [0, m] we have ⎧ ⎨ du(γ1 (s), γ2 ◦ γ1 (s)) = G(γ1 (s), u(γ1 (s), ·))(γ2 ◦ γ1 (s)), s ∈ [t0 , 2t0 − t1 + δ] ds ⎩ u(γ (t ), γ ◦ γ (t )) = u(t , γ (t )) = u(t , a ). 1 0 2 1 0 1 2 1 1 1 (2.10) Thus for each α > 0 and s ∈ [t0 , 2t0 − t1 + δ], (2.10) is equivalent to 0) u(γ1 (s), γ2 ◦ γ1 (s)) = e−α(s−t  s u(t1 , a1 ) + e−α(s−r)G(γ1 (r), u(γ1 (r), ·))(γ2 ◦ γ1 (r))dr t0s + e−α(s−r)αu(γ1 (r), γ2 ◦ γ1 (r))dr.

t0

(2.11)

270

M. L. Diagne et al.

In order to give the formula along the characteristics s → (γ1 (s), γ2 ◦ γ1 (s)) we have to find for each (t, a) ∈ [t0 , t0 + δ] × [0, m] the characteristic curves that pass through (t, a) that is to solve the equations γ1 (s) = t and γ2 ◦ γ1 (s) = a where γ1 and γ2 satisfy (2.4). Therefore we have to find s ∈ [t0 , 2t0 − t1 + δ], a1 ∈ [0, m] and t1 ≥ t0 such that 

γ1 (s) = t ⇔ γ2 ◦ γ1 (s) = a



t1 + s − t0 = t ⇔ γ2 (t1 + s − t0 ) = a



s + t1 = t + t0 θ (t1 + s − t0 , t1 ; a1 ) = a. (2.12)

Using (2.5) and applying θ (t1 , t1 + s − t0 ; ·) to the both side of the equality θ (t1 + s − t0 , t1 ; a1) = a we see that (2.12) is indeed equivalent to 

s + t1 = t + t0 a1 = θ (t1 , t1 + s − t0 ; a).

(2.13)

Setting t1 = t0 our system reduce to find s ∈ [t0 , 2t0 − t1 + δ] = [t0 , t0 + δ] and a1 ∈ [0, m] such that 

s=t ⇔ a1 = θ (t0 , s; a)



s=t a1 = θ (t0 , t; a)

since a1 ≥ 0 the equality make sense a1 = θ (t0 , t; a) only if a ≥ θ (t, t0 ; 0). Indeed by (2.9) we have a < θ (t, t0 ; 0) ⇒ a1 = θ (t0 , t; a) < θ (t0 , t; θ (t, t0 ; 0)) = 0. Therefore for all a ≥ θ (t, t0 ; 0) by setting t1 = t0 , s = t ∈ [t0 , t0 + δ] and a1 = θ (t0 , t; a), the expression (2.11) combined with (2.12) provides 0) u(t, a) = e−α(t−t  t u(t0 , θ(t0 , t; a)) e−α(t−r) G(r, u(r, ·))(θ(r, t; a))dr + t0t e−α(t−r) αu(r, θ(r, t; a))dr, ∀t ∈ [t0 , t0 + δ], a ≥ θ(t, t0 ; 0). +

(2.14)

t0

It remains to obtain an expression for u(t, a) whenever 0 ≤ a < θ (t, t0 ; 0). Indeed in this case since we have 0 = θ (t, t; 0) ≤ a < θ (t, t0 ; 0)

Monotony and Comparison Principle

271

then by the intermediate value theorem and Remark 2.3 there exists a unique τ := τ (t, t0 , a) ∈ [t0 , t] such that a = θ (t, τ ; 0) ⇔ θ (τ, t; a) = 0.

(2.15)

Hence it is clear that a1 = 0, t1 = τ and s = t + t0 − τ ∈ [t0 , t0 + δ] solve (2.13) so that the expression (2.11) rewrites for a < θ (t, t0 ; 0) and t ∈ [t0 , t0 + δ] as u(t, a) = e−α(t −τ )u(τ, 0)  t +t0 −τ + e−α(t +t0−τ −r) G(τ + r − t0 , u(τ + r − t0 , ·)) t0

×(θ (τ + r − t0 , τ ; 0))dr  t +t0 −τ e−α(t +t0−τ −r) αu(τ + r − t0 , θ (τ + r − t0 , τ ; 0))dr +

(2.16)

t0

where τ = τ (t, t0 , a) is uniquely determined by (2.15) with t1 = t0 . Thus by setting Z(t, t0 ) := θ (t, t0 ; 0), ∀t ∈ [t0 , t0 + δ]

(2.17)

and using the boundary conditions, (2.14) and (2.16) can be grouped into the following formula ⎧  t ⎪ −α(t −t0 ) ⎪ e u (θ (t , t; a)) + e−α(t −r)G(r, u(r, ·))(θ (r, t; a))dr ⎪ 0 0 ⎪ ⎪ t 0 ⎪  t ⎪ ⎪ ⎪ ⎪ ⎨ + e−α(t −r)αu(r, θ (r, t; a))dr, a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] t0  t u(t, a) = ⎪ ⎪ −α(t −τ ) F (τ, u(τ, ·)) ⎪ + e e−α(t −r)G(r, u(r, ·))(θ (r, τ ; 0))dr ⎪ ⎪ ϑ(τ, 0) ⎪ τ  ⎪ t ⎪ ⎪ ⎪ ⎩ e−α(t −r)αu(r, θ (r, τ ; 0))dr, a < Z(t, t0 ), t ∈ [t0 , t0 + δ] + τ

(2.18) where τ = τ (t, t0 , a) ∈ [t0 , t] is uniquely determined by (2.15).  Definition 2.4 ([14]) u ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn ) is a mild solution of (1.1) if it satisfies (2.18) with α = 0. Remark 2.5 By the same arguments of Lemma 3.1 and Corollary 3.3 in [14] one knows that u ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn ) is a mild solution of (1.1) if and only if it satisfies (2.18) for some α ∈ R. Lemma 2.6 If t → u(t, θ (t, t0 ; a)) satisfies (2.18) for t ∈ [t0 , t0 +δ] then it satisfies (2.10).

272

M. L. Diagne et al.

Proof Let  t ⎧ ⎪ −α(t −t0 ) u (θ(t , t; a)) + ⎪ e e−α(t −r) G(r, u(r, ·))(θ(r, t; a))dr ⎪ 0 0 ⎪ ⎪ t 0 ⎪  t ⎪ ⎪ ⎪ ⎪ + e−α(t −r) αu(r, θ(r, t; a))dr, a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] ⎨ t0 K α u(t, a) =  t ⎪ ⎪ −α(t −τ ) F (τ, u(τ, ·)) + ⎪ e e−α(t −r) G(r, u(r, ·))(θ(r, τ ; 0))dr ⎪ ⎪ ϑ(τ, 0) ⎪ τ ⎪  t ⎪ ⎪ ⎪ ⎩ e−α(t −r) αu(r, θ(r, τ ; 0))dr, a < Z(t, t0 ), t ∈ [t0 , t0 + δ] + τ

Let u(t, θ (t, t0 ; a)) satisfies (2.18) (i.e. u(t, θ (t, t0 ; a)) = K α u(t, θ (t, t0 ; a))). It follows Corollary 3.4 in [14] that u(t, θ (t, t0 ; a)) = K 0 u(t, θ (t, t0 ; a)). Hence by Corollary 3.1 in [14], u(t, θ (t, t0 ; a)) = K 0 u(t, θ (t, t0 ; a)) satisfies (2.10) (with t0 = t1 and initial value u(t0 , a) = u(t0 , θ (t0 , t0 ; a))).  Definition 2.7 (Non Autonomous Maximal Semiflow) Consider two maps T : [0, +∞) × L1+ ([0, m], Rn ) → (0, +∞] and U : DT → L1+ ([0, m], Rn ) where $ % DT = (t, t0 , ϕ) ∈ [0, +∞)2 × L1+ ([0, m], Rn ) : t0 ≤ t < t0 + T (t0 , ϕ) . We say that U is a maximal non-autonomous semiflow on L1+ ([0, m], Rn ) if U satisfies the following properties (i) T (r, U (r, t0 )ϕ) + r = t0 + T (t0 , ϕ), ∀t0 ≥ 0, ∀ϕ ∈ L1+ ([0, m], Rn ), ∀r ∈ [t0 , t0 + T (t0 , ϕ)) (ii) U (t0 , t0 )ϕ = ϕ for all t0 ≥ 0 (iii) U (t, r)U (r, t0 )ϕ = U (t, t0 )ϕ, ∀t ≥ r, r, t ∈ [t0 , t0 + T (t0 , ϕ)), ϕ ∈ L1+ ([0, m], Rn ) (iv) If T (t0 , u0 ) < +∞ then lim

s→T (t0 ,u0 )−

U (s + t0 , t0 )u0 L1 = +∞.

Using Theorem 2.1 of [15] and Proposition 5.1 of [14] we have the following theorem. Theorem 2.8 Let Assumptions C1–C3 and C4’–C5’ be satisfied. Then there exists a map T : [0, +∞) × L1+ ([0, m], Rn ) → (0, +∞] and a maximal evolution semiflow U : DT → L1+ ([0, m], Rn ) such that for each t0 ≥ 0 and u0 ∈ L1+ ([0, m], Rn ),  U (·, t0 )u0 ∈ C [0, t0 + T (t0 , u0 )), L1+ ([0, m], Rn ) is the unique maximal mild solution of (1.1).

Monotony and Comparison Principle

273

Proof We just need to prove that the U satisfies property (i) since the other properties are obtained from [15]. Let t0 ≥ 0 and ϕ ∈ L1+ ([0, m], Rn ) be given. Let r ∈ [t0 , t0 + T (t0 , ϕ)) be given. Set ϕˆ = U (r, t0 )ϕ. Then by property (iii) we have U (t, t0 )ϕ = U (t, r)ϕ, ˆ ∀t ∈ [r, t0 + T (t0 , ϕ)) ˆ Moreover if we set hence (iii) implies that t0 + T (t0 , ϕ) ≤ r + T (r, ϕ).  u(t, ·) =

U (t, r)ϕˆ if t ∈ [r, r + T (r, ϕ)) ˆ U (t, t0 )ϕ if t ∈ [t0 , r)

then from the definition of ϕˆ it is clear that t ∈ [t0 , r + T (r, ϕ)) ˆ → u(t, ·) is a mild solution of (1.1). Thus the property (iii) in Definition 2.7 implies that r + T (r, ϕ)) ˆ ≤ t0 + T (t0 , ϕ)) and the result follows.  The main results of this paper are the following Theorems 2.9 and 2.10. The proofs of these three theorems are postponed respectively in Sects. 2.1 and 2.2. Theorem 2.9 (Monotony) Let Assumptions C1–C3 and C4’–C5’ be satisfied. Then there exists a map T : [0, +∞) × L1+ ([0, m], Rn ) → (0, +∞] and a maximal evolution semiflow U : DT → L1+ ([0, m], Rn ) such that for each t0 ≥ 0 and u0 , u¯ 0 ∈ L1+ ([0, m], Rn ) with u0 ≤ u¯ 0 we have U (t, t0 )u0 ≤ U (t, t0 )u¯ 0 , ∀t ∈ [t0 , min (T (t0 , u0 ), T (t0 , u¯ 0 ))). The next result concerns comparison principles for system (1.1). Theorem 2.10 (Lower Solution) Let  Assumptions C1–C3 and C4’–C5’ be satisfied. Let t0 ≥ 0, δ > 0 and w ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn be given. Assume that for each β > 0 large enough ⎧  t ⎪ −β(t−t0 ) ⎪ u (θ(t , t; a)) + e−β(t−r) G(r, w(r, ·))(θ(r, t; a))dr e ⎪ 0 0 ⎪ ⎪ t0 ⎪  ⎪ t ⎪ ⎪ ⎪ e−β(t−r) βw(r, θ(r, t; a))dr, a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] + ⎨ t 0  t w(t, a) ≤ ⎪ F (τ, w(τ, ·)) ⎪ ⎪ e−β(t−τ ) e−β(t−r) G(r, w(r, ·))(θ(r, τ ; 0))dr + ⎪ ⎪ ⎪ τ  t ϑ(τ, 0) ⎪ ⎪ ⎪ ⎪ ⎩ + e−β(t−r) βw(r, θ(r, τ ; 0))dr, a < Z(t, t0 ), t ∈ [t0 , t0 + δ] τ

then 0L1 ≤ w(t, ·) ≤ U (t, t0 )u0 , ∀t ∈ [t0 , t0 + min (T (t0 , u0 ), δ)).

(2.19)

274

M. L. Diagne et al.

Theorem 2.11 (Upper Solution) Let  Assumptions C1–C3 and C4’–C5’ be satisfied. Let t0 ≥ 0, δ > 0 and w ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn be given. Assume that for each β > 0 large enough ⎧  t ⎪ −β(t−t0 ) ⎪ u (θ(t , t; a)) + e−β(t−r) G(r, w(r, ·))(θ(r, t; a))dr e ⎪ 0 0 ⎪ ⎪ t0 ⎪  ⎪ t ⎪ ⎪ ⎪ e−β(t−r) βw(r, θ(r, t; a))dr, a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] + ⎨ t 0 w(t, a) ≥  t ⎪ −β(t−τ ) F (τ, w(τ, ·)) ⎪ ⎪ e−β(t−r) G(r, w(r, ·))(θ(r, τ ; 0))dr e + ⎪ ⎪ ϑ(τ, 0) ⎪ τ  ⎪ t ⎪ ⎪ ⎪ ⎩ + e−β(t−r) βw(r, θ(r, τ ; 0))dr, a < Z(t, t0 ), t ∈ [t0 , t0 + δ]

(2.20)

τ

then w(t, ·) ≥ U (t, t0 )u0 ≥ 0L1 , ∀t ∈ [t0 , t0 + min (T (t0 , u0 ), δ)). As consequences of Theorems 2.10 and 2.11 we have the following Propositions. 1 satisfy AssumpProposition 2.12 Let ϑ satisfies Assumption C3. Let F1 , G 2 satisfy C1–C2 and C4’–C5’. Assume in tions C1–C2 and C4–C5 while F2 , G addition that (A1) F1 ≤ F2 in the sense that 0L1 ≤ ϕ ⇒ 0L1 ≤ F1 (t, ϕ) ≤ F2 (t, ϕ), ∀t ≥ 0. (A2) We assume that for each ξ > 0 and σ > 0 there exists α = α(ξ, σ ) > 0 such that 1 (t, ϕ) + αϕ ≤ G 2 (t, ϕ) + αϕ 0L1 ≤ ϕ ⇒ 0L1 ≤ G for all ϕ ∈ L1+ ([0, m], Rn ), with ϕL1 ≤ ξ and t ∈ [t0 , t0 + σ ]. Let t0 ≥ 0 be given and fixed. Then for each uk0 ∈ L1+ ([0, m], Rn ), k = 1, 2 with u10 ≤ u20  and each δ > 0 the mild solutions uk ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn ) , k = 1, 2 of ⎧ k (t, uk (t, ·))(a), a ∈ (0, m), t > t0 ⎪ ∂t uk (t, a) + ∂a (ϑ(t, a)uk (t, a)) = G ⎪ ⎪ ⎪ ⎪ ⎨ ϑ(t, 0)uk (t, 0) = Fk (t, uk (t, ·)), t > t0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u (t , ·) = u ∈ L1 ([0, m], Rn ). k 0 k0 + (2.21)

Monotony and Comparison Principle

275

satisfy 0L1 ≤ u1 (t, ·) ≤ u2 (t, ·), ∀t ∈ [t0 , t0 + δ]. Proof We observe that for k = 1, 2 and for each α > 0 we have 0L1 ≤ uk (t, ·), ∀t ∈ [t0 , t0 + δ] and ⎧  t ⎪ −α(t−t0 ) ⎪ e uk0 (θ (t0 , t; a)) + e−α(t−r) Gk (r, uk (r, ·))(θ (r, t; a))dr ⎪ ⎪ ⎪ t0 ⎪  ⎪ t ⎪ ⎪ ⎪ ⎨ + e−α(t−r) αuk (r, θ (r, t; a))dr, a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] t0  t uk (t, a) = ⎪ ⎪ −α(t−τ ) Fk (τ, uk (τ, ·)) ⎪ e e−α(t−r) Gk (r, uk (r, ·))(θ (r, τ ; 0))dr + ⎪ ⎪ ϑ(τ, 0) ⎪ τ  ⎪ t ⎪ ⎪ ⎪ ⎩ + e−α(t−r) αuk (r, θ (r, τ ; 0))dr, a < Z(t, t0 ), t ∈ [t0 , t0 + δ] τ

Hence setting ξ :=

sup

t ∈[t0 ,t0 +δ]



u1 (t, ·)L1 + u2 (t, ·)L1



we infer from Lemma 2.2 that for each α = α(ξ, δ) > 0 large enough we have 0L1 ≤ G1 (t, u1 (t, ·)) + αu1 (t, ·) ≤ G2 (t, u1 (t, ·)) + αu1 (t, ·), ∀t ∈ [t0 , t0 + δ]. Therefore t → u1 (t, ·) satisfies ⎧  t ⎪ −α(t−t0 ) ⎪ e u (θ (t , t; a)) + e−α(t−r) G2 (r, u1 (r, ·))(θ (r, t; a))dr ⎪ 20 0 ⎪ ⎪ t 0 ⎪  t ⎪ ⎪ ⎪ ⎪ + e−α(t−r) αu1 (r, θ (r, t; a))dr, a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] ⎨ t0 u1 (t, a) ≤  t ⎪ F2 (τ, u(τ, ·)) ⎪ −α(t−τ ) ⎪ e e−α(t−r) G2 (r, u1 (r, ·))(θ (r, τ ; 0))dr + ⎪ ⎪ ϑ(τ, 0) ⎪ τ  t ⎪ ⎪ ⎪ ⎪ ⎩ + e−α(t−r) αu1 (r, θ (r, τ ; 0))dr, a < Z(t, t0 ), t ∈ [t0 , t0 + δ] τ

and the result follows from Theorem 2.11.



276

M. L. Diagne et al.

1 satisfy AssumpProposition 2.13 Let ϑ satisfies Assumption C3. Let F1 , G 2 satisfy C1–C3 and C4’–C5’. Assume in tions C1–C3 and C4–C5 while F2 , G addition that (A1) F1 ≥ F2 in the sense that 0L1 ≤ ϕ ⇒ F1 (t, ϕ) ≥ F2 (t, ϕ) ≥ 0L1 , ∀t ≥ 0. (A2) We assume that for each ξ > 0 and σ > 0 there exists α = α(ξ, σ ) > 0 such that 1 (t, ϕ) + αϕ ≥ G 2 (t, ϕ) + αϕ ≥ 0L1 0L1 ≤ ϕ ⇒ G for all ϕ ∈ L1+ ([0, m], Rn ), with ϕL1 ≤ ξ and t ∈ [t0 , t0 + σ ]. Let t0 ≥ 0 be given and fixed. Then for each uk0 ∈ L1+ ([0, m], Rn ), k = 1, 2 with u10 ≥ u20  and each δ > 0 the mild solutions uk ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn ) , k = 1, 2 of ⎧ k (t, uk (t, ·))(a), a ∈ (0, m), t > t0 ⎪ ∂t uk (t, a) + ∂a (ϑ(t, a)uk (t, a)) = G ⎪ ⎪ ⎪ ⎪ ⎨ ϑ(t, 0)uk (t, 0) = Fk (t, uk (t, ·)), t > t0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ uk (t0 , ·) = uk0 ∈ L1+ ([0, m], Rn ). (2.22) satisfy u1 (t, ·) ≥ u2 (t, ·) ≥ 0L1 , ∀t ∈ [t0 , t0 + δ]. Proof The proof is similar to the proof of Proposition 2.12.



2.1 Proof of Theorem 2.8: Monotony Let us first prove the following lemma that will be used in the proof of Theorems 2.10 and 2.11. Lemma 2.14 For each t ≥ t0 and r ∈ [t0 , t] we have τ (t, t0 , ·) ◦ θ (t, r; 0) = r.

Monotony and Comparison Principle

277

Proof Let t ≥ t0 and r ∈ [t0 , t] be given. Set aˆ = θ (t, r; 0) and observe that by Remark 2.3 we have 0 ≤ aˆ = θ (t, r; 0) ≤ θ (t, t0 ; 0). ˆ ∈ [t0 , t] we have Then by the definition of τ := τ (t, t0 , a) aˆ = θ (t, τ (t, t0 , a); ˆ 0) ⇔ θ (t, r; 0) = θ (t, τ (t, t0 , a); ˆ 0) and since the map s → θ (t, s; 0) is decreasing in [t0 , t] (see Remark 2.3) and τ (t, t0 , a) ˆ ∈ [t0 , t] it follows that r = τ (t, t0 , a). ˆ The proof is complete.  Proof of Theorem 2.8 Let u0 , u¯ 0 ∈ L1+ ([0, m], Rn ) be given such that 0L1 ≤ u0 ≤ u¯ 0 .

(2.23)

Let t0 ≥ 0 be given and fixed. Let ν > 0 such that [0, ν] ⊃ [t0 , t0 + 1] and ξ := 1 + u0 L1 eLϑ ν + u¯ 0 L1 eLϑ ν > 0. Let α := α(ν, ξ ) > 0 such that 0L1 ≤ G(t, ϕ1 ) + αϕ1 ≤ G(t, ϕ2 ) + αϕ2 and 0Rn ≤ F (t, ϕ1 ) ≤ F (t, ϕ2 ) for all t ∈ [0, ν] and ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ) with ϕ1 L1 ≤ ξ and ϕ2 L1 ≤ ξ . Let LG := LG (ξ, ν), LF := LF (ξ, ν) be the constants provided by Lemma 2.2. Define # " <  <  < < < < < < M := max LF (ξ, ν)ξ + sup F t, 0 1 , LG (ξ, ν)ξ + sup G t, 0 1 1 t ∈[0,ν]

L

t ∈[0,ν]

L

L

so that F (t, ϕ) ≤ M, t ∈ [0, ν] and ϕL1 ≤ ξ

(2.24)

G(t, ϕ)L1 ≤ M, t ∈ [0, ν] and ϕL1 ≤ ξ.

(2.25)

and

278

M. L. Diagne et al.

Let  0 < δ < min 1,

1 (3M + 2αξ ) eLϑ ν

 (2.26)

and set 

 E δ = u ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn ) :

= sup

t ∈[t0 ,t0 +δ]

u(t, ·)L1 ≤ ξ .

 Define for each u ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn ) and ψ = u0 , u¯ 0 the operator  t ⎧ ⎪ −α(t −t0 ) ψ(θ(t , t; a)) + ⎪ e e−α(t −r) G(r, u(r, ·))(θ(r, t; a))dr ⎪ 0 ⎪ ⎪ t0 ⎪  ⎪ t ⎪ ⎪ ⎪ + e−α(t −r) αu(r, θ(r, t; a))dr, a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] ⎨ t 0 Kψ (u)(t, a) :=  t ⎪ F (τ, u(τ, ·)) ⎪ ⎪ e−α(t −τ ) + e−α(t −r) G(r, u(r, ·))(θ(r, τ ; 0))dr ⎪ ⎪ ϑ(τ, 0) ⎪ τ  ⎪ ⎪ t ⎪ ⎪ ⎩ e−α(t −r) αu(r, θ(r, τ ; 0))dr, a < Z(t, t0 ), t ∈ [t0 , t0 + δ] + τ

where τ = τ (t, t0 , a) is uniquely determined  by (2.15). Thanks to [15] for each ψ = u0 , u¯ 0 ∈ L1+ ([0, m], Rn ), Kψ maps C [t0 , t0 + δ], L1+ ([0, m], Rn ) into itself. Next we will show that for each k = 1, 2, the operator K maps E δ into itself. To do so we introduce for k = 1, 2 and ψ = u0 , u¯ 0 the operators Kψ,1 (u)(t, τ (t, t0 , a)) ⎧ 0, If a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] ⎪ ⎪ ⎪ t ⎪ ⎨ −α(t −τ ) F (τ, u(τ, ·)) + e−α(t −r)G(r, u(r, ·))(θ (r, τ ; 0))dr e := ϑ(τ, 0) τ  t ⎪ ⎪ ⎪ ⎪ ⎩ + e−α(t −r)αu(r, θ (r, τ ; 0))dr, If a < Z(t, t0 ), t ∈ [t0 , t0 + δ] τ

and Kψ,2 (u)(t, a)

:=

 t ⎧ −α(t−t0 ) ⎪ ⎪ e ψ(θ (t0 , t; a)) + e−α(t−r) G(r, u(r, ·))(θ (r, t; a))dr ⎪ ⎪ ⎨ t0  ⎪ ⎪ ⎪ ⎪ ⎩

+

t

t0

0,

e−α(t−r) αu(r, θ (r, t; a))dr, If a ≥ Z(t, t0 ), t ∈ [t0 , t0 + δ] If a < Z(t, t0 ), t ∈ [t0 , t0 + δ].

Monotony and Comparison Principle

279

Let us now show that Kψ (E δ ) ⊂ E δ for ψ = u0 , u¯ 0 . To this end we note that for each u ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn ) we have for k = 1, 2 

m



Z(t,t0 )

Kψ (u(t, ·))(a)da =

0

Kψ,1 (u)(t, τ (t, t0 , a))da  m + Kψ,2 (u)(t, a)da

0

 =

Z(t,t0 ) θ(t,t0 ;0) θ(t,t ;0)



+

Kψ,1 (u)(t, τ (t, t0 , a))da

θ(t,t0 ;m)

θ(t,t0 ;0)

t0

Kψ,2 (u)(t, a)da

∂θ (t, r; 0) dr Kψ,1 (u)(t, τ (t, t0 , ·)) ◦ θ (t, r; 0) ∂r t  m ∂θ (t, t0; a) da + Kψ,2 (u)(t, ·) ◦ θ (t, t0 ; a) 0   ∂a  t  ∂θ (t, r; 0)   dr = Kψ,1 (u)(t, τ (t, t0 , ·)) ◦ θ (t, r; 0)   ∂r t0  m ∂θ (t, t0; a) + da Kψ,2 (u)(t, ·) ◦ θ (t, t0 ; a) ∂a 0 =

and by using Lemma 2.14 one obtains 

m 0

 Kψ (u(t, ·))(a)da ≤

t

e t0

+

−α(t −r) F (r, u(r, ·)) t



ϑ(r, 0) t

e

−α(t −l)

   ∂θ (t, r; 0)    dr   ∂r

G(l, u(l, ·))(θ (l, r; 0))dl



r   ∂θ (t, r; 0)   dr ×     t  ∂r t + e−α(t −l)αu(l, θ (l, r; 0))dl  t0 r   ∂θ (t, r; 0)   dr  ×   m ∂r ∂θ (t, t0 ; a) da e−α(t −t0) ψ(a) + ∂a  0 m  t −α(t −l) + e G(l, u(l, ·))(θ (l, t0 ; a))dl t0

0

+

t0

∂θ (t, t0 ; a) × da  m ∂at

e−α(t −l)αu(l, θ (l, t0 ; a))dl

0

t0

∂θ (t, t0 ; a) da × ∂a = I1 + I2 + I3 + I4 + I5 + I6 .



280

M. L. Diagne et al.

Next we rewrite I1 , I2 , I3 , I4 , I5 and I6 into a more convenient form. Indeed for I1 and I4 by using (2.6) and (2.7) these rewrite as  I1 =

t

e−α(t −r)F (r, u(r, ·)) exp



t0

r

t

 ∂ϑ(s, θ (s, r; 0)) ds dr ∂a

and 

m

I4 =

e

−α(t −t0 )

 ψ(a) exp

0

t t0

 ∂ϑ(s, θ (s, t0 ; a)) ds da. ∂a

Hence we infer from (2.3) that I1 ≤ (t − t0 ) sup F (r, u(r, ·))eLϑ (t −t0 ) and I4 ≤ ψL1 eLϑ (t −t0) . r∈[t0 ,t ]

To rewrite I2 we use the following change of variables 2 : D2 → 2 (D2 ) defined by       l l l 2 = = r θ (l, r; 0) a with D2 =

   l ∈ R2 : r ∈ [t0 , t], r ≤ l ≤ t r

and 2 (D2 ) =

   l ∈ R2 : l ∈ [t0 , t], 0 ≤ a ≤ θ (l, t0 ; 0) = Z(l, t0 ) . r

Next note that since θ (t, r; 0) = θ (t, l; ·) ◦ θ (l, r; 0) by using a composition derivation combined with (2.7) we have for each r ∈ [0, t] and l ∈ [r, t] ∂θ (t, r; 0) ∂θ (t, l; θ (l, r; 0)) ∂θ (l, r; 0) = × ∂r ∂a ∂r  t  ∂θ (l, r; 0) ∂ϑ(s, θ (s, l; θ (l, r; 0))) = exp ds ∂r ∂a l

Monotony and Comparison Principle

281

providing that    ∂θ (t, r; 0)   dldr e−α(t −l)G(l, u(l, ·))(θ (l, r; 0))   ∂r   t0t r t   ∂θ (l, r; 0)  e−α(t −l)G(l, u(l, ·))(θ (l, r; 0))  =  t0 r  ∂r  t ∂ϑ(s, θ (s, l; θ (l, r; 0))) × exp l ds dldr ∂a

I2 =

 t

t

so that by the formula for the change of variable in double integrals we get I2 =

 t

Z(l,t0 )

e t0

−α(t −l)



t

G(l, u(l, ·))(a) exp

0

l

 ∂ϑ(s, θ (s, l; a)) ds dadl. ∂a

I3 can be treated similarly as I2 to obtain I3 =

 t

Z(l,t0 )

e t0

−α(t −l)



t

αu(l, a) exp

0

l

 ∂ϑ(s, θ (s, l; a)) ds dadl. ∂a

Hence the estimates of I2 and I3 reads as I2 ≤ (t − t0 ) sup G(l, u(l, ·))L1 eLϑ (t −t0) and r∈[t0 ,t ]

I3 ≤ (t − t0 )α sup u(l, ·)L1 eLϑ (t −t0 ) r∈[t0 ,t ]

To rewrite I5 we use the following change of variables 5 : D5 → 5 (D5 ) defined by       l l l 5 = = a σ θ (l, t0 ; a) with D5 = [0, m] × [0, t] and 5 (D5 ) =

   l ∈ R2 : l ∈ [t0 , t], Z(l, t0 ) = θ (l, t0 ; 0) ≤ a < m . σ

282

M. L. Diagne et al.

Thus recalling that θ (t, t0 ; a) = θ (t, l; θ (l, t0 ; a)) and using the composition derivative rule ∂θ (t, l; θ (l, 0; a)) ∂θ (l, 0; a) ∂θ (t, 0; a) = × ∂a ∂a ∂a it follows by a change of variables that 



∂θ(t, l; θ(l, t0 ; a)) ∂θ(l, t0 ; a) × dlda ∂a ∂a t ∂θ(t, l; σ ) e−α(t−l) G(l, u(l, ·))(σ ) dσ dl = ∂a ) t Z(l,t  0t  m 0 ∂θ(t, l; a) e−α(t−l) G(l, u(l, ·))(a) dadl = ∂a t0 Z(l,t0 )

I5 =

m

0

t

e−α(t−l) G(l, u(l, ·))(θ(l, t0 ; a))

t0 m

and we obtain from (2.7) that I5 =

 t t0

m

e−α(t −l)G(l, u(l, ·))(a) exp



Z(l,t0 )

t l

 ∂ϑ(s, θ (s, l; a)) ds dadl. ∂a

Similarly we also have I6 =

 t t0

m

e−α(t −l)αu(l, a) exp

Z(l,t0 )



t l

 ∂ϑ(s, θ (s, l; a)) ds dadl. ∂a

Thus we obtain I5 ≤ (t − t0 ) sup G(l, u(l, ·))L1 eLϑ (t −t0 ) and l∈[t0 ,t ]

I6 ≤ (t − t0 ) sup αu(l, ·)L1 eLϑ (t −t0 ) . l∈[t0 ,t ]

By combining the estimations I1 − I6 , we obtain that for each t ∈ [t0 , t0 + δ] and ψ = u0 , u¯ 0 Kψ (u(t, ·))L1 ≤ (t − t0 ) sup F (r, u(r, ·))eLϑ (t −t0) r∈[t0 ,t ]

+ 2(t − t0 ) supr∈[t0,t ] G(l, u(l, ·))L1 eLϑ (t −t0 ) + 2(t − t0 )α supr∈[t0 ,t ] u(l, ·)L1 eLϑ (t −t0 ) + ψL1 eLϑ (t −t0) so that using (2.24), (2.25) and (2.26) we obtain that sup

t ∈[t0 ,t0 +δ]

Kψ (u(t, ·))L1 ≤ δ (3M + 2αξ ) eLϑ ν + ψL1 eLϑ ν ≤ ξ.

(2.27)

Monotony and Comparison Principle

283

Furthermore using similar arguments it is easy to see that for each u, u¯ ∈ E δ we have sup

t ∈[t0 ,t0 +δ]

Kψ (u(t, ·)) − Kψ (u(t, ¯ ·))L1

≤ δ (3M + 2αξ ) eLϑ ν

sup

t ∈[t0 ,t0 +δ]

u(t, ·) − u(t, ¯ ·)

(2.28)

which means that Kψ (E δ ) ⊂ E δ for ψ = u0 , u¯ 0 and is a contraction on E δ . Therefore by the Banach fixed point Theorem, the exists a unique u ∈ E δ and a unique u¯ ∈ E δ such that ¯ u = Ku0 (u) and u¯ = Ku¯ 0 (u).  Now note that if we define v0 , v¯0 ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn ) by v0 (t, a) = u0 (a) and v¯0 (t, a) = u¯ 0 (a), ∀t ∈ [t0 , t0 + δ], ∀a ∈ [0, m], then by (2.23), (2.27) and (2.28) we have 0L1 ≤ Ku0 (v0 ) ≤ Ku¯ 0 (v¯0 ). Since Kψ (E δ ) ⊂ E δ for ψ = u0 , u¯ 0 , v0 , v¯0 ∈ E δ and K is monotone with respect to its two variables if follows that 0L1 ≤ Ku20 (v0 ) ≤ Ku0 ◦ Ku¯ 0 (v¯0 ) ≤ Ku2¯ 0 (v¯0 ) and by induction 0L1 ≤ Kuk 0 (v0 ) ≤ Kuk¯ 0 (v¯0 ), ∀k ∈ N \ {0}

(2.29)

k , ψ = u0 , u¯ 0 means the kth composition of the operator Kψ . Recalling where Kψ  that C [t0 , t0 + δ], L1+ ([0, m], Rn ) is a normal cone it follows from (2.29) that

0L1 ≤ u = lim Kuk 0 (v0 ) ≤ u¯ = lim Kuk¯ 0 (v¯0 ). k→+∞

k→+∞

We infer from Remark 2.5 that ¯ ∀t ∈ [t0 , t0 + δ]. U (t, t0 )u ≤ U (t, t0 )u, The proof is completed by using the evolution semiflow properties of U to extend the foregoing inequality in the intersection of the maximal interval of existence. 

284

M. L. Diagne et al.

2.2 Proof of Theorem 2.10 and Theorem 2.11: Comparison Principle In this section we will prove the comparison principle. The proof of Theorems 2.10 we shall only prove Theorem 2.10. Let  and 2.10 are similar. Hence w ∈ C [t0 , t0 + δ], L1+ ([0, m], Rn be defined in (2.20). It is sufficient to show that there exists δ > 0 (independent of t0 ) sufficiently small such that w(t, a) ≤ U (t, t0 )(u0 ), ∀t ∈ [t0 , t0 + δ], a ∈ [0, m]. Let ν > 0 large enough such that [0, ν] ⊃ [t0 , t0 + 1] and ξ := 1 +

sup

t ∈[t0 ,t0 +ν]

w(t, ·)L1 eLϑ ν > 0.

Let α := α(ν, ξ ) > 0 be large enough such that (2.20) holds true with β = α as well as 0L1 ≤ G(t, ϕ1 ) + αϕ1 ≤ G(t, ϕ2 ) + αϕ2 and 0Rn ≤ F (t, ϕ1 ) ≤ F (t, ϕ2 ) for all t ∈ [0, ν] and ϕ1 , ϕ2 ∈ L1+ ([0, m], Rn ) with ϕ1 L1 ≤ ξ and ϕ2 L1 ≤ ξ . Let LG := LG (ξ, ν), LF := LF (ξ, ν) be the constants provided by Lemma 2.2. Define " # <  <  < < < < < < M := max LF (ξ, ν)ξ + sup F t, 0L1 , LG (ξ, ν)ξ + sup G t, 0L1 1 t∈[0,ν]

t∈[0,ν]

so that F (t, ϕ) ≤ M, t ∈ [0, ν] and ϕL1 ≤ ξ and G(t, ϕ)L1 ≤ M, t ∈ [0, ν] and ϕL1 ≤ ξ. Let 0 0 and μ+ > 0 such that μ− ≤ μ(a) ≤ μ+ , a.e a ∈ [0, m]. (H2) The function β is continuous from R × R+ into R+ and (1) β(·, a) is a T -periodic mapping from R into R+ , for all a ≥ 0 (2) (maximum size of reproduction) There exists 0 < m0 < m such that β > 0 in R × [0, m0 ) and β(t, a) = 0 for all a ≥ m0 and t ∈ R. (H3) The function ϑ is continuous from R × R+ into R+ and (1) ϑ(·, a) is a T -periodic mapping from R into R+ , for all a ≥ 0 (2) ϑ > 0 in R × [0, m) and ϑ(t, a) = 0 for all a ≥ m and t ∈ R (3) ϑ is differentiable with respect to a ∈ [0, m] and for each a ∈ [0, m], the map t → ϑ(t, a) is continuous from R+ into R+ (4) There exists a constant Lϑ > 0 such that for each t ∈ R+ and a1 , a2 × [0, m] we have ϑ(t, a) < Lϑ and |ϑ(t, a1 ) − ϑ(t, a2 )| ≤ Lϑ |a1 − a2 |. For later use, let us note that system (3.1) rewrite as follow ⎧  u(t, ·)), ⎪ ∂t u(t, a) + ∂a (ϑ(t, a)u(t, a)) = G(t, ⎪ ⎪ ⎪ ⎪ ⎨ ϑ(t, 0)u(t, 0) = F1 (t, u(t, ·)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(0, a) = u (a), ∀a ∈ [0, m], 0

(3.2)

with  ϕ) = −μ(·)ϕ, ∀ϕ ∈ L1 ([0, m], R), ∀t ≥ 0 G(t,

(3.3)

and  +  m  1 m F (t, ϕ) = 1 − ϕ(a)da β(t, a)ϕ(a)da, ∀ϕ ∈ L1 ([0, m], R), ∀t ≥ 0. K 0 0

(3.4)

Monotony and Comparison Principle

287

On the Perturbed Linear Equation Consider the following system ⎧ ∂t p(t, a) + ∂a (ϑ(t, a)p(t, a)) = −μ(a)p(t, a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  m ⎨ ϑ(t, 0)p(t, 0) = β (t, a)p(t, a)da, ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ p(0, ·) = p0 ∈ L1 ([0, m], R)

(3.5)

β (t, a) := (1 − )β(t, a), ∀t ≥ 0, a ≥ 0.

(3.6)

with

The foregoing system rewrites as ⎧  p(t, ·)), ⎪ ∂t p(t, a) + ∂a (ϑ(t, a)p(t, a)) = G(t, ⎪ ⎪ ⎪ ⎪ ⎨ ϑ(t, 0)p(t, 0) = F (t, p(t, ·)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ p(0, a) = p0 (a), ∀a ∈ [0, m],

(3.7)

 defined in (3.3) and with the map G 

m

F (t, ϕ) =

β (t, a)ϕ(a)da, ∀ϕ ∈ L1 ([0, m], R), ∀t ≥ 0.

(3.8)

0

Our aim in this section is to study the asymptotic properties of (3.5). This will be done by using a suitable Volterra integral equation for the offspring ϑ(t, 0)p(t, 0). Therefore the first step will consist of deriving such integral equation. To this aim we rewrite (3.5) in a more convenient form by using the regularity of ϑ with respect to a that is ⎧ ∂t p(t, a) + ϑ(t, a)∂a p(t, a) = − ϑ(t, a)p(t, a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  m ⎨ β (t, a)p(t, a)da, (3.9) ϑ(t, 0)p(t, 0) = ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ p(0, ·) = p0 ∈ L1 ([0, m], R) with  ϑ(t, a) := ∂a ϑ(t, a) + μ(a), ∀t ∈ R, ∀a ∈ [0, m].

(3.10)

288

M. L. Diagne et al.

Let us integrate the new system (3.9) along the characteristics. Let θ : R2 × R+ → R+ be the nonlinear evolution family generated by the non autonomous system dγ2 (s) = ϑ(s, γ2 (s)), s ∈ R, γ2 (t1 ) = a1 ds that is γ2 (t) = θ (t, t1 , a1 ), ∀t ≥ t1 , t, t1 ∈ R. We also consider the ordinary differential equations dγ1 (s) = 1, s > t0 , γ1 (t0 ) = t1 ds so that the parametric curves s → (γ1 (s), γ2 ◦ γ1 (s)) constitute the characteristics for system (3.9). Remark 3.2 Note that due to condition (H 3) in Assumption 3.1, θ : R2 × R+ → R+ is a T -periodic evolution family in the sense that θ (t + T , t0 + T , a) = θ (t, t0 , a), ∀t, t0 ∈ R, ∀a ∈ [0, m]. Using Lemma 2.6 we know that for each t1 ≥ t0 and a1 ∈ [0, m] we have ⎧ ⎨ dq(γ1 (s), γ2 ◦ γ1 (s)) = − ϑ (γ1 (s), γ2 ◦ γ1 (s))p(γ1 (s), γ2 ◦ γ1 (s)), s ∈ [t0 , 2t0 − t1 + δ] ⎩ p(γ (t ), ds 1 0 γ2 ◦ γ1 (t0 )) = p(t1 , a1 ). (3.11)

Therefore for each s ∈ [t0 , 2t0 − t1 + δ]   s   p(γ1 (s), γ2 ◦ γ1 (s)) = exp − ϑ (γ1 (l), γ2 ◦ γ1 (l))dl p(γ1 (t0 ), γ2 ◦ γ1 (t0 )). t0

Hence making the change of variable r = γ1 (l) = t1 + l − t0 and using the initial condition in (3.11) we get   p(γ1 (s), γ2 ◦ γ1 (s)) = exp −

t1 +s−t0

  ϑ(r, θ (r, t1 , a1 ))dr p(t1 , a1 ),

t1

∀s ∈ [t0 , 2t0 − t1 + δ]

Monotony and Comparison Principle

289

so that   p(γ1 (s), γ2 ◦ γ1 (s)) = exp −

t1 +s−t0

  ϑ (r, θ (r, t1 , a1 ))dr p(t1 , a1 ),

t1

∀s ∈ [t0 , 2t0 − t1 + δ]. Using similar arguments as in Sect. 2 that is setting t1 = t0 , s = t ∈ [t0 , t0 + δ] and a1 = θ (t0 , t; a) when a ≥ Z(t, t0 ) = θ (t, t0 ; 0) and a1 = 0, t1 = τ and s = t + t0 − τ ∈ [t0 , t0 + δ] when a < Z(t, t0 ) = θ (t, t0 ; 0) with τ := τ (t, t0 , a) ∈ [t0 , t] the unique solution of a = θ (t, τ ; 0) ⇔ θ (τ, t; a) = 0

(3.12)

we obtain for each t ∈ [t0 , t0 + δ] ⎧   t  ⎪ ⎪  exp − ϑ (r, θ (r, t; a))dr p0 (θ (t0 , t; a)) if a ≥ Z(t, t0 ) ⎨ t 0    p(t, a) = t ⎪ ⎪  ⎩ exp − ϑ(r, θ (r, τ ; 0))dr p(τ, 0) if a < Z(t, t0 ). τ

(3.13)

Volterra Integral Formulation In order to provide a Volterra integral formulation, we split p(t, a) into two components as follow p1 (t, τ (t, t0 , a)) =

⎧ ⎨0

if a ≥ Z(t, t0 )   t   ⎩ exp − ϑ(r, θ (r, τ ; 0))dr p(τ, 0) if a < Z(t, t0 ) τ

and   t  ⎧ ⎨ exp −  ϑ(r, θ (r, t; a))dr p0 (θ (t0 , t; a)) if a ≥ Z(t, t0 ) p2 (t, a) = t0 ⎩ 0 if a < Z(t, t0 ).

290

M. L. Diagne et al.

Note that for each t ≥ 0 we have ϑ(t, 0)p(t, 0) =

 m

β (t, a)p(t, a)da 0 Z(t,t0)  m = β (t, a)p1 (t, τ (t, t0 , a))da + β (t, a)p2 (t, a)da Z(t,t 0)  θ(t,t 0 θ(t,t0 ;0) 0 ;m) β (t, a)p1 (t, τ (t, t0 , a))da + β (t, a)p2 (t, a)da = θ(t,t ;0)

θ(t,t0 ;0)

recalling that from Lemma 2.14 we have τ (t, t0 , ·) ◦ θ (t, r; 0) = r, ∀t ≥ t0 , r ∈ [t0 , t] and making a change of variables we obtain  ϑ(t, 0)p(t, 0) =

t0

∂θ (t, r; 0) dr ∂r t m ∂θ (t, t0; a) da. + β (t, θ (t, t0 ; a))p2 (t, θ (t, t0 ; a)) ∂a 0 β (t, θ (t, r; 0))p1(t, r)

Next we infer from (2.8) that for each t ≥ 0  ϑ(t, 0)p(t, 0) =

t

β (t, θ (t, r; 0))p1(t, r)ϑ(r, 0)    t ∂ϑ(s, θ (s, r; 0)) ds dr × exp r ∂a  m + β (t, θ (t, t0 ; a))p2 (t, θ (t, t0 ; a)) 0    t ∂ϑ(s, θ (s, t0 ; a)) ds da × exp t0 ∂a t0

hence replacing p1 and p2 by their explicit formulas and using (3.10) we obtain the following Volterra integral equation   t  β (t, θ(t, r; 0)) exp − μ(θ(s, r; 0))ds ϑ(r, 0)p(r, 0)dr  r t  t0m + β (t, θ(t, t0 ; a)) exp − μ(θ(s, t0 ; a))ds p0 (a)da, ∀t ≥ t0 .

ϑ(t, 0)p(t, 0) =

 t

0

t0

(3.14)

Monotony and Comparison Principle

291

On the Periodic Renewal Equation The asymptotic behavior of (3.5) will be derived by using properties of the Volterra integral equation (3.14). To that aim will study the following renewal equation 

t

ψ(t) = ψ(t) +

A (t, a)ψ(t − a)da,

∀t ≥ 0,

(3.15)

0

with ψ ∈ C([0, +∞) , R) while we have set for all a ≥ 0   A (t, a) = β (t, θ (t, t − a; 0)) exp −

t

t −a

 μ(θ (s, t − a; 0))ds , ∀t ∈ R (3.16)

where (t, a) → β (t, a) is defined in (3.6). We note that due the fact that θ : R2 × [0, m] → R is a non negative increasing evolution family we have 0 ≤ θ (t, t − a; 0) < θ (t, t − a; m) = m, ∀t ∈ R, ∀a ≥ 0. This last inequality means in one hand that an individual with size in [0, m) cannot reach the maximum size in a finite unit of time. On the other hand an individual with size m cannot grow anymore. Since we are dealing with a population with maximum size m, it is biologically reasonable to assume that at each t ≥ m an individual with size 0 at times t − m (that is m units of time ago) reaches the maximum size of reproduction m0 (see Assumption 3.1). This is precisely stated in the next assumption. Assumption 3.3 We assume that  θ (t, t − m; 0) =

t t −m

ϑ(s, θ (s, t − m; 0))ds ≥ m0 , ∀t ∈ R.

By using the same arguments in Remark 2.3 we know that the map a → θ (t, t−a; 0) is increasing. Thus by Assumption 3.3 we have 0 = θ (t, t − 0; 0) < m0 ≤ θ (t, t − m; 0) so that by the intermediate value theorem there exists a unique τ0 := τ0 (t, m0 ) ∈ (0, m] such that m0 = θ (t, t − τ0 ; 0) ⇔ τ0 := τ0 (t, m0 ). Hence using (H2) in Assumption 3.1 and (3.16) we deduce that A (t, a) > 0, ∀t ∈ R, ∀a ∈ [0, τ0 (t, m0 )) and A (t, a) = 0, ∀a ≥ m ≥ τ0 (t, m0 ). (3.17)

292

M. L. Diagne et al.

We now prove the following properties on the map t → τ0 (t, m0 ). Lemma 3.4 Let Assumptions 3.1 and 3.3 be satisfied. The map t ∈ R → τ0 (t, m0 ) ∈ (0, m] is T -periodic and m* := inf τ0 (t, m0 ) = τ0 (t , m0 ) > 0 t ∈R

for some t ∈ [0, T ]. Proof First note that for each t ∈ R we have (see Remark 3.2) m0 = θ (t, t − τ0 (t, m0 ); 0) = θ (t + T , t + T − τ0 (t + T , m0 ); 0) and by the periodic property of the evolution family θ (t + T , t + T − τ0 (t + T , m0 ); 0) = θ (t, t − τ0 (t + T , m0 ); 0) hence m0 = θ (t, t−τ0 (t, m0 ); 0) = θ (t, t−τ0 (t+T , m0 ); 0) ⇒ τ0 (t+T , m0 ) = τ0 (t, m0 ). Therefore the periodicity of τ0 (·, m0 ) implies that m* = inf τ0 (t, m0 ). t ∈[0,T ]

Let (tn ) ⊂ [0, T ] be a sequence such that lim τ0 (tn , m0 ) = m* .

n→+∞

Since the sequence (tn ) is bounded, there exists a sub-sequence (tnk ) such that lim tnk = t ∈ [0, T ].

k→∞

Therefore using the continuity of the evolution semiflow we have m0 = lim θ (tnk , tnk −τ0 (tnk , m0 ); 0) = θ (t , t −m* ; 0) ⇒ m* = τ0 (t , m0 ) > 0. k→+∞

 For later reference let us observe the following Remark. Remark 3.5 Let a ∈ [0, m]. Note that θ (t − m, 0; a) ≥ 0

∀ t ≥ m.

Monotony and Comparison Principle

293

Using Assumption 3.3 and Remark 2.3, we have for each t ≥ m θ (t, 0; ·) = θ (t, t − m; ·) ◦ θ (t − m, 0; a) ≥ θ (t, t − m; 0) ≥ m0 . In what follows we will verify some assumptions in order to use the renewal theory developed in [30]. The family of kernels {A (t, a) : t ∈ R, a ≥ 0} is compact (then power compact) (See [23, 30]) for each  ∈ [0, 1]. Next we prove that the family of periodic kernels {A (t, a) : t ∈ R, a ≥ 0} is φ-positive for any  ∈ [0, 1]. Proposition 3.6 Let Assumptions 3.1 and 3.3 be satisfied. Let φ : R → R+ be a T -periodic continuous, non negative and non zero function. Let k0 be an integer such that k0 T ≥ m* + T . Then the following properties hold true ¯ 1 ) > 0 for some t1 ∈ [0, T ] and (i) If ψ, ψ¯ ∈ C([0, +∞), R2+ ), ψ(t 

t

ψ(t) = ψ(t) +

A (t, a)ψ(t − a)da,

∀t ≥ 0,  ∈ [0, 1]

0

then there exists ς > 0 such that ψ(t) ≥ ς φ(t), ∀t ∈ [k0 T , (k0 + 2)T ]

(3.18)

and ψ(t) > 0, ∀t ≥ k0 T . (ii) Then there exists c0 > 0 such that for all  ∈ [0, 1] c0 |x| ≥ A (t, a)xω(t ) := inf{c > 0 : |A (t, a)x| ≤ cφ(t)}, for all t ∈ R and a ≥ 0. Proof Proof of (i) Let k0 be given such that k0 T ≥ m* + T . Since ψ is continuous on [0, +∞) and ψ(t1 ) > 0 for some t1 ∈ [0, T ], there exist 0 ≤ t2 < t3 ≤ T such that ψ(t) > 0, for all t ∈ (t2 , t3 ) ⊂ [0, T ]. Then we observe that ψ(t) > 0, ∀t ∈ (t2 , t3 ) %⇒ ψ(t − a) > 0, ∀a ∈ (t − t3 , t − t2 ) and since from (3.17) we have A (t, a) > 0, ∀t ∈ R, a ∈ [0, m* ),

294

M. L. Diagne et al.

it follows by (3.15) that for each t ∈ (t2 , t3 + m* ) ⊃ (t3 , t3 + m* ) 

t

ψ(t) ≥

 A (t, a)ψ(t − a)da ≥

0

t −t2 t −t3

A (t, a)ψ(t − a)da > 0.

Thus by induction it follows that ψ (t) > 0, t ∈ (t3 + km* , t3 + (k + 1)m* ), ∀k ≥ 1 ⇒ ψ (t) > 0, ∀t ≥ k0 T . The proof is completed by setting ς = maximum taken in [k0 T , (k0 + 2)T ].

min ψ(t) > 0 with the minimum and the max φ(t)

Proof of (ii) First note that min φ(t) > 0 where the minimum is taken on [0, T ]. Recalling that from (H1)–(H2) in Assumption 3.1, (3.6) and (3.16) one has |A (t, a)| ≤ β∞ for all t ∈ R and a ≥ 0 with β∞ :=

sup

(t,a)∈[0,T ]×[0,m]

|β(t, a)|.

(3.19)

Therefore one can choose c0 > 0 large enough such that for each x ∈ R \ {0}, t ∈ R and a ≥ 0.   1  x  β∞ A (t, a)  ≤ < min φ(t) c0  |x| c0 so that |A (t, a)x| ≤ c0 |x|φ(t), and the result follows.  Now denote by ET the space of continuous and T -periodic functions defined from R into R. Then due to (H2) in Assumption 3.1 combined together with (3.6) and (3.16) one can define for all λ ∈ R the bounded linear operator Lλ : ET → ET by  (Lλ ψ)(t) :=

+∞ 0

e−λa A (t, a)ψ(t − a)da,

∀ψ ∈ ET .

(3.20)

Monotony and Comparison Principle

295

In order to make use of [30, Remark 2.4] we note that using (3.19) and it is straightforward that |A (t, a) − A0 (t, a)| = b∞ , ∀t ∈ R, a ≥ 0, ∀ ∈ [0, 1] hence lim

sup |A (t, a) − A0 (t, a)| = 0.

→0 t ∈R,a≥0

(3.21)

Before stating next results of this section let us introduce the metric d : R2+ → R+ as in [30] ! d(x, y) := inf |c| : c ∈ R, e−c x ≤ y ≤ ec x

(3.22)

As a consequence of (3.17) and (3.21), Proposition 3.6 and [30, Theorem 2.2, Theorem 2.3] we have the following result. Proposition 3.7 Let Assumptions 3.1 and 3.3 be satisfied. Then for each  ∈ [0, 1] there exists a unique pair λ ∈ R and ψˆ  ∈ ET such that the following hold (i) sup ψˆ  (t) = 1, t ∈R

(ii) Lλ ψˆ  = ψˆ  and ρ(Lλ ) = 1, (iii) if ρ(Lλ ) < ρ(Lλ ) = 1 < ρ(Lλ ) then λ < λ < λ , (iv) lim λ = λ0 and lim ψˆ  (t) = ψˆ 0 (t), ∀t ∈ R, →0

→0

(v) If ψ ∈ C(R+ , R+ ) with ψ(t) = 0 for all t ≥ m and ψ ∈ C(R+ , R+ ) satisfies 

t

ψ(t) =

¯ A (t, a)ψ(t − a)da + ψ(t), ∀t ≥ 0

0

then there exists α ≥ 0 such that eλ t ψ(t) − α ψˆ  (t) → 0 as t → +∞ and α > 0 if ψ¯ = 0. If α > 0 then also lim d(eλ t ψ(t), α ψˆ  (t)) = 0

t →+∞

where d is the metric defined in (3.22). Asymptotic Properties of the Trivial Periodic Solution In order to study the global stability of the trivial periodic solution as well as its instability we will introduce the basic reproductive number R0 as defined in [5]. Hence we set R0 := ρ(L00 ).

296

M. L. Diagne et al.

The proof of the next two lemmas can be obtained following the same arguments of [5]. Lemma 3.8 Let Assumptions 3.1 and 3.3 be satisfied. Assume that R0 < 1. Let ϕ ∈ L1+ ([0, m], R) be given. If we denote by t → pϕ0 (t, ·) the mild solution of (3.5) with  = 0 and initial condition ϕ then  lim pϕ0 (t, ·)L1 t →+∞

m

= lim

t →+∞ 0

pϕ0 (t, a)da = 0.

Lemma 3.9 Let Assumptions 3.1 and 3.3 be satisfied. Assume that R0 > 1. Then there exists 0 > 0 such that for each ϕ = 0L1 ∈ L1+ ([0, m], R) if we denote by t → pϕ (t, ·) the mild solution of (3.5) with  ∈ [0, 0 ] and initial condition ψ then  lim pϕ (t, ·)L1 t →+∞

= lim

m

t →+∞ 0

pϕ (t, a)da = +∞.

The main result of this section is the following. Theorem 3.10 Let Assumptions 3.1 and 3.3 be satisfied. Then the following properties hold true (i) If R0 < 1 then the trivial periodic solution 0L1 of (3.1) is globally asymptotically stable in L1+ ([0, m], R). (ii) If R0 > 1 then the trivial periodic solution 0L1 of (3.1) is unstable. Proof Proof of (i) Let u0 ∈ L1+ ([0, m], R) be given. Denote by t → u(t, ·) the mild  F1 and F0 , solution of (3.1) with initial condition u0 . We note that the maps G, with  = 0 defined respectively in (3.3), (3.4) and (3.8) satisfy the conditions of Proposition 2.12. In particular we can make the identification F1 = F , F2 = F0 and 1 = G 2 = G.  Therefore if we denote by t → pu0 (t, ·) the mild solution of (3.5) G 0 with  = 0 and initial condition u0 then u(t, ·) ≤ pu00 (t, ·), ∀t ≥ 0 and we infer from Lemma 3.8 that  m u(t, a)da = lim u(t, ·)L1 = 0. 0 ≤ lim t →+∞ 0

t →+∞

Proof of (ii) We will argue par contradiction to prove the instability of 0L1 . Assume that there exists u0 ∈ L1+ ([0, m], R) such that u0 = 0L1 and  lim u(t, ·)L1 =

t →+∞

m 0

u(t, a)da = 0

(3.23)

Monotony and Comparison Principle

297

where t → u(t, ·) is the mild solution of (3.1) with initial condition u0 . Therefore there exists t0 ≥ 0 such that 

m

0≤

u(t, a)da ≤ K0 , ∀t ≥ t0 .

0

Define 0L1 ≤ u¯ 0 := u(t0 , ·) = 0L1 Then integrating along the characteristics (3.1) with initial condition u¯ 0 and using Theorem 2.11 show that 

0L1 ≤ pu¯0 (t, ·) ≤ u(t, ·), ∀t ≥ t0 ⇒ 0 ≤ 0

 m 0



pu¯0 (t, a)da ≤ 0

Thus thanks to Lemma 3.9 we have  m   lim pu¯00 (t, a)da = +∞ ⇒ lim t →+∞ 0

t →+∞ 0

which contradict (3.23). The proof is completed.

m

 m 0

u(t, a)da, ∀t ≥ t0 .

u(t, a)da = +∞ 

References 1. A.S. Ackleh, K. Deng, A monotone approximation for a nonlinear nonautonomous sizestructured population model. Appl. Math. Comput. 108(2–3), 103–113 (2000) 2. A. Calsina, J. Saldaña, A model of physiologically structured population dynamics with a nonlinear individual growth rate. J. Math. Biol. 33, 335–364 (1995) 3. J.M. Cushing, A size-structured model for cannibalism. Theor. Popul. Biol. 42, 347–361 (1992) 4. M.L. Diagne, P.I. N’diaye, T. Sari, M.T. Niane, Un modéle Mathématique de la prolifération du Typha. Rev. Afr. Rech. Inform. Math. Appl. 1, 1–8 (2010) 5. M.L. Diagne, O. Seydi, A.A.B. Sy, A two-group age of infection epidemic model with periodic behavioral changes. Discrete Contin. Dynam. Syst. Ser. B 25(6), 2057–2092 (2020) https://doi. org/10.3934/dcdsb.2019202 6. J.Z. Farkas, T. Hagen, Stability and regularity results for a size-structured population model. J. Math. Anal. Appl. 328, 119–136 (2007) 7. X. Fu, D. Zhu, Stability results for a size-structured population model withdelayed birth process. Discrete Contin. Dynam. Syst. Ser. B 18(1), 109–131 (2013) 8. Z. He, H. Wang, Well-posedness analysis for a size-structured model of species in a spacelimited habitat. J. Appl. Anal. Comput. 3(4), 335–342 (2013) 9. Z.-R. He, Q.-J. Xie, H.-T. Wang, Stability analysis for a size-structured model of species in a space-limited habitat. Int. J. Biomath. 9, 1650093 (2016) 10. M. Hirsch, Systems of differential equations which are competitive or cooperative I: limit sets. SIAM J. Appl. Math. 13, 167–179 (1982) 11. K.E. Howard, A size- and maturity-structured model of cell dwarfism exhibiting chaotic behavior. Int. J. Bifurc. Chaos 13(10), 3001–3013 (2003)

298

M. L. Diagne et al.

12. H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, 1st edn. (Springer, Singapore, 2017) 13. E. Kamke, Zur Theorie der Systeme Gewoknlicher Differentialgliechungen, II. Acta Math. 58, 57–85 (1932) 14. N. Kato, Positive global solutions for a general model of size-dependent population dynamics. Abstr. Appl. Anal. 5, 191–206 (2000) 15. N. Kato, H. Torikata, Local existence for a general model of size-dependent population dynamics. Abstr. Appl. Anal. 2, 207–226 (1997) 16. M. Krasnoselskii, Positive Solutions of Operators Equations (Groningen, Nordhoff, 1964) 17. M. Krasnoselskii, The Operator of Translation Along Trajectories of Differential Equations, vol. 19, Translations of Mathematical Monographs (American Mathematical Society, Providence, 1968) 18. P. Magal, S.G. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, 1936. Mathematical Biosciences Subseries (Springer, Berlin, 2008) 19. P. Magal, O. Seydi, F.-B. Wang, Monotone abstract non-densely defined Cauchy problems applied to age structured population dynamic models. J. Math. Anal. Appl. 479, 450–481 (2019) 20. H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations. Pub. Res. Inst. Math. Sci. 15(2), 401–454 (1979) 21. H. Matano, Existence of nontrivial unstable sets for equilibrium of strongly order preserving systems. J. Fac. Sci. Univ. Tokyo 30, 645–673 (1984) 22. M. Müller, Uber das fundamenthaltheorem in der theorie der gewohnlichen differentialglliechungen. Math. Zeit. 26, 619–645 (1926) 23. C. Rebelo, A. Margheri, N. Bacaër, Persistence in some periodic epidemic models with infection age or constant periods of infection. Discrete Contin. Dynam. Syst. Ser. B 19, 1155– 1170 (2014) 24. H.L. Smith, Monotone semiflows generated by functional differential equations. J. Diff. Equ. 66, 420–442 (1987) 25. H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41 (American Mathematical Society, Providence, 1995) 26. H.L. Smith, H. Thieme, Monotone semiflows in scalar non-quasi-monotonic functional differential equations. J. Math. Anal. Appl. 150, 289–306 (1990) 27. H.L. Smith, H. Thieme, Quasiconvergence and stability for strongly order preserving semiflows. SIAM J. Math. Anal. 21, 673–692 (1990) 28. H.L. Smith, H. Thieme, Convergence for strongly order preserving semiflows. SIAM J. Math. Anal. 22, 1081–1101 (1991) 29. H.L. Smith, H. Thieme, Strongly order preserving semiflows generated by functional differential equations. J. Diff. Equ. 93, 332–363 (1991) 30. H. Thieme, Renewal theorems for linear periodic Volterra integral equations. J. Integral Equ. 7, 253–277 (1984) 31. S.L. Tucker, S.O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables. SIAM J. Appl. Math. 48(3), 549–591 (1988) 32. G.F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics. Monographs and Textbooks in Pure and Applied Mathematics, vol. 89 (Marcel Dekker, New York, 1985)

A Boundary Value Problem of Sand Transport Equations: An Existence and Homogenization Results B. K. Thiam, M. A. M. T. Baldé, I. Faye, and D. Seck

Abstract In this paper, we consider degenerate parabolic sand transport equations in a non periodic domain with Neumann boundary condition. We give existence and uniqueness results for the models which are also homogenized. Finally some corrector results are given. Keywords Modeling · Dynamical of dune · PDE · Homogenization · Two scale convergence · Corrector result Mathematics Subject Classification (2010) Primary 35K65, 35B25, 35B10; Secondary 92F05, 86A60

1 Introduction and Results In this paper, we are interested to the well-posedness of models, built and studied in [2], [3], [4] and [14]. They are models valid for short, mean and long term dynamics of dunes in tidal area. Considering the transport flow due to Van Rijn see [15], [5–7]

B. K. Thiam Equipe de Recherche Analyse Non Linéaire et Géométrie, Laboratoire de Mathématiques de la Décision et d’Analyse Numérique, Dakar, Senegal M. A. M. T. Baldé · D. Seck Ecole Doctorale de Mathématiques et Informatique, Laboratoire de Mathématiques de la Décision et d’Analyse Numérique, Université Cheikh Anta Diop de Dakar, Dakar, Senegal e-mail: [email protected]; [email protected] I. Faye () Université Alioune Diop de Bambey, UFR S.A.T.I.C, Bambey, Senegal Ecole Doctorale des Sciences et Techniques et Sciences de la Société, Equipe de Recherche Analyse Non Linéaire et Géométrie, Laboratoire de Mathématiques de la Décision et d’Analyse Numérique, Dakar, Senegal e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. Seck et al. (eds.), Nonlinear Analysis, Geometry and Applications, Trends in Mathematics, https://doi.org/10.1007/978-3-030-57336-2_11

299

300

B. K. Thiam et al.

and [8] Faye et al. [2] showed that 

∂z ∂t



− 1 ∇ · A ∇z = 1 ∇ · C  in [0, T ) × T2

(1.1)

z (0, x) = z0 (x) in T2 ,

is a relevant model for short and mean term dynamics of dunes near the sea bed, where z0 ∈ H 1 (T2 ) is a given function and T2 is the two dimensional torus. The coefficients A and C  are given by A (t, x) = a(1 − bm)ga (|u|), and C  (t, x) = c(1 − bm)gc (|u|)

u |u|

(1.2)

where a > 0, b and c are real numbers. The fields u : [0, T ) × T2 → R2 and m : [0, T ) × T2 → R are respectively the water velocity and the height variation due to the tide. We suppose that they are given by t t m(t, x) = M(t, , x) and u(t, x) = U(t, , x)  

(1.3)

in the short term where U and M are regular functions on R+ × R × T2 such that θ −→ (U(t, θ, x), M(t, θ, x)) is periodic of period 1

(1.4)

and √ √ u A (t, x) = a(1−b m)ga (|u|), and C  (t, x) = c(1−b m)gc (|u|) |u|

(1.5)

where t t t t m(t, x) = M(t, √ , , x) and u(t, x) = U(t, √ , , x)    

(1.6)

in the mean term where U and M are regular functions on R+ × R × R × T2 such that  τ −→ (U(t, τ, θ, x), M(t, τ, θ, x)) (1.7) θ −→ (U(t, τ, θ, x), M(t, τ, θ, x)) is periodic of period 1. The model valid for long term dynamics of dunes is given by 

∂z ∂t

−

1 ∇ 2

 · A ∇z =

1 ∇ 2

· C  in [0, T ) × T2

z (0, x) = z0 (x) in T2 ,

(1.8)

A Boundary Value Problem of Sand Transport Equations

301

where A and C  are given by (1.2) and u and m satisfy t t t m(t, x) = M(t, , x) = M1 ( , x) +  2 M2 (t, , x) and    t t t t u(t, x) = U(t, , x) = U0 ( ) + U1 ( , x) +  2 U2 (t, , x).    

(1.9)

Functions U and M in (1.3) having the same behavior as in (1.6) satisfy the following hypotheses ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂U ∂U ∂U |, | |, | |, |∇ U |, ∂t ∂θ ∂τ ∂M ∂M ∂M |M|, | ∂t |, | ∂θ |, | ∂τ |, |∇ M| are bounded by d, ∀(t, τ, θ, x) ∈ R+ × R × R × T2 , |U (t, τ, θ, x)| ≤ Uthr %⇒ ∂U ∂U ∂U ∂M ∂M ∂M = 0, = 0, = 0, = 0, = 0, = 0, ∂t ∂τ ∂θ ∂t ∂τ ∂θ ∇ M(t, τ, θ, x) = 0 and ∇ U (t, τ, θ, x) = 0, ∃θα < θω ∈ [0, 1] such that ∀ θ ∈ [θα , θω ], we have |U (t, τ, θ, x)| ≥ Uthr |U |, |

(1.10) with ga and gc are positive functions satisfying the following hypotheses ⎧ ⎪ ⎪ ⎨

ga ≥ gc ≥ 0, gc (0) = gc (0) = 0, supu∈R+ |ga (u)| + supu∈R+ |ga (u)| ≤ d, ⎪ supu∈R+ |gc (u)| + supu∈R+ |gc (u)| ≤ d, ⎪ ⎩ ∃ Gt hr > 0, such that u ≥ Ut hr %⇒ ga (u) ≥ Gt hr .

(1.11)

The objective of this paper is to study the well-posedness of the models (1.1) and (1.8) coupled with (1.3) or (1.6) in the cases of short and mean term models or (1.9) in the long term model in a domain  with boundary ∂ containing the two dimensional torus T2 . In other words, we are interested in the existence and the uniqueness of solutions to the following two problems posed in the domain  of class C 1 . 

⎧  ∂z 1 1    ⎪ ⎨ ∂t −  ∇ · A ∇z =  ∇ · C in [0, T ) ×  (1.12) z (0, x) = z0 (x) in , ⎪ ⎩ ∂z ∂n = g on [0, T ) × ∂ and ⎧  ∂z ⎪ ⎨ ∂t − ⎪ ⎩

1 ∇ 2



· A ∇z =

1 ∇ 2

· C  in [0, T ) × 

z (0, x) = z0 (x) in , = g on [0, T ) × ∂

∂z ∂n

where g ∈ L2 ([0, T ), L2 ()) and z0 ∈ L2 ().

(1.13)

302

B. K. Thiam et al.

Then we will show that the solutions of the two problems are bounded independently of . The second step consists in the homogenization and correction results for the two problems mentioned. Which in part will require adding additional assumptions. Contrary to what Faye et al. [2] have proposed, we will have in this work to manage integrals on the edge of the domain which will make calculations a bit difficult. The analysis of these results requires other hypotheses and more particularly, that requiring the null average hypothesis of the dune fields. This hypothesis is translated by a natural condition on the bound of domain  linking the Neumann boundary condition that we are going to impose on the bound and the second member of our model equation. With this hypothesis, we show that each of the equations listed above admits a unique solution depending on the data of the problem. We show in addition that the solution of the problem is bounded by a positive real constant not depending on the parameter . Our first result is given by: Theorem 1.1 Let  be a measurable set of class C 1 . For all T > 0 and  > 0, under assumptions (1.10), (1.11), (1.3) and (1.6) or (1.9) if z0 ∈ H 1 () and g ∈ L2 ([0, T ), H 1 (R2 )), there exists a unique function z ∈ L∞ ([0, T ), H 1 ()) solution to (1.12) or (1.13). This solution satisfies < < < < 0 and  > 0, under assumptions (1.10), (1.11), (1.3)–(1.7) and (1.9), the sequence of solutions (z (t, x)) to (1.12) Two scale converges to a profile 2 U ∈ L∞ ([0, T ], L∞ # (R, L ())) solution to  ∂U ∂θ

 ) = ∇ · C in (0, T ) × R ×  − ∇ · (A∇U ∂U ∂n = g on (0, T ) × R × ∂

(1.16)

A Boundary Value Problem of Sand Transport Equations

303

 and Care given by where A  τ, θ, x) = aga (|U(t, τ, θ, x)|) and C(t,  τ, θ, x) = cgc (|U(t, τ, θ, x)|) U(t, τ, θ, x) . A(t, |U(t, τ, θ, x)|

(1.17) with U given by (1.4) or (1.7). The sequence of solutions (z (t, x)) to (1.13) Two scale converges to a profile 2 U ∈ L∞ ([0, T ], L∞ # (R, L ())) solution to ⎧  ) = ∇ · C in (0, T ) × R ×  ⎨ −∇ · (A∇U ∂U ∂θ = 0 on Θt hr ⎩ ∂U = g on (0, T ) × R × ∂ ∂n

(1.18)

 and Care given by where A  θ, x) = aga (|U0 (θ )|) and C(t,  θ, x) = cgc (|U0 (θ )|) U0 (θ ) . A(t, |U0 (θ )|

(1.19)

with U0 given in (1.9).

2 Existence and Estimates, Proof of Theorem 1.1 The objective of this paper is to study the sand transport models obtained in [2] and posed in a domain  with boundary of class C 1 . They necessitate to impose a boundary condition in the domain. Since the sand does not flow outside , we showed in [14] (see especially (1.7)–(1.9) therein) that the natural boundary condition on ∂ is the Neumann one. But we can also use a Dirichlet and mixed Neumann-Dirichlet condition. In our context, we shall consider a Neumann boundary value problem. Consider  (t, t , x), C  (t, x) = C (t, t , x) A (t, x) = A  

(2.1)

where  (t, θ, x) = a(1 − bM(t; θ, x))ga (|U(t, θ, x)|) A

(2.2)

U(t, θ, x) C (t, θ, x) = c(1 − bM(t, θ, x))gc (|U(t, θ, x)|) |U(t, θ, x)|

(2.3)

304

B. K. Thiam et al.

in short and long terms or  (t, √t , t , x), C  (t, x) = C (t, √t , t , x) A (t, x) = A    

(2.4)

where √  (t, τ, θ, x) = a(1 − b M(t, τ, θ, x))ga (|U(t, τ, θ, x)|) (2.5) A √ U(t, τ, θ, x) C (t, τ, θ, x) = c(1 − b M(t, τ, θ, x))gc (|U(t, τ, θ, x)|) (2.6) |U(t, τ, θ, x)| in mean term where U and M are given in (1.3) and (1.6).  and C satisfy the following hypotheses Under assumptions (1.10) and (1.11), A ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 , C ) is periodic of period 1, τ → (A  ; C ) is periodic of period 1 θ → (A  , C ) is defined on  x → (A  | ≤ γ , |C | ≤ γ , | ∂ A | ≤ γ , | ∂ C | ≤ γ , |A ∂θ ∂θ  | ≤  i γ , |∇ · C | ≤  i γ , |∇ A    A C 1+i γ , | ≤  1+i γ , | ∂∂tC | ≤  1+i γ , | ∂∇∂tA | ≤  1+i γ , | ∂∇· | ∂∂t ∂t | ≤ 

(2.7)

where i = 0 in the case of Eq. (1.12) and i = 1 in the case of Eq. (1.13). Equations (1.12) and (1.13) can be written in the generic form ⎧  ∂z ⎪ ⎨ ∂t − ⎪ ⎩

1 ∇ i



· A ∇z =

1 ∇ i

· C  in [0, T ) × 

z (0, x) = z0 (x) in , = g on [0, T ) × ∂

(2.8)

∂z ∂n

for i = 1 or 2. The case where i = 1 corresponds to the valid model for short and mean term whereas for i = 2 we have the long term one. In the following, we are interested to the existence and uniqueness of solutions to (2.8). This equation is a degenerated perturbed parabolic equation. The degeneracy of the coefficient A , makes that, the resolution of (2.8) can not be done with the classical methods like Lax Milgram or Stampachia theorem. Therefore for the study of the problem, we will first prove, as in [2], the existence of a periodic degenerated parabolic equation that we set out. Before giving the proof of Theorem 1.1, we show that the coefficients of Eq. (2.8), as well as their derivatives, are bounded independently of . Existence for a given  follows from results of Lions [10], [11], [12], Ladyzhenskaya and Solonnikov [9]. But, since our aim is to study the asymptotic behavior of z as  goes to 0, we need estimates which do not depend on . For this, based on the work of Faye et al. [2, 4] and [14], we are going to show the existence of the following Neumann boundary value problems:∀μ > 0, ν > 0 and  > 0, find

A Boundary Value Problem of Sand Transport Equations

305

S ν = S ν (t, τ, θ, x) and Sμν = Sμν (t, τ, θ, x) periodic of period 1 in θ and solution to

 ⎧ ν ν ∂S 1  ⎪ − ∇ · A (t, τ, θ, ·) + ν ∇S =  i ⎪ ∂θ  ⎪ ⎨ 1  ∇ · C (t, τ, θ, ·) in (0, T ) × R × R ×  (2.9) i ⎪ ⎪ S ν (0, 0, 0, x) = z0 (x) in  ⎪ ⎩ ∂Sν ∂n = g on (0, T ) × R × R × ∂ and ⎧ ν ν + ∂ Sμ − ⎪ μS ⎪ μ ⎪ ∂θ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

1 ∇ i

·

∂ Sμν ∂n

   (t, τ, θ, ·) + ν ∇Sμν = A

1 ∇ i

· C (t, τ, θ, ·)

in (0, T ) × R × R ×  Sμν (0, 0, 0, x) = z0 (x) in 

(2.10)

= g on (0, T ) × R × R × ∂,

for i = 0 or 1. In Eqs. (2.9) and (2.10), t is only a parameter. Above and in the sequel, we assume that the following hypothesis holds    + ν)g + C · n dσ = 0. ∀ > 0, ν ≥ 0, μ > 0, (2.11) (A ∂

where n is the outward normal on ∂. The relevance of this hypothesis that we have given will find its justification from the physical point of view by the law of mass conservation in other words, the sedimentary particles in a domain  which will be considered. $ are confined % The set  = A + ν g + C  · n = 0 is non-empty. It mean that we can always find a function χ  regular with the same behavior as  C such that C¯ = C  + χ  , A¯  = A , and (A¯  + ν)g + C¯ · n = 0 on ∂. Indeed : A (t, x) = a(1 − bm)ga (|u|) and C  (t, x) = c(1 − bm)gc (|u|)

u |u|

(2.12)

Then (a(1 − bm)ga (|u|) + ν)g + c(1 − bm)gc (|u|)

u · n + χ  · n = 0 on ∂. |u|

Hence, we get χ  · n = −a(1 − bm)ga (|u|)g − c(1 − bm)gc (|u|) = −(1 − bm)(aga (|u|)g + cgc (|u|)

u · n − νg |u|

u · n ) − νg |u| + ,- .

=cos(u,n)

306

B. K. Thiam et al.

We can write, χ  · n = −(1 − bm)(aga (|u|)g + cgc (|u|) cos(u, n) − νg. Finally , we find χ  such that χ  · n = −(1 − bm)(aga (|u|)g + cgc (|u|) cos(u, n) − νg. and (A¯ + ν)g + C¯ · n = 0 on ∂. $ % For example, in one dimension, let’s set  = (0, 1), ∂ = 0, 1 , a = b = c = 1, ga = gc such that gc (|u(x)|) = x 2 , g = 1 and m = 1. In this case, χ  = X have only 2 values X(0) = X1 and X(1) = X2 . Then  χ · n = X =

X1 = −ν . X2 = −2(1 − ) − ν.

(2.13)

This hypothesis translates a free mean divergence:  ∇· 



  + ν ∇Sμν + C dx = 0. A

It can be considered in order to simplify the expressions of the boundary value  (t, τ, θ, x) + problem (2.10). In fact in the case where the mean divergence of −(A   1  + ν)∇Sμν − C (t, τ, θ, x) over  is not zero, we can add the expression || A ∂  ν g + C · n in the first equation of (2.10). This hypothesis plays an important role  in our method. Actually, we shall construct processes Sμν μ,ν that give the desired results in terms of asymptotic analysis and two scale convergence. Let us focus on existence and uniqueness of S ν and Sμν solutions to (2.9) and (2.10). But before proceeding further on, we point out the following remark. Remark 2.1 We have to notice that, under assumptions (2.2), (1.10) and (1.11), the  , C and its derivatives are bounded on R+ × R × R ×  by a coefficients A i constant  γ where γ not depending on  for i = 1 or 2. Moreover, for all 0 ≤  , C ) is periodic of period 1 and there exists a constant G t hr and  < 1, θ −→ (A θα , θω ∈ [0, 1], θα < θω such that  (t, τ, θ, x) ≥ G t hr , ∀(t, τ, θ, x) ∈ R+ × R × R ×  A

(2.14)

A Boundary Value Problem of Sand Transport Equations

307

and such that ∀ (t, τ, θ, x) ∈ R+ × R × R × 

 (t, τ, θ, x) ≤ G t hr A

⎧   ∂A ∂A ⎪ ⎪ ⎪ (t, τ, θ, x) = 0, (t, τ, θ, x) = 0, ⎪ ⎨ ∂t ∂τ  %⇒  (t, τ, θ, x) = 0, ∂ C (t, τ, θ, x) = 0, ⎪ ∇A ⎪ ⎪ ∂t ⎪ ⎩ ∂ C (t, τ, θ, x) = 0, ∇ · C (t, τ, θ, x) = 0. ∂τ (2.15)

We have the following theorem: Theorem 2.2 Under the same assumptions as in Theorem 1.1 and under assumptions (2.7),(2.11), (2.14) and (2.15), ∀ > 0, ν > 0, μ > 0, there exists a unique Sμν = Sμν (t, τ, θ, x) 1-periodic in θ, solution to (2.10). Moreover there exists constants γ2 , γ3 , γ6 which depends only on , γ , ν,  i , g such that     sup  Sμν (θ, x)dx  = 0,

< < < ν