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Table of contents :
Dedication
Contents
Preface
Acknowledgement
1 Efficiency conditions in vector variational problems involving higher-order derivatives
2 Controlled signomial dynamical systems as constraints in variational control problems
3 Sufficient conditions of efficiency for a class of vector fractional variational problems
4 Hamilton and Hamilton-Jacobi dynamics of higher-order
5 Necessary and sufficient conditions of efficiency in vector fractional variational control problems
6 Duality results involving (ρ, b)-quasiinvexity for a class of vector fractional variational control problems
7 Geometric optimal control problems
8 On a class of variational control problems
9 Weak sharpness of the solution set associated with an integral variational inequality
10 On efficient solutions associated with a class of PDE-constrained vector variational control problems
11 Modified variational control problems and saddle-point optimality criteria
12 On local and global optimal solutions in variational control problems
Bibliography
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Variational Analysis with Applications in Optimisation and Control



Variational Analysis with Applications in Optimisation and Control By

Savin Treanţă

Variational Analysis with Applications in Optimisation and Control By Savin Treanţă This book first published 2019 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2019 by Savin Treanţă All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-3728-5 ISBN (13): 978-1-5275-3728-6

To my children, Constantin and Elena

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 Efficiency conditions in vector variational problems higher-order derivatives 1.1 Introduction and problem description . . . . . . . . . . 1.2 Some preliminary results . . . . . . . . . . . . . . . . . 1.3 Necessary and sufficient conditions of efficiency . . . . .

involving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Controlled signomial dynamical systems as constraints variational control problems 2.1 Controlled signomial dynamical systems . . . . . . . . . . . 2.1.1 Variational and adjoint signomial differential systems 2.2 Necessary conditions of optimality . . . . . . . . . . . . . . 2.3 Conclusions and further developments . . . . . . . . . . . . 3 Sufficient conditions of efficiency for a class of vector variational problems 3.1 Introduction and problem formulation . . . . . . . . . . . 3.2 Some auxiliary results . . . . . . . . . . . . . . . . . . . 3.3 Sufficient efficiency conditions based on (ρ, b)-quasiinvexity 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Hamilton and Hamilton-Jacobi dynamics of higher-order 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hamilton dynamics of second-order . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Scalar optimisation problem governed by multiple integral functional . 4.2.2 Scalar optimisation problem governed by path-independent curvilinear integral functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 On Lagrange and Hamilton dynamics involving multi-time second-order Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Hamilton dynamics of higher-order . . . . . . . . . . . . . . . . . . . . . . . 4.5 Higher-order Hamilton-Jacobi divergence PDE . . . . . . . . . . . . . . . . . vii

1 1 3 4

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viii 5 Necessary and sufficient conditions of efficiency in variational control problems 5.1 Problem formulation . . . . . . . . . . . . . . . . . . 5.2 Necessary efficiency conditions . . . . . . . . . . . . . 5.3 (ρ, b)-invexity for multiple integral functional . . . . . 5.4 Sufficient efficiency conditions . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 6 Duality results involving (ρ, b)-quasiinvexity for fractional variational control problems 6.1 Introduction . . . . . . . . . . . . . . . . . . . . 6.2 Notations and some preliminaries . . . . . . . . . 6.3 Weak, strong and converse duality results . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . .

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a class of variational control problems Introduction . . . . . . . . . . . . . . . . . Notations and preliminary results . . . . . . KT-pseudoinvex control problems . . . . . . Illustrative application . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . .

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7 Geometric optimal control problems 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Geometric optimal control problem with multiple integral cost functional . 7.2.1 Variational and adjoint n-form equations . . . . . . . . . . . . . 7.2.2 Necessary conditions of optimality . . . . . . . . . . . . . . . . . 7.3 Geometric optimal control problem with curvilinear integral cost functional 7.3.1 Adjoint fundamental tensor evolution . . . . . . . . . . . . . . . 7.3.2 Simplified geometric maximum principle . . . . . . . . . . . . . . 7.4 Exterior Euler-Lagrange and Hamilton-Pfaff PDEs . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 On 8.1 8.2 8.3 8.4 8.5

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9 Weak sharpness of the solution set associated with an integral variational inequality 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The problem formulation and preliminaries . . . . . . . . . . . . . . 9.3 Some auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Weak sharpness and minimum principle sufficiency property . . . . . . 9.5 Illustrative application . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ix 10 On efficient solutions associated with a vector variational control problems 10.1 Introduction . . . . . . . . . . . . . . . 10.2 Preliminaries and working hypotheses . . 10.3 V-KT-pseudoinvex control problems . . . 10.4 An illustrative example . . . . . . . . . . 10.5 Conclusions . . . . . . . . . . . . . . . .

class of PDE-constrained . . . . .

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119 119 120 122 128 133

11 Modified variational control problems and saddle-point optimality criteria 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Notations and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Optimality conditions in modified variational control problems . . . . . . . 11.4 Saddle-point optimality criteria . . . . . . . . . . . . . . . . . . . . . . . 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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135 135 136 141 146 152

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12 On local and global optimal solutions in variational 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 12.2 Notations and problem description . . . . . . . . . . . . 12.3 A minimal necessary and sufficient criterion . . . . . . . 12.4 Illustrative application . . . . . . . . . . . . . . . . . . 12.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . Bibliography

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control problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 . 153 . 154 . 156 . 159 . 161 163

xi

Preface The present book focuses on that part of calculus of variations and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. Frequently, we operate in physical problems with a two-time, t = (t1 , t2 ), where one component means the intrinsic time and the other one represents the observer time, not having any preference for one of the two components. In this respect, this book provides the latest developments in multidimensional optimization and optimal control. The results presented in this book are based on the author’s recent contributions. With various examples and applications to complement and substantiate the mathematical developments, the present book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics. The book is organized in twelve chapters: First, Chapter 1 presents necessary and sufficient conditions of efficiency for a class of multiobjective (vector) variational problems involving higher-order derivatives. More precisely, an optimisation problem of minimizing a vector of simple integral functionals subject to higher-order differential equations and/or inequations is investigated. By using the notion of quasiinvexity, sufficient efficiency conditions for a feasible solution are established. In Chapter 2, by using the notions of the variational differential system, adjoint differential system and modified Legendrian duality, necessary optimality conditions for a class of signomial constrained optimal control problems are provided. Chapter 3 presents a study on sufficient efficiency conditions for a class of multidimensional vector ratio optimisation problems, identified by (M F P ), of minimizing a vector of path-independent curvilinear integral functional quotients subject to PDE and/or PDI constraints involving higher-order partial derivatives. Under generalised (ρ, b)-quasiinvexity assumptions, sufficient conditions of efficiency are provided for a feasible solution in (M F P ). Chapter 4, by using a non-standard Legendrian duality, investigates the Hamiltonian dynamics and formulates a Hamilton-Jacobi type divergence PDE involving higher-order Lagrangians. In Chapter 5, necessary and sufficient conditions of efficiency are derived in multiobjective variational control problems which involve multiple integral cost functionals. Under (ρ, b)-quasiinvexity assumptions, sufficient efficiency conditions for a feasible solution are formulated, as well. Chapter 6, by using the concept of invexity associated with multiple integral vector functionals, introduces several results of duality for a class of multiobjective fractional variational control problems involving multiple integral cost functionals. Under (ρ, b)-quasiinvexity assumptions, weak, strong and converse duality results are provided. The main goal of Chapter 7 is to formulate and prove, under simplified hypothesis, a maximum principle in a mathematical framework governed by geometric tools. More precisely, using some techniques of calculus of variations, the notion of adjointness and a geometrical context, necessary optimality conditions are established for two optimal control problems governed by: (i) multiple integral cost functional and (ii) curvilinear integral (mechanical work) cost functional, both

xii subject to fundamental tensor (state variable) evolution as constraint. In both optimisation problems, the control variable is a connection, as well. Finally, as an application of the geometric maximum principle introduced in this chapter, exterior Euler-Lagrange and Hamilton-Pfaff PDEs are obtained. In Chapter 8, a KT-pseudoinvex multidimensional control problem is introduced. More exactly, a new condition on the functions which are involved in a multidimensional control problem is formulated and it is proved that a KT-pseudoinvex multidimensional control problem is characterized such that a Kuhn-Tucker point is an optimal solution. The theoretical results are illustrated with an application, as well. In Chapter 9, under some assumptions and using a dual gap-type functional, weak sharp solutions are investigated for a multidimensional variational inequality governed by convex multiple integral functional. Moreover, a relation between the minimum principle sufficiency property and weak sharpness of a solution set for the considered variational-type inequality is established. In order to give a better insight into the main results, a numerical application is formulated. Chapter 10 introduces a V-KTpseudoinvex multidimensional vector control problem. A new condition on the functionals which are involved in a multidimensional multiobjective (vector) control problem is introduced and it is shown that a V-KT-pseudoinvex multidimensional vector control problem is described so that all Kuhn-Tucker points are efficient solutions. An illustrative application is also presented. In Chapter 11, based on a multidimensional control problem, in short (M CP ), a modified multidimensional variational control problem involving firstorder partial differential equations (PDEs) and inequality-type constraints is introduced. Optimality conditions for this new variational control problem are formulated and proved, as well. Furthermore, under some generalised convexity assumptions, an equivalence is established between an optimal solution of (M CP ) and a saddle-point associated with the Lagrange functional (Lagrangian) corresponding to the modified multidimensional control problem. Also, in order to illustrate the main characterization results and their effectiveness, several applications are presented. Chapter 12 investigates optimality conditions for a class of PDE&PDI-constrained variational control problems. Thus, a minimal criterion for a local optimal solution of the considered PDE&PDI-constrained variational control problem to be its global optimal solution is derived. The theoretical development is supported by a suitable nonconvex optimisation problem.

Acknowledgement My warm thanks go to my wife Elena-Cristina whose love, support and encouragement over the years have made the writing of this book possible.

Bucharest, Romania 2019

Savin Treant¸a˘

Chapter 1 Efficiency conditions in vector variational problems involving higher-order derivatives In this chapter, necessary and sufficient conditions of efficiency are formulated and proved for a class of multiobjective (vector) variational problems involving higher-order derivatives. More precisely, we investigate an optimisation problem of minimizing a vector of simple integral functionals subject to higher-order differential equations and/or inequations. By using the notion of quasiinvexity, sufficient efficiency conditions for a feasible solution are established.

1.1

Introduction and problem description

In this chapter, we extend and further develop some optimisation results regarding the efficiency of a feasible solution for a class of vector non-fractional variational problems. More concretely, we introduce and perform a study on the vector variational problem of minimizing a vector of simple integral functionals, denoted by (M V P ), with higherorder differential equation and inequation constraints. The passing from the first-order derivatives to the higher-order derivatives is not a facile task because it requests specific techniques, a new quasiinvexity concept and an appropriate mathematical framework. Over time, several authors have been interested in the study of vector variational problems by using a generalised convexity/invexity (see [8], [10]-[15], [17], [25], [26], [32], [33], [41], [45], [48]-[50], [56], [58], [60], [66], [68], [80], [82], [84], [87], [88], [94]-[98], [102], [104], [105], [114]-[117], [124], [141], [167]). Also, many of them extended the notion of convexity/invexity and developed a multitime (multidimensional) optimisation theory by using a geometrical language (see [83], [110], [111], [135], [144], [146], [151]-[153]). Let us consider the real interval I := [t0 , t1 ] ⊆ R and f = (fα ) : I × Rn(k+1) → Rp ,

α = 1, p,

(f1 (t, x(t), x(1) (t), ..., x(k) (t)), ..., fp (t, x(t), x(1) (t), ..., x(k) (t))), 1

2

Efficiency conditions in vector variational problems with higher-order derivatives

dk x(t), with k ≥ 1 a fixed natural number. Also, dtk let be given g = (g1 , ..., gm ) : I × Rn(k+1) → Rm , with m < n, and h = (h1 , ..., hr ) : I × Rn(k+1) → Rr , with r < n, two C k+1 -class functions. Assume that the C k+1 -class Lagrangians fα (t, x(t), x(1) (t), ..., x(k) (t)), α = 1, p

a C k+1 -class function, where x(k) (t) :=

generate the simple integral functionals 

t1

Fα (x(·)) :=

fα (t, x(t), x(1) (t), ..., x(k) (t))dt,

α = 1, p.

t0

Let C ∞ ([t0 , t1 ], Rn ) be the space of all functions x : [t0 , t1 ] → Rn of C ∞ -class, with the norm k  x := x∞ + x(β) ∞ . β=1

As usually, for any two vectors u = (u1 , ..., us ) , v = (v1 , ..., vs ) in Rs , we shall consider the following convention u = v ⇔ ui = v i , u < v ⇔ u i < vi ,

u ≤ v ⇔ ui ≤ v i ,

u  v ⇔ u ≤ v,

u = v,

i = 1, s.

Also, we underline that the argument of the considered Lagrangians is a graph χx (t) := (t, x(t), x(1) (t), ..., x(k) (t)). Using these ingredients, we formulate the multiobjective variational problem (M V P ) (a constrained optimisation problem) as  min F (x(·)) = x(·)



t1

f1 (χx (t))dt, ..., t0



t1

fp (χx (t))dt t0

subject to x (·) ∈ F (I), where the set F (I) of all feasible solutions is x ∈ C ∞ (I, Rn ) , g(χx (t)) ≤ 0,

x(tε ) = xε ,

x(β) (tε ) = xβε ,

h(χx (t)) = 0,

t ∈ I,

ε ∈ {0, 1},

β = 1, k − 1.

Th next section introduces some necessary preliminary results which will be used for proving the main results of the present chapter.

Chapter 1

1.2

3

Some preliminary results

In the following, let us start with the case of a single simple integral functional, by considering the following scalar variational problem (SV P ),    t1 min I (x(·)) = X (χx (t)) dt x(·)

t0

subject to x (·) ∈ F (I). Further, consider the auxiliary Lagrange function L(χx (t), p(t), q(t), λ) := λX (χx (t)) + pa (t)ga (χx (t)) + q ζ (t)hζ (χx (t)), with summation over the repeated indices, that allows us to establish necessary conditions of optimality for (SV P ). Theorem 2.1 Consider that the feasible solution x0 of the problem (SV P ) is an optimal solution and the functions X, g, h are C k+1 -class functions. Then there exist a scalar λ and the piecewise smooth functions p(t) and q(t), satisfying ∂L d ∂L (χx0 (t), p(t), q(t), λ) (χx0 (t), p(t), q(t), λ) − ∂x dt ∂x(1) +... + (−1)k

dk ∂L (χx0 (t), p(t), q(t), λ) = 0 dtk ∂x(k)

(higher-order Euler-Lagrange ODEs) p(t)g(χx0 (t)) = 0,

p(t) ≥ 0,

(∀)t ∈ I.

Definition 2.1 The optimal solution x0 (·) of problem (SV P ) is called normal optimal solution if λ = 0. Further, without loss of generality, we can assume that λ = 1. Definition 2.2 A feasible solution x0 (·) ∈ F (I) is called efficient solution  in (M V P) if there exists no other feasible solution x(·) ∈ F (I) such that F (x(·))  F x0 (·) . Consider ρ a real number and b : [C ∞ ([t0 , t1 ], Rn )]k+1 → [0, ∞) a positive functional. Also, consider the notations  b x, x0 , x0(1) , ..., x0(k−1) := bxx0 ,  η t, x, x(1) , ..., x(k−1) , x0(k) := ηtxx0

4

Efficiency conditions in vector variational problems with higher-order derivatives

and a : I × Rn(k+1) → R a real function that determines the simple integral functional 

t1

a (χx (t)) dt.

A(x(·)) = t0

Definition 2.3 The functional A(·) is [strictly] (ρ, b)-quasiinvex at x0 if there exist the vector functions η = (η1 , ..., ηn ), with the property dζ ηtx0 x0 = 0, dtζ

ζ ∈ {0, 1, ..., k − 1},

t ∈ I,

and θ : [C ∞ ([t0 , t1 ], Rn )]k+1 → Rn such that, for any x [x = x0 ], we have 

0





t1

A(x) ≤ A(x ) =⇒ (bxx0

ηtxx0 t0



t1

+bxx0 t0

dηtxx0 ∂a (χx0 (t))dt + ... + bxx0 dt ∂x(1)



∂a (χx0 (t))dt ∂x

t1 t0

dk ηtxx0 ∂a (χx0 (t))dt dtk ∂x(k)

[ 0 and the foregoing inequality can be rewritten as    t1 ∂f ∂g ∂h ηtxx0 λ (χx0 (t)) + p(t) (χx0 (t)) + q(t) (χx0 (t)) dt ∂x ∂x ∂x t0

12

Efficiency conditions in vector variational problems with higher-order derivatives    t1 ∂f dηtxx0 ∂g ∂h λ (1) (χx0 (t)) + p(t) (1) (χx0 (t)) + q(t) (1) (χx0 (t)) dt + dt ∂x ∂x ∂x t0    t1 k d ηtxx0 ∂f ∂g λ (k) (χx0 (t)) + p(t) (k) (χx0 (t)) dt +... + dtk ∂x ∂x t0      t1 k p  d ηtxx0 ∂h q(t) (k) (χx0 (t)) dt < − λl ρ1l + ρ2 + ρ3  θxx0 2 . + k dt ∂x t0 l=1

or, using the formula of integration by parts, we get    t1 ∂f ∂g ∂h ηtxx0 λ (χx0 (t)) + p(t) (χx0 (t)) + q(t) (χx0 (t)) dt ∂x ∂x ∂x t0   ∂f ∂g ∂h +ηtxx0 λ (1) (χx0 (t)) + p(t) (1) (χx0 (t)) + q(t) (1) (χx0 (t)) |tt10 ∂x ∂x ∂x    t1 d ∂f ∂g ∂h λ − ηtxx0 (χx0 (t)) + p(t) (1) (χx0 (t)) + q(t) (1) (χx0 (t)) dt dt ∂x(1) ∂x ∂x t0    t1 dk ∂f ∂g k +... + (−1) ηtxx0 k λ (k) (χx0 (t)) + p(t) (k) (χx0 (t)) dt dt ∂x ∂x t0      t1 p k  d ∂h +(−1)k ηtxx0 k q(t) (k) (χx0 (t)) dt < − λl ρ1l + ρ2 + ρ3  θxx0 2 . dt ∂x t0 l=1 Considering the boundary conditions x(tε ) = xε , x(β) (tε ) = xβε , ε = 0, 1, β = 1, k − 1 dζ ηtx0 x0 = 0, ζ ∈ (see x(tε ) = xε = x0 (tε ), x(β) (tε ) = xβε = x0(β) (tε )), and knowing that dtζ {0, 1, 2, ..., k − 1}, (∀) t ∈ I (see Definition 2.3), the previous inequality becomes    t1 ∂f ∂g ∂h ηtxx0 λ (χx0 (t)) + p(t) (χx0 (t)) + q(t) (χx0 (t)) dt ∂x ∂x ∂x t0   d ∂f ∂g ∂h λ ηtxx0 (χx0 (t)) + p(t) (1) (χx0 (t)) + q(t) (1) (χx0 (t)) dt − dt ∂x(1) ∂x ∂x t0    t1 dk ∂f ∂g k +... + (−1) ηtxx0 k λ (k) (χx0 (t)) + p(t) (k) (χx0 (t)) dt dt ∂x ∂x t0      t1 p  dk ∂h k 1 2 3 +(−1) ηtxx0 k q(t) (k) (χx0 (t)) dt < − λl ρl + ρ + ρ  θxx0 2 . dt ∂x t0 l=1 

t1

By using Theorem 3.1, it results  0 0 if xi > 0, i = 1, n, and x ≥ 0 if i n x ≥ 0, i = 1, n. The set R+ = {x ∈ Rn : x ≥ 0} is said to be the positive orthant and, most of the time, we shall use the open positive orthant Pn = {x ∈ Rn : x > 0}. On the set Pn , we consider the distinct monomials of the form v k = v k (x) = (x1 )α1k · · · (xn )αnk , k = 1, m, where αik are real numbers. If aik are real numbers, then the functions aik v k , with summation upon k, are called signomials. The controlled signomial dynamical systems 15

16

Controlled signomial dynamical systems as constraints in control problems

are defined as follows: x˙ i (t) = aik v k (x(t), u(t)) , 1 α1k

1 γ1k

i = 1, n,

where v (x, u) := (x ) · · · (x ) (u ) · · · (u ) , αik , γβk ∈ R, k = 1, m, i = 1, n, β = 1, r, t ∈ I ⊆ R, and Pr  u = (uβ ), β = 1, r, is a control. In the following, let us consider an optimal control problem based on a simple integral cost functional, constrained by a controlled signomial dynamical system:  t0 max I (u(·)) = X (x(t), u(t)) dt (1.1) k

n αnk

u(·),xt0

r γrk

0

subject to x˙ i (t) = aik v k (x(t), u(t)) , u(t) ∈ U, ∀t ∈ [0, t0 ];

i = 1, n, k = 1, m

x(0) = x0 , x(t0 ) = xt0 .

(1.2) (1.3)

In the aforementioned optimal control problem we used the following terminology and notations: t ∈ [0, t0 ] is parameter of evolution, or the time; [0, t0 ] ⊂ R+ is the time interval ; x : [0, t0 ] → Pn , x(t) = (xi (t)), i = 1, n, is a C 2 -class function, called state vector ; u : [0, t0 ] → Pr , u(t) = (uβ (t)), β = 1, r, is a continuous control vector ; U is the set of all admissible controls; the running cost X (x(t), u(t)) is a C 1 -class function, called autonomous Lagrangian. Through this chapter, the summation over the repeated indices is assumed. Further, we introduce the Lagrange multiplier p(t) = (pi (t)), also called co-state variable (vector), and a new Lagrange function

L (x(t), u(t), p(t)) = X (x(t), u(t)) + pi (t) aik v k (x(t), u(t)) − x˙ i (t) . In this way, we change the initial optimal control problem into a free optimisation problem  t0 max L (x(t), u(t), p(t)) dt u(·),xt0

0

subject to u(t) ∈ U, p(t) ∈ P, ∀t ∈ [0, t0 ] x(0) = x0 , x(t0 ) = xt0 , where P is the set of co-state variables, which will be defined later. The control Hamiltonian H (x(t), u(t), p(t)) = X (x(t), u(t)) + aik pi (t)v k (x(t), u(t)) , or, equivalently, H = L + pi x˙ i

(modified Legendrian duality)

permits us to rewrite the previous optimal control problem as follows  t0

max H (x(t), u(t), p(t)) − pi (t)x˙ i (t) dt u(·),xt0

0

Chapter 2

17 subject to u(t) ∈ U, p(t) ∈ P, ∀t ∈ [0, t0 ] x(0) = x0 , x(t0 ) = xt0 .

2.1.1

Variational and adjoint signomial differential systems

Let us suppose that (1.2) is satisfied. Fix the control u(t) and a corresponding solution x(t) of (1.2). Let x(t, ε) be a differentiable variation of the state variable x(t), fulfilling x˙ i (t, ε) = aik v k (x(t, ε), u(t)) x(t, 0) = x(t),

i = 1, n.

Denote by y i (t) := xiε (t, 0). Taking the partial derivative with respect to ε, evaluating at ε = 0, we obtain the following system y˙ i (t) = aik vxkj (x(t), u(t)) · y j (t), called variational signomial differential system. The differential system p˙j (t) = −aik pi (t)vxkj (x(t), u(t)) is called the adjoint signomial differential system of the previous variational differential system since the scalar product pi (t) · y i (t) is a first integral of the two systems. Indeed, we have

d pi (t) · y i (t) = 0. dt

2.2

Necessary conditions of optimality

 Let uˆ(t) = uˆβ (t) , β = 1, r, be a continuous control vector defined on the closed interval [0, t0 ], with uˆ(t) ∈ IntU, which is an optimal point for the aforementioned control problem. Consider u(t, ε) = uˆ(t) + εh(t) a variation of the optimal control vector uˆ(t), where h is an arbitrary continuous vector function. We have uˆ(t) ∈ IntU and, since a continuous function on a compact interval [0, t0 ] is bounded, there exists εh > 0 such that u(t, ε) = uˆ(t) + εh(t) ∈ IntU, ∀ |ε| < εh . This ε is a ”small” parameter used in our variational arguments. Define x(t, ε) as the state variable corresponding to the control variable u(t, ε), i.e., x˙ i (t, ε) = aik v k (x(t, ε), u(t, ε)) ,

i = 1, n, ∀t ∈ [0, t0 ]

and x(0, ε) = x0 . As well, consider (for |ε| < εh ) the function (integral with parameter)  t0 X (x(t, ε), u(t, ε)) dt. I(ε) := 0

18

Controlled signomial dynamical systems as constraints in control problems

Since uˆ(t) is an optimal control variable we get I(0) ≥ I(ε), ∀ |ε| < εh . Also, for any continuous vector function p(t) = (pi )(t) : [0, t0 ] → Rn , we have  t0

pi (t) aik v k (x(t, ε), u(t, ε)) − x˙ i (t, ε) dt = 0. 0

The variations involve L (x(t, ε), u(t, ε), p(t)) = X (x(t, ε), u(t, ε))

+pi (t) aik v k (x(t, ε), u(t, ε)) − x˙ i (t, ε) and the associated function (integral with parameter)  t0 L (x(t, ε), u(t, ε), p(t)) dt. I(ε) = 0

Now, assume that the co-state variable p(t) = (pi (t)) is of C 1 -class. The control Hamiltonian with variations H (x(t, ε), u(t, ε), p(t)) = X (x(t, ε), u(t, ε)) + aik pi (t)v k (x(t, ε), u(t, ε)) changes the above integral with parameter as follows  t0

H (x(t, ε), u(t, ε), p(t)) − pi (t)x˙ i (t, ε) dt. I(ε) = 0

Differentiating with respect to ε, evaluating at ε = 0, and using the formula of integration by parts, it follows  t0  [Hxj (x(t), uˆ(t), p(t)) + p˙j (t)] · xjε (t, 0)dt I (0) = 0



t0

+ 0

 Huβ (x(t), uˆ(t), p(t)) · hβ (t)dt − pi (t) · xiε (t, 0) |t00 ,

where x(t) is the state variable corresponding to the optimal control  β variable uˆ(t). We  must have I (0) = 0, for any continuous vector function h(t) = h (t) , β = 1, r. Also, the functions xiε (t, 0) solve the following Cauchy problem ∇t xiε (t, 0) = aik vxk (x(t, 0), u(t)) · xε (t, 0) + aik vuk (x(t, 0), u(t)) · h(t) t ∈ [0, t0 ],

xε (0, 0) = 0.

Consequently, we obtain ∂H (x(t), uˆ(t), p(t)) = 0, ∂uβ

∀t ∈ [0, t0 ].

(2.1)

Chapter 2

19

Using the adjoint differential system introduced in Sect. 1.1, we define P as the set of solutions for the following problem p˙j (t) = −

∂H (x(t), uˆ(t), p(t)) , ∂xj

pj (t0 ) = 0, ∀t ∈ [0, t0 ].

(2.2)

Moreover, we get x˙ j (t) =

∂H (x(t), uˆ(t), p(t)) , ∂pj

x(0) = x0 , ∀t ∈ [0, t0 ].

(2.3)

Remark 2.1 The algebraic system (2.1) describes the critical points of the control Hamiltonian H with respect to the control vector u = (uβ ). Now, taking into account the previous computations, we are able to formulate the main result of this chapter. Theorem 2.1 (Simplified maximum principle) Consider that the problem of maximizing the functional (1.1), subject to the signomial constraints (1.2) and to the conditions (1.3), with X, v k of C 1 -class, ∈ IntU determining the opti an interior solution uˆ(t)  i has 1 mal state variable x(t) = x (t) . Then there exists the C -class co-state variable p = (pi ), defined on the closed interval [0, t0 ], such that the relations (1.2), (2.1), (2.2) and (2.3) are satisfied. Further, by using the new Lagrange function L and the above mentioned theorem, the following result is obvious. Corollary 2.1 If the problem of maximizing the functional (1.1), subject to the signomial constraints (1.2) and to the conditions (1.3), with X, v k of C 1 -class,  has an interior solution uˆ(t) ∈ IntU determining the optimal state variable x(t) = xi (t) , then there exists the C 1 -class co-state variable p = (pi ), defined on the closed interval [0, t0 ], such that x˙ i (t) = aik v k (x(t), u(t)) , i = 1, n, k = 1, m and the following Euler-Lagrange ODEs associated with the Lagrangian L ∂L d ∂L − = 0, β = 1, r ∂uβ dt ∂ u˙ β ∂L d ∂L ∂L d ∂L − = 0, − = 0, i = 1, n ∂xi dt ∂ x˙ i ∂pi dt ∂ p˙i are fulfilled.

2.3

Conclusions and further developments

In this chapter, by using the concepts of the variational differential system, adjoint differential system and modified Legendrian duality, we have provided a simplified maximum principle associated with a signomial constrained optimal control problem. An immediate perspective of the present chapter is the formulation of Euler-Lagrange and Hamilton ODEs, with important applications in Optimisation Theory and Mechanics.

Chapter 3 Sufficient conditions of efficiency for a class of vector fractional variational problems This chapter presents a study on sufficient efficiency conditions for a class of multidimensional vector ratio optimisation problems, identified by (M F P ), of minimizing a vector of path-independent curvilinear integral functional quotients subject to PDE and/or PDI constraints involving higher-order partial derivatives. Under generalised (ρ, b)-quasiinvexity assumptions, sufficient conditions of efficiency are provided for a feasible solution in (M F P ).

3.1

Introduction and problem formulation

It is very well known that the concept of convexity does no longer suffice for many mathematical models coming from Engineering, Economics, Decision Sciences, and Mechanics. In consequence, Hanson [52] introduced a significant generalisation of convexity, called invexity. A generalisation of invexity is the notion of preinvexity, introduced by Weir and Mond [166]. In this respect, for more contributions and various approaches, the reader is directed, for instance, to [62], [11], [99], [132], [5], [83], [135] and [153]. Moreover, a generalisation of convexity on manifolds has been formulated in [159], [120] and [109]. Other approaches have been well documented in [19], [1] and [151]. In this chapter, the goal is to establish some results associated with the nonlinear optimisation theory on higher-order jet bundles, which extend and further develop a part of the results derived in the following research works: [57], [131], [88], [167], [115], [116], [73], [33], [142], [144], [153], [147], [150] and [146]. More concretely, in this chapter, we are looking for sufficient efficiency conditions in the following multidimensional multiobjective fractional variational problem  1  F (x(·)) F 2 (x(·)) F r (x(·)) (M F P ) min , , ..., r x(·) W 1 (x(·)) W 2 (x(·)) W (x(·)) 21

22

Sufficient conditions of efficiency for vector fractional variational problems subject to x (·) ∈ F (Ωt0 ,t1 ),

where the mathematical tools used are given briefly below (for more details, see Treant¸˘a [146]): • the path-independent curvilinear integral functionals    l F (x(·)) := fβl χxα1 ...αs−1 (t) dtβ , l = 1, r, β = 1, m, Γt0 ,t1

 l

W (x(·)) := Γt0 ,t1

  wβl χxα1 ...αs−1 (t) dtβ > 0,

l = 1, r, β = 1, m,

generated by the (higher-order) closed Lagrange 1-form densities of C ∞ -class  fβ = fβl : J s−1 (T, M ) → Rr , l = 1, r, β = 1, m,  wβ = wβl : J s−1 (T, M ) → Rr ,

l = 1, r, β = 1, m;

• the notations  χxα1 ...αs−1 (t) := t, x(t), xα1 (t), ..., xα1 α2 ...αs−1 (t) ,

t ∈ Ωt0 ,t1 ,

∂x ∂ s−1 x (t), ..., x (t) := (t), and α α ...α 1 2 s−1 ∂tα1 ∂tα1 ∂tα2 ...∂tαs−1  αj ∈ {1, 2, ..., m}, j = 1, s − 1, x = (x1 , ..., xn ) = xi , i = 1, n;

with xα1 (t) :=

• t = (tβ ), β = 1, m, and x = (xi ), i = 1, n, are the local coordinates on the Riemannian manifolds (T, h) and (M, g), respectively; in addition, M is a complete manifold; • Γt0 ,t1 represents a piecewise C s−1 -class curve joining the diagonally opposite points  t0 = t10 , ..., tm and t1 = t11 , ..., tm of the hyper-parallelepiped Ωt0 ,t1 ⊂ Rm ; 0 1 • throughout this chapter, there are used the customary relations between two vectors of the same dimension; • the set F (Ωt0 ,t1 ) of all feasible solutions in (M F P ) is     x ∈ C ∞ (Ωt0 ,t1 , M ) , g χxα1 ...αs−1 (t) ≤ 0, h χxα1 ...αs−1 (t) = 0, t ∈ Ωt0 ,t1 x(tξ ) = xξ , xα1 ...αj (tξ ) = x˜α1 ...αj ξ , αζ ∈ {1, ..., m}, ζ, j = 1, s − 2, ξ ∈ {0, 1}, where

  g χxα1 ...αs−1 (t) ≤ 0,

  h χxα1 ...αs−1 (t) = 0,

t ∈ Ωt0 ,t1 ,

are partial differential inequations (PDIs), respectively partial differential equations (PDEs) of evolution, generated by the C ∞ -class Lagrange matrix densities  g = gab : J s−1 (T, M ) → Rpq , a = 1, q, b = 1, p, p < n,

Chapter 3

23  h = hba : J s−1 (T, M ) → Rde ,

a = 1, e, b = 1, d, d < n,

and C ∞ (Ωt0 ,t1 , M ) := {x : Ωt0 ,t1 → M ; x of C ∞ − class}    is equipped with the distance d x, x0 = d x(·), x0 (·) = sup dg x(t), x0 (t) , where t∈Ω  0 dg x(t), x (t) is geodesic distance in (M, g). Also, in this chapter, we shall use the multi-index notation (see Saunders [125]). Saunders defines a multi-index as an m-tuple I of natural numbers. Its components are denoted I(α), where α is an ordinary index, 1 ≤ α ≤ m. For instance, the multi-index 1α is defined as follows: 1α (α) = 1, 1α (β) = 0 for α = β. Define on components the addition and the substraction of the multi-indexes (although the result of a substraction might not be a multi-index): (I ± J)(α) = I(α) ± J(α). We call the length of a multi-index m m   the following number | I |= I(α), and its factorial is I! = (I(α))!. The number of α=1

α=1

distinct indices represented by {α1 , α2 , ..., αk }, αj ∈ {1, 2, ..., m}, j = 1, k, is n(α1 , α2 , ..., αk ) :=

3.2

| 1α1 + 1α2 + ... + 1αk |! . (1α1 + 1α2 + ... + 1αk )!

Some auxiliary results

To make complete our presentation, we recall and introduce some definitions and preliminary results. Definition 2.1 A feasible solution x0 (·) ∈ F (Ωt0 ,t1 ) of the problem (M F P ) is called efficient solution  0 if there exists no other feasible solution x(·) ∈ F (Ωt0 ,t1 ) such that K (x(·))  K x (·) , where ⎛ ⎜ ⎜ Γt ,t K (x(·)) := ⎜  0 1 ⎝ Γt0 ,t1

fβ1 wβ1





χxα1 ...αs−1 (t) dt 

 β

 χxα1 ...αs−1 (t) dt

, ..., 

Γt0 ,t1

β Γt0 ,t1

fβr





⎞ β

χxα1 ...αs−1 (t) dt ⎟ ⎟ ⎟.   r β⎠ wβ χxα1 ...αs−1 (t) dt

In Treant¸a˘ [146], the following result is proved: if x0 (·) ∈ F (Ωt0 ,t1 ) is [normal] efficient solution of the problem (M F P ), then there exist the multipliers λ ∈ Rr , μ and ν such that the following conditions are fulfilled: r  c=1

 λc

  ∂wβc  ∂fβc  c χx0α1 ...αs−1 (t) − R0 χx0α1 ...αs−1 (t) ∂x ∂x

+μβ (t)

  ∂g  ∂h  χx0α1 ...αs−1 (t) + νβ (t) χx0α1 ...αs−1 (t) ∂x ∂x

(2.1)

24

Sufficient conditions of efficiency for vector fractional variational problems

r c      ∂fβc  ∂w β − Dα 1 λc χx0α1 ...αs−1 (t) − R0c χx0α1 ...αs−1 (t) ∂x ∂x α α 1 1 c=1    ∂g  ∂h  χx0α1 ...αs−1 (t) + νβ (t) χx0α1 ...αs−1 (t) − Dα1 μβ (t) ∂xα1 ∂xα1

r c    ∂f 1 β Dαs−1 λc χx0α1 ...αs−1 (t) +... + (−1)s−1 1 ...αs−1 n(α1 , ..., αs−1 ) ∂x α ...α 1 s−1 c=1

r c    ∂w 1 β Dαs−1 +(−1)s λc R0c χx0α1 ...αs−1 (t) 1 ...αs−1 n(α1 , ..., αs−1 ) ∂x α ...α 1 s−1 c=1    1 ∂g s−1 s−1 +(−1) μβ (t) D χx0α1 ...αs−1 (t) n(α1 , ..., αs−1 ) α1 ...αs−1 ∂xα1 ...αs−1    1 ∂h s−1 s−1 +(−1) νβ (t) D =0 χx0α1 ...αs−1 (t) n(α1 , ..., αs−1 ) α1 ...αs−1 ∂xα1 ...αs−1 (higher − order Euler − Lagrange P DEs), β = 1, m   μβ (t)g χx0α1 ...αs−1 (t) = 0, μβ (t) ≥ 0, t ∈ Ωt0 ,t1 , β = 1, m λ ≥ 0,

et λ = 1,

et := (1, 1, ..., 1) ∈ Rr .

Further, we shall introduce a generalised (ρ, b)-quasiinvexity associated with the aforementioned optimisation problem involving path-independent curvilinear integral functionals. Let ρ be a real number and b : [C ∞ (Ωt0 ,t1 , M )]2s → [0, ∞) a functional. In the following, we consider the notations:  b x(·), xα1 (·), . . . , xα1 ...αs−1 (·), x0 (·), x0α1 (·), . . . , x0α1 ...αs−1 (·) := bxx0  η t, x(t), xα1 (t), . . . , xα1 ...αs−1 (t), x0 (t), x0α1 (t), . . . , x0α1 ...αs−1 (t) := ηtxx0 ,

t ∈ Ωt0 ,t1 .

Also, let a = (aβ ) : J s−1 (T, M ) → Rm be a closed Lagrange 1-form that determines the following path-independent curvilinear integral functional    A(x(t)) = aβ χxα1 ...αs−1 (t) dtβ . Γt0 ,t1

Definition 2.2 The functional A(x) is [strictly] (ρ, b)-quasiinvex at x0 if there exist the vector functions η = (η1 , ..., ηn ), with the property ηtx0 x0 = 0, Dα1 ηtx0 x0 = 0, · · · , Dαs−2 η 0 0=0 1 ...αs−2 tx x αζ ∈ {1, ..., m},

ζ = 1, s − 2,

t ∈ Ωt0 ,t1 ,

Chapter 3

25

and θ : [C ∞ (Ωt0 ,t1 , M )]2s → Rn such that, for any x [x = x0 ], we have the following implication:

A(x) ≤ A(x0 )     ∂aβ  ∂aβ  ηtxx0 χx0α1 ...αs−1 (t) + (Dα1 ηtxx0 ) χx0α1 ...αs−1 (t) dtβ =⇒ [bxx0 ∂x ∂x α Γt0 ,t1 1      1 ∂aβ s−1 Dα1 ...αs−1 ηtxx0 +... + bxx0 χx0α1 ...αs−1 (t) dtβ n (α , ..., α ) ∂x 1 s−1 α ...α Γt0 ,t1 1 s−1 [, we say that  0 with H is strictly (ρ, b)-invex at x , u0 with respect  0to 0η, ξ and d; (i”) in the above inequality, with  (x, u) = x , u , if we replace ≥ with , we say that H is strong (ρ, b)-invex at x0 , u0 with respect to η, ξ and d; (ii) If there exist  η : Ω × (M × U )2 → Rn , η = η t, x(t), u(t), x0 (t), u0 (t) of C 1 -class with η|∂Ω = 0, and ξ : Ω × (M × U )2 → Rk ,

 ξ = ξ t, x(t), u(t), x0 (t), u0 (t)

of C 0 -class with ξ|∂Ω = 0, such that for any (x, u) ∈ X × U:  H (x, u) ≤ H x0 , u0 =⇒

Chapter 5 

0

0

b x, u, x , u



59



Ω

 

hx t, x0 (t), x0α (t), u0 (t) η + hxα t, x0 (t), x0α (t), u0 (t) Dη dv 

0

0

+b x, u, x , u



 Ω

 hu t, x0 (t), x0α (t), u0 (t) ξdv

 ≤ −ρb x, u, x0 , u0 d2 (x, u), (x0 , u0 ) ,  then H is called (ρ, b)-quasiinvex at x0 , u0 withrespect to η, ξ and d; (ii’) if, in the same hypotheses, with (x, u) = x0 , u0 , we have  H (x, u) ≤ H x0 , u0 =⇒ 



0

0

b x, u, x , u





Ω

 

hx t, x0 (t), x0α (t), u0 (t) η + hxα t, x0 (t), x0α (t), u0 (t) Dη dv 

0

0

+b x, u, x , u



 Ω



 hu t, x0 (t), x0α (t), u0 (t) ξdv

 < −ρb x, u, x0 , u0 d2 (x, u), (x0 , u0 ) ,  then H is called strictly (ρ, b)-quasiinvex at x0 , u0 with respect to η, ξ and d; (iii) If there exist  η : Ω × (M × U )2 → Rn , η = η t, x(t), u(t), x0 (t), u0 (t) of C 1 -class with η|∂Ω = 0, and ξ : Ω × (M × U )2 → Rk ,

 ξ = ξ t, x(t), u(t), x0 (t), u0 (t)

of C 0 -class with ξ|∂Ω = 0, such that for any (x, u) ∈ X × U:  H (x, u) = H x0 , u0 =⇒ 

0

0

b x, u, x , u





Ω

 

hx t, x0 (t), x0α (t), u0 (t) η + hxα t, x0 (t), x0α (t), u0 (t) Dη dv 

0

0

+b x, u, x , u



 Ω

 hu t, x0 (t), x0α (t), u0 (t) ξdv

 = −ρb x, u, x0 , u0 d2 (x, u), (x0 , u0 ) ,  then H is called monotonic (ρ, b)-quasiinvex at x0 , u0 with respect to η, ξ and d. Examples 3.1 Consider   x : [0, 1] → M, x(t) = x1 (t), x2 (t) , u : [0, 1] → U, u(t) = u1 (t), u2 (t) 

piecewise smooth, respectively piecewise continuous, functions defined on the real interval [0, 1]. Let p = 1 and h : [0, 1] × M × U → R a continuously differentiable function.

60

Conditions of efficiency in vector fractional variational control problems 1. The following functional



1

H (x, u) =

h (t, x(t), u(t)) dt 0

 is, as it can be verified, (ρ, 1)-quasiinvex, for ρ ≤ 0 and any distance function d, at x0 , u0 with respect to   η t, x(t), u(t), x0 (t), u0 (t) = η1 (t, x(t), u(t), x0 (t), u0 (t)), η2 (t, x(t), u(t), x0 (t), u0 (t))   ∂h  0   0 0 ∂h  0 0 0 t, x (t), u (t) , 2 t, x (t), u (t) = H (x, u) − H x , u ∂x1 ∂x and   ξ t, x(t), u(t), x0 (t), u0 (t) = ξ1 (t, x(t), u(t), x0 (t), u0 (t)), ξ2 (t, x(t), u(t), x0 (t), u0 (t))   ∂h  0   0 0 ∂h  0 0 0 t, x (t), u (t) , 2 t, x (t), u (t) . = H (x, u) − H x , u ∂u1 ∂u 2. In the same hypotheses, the functional  1 h (t, x(t), ˙ u(t)) dt H (x, u) = 0

 is, as it can be verified, (ρ, 1)-quasiinvex at x0 , u0 with respect to   η t, x(t), u(t), x0 (t), u0 (t) = η1 (t, x(t), u(t), x0 (t), u0 (t)), η2 (t, x(t), u(t), x0 (t), u0 (t))     0 0 ∂h  0 ∂h  0 0 0 D 1 t, x˙ (t), u (t) , D 2 t, x˙ (t), u (t) = − H (x, u) − H x , u ∂ x˙ ∂ x˙ and   ξ t, x(t), u(t), x0 (t), u0 (t) = ξ1 (t, x(t), u(t), x0 (t), u0 (t)), ξ2 (t, x(t), u(t), x0 (t), u0 (t))   ∂h  0   0 0 ∂h  0 0 0 t, x˙ (t), u (t) , 2 t, x˙ (t), u (t) , = H (x, u) − H x , u ∂u1 ∂u for ρ ≤ 0 and any distance function d, where D is the total derivative operator. The previous two examples can be easily extended to n-dimensional vector valued functions and, by using normal coordinates, to the multi-time case. 3. The ”negative” Boltzmann-Shannon functional  H (x, u) = [x(t) + u(t)] ln [x(t) + u(t)] dv, (x, u) ∈ X × U [0,1]m

 is, as it can be verified, (ρ, 1)-quasiinvex at x0 , u0 ∈ X × U, for ρ ≤ 0 and any distance function d, with respect to   η x(t), u(t), x0 (t), u0 (t) = ξ x(t), u(t), x0 (t), u0 (t)

Chapter 5

61  =

where

5.4

(H (x, u) − H (x0 , u0 )) [1 + ln (x0 (t) + u0 (t))] , 0, ( X = x : [0, 1]m ⊂ Rm → R+ , ( U = u : [0, 1]m ⊂ Rm → R+ ,

t ∈ Int(Ω) t ∈ ∂Ω,

) x(·) of C 0 − class , ) u(·) of C 0 − class .

Sufficient efficiency conditions

In this section, we shall establish sufficient efficiency conditions in multi-time vector and vector ratio control problems governed by multiple integral cost functionals, under (ρ, b)quasiinvexity assumptions. The definition of (ρ, b)-quasiinvexity, introduced in the previous section (see, also, [41] and [87]), helps us to formulate and prove the results included in this section. Further, all the ρ in this chapter are real scalars and the indices in ρ2 , ρ3 , ρ2 do not have the role of powers. Theorem 4.1 (Sufficient efficiency conditions for (V CP )) Let (x0 , u0 ) ∈ D be a feasible solution of the control problem (V CP ) and θ = (θr ), μ(t) and λ(t), with μ(t) = (μβ (t)), λ(t) = (λαi (t)) piecewise smooth functions fulfilling Theorem 2.2. Assume that the following conditions are true: a) each functional Fr (x, u) = fr (t, x(t), u(t)) dv, r ∈ {1, . . . , p}, is (ρ1r , b)-quasiinvex Ω

at (x0 , u0 ) with respect to η, ξ and d;   ∂xi α i b) the functional X(x, u) = λi (t) Xα (t, x(t), u(t)) − α (t) dv is monotonic (ρ2 , b)∂t Ω quasiinvex at (x0 , u0 ) with respect to η, ξ and d;  c) the functional Y (x, u) = Ω

μβ (t)Yβ (t, x(t), u(t)) dv is (ρ3 , b)-quasiinvex at (x0 , u0 )

with respect to η, ξ and d; d) one of the functionals given in a), c) is strictly (ρ, b)-quasiinvex at (x0 , u0 ) with respect to η, ξ and d, where ρ = ρ1r or ρ3 ; e) θr ρ1r + ρ2 + ρ3 ≥ 0 (ρ1r , ρ2 , ρ3 ∈ R). Then the point (x0 , u0 ) is an efficient solution for (V CP ). Proof. Let assume that (x0 , u0 ) is not an efficient solution for (V CP ). For r = 1, p, consider the following non-empty set ( ) S = (x, u)|Fr (x, u) ≤ Fr (x0 , u0 ), X(x, u) = X(x0 , u0 ), Y (x, u) ≤ Y (x0 , u0 ) . Using a), for (x, u) ∈ S and r = 1, p, we get Fr (x, u) ≤ Fr (x0 , u0 ) =⇒ 

0

0

b x, u, x , u





Ω

  (fr )x t, x0 (t), u0 (t) η + (fr )u t, x0 (t), u0 (t) ξ dv

62

Conditions of efficiency in vector fractional variational control problems   ≤ −ρ1r b x, u, x0 , u0 d2 (x, u), (x0 , u0 ) ,

or, multiplying by θr ≥ 0 and making summation over r = 1, p, we obtain [see (θr fr )x := ∂(θr fr ) ∂(θr fr ) , (θr fr )u := ] ∂x ∂u  r    0 0 (θ fr )x t, x0 (t), u0 (t) η + (θr fr )u t, x0 (t), u0 (t) ξ dv (4.1) b x, u, x , u Ω

  ≤ −(θr ρ1r )b x, u, x0 , u0 d2 (x, u), (x0 , u0 ) .

For (x, u) ∈ S, the equality X(x, u) = X(x0 , u0 ) holds and, according to b), it follows  α  

0 0 λi (t)(Xαi )x t, x0 (t), u0 (t) η − λα (t)Dα η dv b x, u, x , u (4.2) Ω



0

0

+b x, u, x , u



 Ω

 λαi (t)(Xαi )u t, x0 (t), u0 (t) ξdv

 = −ρ b x, u, x0 , u0 d2 (x, u), (x0 , u0 ) . 

2

Also, the inequality Y (x, u) ≤ Y (x0 , u0 ), (x, u) ∈ S, gives (see c))  β    0 0 μ (t)(Yβ )x t, x0 (t), u0 (t) η + μβ (t)(Yβ )u t, x0 (t), u0 (t) ξ dv b x, u, x , u Ω

(4.3)

  ≤ −ρ3 b x, u, x0 , u0 d2 (x, u), (x0 , u0 ) .

Making the sum (4.1) + (4.2) + (4.3), side by side, of the previous relations and taking into account d), we have     0 0 b x, u, x , u η (θr fr )x t, x0 (t), u0 (t) + λαi (t)(Xαi )x t, x0 (t), u0 (t) dv Ω

0

0





β 

μ (t)(Yβ )x t, x0 (t), u0 (t) η − λα (t)Dα η dv +b x, u, x , u Ω     ξ (θr fr )u t, x0 (t), u0 (t) + λαi (t)(Xαi )u t, x0 (t), u0 (t) dv +b x, u, x0 , u0 Ω    0 0 μβ (t)(Yβ )u t, x0 (t), u0 (t) ξdv +b x, u, x , u 

Ω

  + ρ + ρ )b x, u, x0 , u0 d2 (x, u), (x0 , u0 ) .  The previous inequality implicates b x, u, x0 , u0 > 0 and it can be rewritten as    η (θr fr )x t, x0 (t), u0 (t) + λαi (t)(Xαi )x t, x0 (t), u0 (t) dv
0 (3.18) is not as well satisfied. Consequently, we obtain    T η t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) , ξ t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) = 0 (3.19) for all vector functions   η = ηi t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) , i = 1, n, (3.20) of C 1 -class with η|∂Ω = 0, and   ξ = ξj t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) ,

j = 1, k,

(3.21)

of C 0 -class with ξ|∂Ω = 0. Let us consider ξ t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) = 0, ∀t ∈ Ω, and fix it as argument of T . Thus,   (3.22) T η t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) , 0 = 0, or, equivalently   0    fx t, x (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) + μβ (t) (Yβ )x t, x0 (t), u0 (t) ηdv Ω

− or, 

Ω

(3.23)

 Ω

[λα (t)Dα η] dv = 0,

    fx t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) + μβ (t) (Yβ )x t, x0 (t), u0 (t) ηdv (3.24)

 + Ω

[Dα λα (t)] ηdv = 0,

is fulfilled, for all vector functions   η = ηi t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) ,

i = 1, n,

(3.25)

of C 1 -class with η|∂Ω = 0. Using the generalised Dubois-Raymond’s Lemma (see Alekseev et al. [3]), we have    fx t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) (3.26)  +μβ (t) (Yβ )x t, x0 (t), u0 (t) + Dα λα (t) = 0.

100

On a class of variational control problems  In a similar manner, if we fix η t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) = 0, ∀t ∈ Ω, as argument of T , we obtain   T 0, ξ t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) = 0, (3.27) or, equivalently



Ω

   fu t, x0 (t), u0 (t) + λαi (t) Xαi u t, x0 (t), u0 (t) ξdv 



+ Ω

(3.28)

 μβ (t) (Yβ )u t, x0 (t), u0 (t) ξdv = 0

is fulfilled, for all vector functions   ξ = ξj t, x(t), u(t), x0 (t), u0 (t), λ(t), μ(t) ,

j = 1, k,

(3.29)

of C 0 -class with ξ|∂Ω = 0. It follows (using the generalised Dubois-Raymond’s Lemma)    (3.30) fu t, x0 (t), u0 (t) + λαi (t) Xαi u t, x0 (t), u0 (t)  +μβ (t) (Yβ )u t, x0 (t), u0 (t) = 0. Since x0 , u0 , λ, μ verify the conditions (2.16), (3.26), (3.30), then (x0 , u0 ) ∈ D is a Kuhn-Tucker point, and, by hypothesis,  it is an optimal solution for (M CP ), which stands in contradiction to F (x, u) − F x0 , u0 < 0. Therefore, the multidimensional scalar control problem (M CP ) is KT-pseudoinvex and the proof is complete.  Now, by using Theorems 3.1 and 3.2, we derive the following result: Theorem 3.3 The multidimensional scalar control problem (M CP ) is KT-pseudoinvex if and only if all Kuhn-Tucker points are optimal solutions in (M CP ). In this way, the KT-pseudoinvexity of the multidimensional scalar control problem (M CP ) is necessary and sufficient such that its Kuhn-Tucker points are optimal solutions in (M CP ). Theorem 3.3 extends a characterization result of optimal solutions in (singletime) variational and control problems established by Arana-Jim´enez et al. [11, 10] and, as well, generalizes previous theorems formulated by Martin [77] and Osuna-G´omez et al. [105] for (single-time) mathematical programming problems. Let us remark that if constraints (2.8) and (2.9) are removed from (M CP ), the definition of KT-pseudoinvexity for (M CP ) reduces to the definition of pseudoinvexity for F , and, by applying Theorem 3.3, we get the following characterization result: Corollary 3.1 F is pseudoinvex if and only if all critical points of F are global minimums. Now, using Corollary 3.1 and the characterization result given by Mond and Smart [91] for the class of invex functions, we obtain: Corollary 3.2 F is invex if and only if F is pseudoinvex.

Chapter 8

8.4

101

Illustrative application

To illustrate the nature of the results introduced in this chapter, we formulate an example of a KT-pseudoinvex multidimensional control problem and, moreover, we prove that the invexity of the functions involved in this problem is not verified (see Mond and Smart [91]). In this regard, denote by Ωt0 ,t1 a rectangle (in cases, a square) fixed by  1particular 1 2 2 2 the diagonal opposite points t0 = t0 , t0 and t1 = t1 , t1 in R . Consider the following multidimensional scalar control problem:  min −x(t)dt1 dt2 (4.1) (x,u)

Ω0,20

subject to ∂x (t) = cos uα (t), α = 1, 2 ∂tα π π uα (t) − ≤ 0, −uα (t) − ≤ 0, α = 1, 2 2 2 x(0) = x(0, 0) = 0, x(20) = x(20, 20) = 10.

(4.2) (4.3) (4.4)

This problem means to find the optimal control u(t) = (u1 (t), u2 (t)) [that determines the state variable x(t)] to bring the (PDEs) dynamical system (4.2) from the origin x(0, 0) = 0, at two-time (t1 , t2 ) = (0, 0), to the terminal point x(20, 20) = 10, at two-time (t1 , t2 ) = (20, 20), such as to minimize the objective functional. Here, we have: f : Ω0,20 × R × R2 → R,

Xα : Ω0,20 × R × R2 → R,

α = 1, 2

(4.5)

Y : Ω0,20 × R × R2 → R4 ,

(4.6)

f (t, x(t), u(t)) = −x(t)

(4.7)

with X1 (t, x(t), u(t)) = cos u1 (t), X2 (t, x(t), u(t)) = cos u2 (t) (4.8)   π π π π , (4.9) Y (t, x(t), u(t)) = u1 (t) − , −u1 (t) − , u2 (t) − , −u2 (t) − 2 2 2 2 continuously differentiable functions. Let D be the set of all feasible solutions (domain) for the previous scalar control problem. The complete integrability condition associated to Xα , α = 1, 2, imposes ∂u1 ∂u2 (t) sin u1 (t) = 1 (t) sin u2 (t). 2 ∂t ∂t

(4.10)

If (x, u) ∈ D is a Kuhn-Tucker point in previous control problem, then there exist  the 4 1 2 3 4 the multipliers μ : Ω → R , μ(t) = μ (t), μ (t), μ (t), μ (t) ≥ 0, and λ : Ω0,20 → 0,20  1 2 2 R , λ(t) = λ (t), λ (t) , such that: −1 +

∂λ2 ∂λ1 (t) + (t) = 0 ∂t1 ∂t2

(4.11)

102

On a class of variational control problems

(4.12) −λ1 (t) sin u1 (t) + μ1 (t) − μ2 (t) = 0, −λ2 (t) sin u2 (t) + μ3 (t) − μ4 (t) = 0 π π (4.13) μ1 (t)[u1 (t) − ] = 0, −μ2 (t)[u1 (t) + ] = 0 2 2 π π μ3 (t)[u2 (t) − ] = 0, −μ4 (t)[u2 (t) + ] = 0. (4.14) 2 2 Taking into account that μ,(with μ ≥ 0, and λ are piecewise ) smooth functions, by 1 2 2 1 2 direct computation, for M := (t , t ) ∈ R+ | t + t − 10 ≤ 0 , it is obtained that the Kuhn-Tucker points are (¯ x, u¯1 ), (¯ x, u¯2 ), (¯ x, u¯3 ) and (¯ x, u¯4 ), where  1 t∈M t + t2 , 1 2 (4.15) x¯(t) = x¯(t , t ) = 10, t ∈ [0, 20]2 \ M  and u¯j (t) = u¯1j (t), u¯2j (t) , j = 1, 4, with 

=

u¯21 (t)

=

0, π , 2

t∈M , t ∈ [0, 20]2 \ M



t∈M t ∈ [0, 20]2 \ M (4.16)   0, t∈M 0, t∈M , u¯14 (t) = −¯ u23 (t) = u24 (t) = u¯13 (t) = −¯ π π 2 , t ∈ [0, 20] \ M t ∈ [0, 20]2 \ M. −2, 2 (4.17) Moreover, we have that the points (¯ x, u¯1 ), (¯ x, u¯2 ), (¯ x, u¯3 ), (¯ x, u¯4 ) are optimal solutions of the previous scalar control problem. Thus, we get that all Kuhn-Tucker points are optimal solutions. Consequently, by Theorem 3.2, the foregoing scalar control problem is a KT-pseudoinvex control problem.    ∂xi α i However, the functional λi (t) Xα (t, x(t), u(t)) − α (t) dv is not invex (for ex∂t Ω ample, we take λ(t) = 1), and, consequently, it is not verified that       ∂xi α i β f (t, x, u) dv, λi Xα (t, x, u) − α dv, μ Yβ (t, x, u) dv (4.18) ∂t Ω Ω Ω u¯11 (t)

u¯12 (t)

=

u¯22 (t)

=

0,

− π2 ,

is invex, for any μ ≥ 0 and λ (Mond and Smart [91] requires it in order for a Kuhn-Tucker critical point is an optimal solution).

8.5

Conclusions

In this chapter, we have introduced a condition of invexity that is characterized such that all Kuhn-Tucker points are optimal solutions in (M CP ). More precisely, we have established that KT-pseudoinvexity condition is necessary and sufficient in order for all Kuhn-Tucker points are optimal solutions in (M CP ). As well, as a consequence, we have proved that the class of invex functions or functionals is equivalent to the class of pseudoinvex functions or functionals. Also, the theoretical results have been illustrated with an application.

Chapter 9 Weak sharpness of the solution set associated with an integral variational inequality In this chapter, under some assumptions and using a dual gap-type functional, weak sharp solutions are investigated for a multidimensional variational inequality governed by convex multiple integral functional. Moreover, a relation between the minimum principle sufficiency property and weak sharpness of a solution set for the considered variational-type inequality is established. In order to give a better insight into the main results, a numerical application is formulated, as well.

9.1

Introduction

Nowadays, nonlinear and variational analysis, with many applications in optimisation and control theory, have become an important mathematical tool. Starting with the research works of Burke and Ferris [29], Patriksson [106] and following Marcotte and Zhu [76], the variational-type inequalities have been strongly studied and investigated by using the notion of a weak sharp solution. In this regard, Hu and Song [55] introduced the notion of a weak sharp set of solutions for a variational-type inequality problem in a reflexive, strictly convex and smooth Banach space. Alshahrani et al. [4], by using gap functions, studied the minimum and maximum principle sufficiency properties associated with nonsmooth variational inequalities. Also, weakly sharp solutions of a variational inequality have been described in terms of its primal gap function by Liu and Wu [72]. As well, under some assumptions, a necessary optimality condition for the local weak sharp efficient solution of a constrained multiobjective optimisation problem was proved by Zhu [175]. In this chapter, motivated and inspired by the ongoing research in this area and taking into account some techniques developed in Clarke [34], Treant¸˘a [150, 135], Jayswal and Singh [61] and Mititelu and Treant¸a˘ [83], we develop a mathematical framework on continuous-time variational inequalities governed by convex multiple integral functionals 103

104

Weak sharpness of the solution set for an integral variational inequality

and, under some hypotheses and using a dual gap-type functional, provide some characterizations of their solution set. Of course, the multiple integrals in the calculus of variations have been considered and studied so far, but, they have been very little exploited in the scalar and vector variational inequalities context. Also, the new class of integral variational inequalities, with multiple parameter of evolution, requests specific mathematical tools such as variational (functional) derivative associated with a multiple integral functional.

9.2

The problem formulation and preliminaries

In order to introduce our study, we start with the following notations and working hypotheses:  Ω ⊂ Rm is a compact domain in Rm and the point Ω  t = (tα ), α = 1, m, is a multiple parameter of evolution;  denote by dv = dt1 · · · dtm the volume element on Rm ⊃ Ω;  let X be the space of piecewise smooth functions x : Ω ⊂ Rm → Rn , endowed with the inner product  x, y = x(t) · y(t)dv, ∀x, y ∈ X Ω

and the induced norm;  denote by X a nonempty, closed and convex subset of X , defined as ) ( X = x ∈ X : x(t) ∈ E ⊂ Rn , x|∂Ω = ϕ = given ;  throughout this chapter, x, y, xα are the simplified notations for x(t), y(t), xα (t); ∂x ∂f ∂f , fxα := . also, denote xα := α , fx := ∂t ∂x ∂xα In the following, consider the real valued continuously differentiable functions f, g, h : J 1 (Rm , Rn ) → R (see J 1 (Rm , Rn ) as the first-order jet bundle associated to Rm and Rn ) and, for x ∈ X , define the following scalar multiple integral functionals:  f (t, x, xα ) dv, F : X → R, F (x) = Ω

 G : X → R,

G(x) = Ω

g (t, x, xα ) dv,

 H : X → R,

H(x) = Ω

h (t, x, xα ) dv.

Definition 2.1 The multiple integral functional F : X → R, F (x) =

 Ω

f (t, x, xα ) dv, is

called convex on X if for any x, y ∈ X :  [fx (t, y, yα ) (x − y) + fxα (t, y, yα ) Dα (x − y)] dv, F (x) − F (y) ≥ Ω

Chapter 9

105

where Dα denotes the total derivative operator. δF Definition 2.2 The variational (functional) derivative of the scalar functional F : δx  X → R, F (x) = f (t, x, xα ) dv, is defined as Ω

δF = fx (t, x, xα ) − Dα fxα (t, x, xα ) ∈ X δx (see summation over the repeated indices) and satisfies the following relation  δF δF F (x + εψ) − F (x)  , ψ = (t) · ψ(t)dv = lim , ∀ψ ∈ X , ψ|∂Ω = 0. ε→0 δx ε Ω δx Important note. During this chapter, it is assumed that the inner product between the variational derivative of a scalar functional and an element ψ ∈ X is accompanied by the condition ψ|∂Ω = 0. Now, taking into account the mathematical tools previously introduced, we are in a position to formulate the following multidimensional variational-type inequality problem: find y ∈ X such that  (M V IP ) [fx (t, y, yα ) (x − y) + fxα (t, y, yα ) Dα (x − y)] dv ≥ 0, Ω

for any x ∈ X . The dual multidimensional variational-type inequality problem associated to (M V IP ) is formulated as follows: find y ∈ X such that  (DM V IP ) [fx (t, x, xα ) (x − y) + fxα (t, x, xα ) Dα (x − y)] dv ≥ 0, Ω

for any x ∈ X . Denote by X ∗ and X∗ the solution set of (M V IP ) and (DM V IP ), respectively, and assume they are nonempty. Remark 2.1 As it can be easily seen, the aforementioned multidimensional variationaltype inequality problems can be formulated as follows: find y ∈ X such that (M V IP )



δF , x − y ≥ 0, δy

∀x ∈ X ,

respectively, find y ∈ X such that (DM V IP )



δF , x − y ≥ 0, δx

∀x ∈ X .

In order to find X ∗ , we introduce the following gap-type multiple integral functionals.

106

Weak sharpness of the solution set for an integral variational inequality

Definition 2.3 For x ∈ X , the primal gap-type multiple integral functional associated to (M V IP ) is defined as  G(x) = max [fx (t, x, xα ) (x − y) + fxα (t, x, xα ) Dα (x − y)] dv y∈X

Ω

and, similarly, the dual gap-type multiple integral functional associated to (M V IP ) is defined as  H(x) = max [fx (t, y, yα ) (x − y) + fxα (t, y, yα ) Dα (x − y)] dv. y∈X

Ω

From now onwards, for x ∈ X , consider the following notations:    A(x) = z ∈ X : G(x) = [fx (t, x, xα ) (x − z) + fxα (t, x, xα ) Dα (x − z)] dv , Ω

 Z(x) =



 z ∈ X : H(x) =

Ω

[fx (t, z, zα ) (x − z) + fxα (t, z, zα ) Dα (x − z)] dv .

Remark 2.2 By using the above notations, we can observe the following: δF δF (i) G(x) = max , x − y, H(x) = max , x − y; y∈X δy y∈X δx   δF δF δF (ii) A(x) = arg max , x − y = arg max − , y , where arg max , x − y y∈X δx y∈X y∈X δx δx δF denotes the (possibly empty) solution set of max , x − y; y∈X δx δF (iii) Z(x) = arg max , x − y; y∈X δy δF (iv) if A(x) = ∅, then G(x) = sup , x − y; similarly, if Z(x) = ∅, then H(x) = y∈X δx δF sup , x − y. y∈X δy In order to formulate and prove the main results of this chapter, in accordance with Marcotte and Zhu [76], we introduce several relevant concepts. Definition 2.4 The polar set X ◦ associated to X is defined as follows ( ) X ◦ = y ∈ X : y, x ≤ 0, ∀x ∈ X . Definition 2.5 The projection of a point x ∈ X onto the set X is defined as projX x = arg min  x − y  . y∈X

Definition 2.6 The normal cone to X at x ∈ X is defined as ( ) NX (x) = y ∈ X : y, z − x ≤ 0, ∀z ∈ X ,

x ∈ X,

Chapter 9

107 NX (x) = ∅,

x ∈ X

and the tangent cone to X at x ∈ X is TX (x) = [NX (x)]◦ . Remark 2.3 Taking into account the definition of normal cone at x ∈ X , we notice that: δF x∗ ∈ X ∗ ⇐⇒ − ∗ ∈ NX (x∗ ). δx

9.3

Some auxiliary results

In this section, we will formulate and demonstrate several results which will be useful for establishing the main results of this chapter.  f (t, x, xα ) dv is convex on X . Proposition 3.1 Assume the scalar functional F (x) = Ω

Then: (i) for any x1 , x2 ∈ X ∗ , it follows   2 2 1 

fx t, x , xα (x − x2 ) + fxα t, x2 , x2α Dα (x1 − x2 ) dv = 0; Ω

(ii) the inclusion X ∗ ⊂ X∗ is true. Proof. (i) By x1 ∈ X ∗ , we get   1 1 

fx t, x , xα (x − x1 ) + fxα t, x1 , x1α Dα (x − x1 ) dv ≥ 0, Ω

∀x ∈ X .

Since x2 ∈ X ∗ ⊂ X , the previous inequality is rewritten as follows   1 1 2 

fx t, x , xα (x − x1 ) + fxα t, x1 , x1α Dα (x2 − x1 ) dv ≥ 0. Ω



By hypothesis, the scalar functional F (x) = Ω

quently, it results 1

2



F (x ) − F (x ) ≥ and 2

1

F (x ) − F (x ) ≥

Ω



Ω

f (t, x, xα ) dv is convex on X . Conse-

 

fx t, x2 , x2α (x1 − x2 ) + fxα t, x2 , x2α Dα (x1 − x2 ) dv

(3.2)

 

fx t, x1 , x1α (x2 − x1 ) + fxα t, x1 , x1α Dα (x2 − x1 ) dv.

(3.3)

Making the summation (3.2) + (3.3) and using (3.1), we obtain   2 2 1 

fx t, x , xα (x − x2 ) + fxα t, x2 , x2α Dα (x1 − x2 ) dv ≤ 0. Ω

(3.1)

(3.4)

108

Weak sharpness of the solution set for an integral variational inequality Similarly as above, by x2 ∈ X ∗ , we can write   2 2 1 

fx t, x , xα (x − x2 ) + fxα t, x2 , x2α Dα (x1 − x2 ) dv ≥ 0.

(3.5)

Ω

Now, taking into account (3.4) and (3.5), the proof is complete. (ii) By x∗ ∈ X ∗ , it results  [fx (t, x∗ , x∗α ) (x − x∗ ) + fxα (t, x∗ , x∗α ) Dα (x − x∗ )] dv ≥ 0, Ω

∀x ∈ X .

(3.6)

As well, the convexity property on X of the scalar functional F (x) (see the summation (3.2) + (3.3)) implies   1 1 1 

fx t, x , xα (x − x2 ) + fxα t, x1 , x1α Dα (x1 − x2 ) dv (3.7) Ω

 ≥

Ω

 

fx t, x2 , x2α (x1 − x2 ) + fxα t, x2 , x2α Dα (x1 − x2 ) dv,

By using the relations (3.6) and (3.7), we get  [fx (t, x, xα ) (x − x∗ ) + fxα (t, x, xα ) Dα (x − x∗ )] dv ≥ 0, Ω

∀x1 , x2 ∈ X .

∀x ∈ X ,

and the proof is complete.  δF implies X∗ ⊂ X ∗ . Remark 3.1 The continuity property of the variational derivative δx By Proposition 3.1, we can conclude X ∗ = X∗ . As well, the solution set X∗ associated to (DM V IP ) is convex and, consequently, the solution set X ∗ associated to (M V IP ) is a convex set. Proposition 3.2 Assume the scalar functional H(x) is differentiable on X . Then, for any x, v ∈ X , y ∈ Z(x), the following ineguality 

δH δF , v ≥  , v δx δy

is true. Proof. By Definition 2.3, for x ∈ X , we have  H(x) = max [fx (t, y, yα ) (x − y) + fxα (t, y, yα ) Dα (x − y)] dv y∈X

Ω

and, in accordance with Remark 2.2, we obtain H(x) = max y∈X

δF , x − y, δy

∀x ∈ X ,

Chapter 9

109

or, obviously, H(x) = 

δF , x − y, δy

∀y ∈ Z(x).

(3.8)

Moreover, for any y ∈ X , z ∈ X , the inequality H(z) ≥ 

δF , z − y δy

(3.9)

is true and, using (3.8) and (3.9), it follows H(z) − H(x) ≥ 

δF , z − x, δy

∀y ∈ Z(x),

∀x, z ∈ X .

For z = x + λv ∈ X , with λ > 0, the aforementioned inequality becomes H(x + λv) − H(x) ≥ 

δF , λv, δy

∀y ∈ Z(x),

∀x, v ∈ X ,

∀y ∈ Z(x),

∀x, v ∈ X .

or, by dividing both sides with λ > 0, we get δF H(x + λv) − H(x) ≥  , v, λ δy

Further, by taking the limit for λ → 0 of the previous inequality and using Definition 2.2, the proof is complete.  Proposition 3.3 Assume the scalar functional H(x) is differentiable on X ∗ and the scalar functional F (x) is convex on X . Also, for any x∗ ∈ X ∗ , v ∈ X , z ∈ Z(x∗ ), the following implication δH δH δF δF  ∗ , v ≥  , v =⇒ ∗ = δx δz δx δz ∗ ∗ ∗ ∗ is true. Then Z(x ) = X , ∀x ∈ X . Proof. ”⊂” Consider z ∈ Z(x∗ ). In consequence, it follows  ∗ [fx (t, z, zα ) (x∗ − z) + fxα (t, z, zα ) Dα (x∗ − z)] dv, H(x ) = Ω

x∗ ∈ X ∗ .

(3.10)

By hypothesis, the scalar functional F (x) is convex on X and x∗ ∈ X ∗ . According to Proposition 3.1 and Remark 3.1, we get x∗ ∈ X∗ , that is  [fx (t, x, xα ) (x − x∗ ) + fxα (t, x, xα ) Dα (x − x∗ )] dv ≥ 0, (3.11) Ω

for any x ∈ X . By (3.10) and (3.11), it results H(x∗ ) = 0, ∀x∗ ∈ X ∗ , or equivalently,  [fx (t, z, zα ) (x∗ − z) + fxα (t, z, zα ) Dα (x∗ − z)] dv = 0, x∗ ∈ X ∗ . (3.12) Ω

110

Weak sharpness of the solution set for an integral variational inequality

Taking into account (3.12), for any x ∈ X , we obtain  [fx (t, z, zα ) (x − z) + fxα (t, z, zα ) Dα (x − z)] dv 

(3.13)

Ω

= Ω

[fx (t, z, zα ) (x − x∗ ) + fxα (t, z, zα ) Dα (x − x∗ )] dv.

Further, by definition of dual gap-type multiple integral functional H(x) associated to (M V IP ), for any λ ∈ [0, 1] and x ∈ X , we can write as follows H(x∗ + λ(x − x∗ )) − H(x∗ ) λ

 ≥

[fx (t, x∗ , x∗α ) (x − x∗ ) + fxα (t, x∗ , x∗α ) Dα (x − x∗ )] dv.

Ω

By taking the limit for λ → 0 of the previous inequality and using Definition 2.2, we get  δH ∗  ∗,x − x  ≥ [fx (t, x∗ , x∗α ) (x − x∗ ) + fxα (t, x∗ , x∗α ) Dα (x − x∗ )] dv. (3.14) δx Ω According to Proposition 3.2 and using the hypothesis, we conclude (3.14) becomes δF  , x − x∗  ≥ δz or, equivalently,

 



Ω

Ω

 Ω

δH δF . Therefore, = ∗ δx δz

[fx (t, x∗ , x∗α ) (x − x∗ ) + fxα (t, x∗ , x∗α ) Dα (x − x∗ )] dv,

[fx (t, z, zα ) (x − x∗ ) + fxα (t, z, zα ) Dα (x − x∗ )] dv [fx (t, x∗ , x∗α ) (x − x∗ ) + fxα (t, x∗ , x∗α ) Dα (x − x∗ )] dv.

Combining (3.13) and (3.15), it follows  [fx (t, z, zα ) (x − z) + fxα (t, z, zα ) Dα (x − z)] dv Ω

 ≥ ∗

Ω

[fx (t, x∗ , x∗α ) (x − x∗ ) + fxα (t, x∗ , x∗α ) Dα (x − x∗ )] dv.



Since x ∈ X , the previous inequality implies  [fx (t, z, zα ) (x − z) + fxα (t, z, zα ) Dα (x − z)] dv ≥ 0, Ω

involving z ∈ X ∗ and, in consequence, Z(x∗ ) ⊂ X ∗ .

∀x ∈ X ,

(3.15)

Chapter 9

111

”⊃” Consider z, x∗ ∈ X ∗ . By Proposition 3.1, we get  [fx (t, z, zα ) (x∗ − z) + fxα (t, z, zα ) Dα (x∗ − z)] dv = 0. Ω

Since H(x∗ ) = 0, ∀x∗ ∈ X ∗ , it results  ∗ H(x ) = [fx (t, z, zα ) (x∗ − z) + fxα (t, z, zα ) Dα (x∗ − z)] dv, Ω

involving z ∈ Z(x∗ ). The proof is now complete. 

9.4

Weak sharpness and minimum principle sufficiency property

In this section, using the auxiliary results established in the previous section, weak sharp solutions are investigated for the considered multidimensional variational-type inequality governed by convex multiple integral functional. More precisely, in accordance with Ferris and Mangasarian [42], following Marcotte and Zhu [76], the weak sharpness property of the solution set X ∗ for (M V IP ) is studied. In this regard, two characterization results are formulated and proved. Definition 4.1 The solution set X ∗ associated to (M V IP ) is called weakly sharp if   , δF ◦ − ∗ ∈ int [TX (u) ∩ NX ∗ (u)] , ∀x∗ ∈ X ∗ , δx u∈X ∗ or, equivalently, there exists a positive number γ > 0 such that γB ⊂

δF + [TX (x∗ ) ∩ NX ∗ (x∗ )]◦ , δx∗

∀x∗ ∈ X ∗ ,

where int(S) stands for interior of the set S and B denotes the open unit ball in X . Lemma 4.1 There exists a positive number γ > 0 such that γB ⊂ if and only if 

δF + [TX (y) ∩ NX ∗ (y)]◦ , δy

δF , z ≥ γ  z , δy

∀y ∈ X ∗

(4.1)

∀z ∈ TX (y) ∩ NX ∗ (y).

(4.2)

Proof. Relation (4.1) is equivalent with γb −

δF ∈ [TX (y) ∩ NX ∗ (y)]◦ , δy

∀y ∈ X ∗ , ∀b ∈ B,

112

Weak sharpness of the solution set for an integral variational inequality

or

δF , z ≤ 0, ∀y ∈ X ∗ , ∀b ∈ B, ∀z ∈ TX (y) ∩ NX ∗ (y). δy z , z = 0, the previous inequality becomes (4.2). Considering B  b = z Conversely, if relation (4.2) holds, then there exists a positive number γ > 0 such that γb −

γb − ≤ γ  z  −γ  z = 0, that is γb −

δF , z ≤ 0, δy

δF δF , z = γb, z −  , z δy δy ∀y ∈ X ∗ , ∀b ∈ B, ∀z ∈ TX (y) ∩ NX ∗ (y), ∀y ∈ X ∗ , ∀b ∈ B, ∀z ∈ TX (y) ∩ NX ∗ (y),

or, equivalently, γb −

δF ∈ [TX (y) ∩ NX ∗ (y)]◦ , δy

∀y ∈ X ∗ , ∀b ∈ B,

which implies (4.1) and the proof is complete.  Theorem 4.1 Assume the scalar functional H(x) is differentiable on X ∗ and the scalar functional F (x) is convex on X . Also, for any x∗ ∈ X ∗ , v ∈ X , z ∈ Z(x∗ ), the following implication δH δF δF δH  ∗ , v ≥  , v =⇒ ∗ = δx δz δx δz δF is true and ∗ is constant on X ∗ . Then X ∗ is weakly sharp if and only if there exists a δx positive number γ > 0 such that H(x) ≥ γd(x, X ∗ ),

∀x ∈ X ,

where d(x, X ∗ ) = min∗  x − y . y∈X

Proof. ”=⇒” Consider X ∗ is weakly sharp. Consequently, by Definition 4.1, it follows   , δF ◦ ∈ int [TX (u) ∩ NX ∗ (u)] , ∀y ∈ X ∗ , − δy u∈X ∗ or, by Lemma 4.1, there exists a positive number γ > 0 such that (4.1) (or (4.2)) is fulfilled. Further, taking into account the convexity property of the solution set X ∗ associated to (M V IP ) (see Remark 3.1), it results projX ∗ (x) = yˆ ∈ X ∗ ,

∀x ∈ X

Chapter 9

113

y ) ∩ NX ∗ (ˆ y ). By and, following Hiriart-Urruty and Lemar´echal [53], we get x − yˆ ∈ TX (ˆ hypothesis and Lemma 4.1, we get 

δF , x − yˆ ≥ γ  x − yˆ = γd(x, X ∗ ), δ yˆ

or, equivalently,  [fx (t, yˆ, yˆα ) (x − yˆ) + fxα (t, yˆ, yˆα ) Dα (x − yˆ)] dv ≥ γd(x, X ∗ ), Ω

Since

∀x ∈ X .

(4.3)

 H(x) ≥

Ω

[fx (t, yˆ, yˆα ) (x − yˆ) + fxα (t, yˆ, yˆα ) Dα (x − yˆ)] dv,

by using (4.3), we obtain

H(x) ≥ γd(x, X ∗ ),

∀x ∈ X ,

∀x ∈ X .

”⇐=” Consider there exists a positive number γ > 0 such that H(x) ≥ γd(x, X ∗ ),

∀x ∈ X .

Obviously, for any y ∈ X ∗ , the case TX (y) ∩ NX ∗ (y) = {0} involves [TX (y) ∩ NX ∗ (y)]◦ = X and, consequently, γB ⊂

δF + [TX (y) ∩ NX ∗ (y)]◦ , δy

∀y ∈ X ∗

is trivial. In the following, let 0 = u ∈ TX (y) ∩ NX ∗ (y) involving there exists a sequence uk converging to u with y + tk uk ∈ X (for some sequence of positive numbers {tk } decreasing to zero), such that d(y + tk uk , X ∗ ) ≥ d(y + tk uk , Hu ) =

tk u, uk  , u

(4.4)

( ) where Hu = x ∈ X : u, x − y = 0 is a hyperplane passing through y and orthogonal to u. By hypothesis and (4.4), it follows H(y + tk uk ) ≥ γ

tk u, uk  , u

or, equivalently (H(y) = 0, ∀y ∈ X ∗ ), H(y + tk uk ) − H(y) u, uk  . ≥γ tk u

(4.5)

114

Weak sharpness of the solution set for an integral variational inequality

Further, by taking the limit for k → ∞ in (4.5) and using a classical result of functional analysis, we get H(y + λu) − H(y) ≥ γu, (4.6) lim λ→0 λ where λ > 0. By Definition 2.2, the inequality (4.6) can be rewritten as 

δH , u ≥ γu. δy

(4.7)

Now, taking into account the hypothesis and (4.7), for any b ∈ B, it results γb −

δH δF , u = γb, u −  , u ≤ γu − γu = 0 δy δy

and, therefore γB ⊂

δF + [TX (y) ∩ NX ∗ (y)]◦ , δy

∀y ∈ X ∗

and the proof is complete.  Remark 4.1 (i) The weak sharpness property of the solution set associated to the scalar variational problem min H(x) x∈X

is described by the inequality (H(y) = 0, ∀y ∈ X ∗ ) H(x) − H(x∗ ) ≥ γd(x, X ∗ ), formulated in Theorem 4.1. (ii) If the condition

H(x) ≥ γd(x, X ∗ ),

∀x ∈ X , x∗ ∈ X ∗ ,

∀x ∈ X

is fulfilled, the function H provides an error bound for the distance from a feasible point and the solution set X ∗ . The positive constant γ is called the modulus of sharpness for the solution set X ∗ . The second characterization of weak sharpness for X ∗ implies the notion of minimum principle sufficiency property, introduced by Ferris and Mangasarian [42]. Definition 4.2 The multidimensional variational-type inequality problem (M V IP ) satisfies minimum principle sufficiency property if A(x∗ ) = X ∗ , for any x∗ ∈ X ∗ . Lemma 4.2 The following inclusion arg maxx, y ⊂ X ∗ is fulfilled for any y∈X

 x ∈ int

, u∈X ∗

 [TX (u) ∩ NX ∗ (u)]◦

= ∅.

Chapter 9

115

Proof. Let y ∈ X \ X ∗ . By the convexity property of X ∗ (see Remark 3.1), it results projX ∗ (y) = yˆ ∈ X ∗ and, following Hiriart-Urruty and Lemar´echal [53], we get y − yˆ ∈ TX (ˆ y ) ∩ NX ∗ (ˆ y ). There exists a positive number α > 0 such that 

,

for any x ∈ int

x + v, y − yˆ < 0,  [TX (u) ∩ NX ∗ (u)]◦

∀v ∈ αB,

= ∅, or, equivalently,

u∈X ∗

 for any x ∈ int

x, y < x, yˆ − v, y − yˆ, 

,

∀v ∈ αB,

= ∅. For v = α

y − yˆ ∈ αB, the previous  y − yˆ 

x, y < x, yˆ − α  y − yˆ , 

(4.8)

[TX (u) ∩ NX ∗ (u)]◦

u∈X ∗

inequality becomes  for any x ∈ int

,

[TX (u) ∩ NX ∗ (u)]◦

= ∅. By (4.8), we conclude

u∈X ∗

y ∈ arg maxx, y, y∈X

 that is arg maxx, y ⊂ X ∗ , for any x ∈ int y∈X

,

 [TX (u) ∩ NX ∗ (u)]



= ∅. The proof is

u∈X ∗

complete. 

Theorem 4.2 If the solution set X ∗ associated to (M V IP ) is weakly sharp and the scalar functional F (x) is convex on X , then (M V IP ) satisfies minimum principle sufficiency property. Proof. By Definition 4.2, (M V IP ) satisfies minimum principle sufficiency property if A(x∗ ) = X ∗ , for any x∗ ∈ X ∗ . Since X ∗ is weakly sharp, by Definition 4.1 we get   , δF ◦ [TX (u) ∩ NX ∗ (u)] , ∀x∗ ∈ X ∗ − ∗ ∈ int δx u∈X ∗ and, according to Lemma 4.2, it results arg max− y∈X

δF , y ⊂ X ∗ ⇐⇒ A(x∗ ) ⊂ X ∗ . δx∗

(4.9)

Further, let z ∈ X ∗ . For x∗ ∈ X ∗ , in accordance with Proposition 3.1, we have  [fx (t, x∗ , x∗α ) (z − x∗ ) + fxα (t, x∗ , x∗α ) Dα (z − x∗ )] dv = 0. (4.10) Ω

116

Weak sharpness of the solution set for an integral variational inequality

Taking into account (4.10), for any y ∈ X , it follows  [fx (t, x∗ , x∗α ) (z − y) + fxα (t, x∗ , x∗α ) Dα (z − y)] dv 

(4.11)

Ω

= Ω

[fx (t, x∗ , x∗α ) (x∗ − y) + fxα (t, x∗ , x∗α ) Dα (x∗ − y)] dv.

Since x∗ ∈ X ∗ , relation (4.11) provides  [fx (t, x∗ , x∗α ) (z − y) + fxα (t, x∗ , x∗α ) Dα (z − y)] dv ≤ 0, Ω

∀y ∈ X ,

that is z ∈ A(x∗ ) and, consequently, X ∗ ⊂ A(x∗ ).

(4.12)

By using (4.9) and (4.12), the proof is complete.  Theorem 4.3 Assume the scalar functional H(x) is differentiable on X ∗ and the scalar functional F (x) is convex on X . Also, for any x∗ ∈ X ∗ , v ∈ X , z ∈ Z(x∗ ), the following implication δH δF δF δH  ∗ , v ≥  , v =⇒ ∗ = δx δz δx δz δF is true and ∗ is constant on X ∗ . Then (M V IP ) satisfies the minimum principle suffiδx ciency property if and only if X ∗ is weakly sharp. Proof. ”=⇒” Let (M V IP ) satisfies minimum principle sufficiency property. In consequence, A(x∗ ) = X ∗ , for any x∗ ∈ X ∗ . Obviously, for x∗ ∈ X ∗ and x ∈ X , we obtain  [fx (t, x∗ , x∗α ) (x − x∗ ) + fxα (t, x∗ , x∗α ) Dα (x − x∗ )] dv. (4.13) H(x) ≥ Ω

δF , x, x ∈ X , we have A(x∗ ) the solution set δx∗ for min P (x). In accordance with Remark 4.1, we can write In the following, considering P (x) =  x∈X

P (x) − P (x) ≥ γd(x, A(x∗ )), or, 

∀x ∈ X , x ∈ A(x∗ ),

δF , x − x∗  ≥ γd(x, X ∗ ), ∗ δx

∀x ∈ X ,

or, equivalently,  [fx (t, x∗ , x∗α ) (x − x∗ ) + fxα (t, x∗ , x∗α ) Dα (x − x∗ )] dv ≥ γd(x, X ∗ ), Ω

By (4.13), (4.14) and Theorem 4.1, we get X ∗ is weakly sharp. ”⇐=” This implication is a consequence of Theorem 4.2. 

∀x ∈ X . (4.14)

Chapter 9

9.5

117

Illustrative application

In this section, we illustrate the effectiveness of the main results established in the previous section. In this regard, we denote by Ω0,2 a square fixed by the diagonally opposite points 0 = (0, 0) and 2 = (2, 2) in R2 . Also, let ( ) X = x : Ω0,2 ⊂ R2 → [−1, 4] : x = piecewise smooth function , ) ( X = x ∈ X : x(t) ∈ [0, 1] ⊂ R, x(0) = x(0, 0) = 0, x(2) = x(2, 2) = 0 and the real valued continuously differentiable function f : J 1 (R2 , R) → R,

f (t, x, xα ) = x2 + 4x.

Let us consider the following multidimensional variational-type inequality problem: find y ∈ X such that  (BV IP ) [fx (t, y, yα ) (x − y) + fxα (t, y, yα ) Dα (x − y)] dt1 dt2 ≥ 0, Ω0,2

for any x ∈ X . By direct computation, the dual gap-type multiple integral functional  H : X → R, H(x) = h (t, x, xα ) dt1 dt2 Ω0,2

is the following  H(x) = max y∈X

Ω0,2

[fx (t, y, yα ) (x − y) + fxα (t, y, yα ) Dα (x − y)] dt1 dt2



1

= max

2

(2y + 4)(x − y)dt dt =

⎧ ⎪ ⎪ ⎨

 

4xdt1 dt2 , Ω0,2

−1 ≤ x < 2

(x + 2)2 1 2 dt dt , 2 ≤ x ≤ 4. 2 Ω0,2  As well, the scalar functional F : X → R, F (x) = f (t, x, xα ) dt1 dt2 , is convex on X : y∈X

Ω0,2

Ω0,2

 F (x) − F (y) −

⎪ ⎪ ⎩

Ω0,2

[fx (t, y, yα ) (x − y) + fxα (t, y, yα ) Dα (x − y)] dt1 dt2



= Ω0,2

(x − y)2 dt1 dt2 ≥ 0,

∀x, y ∈ X .

As can be easily seen, we get ) ( X ∗ = y : Ω0,2 ⊂ R2 → [0, 1] : y(t) = 0, ∀t ∈ Ω0,2 ;

118

Weak sharpness of the solution set for an integral variational inequality

δF = 2x + 4. δx Obviously, the scalar functional H(x) is differentiable on X ∗ and, for any x ∈ X , there exists a positive number γ > 0 such that  4xdt1 dt2 ≥ γd(x, X ∗ ). H(x) = Z(x∗ ) = X ∗ ,

∀x∗ ∈ X ∗ ;

Ω0,2

Further, following the same steps as in Theorem 4.1, it results that X ∗ is weakly sharp with the positive modulus γ. Also, by Theorems 4.2 and 4.3, it follows that (BV IP ) satisfies minimum principle sufficiency property (i.e., A(x∗ ) = X ∗ , for any x∗ ∈ X ∗ ).

9.6

Conclusions

In this chapter, by using some concepts of convex and functional analysis, we have investigated weak sharp solutions associated with a scalar variational problem by using a variational-type inequality governed by convex multiple integral functional. Moreover, under some hypotheses, an equivalence between the minimum principle sufficiency property and weak sharpness property associated with solution set of the considered scalar variational problem has been established. In order to illustrate the main results, a numerical application was formulated, as well.

Chapter 10 On efficient solutions associated with a class of PDE-constrained vector variational control problems In this chapter, we introduce a V-KT-pseudoinvex multidimensional vector control problem. More precisely, we formulate a new condition on the functionals which are involved in a multidimensional multiobjective (vector) control problem and we prove that a V-KTpseudoinvex multidimensional vector control problem is described so that all Kuhn-Tucker points are efficient solutions. Also, the theoretical results derived in this chapter are illustrated with an application.

10.1

Introduction

In the last decade, the multidimensional variational and control problems have been intensively studied (see, for instance, [160], [144], [145], [147], [150], [60], [83] and [165]), having many important applications in various research areas such as Physics, Mechanics, Mathematical Statistics, Engineering Design, Portofolio Selection, and Game Theory. The finding of solutions in scalar and vector control problems means the study of efficiency conditions and of functionals (functions) that are involved. Quite recently, Treant¸a˘ and Arana-Jim´enez [151], [152] have improved the result formulated by Mond and Smart [91], in accordance with Martin [77], following Arana-Jim´enez et al. [10], by introducing the notion of a KT-pseudoinvexity (associated with a scalar control problem of minimizing a multiple integral cost functional subject to nonlinear equality and inequality constraints involving first-order partial derivatives) as a necessary and sufficient condition in order for all Kuhn-Tucker points are optimal solutions. In addition, as a consequence, they have proved that the class of invex functions or functionals is equivalent to the class of pseudoinvex functions or functionals. In this chapter, motivated and inspired by the on going research in this area, we extend the KT-pseudoinvexity associated with a multidimensional scalar control problem to the 119

120

Efficient solutions for PDE-constrained vector variational control problems

multiobjective case. Thus, we improve the characterization results for efficient solutions in scalar and multiobjective control problems considered in Mond and Hanson [90], Preda [117], and Arana-Jim´enez et al. [10]. In this regard, we introduce a new class of control problems governed by multiple integrals and m-flow type PDE constraints. More exactly, we consider a multidimensional control problem of minimizing a vector of multiple integral cost functionals subject to nonlinear equality and inequality constraints involving firstorder partial derivatives. Also, by introducing a new condition on the involved functionals, we define the V-KT-pseudoinvex multidimensional vector control problems and we prove that such control problems are described so that all Kuhn-Tucker points are efficient solutions. This chapter is based on original results obtained in paper [155].

10.2

Preliminaries and working hypotheses

Let us introduce our study problem by considering the following notations, mathematical tools and working hypotheses: • two Riemannian manifolds: an n-dimensional complete Riemannian manifold (M, g) and a Riemannian manifold (T, h) of dimension m; • consider t = (tα ), α = 1, m, and x = (xi ), i = 1, n, the local coordinates on (T, h) and (M, g), respectively; • let Ω ⊂ T be a compact domain in T and t = (tα ) ∈ Ω ⊂ T a multi-parameter of evolution or√a multi-time; • dv := det h dt1 ∧ dt2 ∧ . . . ∧ dtm represents the volume element on T ⊃ Ω and, in order to simplify the writing for our control problem formulation, it was assumed T = Rm , without loss the generality (hence, det h = 1); • for U ⊂ Rk and P := Ω × M × U , define the following continuously differentiable functions f = (f1 , . . . , fp ) = (fr ) : P → Rp , r = 1, p  X = Xαi : P → Rnm , i = 1, n, α = 1, m Y = (Y1 , . . . , Yq ) = (Yβ ) : P → Rq ,

β = 1, q;

• the following convention for equalities and inequalities will be used throughout the chapter: u = v ⇔ ui = v i , u ≤ v ⇔ ui ≤ v i , u < v ⇔ u i < vi ,

u  v ⇔ u ≤ v, u = v,

i = 1, p,

for any two p-tuples u = (u1 , ..., up ) , v = (v1 , ..., vp ) in Rp ; • denote by X the smooth state functions x : Ω ⊂ T → of piecewise   M endowed  space 0 0 0 with the distance d x, x = d x(·), x (·) = sup dg x(t), x (t) , where dg x(t), x0 (t) is t∈Ω

geodesic distance in (M, g), and by U the space of piecewise continuous control functions u : Ω ⊂ T → U equipped with the uniform norm  · ∞ .

Chapter 10

121

Taking into account the previous mathematical data, we consider the following Multidimensional Vector Control Problem:    f (t, x(t), u(t)) dv (M V CP ) min F (x, u) = (x,u)

Ω

subject to ∂xi (t) = Xαi (t, x(t), u(t)) , i = 1, n, α = 1, m, t ∈ Ω ∂tα Y (t, x(t), u(t)) ≤ 0, t ∈ Ω

(2.2)

x(t)|∂Ω = ϕ(t) = given,

(2.3)

(2.1)



where

f (t, x(t), u(t)) dv

F (x, u) = Ω

 := Ω





f1 (t, x(t), u(t)) dv, . . . ,

Ω

fp (t, x(t), u(t)) dv

:= (F1 (x, u), ..., Fp (x, u)) . Working hypotheses: • the continuously differentiable functions  Xα = Xαi : P → Rn ,

i = 1, n, α = 1, m

satisfy the closeness conditions (complete integrability conditions) Dη Xαi = Dα Xηi ,

α, η = 1, m, α = η, i = 1, n,

where Dη is the total derivative operator; • the set of all feasible solutions (domain) for (M V CP ) is defined by D := {(x, u) |x = x(·) ∈ X , u = u(·) ∈ U satisf ying (2.1), (2.2), (2.3)} .

Let us notice that the partial differential equations (PDEs) of evolution, respectively the inequations, formulated in (2.1) and (2.2), are generated by the following Lagrange densities,  Xα = Xαi : P → Rn , i = 1, n, α = 1, m Y = (Yβ ) : P → Rq ,

β = 1, q

and the control variable u(t) is related to the state variable x(t) via the state PDEs given in (2.1).

122

Efficient solutions for PDE-constrained vector variational control problems

Definition 2.1 (Mititelu and Treant¸˘a [83]) A feasible solution (x0 , u0 ) ∈ D in the multidimensional vector control problem (M V CP ) is called efficient solution if there exists no other feasible solution (x, u) ∈ D such that F (x, u)  F (x0 , u0 ). Further, in accordance with Mititelu and Treant¸˘a [83], following Treant¸a˘ and AranaJim´enez [151, 152], we introduce the notion of Kuhn-Tucker point associated with (M V CP ). Definition 2.2 The point (x0 , u0 ) ∈ D is said to be a Kuhn-Tucker point if there exist a scalar vector θ = (θr ) ∈ Rp and the piecewise smooth functions μ(t) = (μβ (t)) ∈ Rq , λ(t) = (λαi (t)) ∈ Rnm fulfilling the following conditions (see summation over the repeated indices!) θr

∂fr  0 ∂Xαi  0 0 α t, x t, x (t), u0 (t) (t), u (t) + λ (t) i i i ∂x ∂x

∂λαi ∂Yβ  0 0 t, x (t), u (t) + α (t) = 0, i = 1, n ∂xi ∂t ∂fr  ∂X i  θr j t, x0 (t), u0 (t) + λαi (t) jα t, x0 (t), u0 (t) ∂u ∂u ∂Yβ  +μβ (t) j t, x0 (t), u0 (t) = 0, j = 1, k ∂u  μβ (t)Yβ t, x0 (t), u0 (t) = 0 (no summation)

(2.4)

+μβ (t)

θ = 0,

(θ, μ(t))  0,

(2.5)

(2.6) (2.7)

for all t ∈ Ω, except at discontinuities. Definition 2.3 (Mititelu and Treant¸a˘ [83]) A feasible solution (x0 , u0 ) ∈ D in (M V CP ) is said to be normal efficient solution if the conditions (2.4) − (2.7), formulated in Definition 2.2, hold for θ  0 and et θ = 1, where et = (1, . . . , 1) ∈ Rp . The next result establishes the neccesary conditions of efficiency for a feasible solution in (M V CP ). Theorem 2.1 (Mititelu and Treant¸a˘ [83]) Under constraint qualification assumptions, if (x0 , u0 ) ∈ D is an efficient solution in (M V CP ), then (x0 , u0 ) ∈ D is a Kuhn-Tucker point.

10.3

V-KT-pseudoinvex control problems

In this section, in accordance with Mond and Smart [91], Martin [77], following Treant¸˘a and Arana-Jim´enez [151], [152], we contribute an invexity condition that is characterized such that all Kuhn-Tucker points are efficient solutions in (M V CP ). More exactly, we introduce the notion of a V-KT-pseudoinvexity associated with the multidimensional vector control problem (M V CP ).

Chapter 10

123

In the sequel, let us consider the following vector functional  p f (t, x(t), u(t)) dv, F : X × U → R , F (x, u) = Ω

where f : P → Rp is a continuously differentiable function. Definition 3.1 The multidimensional vector control problem (M V CP ) is said to be V-KT-pseudoinvex at (x0 , u0 ) ∈ D if for all θ = (θr ) ∈ Rp , μ : Ω → Rq , which satisfy (2.6), (2.7), and λ : Ω → Rnm piecewise smooth functions, there exist η : Ω × (M × U )2 × Rp × Rnm × Rq → Rn ,  η = η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t)   = ηi t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , i = 1, n, of C 1 -class with η|∂Ω = 0, and ξ : Ω × (M × U )2 × Rp × Rnm × Rq → Rk ,  ξ = ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t)   = ξj t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , j = 1, k, of C 0 -class with ξ|∂Ω = 0, such that for all (x, u) ∈ D the inequality  F (x, u) − F x0 , u0  0 involves



Ω

   θr (fr )x t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) ηdv 



+ Ω





+ Ω



β

0

0

μ (t) (Yβ )x t, x (t), u (t)



 ηdv −

Ω

[λα (t)Dα η] dv

   θr (fr )u t, x0 (t), u0 (t) + λαi (t) Xαi u t, x0 (t), u0 (t) ξdv 



+ Ω

 μβ (t) (Yβ )u t, x0 (t), u0 (t) ξdv < 0.

Definition 3.2 The multidimensional vector control problem (M V CP ) is said to be V-KT-pseudoinvex if it is V-KT-pseudoinvex for all (x0 , u0 ) ∈ D. The next result proves that V-KT-pseudoinvexity of (M V CP ) is a sufficient condition in order for a Kuhn-Tucker point is an efficient solution in (M V CP ). Theorem 3.1 If (M V CP ) is V-KT-pseudoinvex, then all Kuhn-Tucker points are efficient solutions in (M V CP ).

124

Efficient solutions for PDE-constrained vector variational control problems

Proof. Let (x0 , u0 ) ∈ D be a Kuhn-Tucker point. Therefore, by applying Definition 2.2, there exist a scalar vector θ = (θr ) ∈ Rp and the piecewise smooth functions μ : Ω → Rq and λ : Ω → Rnm which satisfy (2.4) − (2.7). By hypothesis, the multidimensional vector control problem (M V CP ) is V-KT-pseudoinvex. Thus, for all (x, u) ∈ D, and for μ (which satisfy (2.6), (2.7)) and λ there exist η : Ω × (M × U )2 × Rp × Rnm × Rq → Rn ,  η = η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t)   = ηi t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , i = 1, n, of C 1 -class with η|∂Ω = 0, and ξ : Ω × (M × U )2 × Rp × Rnm × Rq → Rk ,  ξ = ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t)   = ξj t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , j = 1, k, of C 0 -class with ξ|∂Ω = 0, which satisfy Definition 3.1. By direct computations, we get Dα [ηλα (t)] = λα (t)Dα η + ηDα λα (t),    α α ηDα λ (t)dv = Dα [ηλ (t)] dv − [λα (t)Dα η] dv Ω

Ω

Ω

and, by applying the condition η|∂Ω = 0 and the flow-divergence formula, it follows   α Dα [ηλ (t)] dv = [ηλα (t)] ndσ = 0, Ω

∂Ω

where n = (nα ), α = 1, m, is the normal unit vector to the hypersurface ∂Ω. Hence, we obtain   α ηDα λ (t)dv = − [λα (t)Dα η] dv. Ω

Ω

Taking into account the above calculation, we find  r    θ (fr )x t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) ηdv Ω





+ 



+ Ω

Ω



β

0

0

μ (t) (Yβ )x t, x (t), u (t)



 ηdv −

Ω

[λα (t)Dα η] dv

   θr (fr )u t, x0 (t), u0 (t) + λαi (t) Xαi u t, x0 (t), u0 (t) ξdv 



+ Ω

 μβ (t) (Yβ )u t, x0 (t), u0 (t) ξdv

Chapter 10

125





= Ω

   θr (fr )x t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) ηdv





+ 



+ Ω

Ω



β

0

0

μ (t) (Yβ )x t, x (t), u (t)





ηdv + Ω

[Dα λα (t)] ηdv

   θr (fr )u t, x0 (t), u0 (t) + λαi (t) Xαi u t, x0 (t), u0 (t) ξdv 



+ Ω

 μβ (t) (Yβ )u t, x0 (t), u0 (t) ξdv = 0,

where we used and (2.5). Since (M V CP ) is V-KT-pseudoinvex,0 we0 obtain that  0(2.4) 0 F (x, u) − F x , u  0 is not verified, for any (x, u) ∈ D. Thus, (x , u ) ∈ D is an efficient solution in (M V CP ) and the proof is complete.  The next result provides a characterization of V-KT-pseudoinvex multidimensional vector control problem. Also, it shows that V-KT-pseudoinvexity of (M V CP ) is not only a sufficient condition in order for a Kuhn-Tucker point is an efficient solution in (M V CP ), but it is a necessary condition. Theorem 3.2 If all Kuhn-Tucker points are efficient solutions in (M V CP ), then the multidimensional vector control problem (M V CP ) is V-KT-pseudoinvex. Let (x, u), (x0 , u0 ) ∈ D be two feasible points in (M V CP ), with F (x, u) −  Proof. F x0 , u0  0, θ = (θr ) ∈ Rp , μ (which verifies (2.6), (2.7)) and λ piecewise smooth functions. We look for two vector valued functions η : Ω × (M × U )2 × Rp × Rnm × Rq → Rn ,  η = η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t)   = ηi t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , i = 1, n, of C 1 -class with η|∂Ω = 0, and ξ : Ω × (M × U )2 × Rp × Rnm × Rq → Rk ,  ξ = ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t)   = ξj t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , j = 1, k, of C 0 -class with ξ|∂Ω = 0, such that    T η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) < 0, where    T η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) :=  r    θ (fr )x t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) ηdv Ω

126

Efficient solutions for PDE-constrained vector variational control problems   β  0 0 μ (t) (Yβ )x t, x (t), u (t) ηdv − [λα (t)Dα η] dv + Ω





+ Ω

Ω







 i

θr (fr )u t, x0 (t), u0 (t) + λαi (t) Xα 



+ Ω

u

t, x0 (t), u0 (t)



ξdv

 μβ (t) (Yβ )u t, x0 (t), u0 (t) ξdv.

By reductio ad absurdum, suppose that    T η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) < 0 is not verified for any  η = η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) ,  ξ = ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , satisfying the above conditions. Taking   −η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , −ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) as arguments of T in the previous inequality, we get that    T η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) > 0 is not as well satisfied for any η and ξ. Consequently, we obtain    T η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) = 0 for all vector valued functions   η = ηi t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) ,

i = 1, n,

of C 1 -class with η|∂Ω = 0, and   ξ = ξj t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) ,

j = 1, k,

of C 0 -class with ξ|∂Ω =  0. Let us consider ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) = 0, ∀t ∈ Ω, and fix it as argument of T . Thus,   T η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) , 0 = 0, or, equivalently 

Ω

   θr (fr )x t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) ηdv

Chapter 10





+ Ω

or,





β

0

0

μ (t) (Yβ )x t, x (t), u (t)



127

 ηdv −

Ω

[λα (t)Dα η] dv = 0,



   θr (fr )x t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t) ηdv Ω   β  0 0 μ (t) (Yβ )x t, x (t), u (t) ηdv + [Dα λα (t)] ηdv = 0, + Ω

Ω

is fulfilled, for all vector valued functions   η = ηi t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) ,

i = 1, n,

of C 1 -class with η|∂Ω = 0. Using the generalised Dubois-Raymond’s Lemma (see Alekseev et al. [3]), we have    (3.1) θr (fr )x t, x0 (t), u0 (t) + λαi (t) Xαi x t, x0 (t), u0 (t)  +μβ (t) (Yβ )x t, x0 (t), u0 (t) + Dα λα (t) = 0.  In a similar manner, if we fix η t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) = 0, ∀t ∈ Ω, as argument of T , we obtain   T 0, ξ t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) = 0, or, equivalently 

Ω

   θr (fr )u t, x0 (t), u0 (t) + λαi (t) Xαi u t, x0 (t), u0 (t) ξdv 



+ Ω

 μβ (t) (Yβ )u t, x0 (t), u0 (t) ξdv = 0

is fulfilled, for all vector valued functions   ξ = ξj t, x(t), u(t), x0 (t), u0 (t), θ, λ(t), μ(t) ,

j = 1, k,

of C 0 -class with ξ|∂Ω = 0. It follows (using the generalised Dubois-Raymond’s Lemma)    θr (fr )u t, x0 (t), u0 (t) + λαi (t) Xαi u t, x0 (t), u0 (t) (3.2)  +μβ (t) (Yβ )u t, x0 (t), u0 (t) = 0. Since x0 , u0 , θ, λ, μ verify the conditions (2.6), (2.7), (3.1), (3.2), then (x0 , u0 ) ∈ D is a Kuhn-Tucker point, and, by hypothesis,  0 it 0is an efficient solution for (M V CP ), which stands in contradiction to F (x, u) − F x , u  0. Therefore, the multidimensional vector control problem (M V CP ) is V-KT-pseudoinvex and the proof is complete.  Further, taking into account Theorems 3.1 and 3.2, we derive that V-KT-pseudoinvexity of the multidimensional vector control problem (M V CP ) is a necessary and sufficient condition in order for its Kuhn-Tucker points are efficient solutions in (M V CP ). The next

128

Efficient solutions for PDE-constrained vector variational control problems

theorem extends some characterization results of efficient solutions in scalar and vector control problems established by Arana-Jim´enez et al. [10] and Treant¸˘a and Arana-Jim´enez [151, 152]. Theorem 3.3 The control problem (M V CP ) is V-KT-pseudoinvex if and only if all Kuhn-Tucker points are efficient solutions in (M V CP ). Proof. By applying Theorems 3.1 and 3.2, the proof is complete. 

10.4

An illustrative example

In order to illustrate the effectiveness of our main result (see Theorem 3.3), we present an example of a V-KT-pseudoinvex bidimensional vector control problem. In this regard, we take T = R2 , M = U = R (see Preliminaries) and denote by Ωt0 ,t1 a rectangle 1 2 and = t , t (in particular cases, a square), fixed by the diagonally opposite points t 0 0 0  t1 = t11 , t21 in T = R2 . Consider the following Bidimensional Vector Control Problem:   (u(t) − 5)2 , u2 (t) dt1 dt2 (BV CP ) min (x,u)

Ω0,2

subject to ∂x (t) = 2 − u(t), ∂t1 ∂x (t) = 2 − u(t), ∂t2 16 − x2 (t) ≤ 0, x(0) = x(0, 0) = 4,

 t = t1 , t2 ∈ Ω0,2

(4.1)

 t = t1 , t2 ∈ Ω0,2  t = t1 , t2 ∈ Ω0,2

(4.2) (4.3)

x(2) = x(2, 2) = 10.

(4.4)

The previous bidimensional bi-objective control problem means to find the optimal control u : Ω0,2 → R [that determines the state function x : Ω0,2 → R] to bring the (PDEs) dynamical system (4.1) − (4.2) from the initial point x(0) = x(0, 0) = 4, at two-time (t1 , t2 ) = (0, 0), to the terminal point x(2) = x(2, 2) = 10, at two-time (t1 , t2 ) = (2, 2), such as to minimize the objective vector functional. We assume in our control problem that we have interest only for affine state functions (this assumption is essential in our subsequent considerations). Also, let us notice that Ω0,2 = [0, 2] × [0, 2] = [0, 2]2 . In our previous application, we have: f : Ω0,2 × R × R → R2 ,

Xα : Ω0,2 × R × R → R,

α = 1, 2,

Y : Ω0,2 × R × R → R, with  f (t, x(t), u(t)) = (f1 (t, x(t), u(t)), f2 (t, x(t), u(t))) = (u(t) − 5)2 , u2 (t) ,

Chapter 10

129 X1 (t, x(t), u(t)) = X2 (t, x(t), u(t)) = 2 − u(t), Y (t, x(t), u(t)) = 16 − x2 (t),

continuously differentiable functions. Let D be the set of all feasible solutions (domain) for (BV CP ). The complete integrability condition associated to Xα , α = 1, 2, imposes ∂u ∂u (t) = 2 (t). 1 ∂t ∂t Let (x, u) ∈ D be a feasible solution in (BV CP ). In accordance with Definition 2.2, vector θ = the1 point (x,2 u) ∈ D is said to be a Kuhn-Tucker point if there exist a scalar 2 1 θ ,θ ∈ smooth functions μ = μ(t) ∈ R, λ(t) = (λ (t), λ2 (t)) ∈  1R 2 and the piecewise 2 2 R , t = t , t ∈ Ω0,2 = [0, 2] , fulfilling the following conditions: −2μ(t)x(t) +

∂λ2 ∂λ1 (t) + (t) = 0, ∂t1 ∂t2

(4.5)

2θ1 u(t) − 10θ1 + 2θ2 u(t) − λ1 (t) − λ2 (t) = 0,

(4.6)

μ(t)[16 − x2 (t)] = 0,  θ = θ1 , θ2 = 0, (θ, μ(t))  0,

(4.7) (4.8)

for all t ∈ Ω0,2 , except at discontinuities. Remark 4.1 Let us denote byInt Ω 0,2 the interior of the set Ω0,2 and by V (t0 ) 1 2 of the point t0 = t0 , t0 ∈ Int Ω0,2 . Assume that there exists t0 = a 1neighborhood t0 , t20 ∈ Int Ω0,2 such that μ(t0 ) > 0. Taking into account that μ is a piecewise smooth function (hence, continuous function), it follows that μ(t) > 0, ∀t ∈ V (t0 ). By (4.7), we get x(t) = ±4, t ∈ V (t0 ). Using the continuity property of x, we obtain x(t) = 4 or x(t) = −4, ∀t ∈ V (t0 ). From (4.1) and (4.2), we have that u(t) = 2, ∀t ∈ V (t0 ). By (4.5), we find ∂λ1 ∂λ2 (t) + 2 (t) = ±8μ(t), t ∈ V (t0 ) (∗) ∂t1 ∂t and using (4.6), we get λ1 (t) + λ2 (t) = −6θ1 + 4θ2 ,

t ∈ V (t0 ),

(∗∗)

which implies that λ1 +λ2 is a constant function on V (t0 ). As it can be easily checked, the equations (∗) and (∗∗) (where λ(t) = (λ1 (t), λ2 (t)) and μ(t) > 0) admit more solutions. But, since we could consider that V (t0 ) is even Int Ω0,2 , taking into account that x is piecewise smooth function (therefore, continuous function), we have that the boundary condition x(2) 2) = 10 is not satisfied. Consequently, it is not possible that there  1= 2x(2, exists t0 = t0 , t0 ∈ Int Ω0,2 such that μ(t0 ) > 0. Another way to prove this fact is the following: the equations (∗) and (∗∗) imply that λ(t) = (λ1 (t), λ2 (t)) is not a constant function and this stands in contradiction to the assumption of affine function for x. Therefore, we shall consider μ(t) = 0, ∀t ∈ Ω0,2 .

130

Efficient solutions for PDE-constrained vector variational control problems For μ(t) = 0, ∀t ∈ Ω0,2 , by (4.5) we get

∂λ1 ∂λ2 (t) + (t) = 0, t ∈ Ω0,2 , (4.9) ∂t1 ∂t2  which admits the trivial solution λ(t) = λ1 (t), λ2 (t) = (c1 , c2 ), where c1 , c2 are real constants. Using (4.6), we find λ1 (t) + λ2 (t) = 2θ1 u(t) − 10θ1 + 2θ2 u(t) = c1 + c2 ,

t ∈ Ω0,2 ,

which implies

c1 + c2 + 10θ1 , t ∈ Ω0,2 , 2θ1 + 2θ2 that is, u is a constant function on Ω0,2 . Taking into account (4.1) and (4.2), we have that   c1 + c2 + 10θ1  1 x(t) = 2 − t + t2 + c, c ∈ R, t ∈ Ω0,2 . 1 2 2θ + 2θ u(t) =

Since x(0) = x(0, 0) = 4 and x(2) = x(2, 2) = 10, it follows that c = 4 and c1 + c2 = θ2 − 9θ1 . Consequently, we obtain x¯(t) =

3 1 t + t2 + 4, 2

1 u¯(t) = , 2

t ∈ Ω0,2

 as a Kuhn-Tucker point associated with (BV CP ) since, by considering: θ = θ1 , θ2 = 0  1 and non-negative, λ(t) = λ (t), λ2 (t) = (c1 , c2 ), with c1 , c2 real constants satisfying c1 + c2 = θ2 − 9θ1 , and μ(t) = 0, ∀t ∈ Ω0,2 , the conditions (4.5) − (4.8) are fulfilled. 1 3 1 t + t2 +4 and u¯(t) = , is an efficient Further, let us show that (¯ x, u¯), with x¯(t) = 2 2 solution for (BV CP ). In this sense, let (x, u) ∈ D. We have that  2   2   1 1 2 F (x, u) − F (¯ x, u¯) = − 5 , u2 (t) − (u(t) − 5) − dt1 dt2  0 2 2 Ω0,2 if and only if ⎧  ⎪ ⎪ ⎨

2

1

(u(t) − 5) dt dt ≤ 81 Ω0,2  ⎪ ⎪ u2 (t)dt1 dt2 < 1 ⎩ Ω0,2

⎧  ⎪ ⎪ ⎨

2

or

(u(t) − 5)2 dt1 dt2 < 81 Ω0,2  ⎪ ⎪ u2 (t)dt1 dt2 ≤ 1. ⎩ Ω0,2

∂x ∂x Since (x, u) ∈ D, by (4.1) (or (4.2)) we get u(t) = 2 − 1 (t) (or u(t) = 2 − 2 (t)) and, ∂t ∂t by direct computation, we obtain   2 2    ∂x ∂x 2 1 2 1 2 (u(t) − 5) dt dt = (t) + 3 dt dt (or (t) + 3 dt1 dt2 ) 1 2 ∂t ∂t Ω0,2 Ω0,2 Ω0,2

Chapter 10

131 

 = 36 + Ω0,2

or



 36 + Ω0,2

if and only if



 Ω0,2

or



 Ω0,2

∂x (t) ∂t1

∂x (t) ∂t2

∂x (t) ∂t1 ∂x (t) ∂t2

 ∂x + 6 1 (t) dt1 dt2 ≤ ( 0, such that Br0 (x0 , u0 ) ⊂ X × U . Let β ∈ Q \ Q(x0 ,u0 ) . Consequently, it follows that Gβ (x0 , u0 ) < 0 and, by using the continuity property of the functionals Gβ , there exist rβ ∈ R, rβ > 0, such that Brβ (x0 , u0 ) ⊂ X × U and Gβ (y, w) < 0, Since

∀(y, w) ∈ Brβ (x0 , u0 ).

  lim+  γ (x, u), (x0 , u0 ) (τ ) = γ (x, u), (x0 , u0 ) (0) = 0,

τ →0

(3.5) (3.6)

therefore, for any r ∈ R, r > 0, there exists τ0 ∈ (0, 1) such that   γ (x, u), (x0 , u0 ) (τ ) < min{r, r0 , rβ }, β ∈ Q \ Q(x0 ,u0 ) , τ ∈ [0, τ0 ]. (3.7)  τ0 For (z, μ) = (x0 , u0 )+γ (x, u), (x0 , u0 ) ( ), we get (z, μ) ∈ Brβ (x0 , u0 ) ⊂ X ×U , β ∈ 2 Q \ Q(x0 ,u0 ) , and, by using (3.5), it follows Gβ (z, μ) < 0,

β ∈ Q \ Q(x0 ,u0 ) .

(3.8)

Chapter 12

157

By hypothesis (3.2), we obtain   τ0  Gβ (z, μ) = Gβ (x0 , u0 ) + γ (x, u), (x0 , u0 ) ( ) = 0, 2

β ∈ Q(x0 ,u0 ) .

(3.9)

Taking into account (3.4), (3.8) and (3.9), it follows that (z, μ) ∈ F and W (z, μ) < W (x0 , u0 ), that is, (x0 , u0 ) is not a local optimal solution of (P ). Consequently, we obtain a contradiction and the proof is complete.  Now, we prove that the previous condition is not only sufficient but also necessary such that any local optimal solution associated with the variational control problem (P ) is its global optimal solution. Theorem 3.2 If any local optimal solution associated with the considered constrained variational control problem (P ) is also its global optimal solution, then, for any (x, u), (y, w) ∈  2 F, with W (x, u) − W (y, w) < 0, there exists γ : X × U → S, with S = {s : [0, 1] → X × U | (∃) lim+  s(τ ) = s(0) = 0}, such that (y, w) + γ ((x, u), (y, w)) (τ ) ∈ X × U , τ →0

and, for all τ ∈ (0, 1), it is verified γ ((x, u), (y, w)) (τ )|∂Θ = (0, u) and: W ((y, w) + γ ((x, u), (y, w)) (τ )) < W (y, w), Gβ ((y, w) + γ ((x, u), (y, w)) (τ )) ≤ 0,

β ∈ Q.

(3.10) (3.11)

Proof. Assume that any local optimal solution of (P ) is also global. In accordance with (3.10), consider (x, u), (y, w) ∈ F such that W (x, u) − W (y, w) < 0.

(3.12)

Since X × U is an open set, there exists r1 ∈ R, r1 > 0, such that Br1 (y, w) ⊂ X × U . W (y, w) − W (x, u) > 0, there By using the continuity property of F on X × U , for  := 2 exists r2 ∈ (0, r1 ) such that  W (z, μ) − W (y, w) < ,

∀(z, μ) ∈ Br2 (y, w).

(3.13)

Further, for any r ∈ (0, r2 ), define the following variational control problem associated to (P )  (Pr ) min W (z, μ) (3.14) (z,μ)

subject to

(z, μ) ∈ cl (Br (y, w)) ∩ F.



(3.15)



Denote by Fr the set of optimal solutions associated to (Pr ). Since (cl (Br (y, w)) ∩ F) ⊂ X × U

(3.16)

158

On local and global optimal solutions in variational control problems

 is a compact subset of X˜ × U˜ , it follows that Fr = ∅. Now, let us consider the following two cases. (i) There exists r∗ ∈ (0, r2 ) such that (z ∗ , μ∗ ) ∈ int (cl (Br∗ (y, w)) ∩ F) and (z ∗ , μ∗ ) ∈  Fr∗ . This case involves (z ∗ , μ∗ ) is a local optimal solution of (P ) and, by hypothesis, it is a global optimal solution for (P ). But, in accordance with (3.13) and using the inequality (3.12), we get

W (x, u) < W (y, w) −  = W (y, w) − W (z ∗ , μ∗ ) + W (z ∗ , μ∗ ) − 

(3.17)

<  + W (z ∗ , μ∗ ) −  = W (z ∗ , μ∗ ), which means that (z ∗ , μ∗ ) is not a global optimal solution of (P ) and, therefore, we obtain a contradiction. In consequence, we shall consider the next case.  (ii) For any r∗ ∈ (0, r2 ), if (z ∗ , μ∗ ) ∈ Fr∗ , then (z ∗ , μ∗ ) ∈ (cl (Br∗ (y, w)) ∩ F) \ int (cl (Br∗ (y, w)) ∩ F) ,

(3.18)

0