Stability and Controls Analysis for Delay Systems 9780323997928

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Stability and Controls Analysis for Delay Systems

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Stability and Controls Analysis for Delay Systems

JinRong Wang ˇ Michal Feckan Mengmeng Li

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2023 Elsevier Inc. All rights reserved. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-99792-8 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Glyn Jones Editorial Project Manager: Elaine Desamero Production Project Manager: Fahmida Sultana Cover Designer: Vicky Pearson Esser Typeset by VTeX

Contents

Preface Acknowledgments

1.

Introduction

2.

Delay systems 2.1 Finite time stability 2.2 Controllability 2.3 Iterative learning control

3.

105 119 140

Fractional delay systems 5.1 Finite time stability and controllability for Caputo type 5.2 Finite time stability and controllability for Riemann–Liouville type

6.

59 74 90

Impulsive delay systems 4.1 Asymptotical stability 4.2 Finite time stability 4.3 Controllability

5.

5 28 45

Oscillating delay systems 3.1 Finite time stability 3.2 Controllability 3.3 Iterative learning control

4.

vii ix

155 195

Difference delay systems 6.1 Controllability 6.2 Iterative learning control for fixed trial lengths 6.3 Iterative learning control for varying trial lengths

221 239 252 v

vi Contents

7.

Stochastic delay systems 7.1 Controllability for first order systems 7.2 Controllability for oscillating systems

269 288

References

307

Index

313

Preface

It is well known that stability and control analysis of delay systems has received much attention in the fields of mathematics and automatic control. Various mathematical methods as well as stability and controllability concepts are explored to deal with such problems. In recent decades, there have been few developments in seeking explicit formulas of solutions to delay differential/discrete equations by introducing continuous/discrete delayed exponential matrices. One of the biggest advantages of continuous/discrete delayed exponential matrices is to transfer the classical idea of representing the solution of linear ordinary differential equations to linear delay differential/discrete equations. Stability analysis and mathematical control theory are important areas of research for classical differential delay systems. Various mathematical methods as well as new stability concepts are explored to deal with such problems. However, there is no book that uses delayed exponential matrices to deal with the stability, controllability, and iterative learning control of delay systems such as first order systems, oscillating systems, impulsive systems, fractional systems, difference systems, and stochastic systems. This was the main motivation to write this book. This book is devoted to the study of the stability, controllability, and iterative learning control of delay systems of several kinds, such as first order systems, oscillating systems, impulsive systems, fractional systems, difference systems, and stochastic systems in the fields of physics, biology, population dynamics, ecology, and economics, which have not been presented in other books on conventional fields. The delayed exponential matrix function approach is widely used to derive the representation of the solutions, stability, and controllability. Iterative learning control designs are also established, which can be regarded as a way to find the control function. The broad variety of achieved results with rigorous proofs and many numerical examples makes this book unique. This monograph is useful for researchers and graduate students to study stability and control for delay systems. It may also be used as seminars and advanced graduate courses of pure and applied mathematics and related disciplines. We would like to thank Professors A. Debbouche, J. Diblík, Z.S. Hou, M. Pospíšil, D. O’Regan, X. Ruan, D. Shen, J.J. Trujillo, W. Wei, and Y. Zhou vii

viii Preface

for their support. We also wish to express appreciation to our graduate students C.B. Liang, Z.J. Luo, D.H. Luo, and W.Z. Qiu for their help. Guiyang and Bratislava JinRong Wang Michal Feˇckan Mengmeng Li 2022

Acknowledgments

We acknowledge with gratitude the support of the National Natural Science Foundation of China (12161015), the Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), the Major Research Project of Innovation Group of the Guizhou Education Department ([2018]012), the Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), the Major Project of Guizhou Postgraduate Education and Teaching Teform (YJSJGKT[2021]041), the Slovak Research and Development Agency (Contract No. APVV-18-0308), and the Slovak Grant Agency VEGA (No. 1/0358/20 and No. 2/0127/20).

ix

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Chapter 1

Introduction It is well known that delay differential equations arise naturally in economics, physics, and control problems. It is an interesting task to develop the idea of Duhamel’s principle in classical linear ordinary differential equations (ODEs) to seek the explicit representation of solutions [1] to delay discrete/differential systems. In fact, it is not an easy task to construct a fundamental matrix for linear differential delay systems, even for a simple first order delay system x(t) ˙ = Ax(t) + Bx(t − τ ), t ≥ 0, with initial condition x(t) = ϕ(t), t ∈ [−τ, 0], τ > 0, where A, B are suitable constant matrices. Khusainov and Shuklin in [1] introduced the delayed matrix exponential eτB· : R → Rn [1, Definition 0.3] and derived an explicit formula of solutions to such linear differential delay systems with AB = BA. Diblík and Khusainov [2] adopted the idea to construct the discrete matrix delayed exponential, which was used to derive an explicit formula of solutions to a discrete delay system. There is rapid development in seeking explicit formulas of solutions to delay differential/discrete equations by introducing various continuous/discrete delayed exponential matrices (see, for example, [3–18]). One of the biggest advantages of continuous/discrete delayed exponential matrices is the possibility to transfer the classical idea of representing the solution of linear ODEs to linear delay differential/discrete equations. This provides a new idea and approach to study the representation of solutions, asymptotic stability, finite time stability, controllability, and iterative learning control (ILC) for various kinds of linear continuous/discrete delay systems, for example, oscillating delay systems, impulsive delay systems, fractional delay systems, and stochastic delay systems. Stability analysis is one of the most important issues in control systems. In general, the Lyapunov method is used to deal with the asymptotic stability of trivial solutions. However, there exists a stable system which yields undesirable transient performance at one fixed point in practical applications. For this reason, it is necessary to consider the boundedness of system trajectories over a given finite time interval from the engineering point of view instead of determining the long-time asymptotical behavior for the system trajectory from the mathematical point of view. As a result, the concept of finite time stability (the boundedness of system trajectories) is offered to characterize the behavior of dynamical systems. The finite time stability concept, introduced by Dorato [19], of delay differential equations arises from the fields of multibody mechanics, automatic engines, and physiological systems. Finite time stability means that the system Stability and Controls Analysis for Delay Systems. https://doi.org/10.1016/B978-0-32-399792-8.00007-4 Copyright © 2023 Elsevier Inc. All rights reserved.

1

2 Stability and Controls Analysis for Delay Systems

state does not exceed a certain bound for a given finite time interval and seems more appropriate from practical considerations. It is remarkable that Weiss and Infante [20–22] give sufficient conditions for finite time stability of nonlinear systems by introducing the suitable Lyapunov functionals. A solution of an equation or a state of a system is said to be finite time stable if it does not exceed a certain threshold during a fixed finite time interval. Obviously, finite time stability is very different from the classical exponential stability, which deals with equations operated in the whole infinite time interval. Concerning finite time stability, Ulam’s stability and stable manifolds of linear systems, impulsive systems and fractional systems, fundamental matrix methods, linear matrix inequalities, algebraic inequalities, and integral inequalities are often used to deal with this issue. For more recent contributions, one can see [23–37]. The concept of controllability was first introduced by Kalman in the early 1960s. Some very important conclusions about the behavior of linear and nonlinear dynamical systems have been obtained. In the field of engineering technology, there are many real controlled systems with obvious delay effects, such as vehicle active suspension systems, rocket engine combustion systems, and other mechanical systems. The delay effect leads to the evolution of the system state with time depending on not only the current state of the system, but also the state of the system for a certain period of time in the past. Time-delay systems are one of the main tools to study delay effects. Thus, the situation will be much more complicated for time-independent delay differential controlled systems, or even for linear differential systems with pure delay with one input. In fact, in contrast to the theory of classical linear differential/difference controlled systems, when driving these delay systems to rest, one is required to control not only the value of the state at the final time, but also the memory accumulated with an aftereffect. In addition, the representation of the solution is not easy to characterize without knowing the fundamental matrix of a homogeneous delay differential system. As a result, various classes of control methods for delay differential systems were considered and different variants of controllability were developed in the past decades. Khusainov and Shuklin [38] initially studied relative controllability of linear differential systems with pure delay via the associated delayed matrix exponential. Thereafter, Khusainov et al. [5] studied the Cauchy problem for a second order linear differential equation with pure delay. Next, in [4], relative controllability of linear discrete systems with a single constant delay was initially studied using the so-called discrete delayed matrix exponential, where an equivalent condition was stated for an initial value problem to have a control function. In addition, one of the control functions was found. Moreover, Pospíšil et al. [39] extended the study of controllability to linear discrete pure delay systems with constant coefficients and multiple control functions and derived a representation of solution in the form of a matrix polynomial using the Z-transform [18] to a system of nonhomogeneous linear difference equations with any finite number of constant delays and linear parts given by pairwise permutable matrices. Var-

Introduction Chapter | 1

3

ious criteria of relative controllability for linear discrete pure delay systems are presented and the associated control functions are also constructed. In 1978, Uchiyama [40] proposed the concept of ILC from the viewpoint of human learning ability to deal with robotic systems. In the past decades, various types of iterative updating laws with uniform trial lengths have been widely used to deal with the issues in robotics, process control, and biological systems. A large amount of literature on ILC has been published for various types of systems such as discrete time systems, fractional differential systems, impulsive hybrid systems, distributed parameter systems, and networked stochastic systems. After reviewing the previous literature, we observe a restriction that the length of process operation and the desired trajectory are invariant in different iterations. Obviously, this condition may limit the application range of ILC. In fact, the case of operation terminating early does exist [41], for example, the functional electrical stimulation for upper limb movement [42]. This case strongly motivated us to consider ILC under randomly iteration-varying length environments. For more recent contributions on this hot topic, one can refer to [43–55] and reference therein. It is remarkable that the ILC problem of a delay spring–mass–damper system, which has received much attention, is very important in many practical mechanical systems [56]. ILC, which has become an important topic in modern data-driven control, is proposed in [57,58]. Note that ILC problems for linear discrete delayed controlled systems have been studied extensively by transferring to analyze a constructed Roesser model. For more related contributions, one can refer to [59–69] and reference therein. This monograph is devoted to the rapidly developing research area of the stability, controllability, and ILC theory of delay difference/differential controlled systems. Such basic theory should be the starting point for further research concerning the optimal control, numerical analysis, and applications of other types of delay systems. The book is divided into seven chapters. Chapter 1 gives the introduction. In Chapter 2, we present the finite time stability, controllability, and ILC for first order delay differential systems. Chapter 3 is devoted to the study of the same issues in Chapter 2 for second order oscillating delay systems. In Chapter 4, we study asymptotical stability, finite time stability, and controllability for impulsive delay systems. In Chapter 5, we extend some content in Chapter 1 to fractional delay systems by introducing impulsive delay fundamental matrices. Chapter 6 is devoted to establishing the same issues as in Chapter 2 for difference delay systems. Chapter 7 further establishes the related controllability results for first order and second order stochastic delay systems. Key features of this book are stated here. We present the representation and stability of solutions via the delayed exponential matrix function approach. We give useful sufficient conditions to guarantee controllability. We discuss ILC design and focus on new systems such as first order systems, oscillating systems, impulsive systems, fractional systems, difference systems, and stochastic systems occurring in various fields.

4 Stability and Controls Analysis for Delay Systems

This book is useful for researchers and graduate students for research, seminars, and advanced graduate courses in pure and applied mathematics, engineering, and related disciplines.

Chapter 2

Delay systems 2.1 Finite time stability 2.1.1 Finite time stability for linear delay differential systems When reviewing the previous literature dealing with finite time stability problems for delay systems, we observe the following facts: (i) Delay system x(t) ˙ = Ax(t) + Bx(t − τ ), t > 0, is considered mostly as an integral system, where A, B are suitable matrices. (ii) A uniform transition matrix associated with A, B is not computed directly and the structure of solution x(t) is not well characterized on every subinterval [nτ, (n + 1)τ ], n ∈ N. (iii) The Gronwall inequality, linear matrix inequality, and Lyapunov function methods are used to obtain finite time stability results. Recently, Khusainov and Shuklin [1] introduced the delayed exponential matrix eτB1 t , B1 = e−Aτ B, to give a representation of a solution of a linear system with AB = BA and one delay term. Now, we are concerned with the following issue: Can we apply the delayed exponential matrix in [1] to establish a direct method to find the sufficient conditions to guarantee finite time stability? If so, we will provide another direct method to find some conditions to make the desired delay system finite time stable, which is a supplement of the current methods in some sense. Thus, in this part, we study finite time stability of the following linear homogeneous delay differential system:  x(t) ˙ = Ax(t) + Bx(t − τ ), t ∈ J := [0, T ], (2.1) x(t) = ϕ(t), − τ ≤ t ≤ 0, τ > 0, where x ∈ Rn and T denotes the prefixed iteration domain length. Without loss of generality, we set T = nτ and n ∈ N, ϕ ∈ Cτ1 := C 1 ([−τ, 0], Rn ), and the n × n matrices A and B satisfy AB = BA. According to [1, Remark 1.3], any solution of system (2.1) has the form  x(t) = X0 (t)e

Aτ

ϕ(−τ ) +

0

−τ

X0 (t − τ − s)eAτ [ϕ  (s) − Aϕ(s)]ds,

(2.2)

where X0 (t) = eAt eτB1 t , B1 = e−Aτ B Stability and Controls Analysis for Delay Systems. https://doi.org/10.1016/B978-0-32-399792-8.00008-6 Copyright © 2023 Elsevier Inc. All rights reserved.

(2.3) 5

6 Stability and Controls Analysis for Delay Systems

is called the fundamental matrix of (2.2). The delayed exponential matrix eτB1 t is defined by ⎧ Θ, t < −τ, ⎪ ⎪ ⎪ ⎨ I, − τ ≤ t < 0, eτB1 t = ⎪ ⎪ (t − (k − 1)τ )k ⎪ ⎩I + B1 t + · · · + B1k , (k − 1)τ ≤ t < kτ, k = 1, 2, · · · , k! (2.4) where Θ and I denote the zero and identity matrices, respectively. Let Rn be the n-dimensional vector space, let Rn×n denote the n × n matrix with real value elements, let X  denote the transpose of the matrix X, and let λmax (X) denote the maximum eigenvalue of the matrix X. Denote by C(J, Rn ) the Banach space of vector-valued continuous functions from J → Rn endowed with the norm x = max x(t) for a norm  ·  on Rn . We introt∈J

duce a space C 1 (J, Rn ) = {x ∈ C(J, Rn ) : x˙ ∈ C(J, Rn )}. For A : Rn → Rn , we consider its matrix norm A = max Ax generated by  · . In addix=1

tion, we note ϕ = max ϕ(θ ). The λ-norm on C(J, Rn ) is represented as θ∈[−τ,0]  xλ = sup e−λt x(t) , λ > 0. t∈J

Definition 2.1. (see [19] or [70, Definition 1]) System (2.1) is finite time stable with respect to {0, J, α, β, τ } if and only if ϕ  (t)ϕ(t) < α, ∀ t ∈ [−τ, 0], implies x  (t)x(t) < β, ∀ t ∈ J, where α is a real positive number and β ∈ R with β > α. Definition 2.2. (see [70, Definition 2]) System (2.1) is finite time stable with respect to {0, J, α, β, τ } if and only if ϕ2 < α implies x(t)2 < β, ∀ t ∈ J. Remark 2.1. Definition 2.1 coincides with Definition 2.2 when  ·  is a Euclidean norm. Definition 2.3. (see [71] or [72]) The matrix measure (or matrix logarithmic norm) of X is defined as μ(X) = lim

h→0+

I + hX − 1 . h

In particular, if  ·  denotes the 2-norm of a matrix, then 1 μ(X) = λmax (X + X  ). 2

7

Delay systems Chapter | 2

This norm is called the Lozinski matrix norm and this norm was used to characterize stability and error bounds in the numerical integration of ODEs [73]. Lemma 2.1. (see [6, Lemma 3]) For all t ∈ R, eτBt  ≤ eB(t+τ ) . Lemma 2.2. (see [1, Lemma 4]) For all t ∈ R, d Bt e = BeτB(t−τ ) . dt τ Lemma 2.3. (see [74, Lemma 1]) For any positive symmetric constant matrix W ∈ Rn×n , scalars a, b satisfying a < b, and a vector function f : [a, b] → Rn such that the integrations are well defined, we have

 b   b   b f (s)ds W f (s)ds ≤ (b − a) f  (s)Wf (s)ds. a

a

a

Lemma 2.4. (see [75] or [76, Lemma 2]) For any real square matrix W ∈ Rn×n and scalar variable t, the expression 

λmax (eW t eW t ) ≤ e2μ(W )t holds, where μ(W ) is the matrix measure of the matrix W defined in Definition 2.3. Lemma 2.5. (see [76, Lemma 3]) For any vectors u, v ∈ Rn and symmetric positive definite matrix Rn×n Γ = Γ −1 > 0, the following inequalities hold: 2u(t) v(t) ≤ u (t)Γ u(t) + v  (t)Γ −1 v(t) and −2u(t) v(t) ≤ u (t)Γ u(t) + v  (t)Γ −1 v(t). The finite time stability results are presented by providing an approach which is different from previous methods.

2.1.1.1 Finite time stability results based on Definition 2.1 Theorem 2.1. System (2.1) is finite time stable with respect to {0, J, α, β, τ } if √ β X0 (t) < Aτ √ , ∀ t ∈ J, (2.5) e  N (α) where X0 (t) is given in (2.3), α, β are defined in Definition 2.1, and √ N (α) := α + 2 α



0

−τ

ϕ  (s) − Aϕ(s)ds +



0 −τ

ϕ  (s) − Aϕ(s)ds

2 . (2.6)

8 Stability and Controls Analysis for Delay Systems

Proof. Let x be the solution of system (2.1) satisfying (2.2). Denote a(t) := 0 X0 (t)eAτ , ϕ(−τ ) ∈ Rn , and d(t) := −τ X0 (t − τ − s)eAτ [ϕ  (s) − Aϕ(s)]ds ∈ Rn . Clearly, a  (t)d(t) = d  (t)a(t). For any t ∈ J , one can get

 x(t) x(t) = X0 (t)eAτ ϕ(−τ ) +

0

−τ

X0 (t − τ − s)eAτ [ϕ  (s) − Aϕ(s)]ds

 × X0 (t)eAτ ϕ(−τ ) +

0

−τ



X0 (t − τ − s)eAτ [ϕ  (s) − Aϕ(s)]ds



= (eAτ ϕ(−τ )) X0 (t)X0 (t)eAτ ϕ(−τ ) + a  (t)d(t) + d  (t)a(t) + d  (t)d(t) ≤ λm (t)(eAτ ϕ(−τ )) eAτ ϕ(−τ ) + 2a  (t)d(t) + d(t)2 ,

(2.7)

where λm (t) = maxt∈J ρ{X0 (t)X0 (t)} and ρ{Y } denotes the spectrum of Y . For the term a  (t)d(t), we have a  (t)d(t) = (eAτ ϕ(−τ )) X0 (t) 

≤ ϕ(−τ ) e

Aτ



0 −τ

X0 (t − τ − s)eAτ [ϕ  (s) − Aϕ(s)]ds

X0 (t)



0

−τ

X0 (t − τ − s)eAτ ϕ  (s) − Aϕ(s)ds. (2.8)

Let η = t − τ − s, s ∈ [−τ, 0]. Then, η ∈ [t − τ, t] ⊆ [T − τ, T ] ⊆ J . Thus, X0 (η) |η∈[t−τ,t] ≤ X0 (t) |t∈[T −τ,T ] ≤ X0 (t) |t∈J .

(2.9)

Linking (2.7) and (2.8) via (2.9) we obtain x(t) x(t) ≤ λm (t)(eAτ ϕ(−τ )) eAτ ϕ(−τ ) 

+ 2ϕ(−τ ) e

Aτ

X0 (t)X0 (t)eAτ 



+ X0 (t) e 2



Aτ 2

0 −τ



ϕ  (s) − Aϕ(s)ds

0

−τ 2

ϕ  (s) − Aϕ(s)ds

.

Using the fact that λm (t) ≤ X0 (t)X0 (t) ≤ X0 (t)X0 (t) = X0 (t)2 , by (2.10), we obtain x(t) x(t) ≤ X0 (t)2 eAτ 2 ϕ  (−τ )ϕ(−τ )

(2.10)

Delay systems Chapter | 2

+ 2ϕ(−τ ) X0 (t)2 eAτ 2

 + X0 (t)2 eAτ 2

0

−τ



0

−τ

9

ϕ  (s) − Aϕ(s)ds

ϕ  (s) − Aϕ(s)ds

2 .

(2.11)

  √ Next, we choose ϕ (t)ϕ(t) < α, ∀ t ∈ [−τ, 0], which implies ϕ (−τ ) < α. Finally, applying condition (2.5) to the above inequality, (2.11) reduces to

x(t) x(t) ≤ X0 (t)2 eAτ 2 N (α) < β, where N (α) is given in (2.6). According to Definition 2.1, system (2.1) is finite time stable. Theorem 2.2. System (2.1) is finite time stable with respect to {0, J, α, β, τ } if eM(t+τ )
0;

(2.25)

(ii) equations with a nonlinearity depending on delay: 

x(t) ˙ = Ax(t) + Bx(t − τ ) + F (x(t), x(t − τ )), t ∈ J, x(t) = ϕ(t), − τ ≤ t ≤ 0, τ > 0,

(2.26)

where x ∈ C 1 ([−τ, T ], Rn ), f ∈ C(Rn , Rn ), and F ∈ C(Rn × Rn , Rn ). According to [1, Corollary 2.2], if AB = BA, then any solution of (2.25) has the form  0 X0 (t − τ − s)eAτ [ϕ  (s) − Aϕ(s)]ds x(t) = X0 (t)eAτ ϕ(−τ ) + −τ  t X0 (t − τ − s)eAτ f (x(s))ds, (2.27) + 0

Delay systems Chapter | 2

17

and any solution of (2.26) has the form  0 x(t) = X0 (t)eAτ ϕ(−τ ) + X0 (t − τ − s)eAτ [ϕ  (s) − Aϕ(s)]ds −τ  t X0 (t − τ − s)eAτ F (x(s), x(s − τ ))ds, (2.28) + 0

where X0 (t) = eAt eτB1 t , B1 = e−Aτ B is called the fundamental matrix of (2.25) and (2.26), and the delayed exponential matrix eτB1 t is defined in (2.4). Integral inequalities play an important role in finite time stability analysis of the solutions to differential equations. Next, the finite time stability results are presented by virtue of the delayed exponential matrix eτB1 t , B1 = e−Aτ B, and the above two powerful Gronwall inequalities. We give the following hypotheses: [H0 ]: A and B are permutation matrices, that is, AB = BA. [H1 ]: For f ∈ C(Rn , Rn ), there exists P > 0 such that f (x) ≤ P x for all x ∈ Rn . [H2 ]: For F ∈ C(Rn × Rn , Rn ), there exists Q > 0 such that F (x, y) ≤ Q(x + y) for all x, y ∈ Rn .

2.1.2.1 Finite time stability results for system (2.25) Theorem 2.6. The solution of (2.25) is finite time stable with respect to {0, J, α, β, τ } if [H0 ], [H1 ], and the inequality √ β e(N +P )t < , ∀ t ∈ J, (2.29) 0 √ N τ −N s e α + −τ e ϕ  (s) − Aϕ(s)ds hold, where N = A + B1  and α, β are defined in Definition 2.2. Proof. Note that by [H0 ], the solution of (2.25) can be formulated as (2.27). Taking the norm for (2.27) via Lemma 2.1, we obtain x(t) ≤ eA(t+τ ) eB1 (t+τ ) ϕ(−τ )  0 + eA(t−s) eB1 (t−s) ϕ  (s) − Aϕ(s)ds −τ  t eA(t−s) eB1 (t−s) f (x(s))ds. + 0

By assumption [H1 ] and inequality (2.30), we obtain  x(t) ≤ e

N (t+τ )

ϕ(−τ ) +

0

−τ

eN (t−s) ϕ  (s) − Aϕ(s)ds

(2.30)

18 Stability and Controls Analysis for Delay Systems

 +

t

eN (t−s) P x(s)ds.

(2.31)

0

If we denote u(t) = e−N t x(t), (2.31) can be simplified to  0  t u(t) ≤ eN τ ϕ(−τ ) + e−N s ϕ  (s) − Aϕ(s)ds + P u(s)ds, −τ 0  t P u(s)ds, t ∈ J, (2.32) ≤ a(t) + 0

where  a(t) = eN τ ϕ +

0 −τ

e−N s ϕ  (s) − Aϕ(s)ds, t ∈ J.

(2.33)

It is obvious that a(t) ≥ 0. In addition, u(t) ≤ eN τ x(t) = eN τ ϕ(t) ≤ eN τ ϕ ≤ a(t), t ∈ [−τ, 0).

(2.34)

Now we set ϕ(t) = a(t). Obviously, a(0) = ϕ(0). Observing (2.32) and (2.34), applying the Gronwall inequality (see [78]), one can get u(t) ≤ a(t)eP t . By (2.29), we infer that x(t) ≤ a(t)e(N +P )t

 Nτ = e ϕ +

√ < eN τ α +  < β, t ∈ J.

0

−τ  0 −τ

e

−N s





ϕ (s) − Aϕ(s)ds e(N +P )t

 e−N s ϕ  (s) − Aϕ(s)ds e(N +P )t

That is, x(t)2 < β, ∀ t ∈ J . The proof is completed. Remark 2.2. In Theorem 2.6, condition (2.29) involves ϕ  , which seems a slightly stronger requirement than ϕ. In order to weaken (2.29), we expect to obtain a general formula for (2.27) without ϕ  . By using integration by parts and Lemma 2.2, via [H0 ], we have  0 eA(t−τ −s) eτB1 (t−τ −s) eAτ ϕ  (s)ds −τ

= eAt eτB1 (t−τ ) ϕ(0) − eA(t+τ ) eτB1 t ϕ(−τ )  0 + [AeA(t−s) eτB1 (t−τ −s) + BeA(t−τ −s) eτB1 (t−2τ −s) ]ϕ(s)ds. −τ

(2.35)

Delay systems Chapter | 2

19

Combining (2.35) and (2.27), we obtain  0 x(t) = eAt eτB1 (t−τ ) ϕ(0) + BeA(t−τ −s) eτB1 (t−2τ −s) ϕ(s)ds −τ  t A(t−τ −s) B1 (t−τ −s) Aτ e eτ e f (x(s))ds, t ≥ 0. +

(2.36)

0

Clearly, (2.36) does not need ϕ  to exist. Theorem 2.7. The solution of (2.25) is finite time stable with respect to {0, J, α, β, τ } if [H0 ], [H1 ], and the inequality  γe

β , ∀ t ∈ J, α

(2.37)

 B −N τ (1 − e ,1 + ) . N

(2.38)

(N +P )t


0, with AB = BA. This admits one can apply the classical ideas of the variation of constants method via the principle of superposition for ODEs to seek an explicit simple formula of solutions to (2.53). The simple form of (2.54) allows to find a simple criterion of relative controllability. Moreover, we also give some details to show that a model of population

Delay systems Chapter | 2

31

dynamics with delayed birthrates and delayed logistic terms can be turned into delay differential equations with permutable matrices. Definition 2.4. (see [38, Definition 4])System (2.52) is called relatively controllable if for an arbitrary initial vector function ϕ ∈ C 1 ([−τ, 0], Rn ), the final state of the vector x1 ∈ Rn , and time t1 , there exists a control u ∈ L2 (J, Rn ) such that system (2.52) has a solution x ∈ C([−τ, t1 ], Rn ) that satisfies the boundary conditions x(t) = ϕ(t), −τ ≤ t ≤ 0, and x(t1 ) = x1 .

2.2.1.1 Relative controllability for linear delay differential systems Let f (t, x(t)) ≡ 0 = (0, · · · , 0) , t ∈ J , i.e., system (2.52) reduces to the fol   n

lowing linear delay controlled system with linear parts defined by pairwise permutable matrices:  x(t) ˙ = Ax(t) + Bx(t − τ ) + Cu(t), t ∈ J, (2.55) x(t) = ϕ(t), − τ ≤ t ≤ 0. Next, we introduce a notation of a delay Grammian matrix, an extension of the classical Grammian matrix for linear differential systems, as follows:  t1 B  (t1 −τ −s) A (t1 −s) eA(t1 −s) eτB1 (t1 −τ −s) CC  eτ 1 e ds. (2.56) Wτ [0, t1 ] = 0

Theorem 2.12. System (2.55) is relatively controllable if and only if Wτ [0, t1 ] defined in (2.56) is nonsingular. Proof. Sufficiency. Since Wτ [0, t1 ] is nonsingular, its inverse Wτ−1 [0, t1 ] is well defined. One can select a control function as follows: B  (t1 −τ −t) A (t1 −t)

u(t) = C  eτ 1

e

Wτ−1 [0, t1 ]η,

(2.57)

where η = x1 − eA(t1 +τ ) eτB1 t1 ϕ(−τ ) −



0

−τ

eA(t1 −s) eτB1 (t1 −τ −s) [ϕ  (s) − Aϕ(s)]ds, (2.58)

where the vector x1 ∈ Rn is arbitrary before it is chosen. Inserting (2.57) in (2.54) (with f (·, x) = 0), we have  0 eA(t1 −s) eτB1 (t1 −τ −s) [ϕ  (s) − Aϕ(s)]ds x(t1 ) = eA(t1 +τ ) eτB1 t1 ϕ(−τ ) + −τ  t1 B  (t1 −τ −s) A (t1 −s) A(t1 −s) B1 (t1 −τ −s) e eτ CC  eτ 1 e ds · Wτ−1 [0, t1 ]η. + 0

(2.59)

32 Stability and Controls Analysis for Delay Systems

Linking (2.56) and (2.59) via (2.58), it is not difficult to derive  0 A(t1 +τ ) B1 t1 eτ ϕ(−τ ) + eA(t1 −s) eτB1 (t1 −τ −s) [ϕ  (s) − Aϕ(s)]ds + η x(t1 ) = e −τ

= x1 .

(2.60)

The boundary condition x(t) = ϕ(t), −τ ≤ t ≤ 0, holds by Lemma 2.6. Combining with formula (2.60) via Definition 2.4, system (2.52) is relatively controllable. Necessity. We prove our result by contradiction. Assume Wτ [0, t1 ] is singular, i.e., there exists at least one nonzero state x˜ ∈ Rn such that x˜  Wτ [0, t1 ]x˜ = 0. Furthermore, we obtain 0 = x˜  Wτ [0, t1 ]x˜  t1 B  (t1 −τ −s) A (t1 −s) = x˜  eA(t1 −s) eτB1 (t1 −τ −s) CC  eτ 1 e xds ˜ 0  t1    = x˜  eA(t1 −s) eτB1 (t1 −τ −s) C x˜  eA(t1 −s) eτB1 (t1 −τ −s) C ds 0  t1     A(t1 −s) B1 (t1 −τ −s) 2 = eτ C  ds, x˜ e 0

which implies that x˜  eA(t1 −s) eτB1 (t1 −τ −s) C = 0 , ∀ s ∈ J.

(2.61)

Since system (2.55) is relatively controllable, according to Definition 2.4, there exists a control u1 (t) that drives the initial state to zero at t1 , that is,  0 A(t1 +τ ) B1 t1 x(t1 ) = e eτ ϕ(−τ ) + eA(t1 −s) eτB1 (t1 −τ −s) [ϕ  (s) − Aϕ(s)]ds −τ  t1 eA(t1 −s) eτB1 (t1 −τ −s) Cu1 (s)ds = 0. (2.62) + 0

Similarly, there also exists a control u(t) ˜ that drives the initial state to the state x˜ at t1 , i.e.,  0 x(t1 ) = eA(t1 +τ ) eτB1 t1 ϕ(−τ ) + eA(t1 −s) eτB1 (t1 −τ −s) [ϕ  (s) − Aϕ(s)]ds −τ  t1 A(t1 −s) B1 (t1 −τ −s) e eτ C u(s)ds ˜ + 0

= x. ˜

(2.63)

Delay systems Chapter | 2

33

Then by (2.62) and (2.63), we have  x˜ =

t1

0

eA(t1 −s) eτB1 (t1 −τ −s) C[u(s) ˜ − u1 (s)]ds.

(2.64)

Multiplying both sides of (2.64) by x˜ T , we obtain x˜  x˜ =

 0

t1

x˜  eA(t1 −s) eτB1 (t1 −τ −s) C[u(s) ˜ − u1 (s)]ds.

Note that by (2.61), one can obtain x˜  x˜ = 0, i.e., x˜ = 0, which conflicts with x˜ being nonzero. Thus, the delay Grammian matrix Wτ [0, t1 ] is nonsingular. The proof is finished. Remark 2.3. In Theorem 2.12, if B = Θ, then B1 = Θ and therefore eτB1 · = I . Then, the delay Grammian matrix Wτ [0, t1 ] given in (2.56) reduces to the classical Grammian matrix for linear systems. Hence Theorem 2.12 contains the classical controllability result for linear systems. Concerning on the case of A = Θ, some related interesting results for linear continuous delay systems have been reported in [82, Theorem 3] and some results for linear discrete delay systems have been reported in [2,3,17]. It is remarkable that a criterion of relative controllability, expressed using a defining equation, has been obtained in [83,84]. From a practical point of view, it is more convenient than Theorem 2.12, which presents other theoretical criteria to guarantee (2.55) is relatively controllable and also enriches the results in this field.

2.2.1.2 Relative controllability for semilinear delay differential systems Consider f (t, x(t)) ≡ 0, t ∈ J . We need the following hypothesis: [H1 ]: The operator W : L2 (J, Rn ) → Rn defined by  Wu = 0

t1

eA(t1 −s) eτB1 (t1 −τ −s) Cu(s)ds

has an inverse operator W −1 which takes values in L2 (J, Rn )/ ker W . Then we set M = W −1 Lb (Rn ,L2 (J,Rn )/ ker W ) . Remark 2.4. Clearly, W must be surjective to satisfy [H1 ]. On the other hand, if W is surjective, then we can define an inverse W −1 : Rn → L2 (J, Rn )/ ker W . We propose its natural construction as follows. Let (·, ·) denote the Euclidean scalar product in Rn . Then L2 (J, Rn ) and Rn are Hilbert spaces. So we can use ker W = imW ∗ ⊥ and imW = ker W ∗ ⊥ . We need to find W  . Set W (·) =

34 Stability and Controls Analysis for Delay Systems B (t1 −τ −·)

eA(t1 −·) eτ 1

C. For any w ∈ Rn and u ∈ L2 (J, Rn ) we derive

 t1   t1 (W u, w) = W (s)u(s)ds, w = (u(s), W (s) w)ds, 0

0

which gives W ∗ w = W (s) w. Thus ker W ∗ = {0} if and only if  t1 W (s) w2 ds = 0 0

∈ Rn .

for any 0 = w But  t1  t1 W (s) w2 ds = (W (s) w, W (s) w)ds 0 0  t1 = (W (s)W (s) w, w)ds = (Wτ [0, t1 ]w, w).

(2.65)

0

So the surjectivity of W is equivalent to the regularity of Wτ [0, t1 ], and we assume this. To solve W u = v, u ∈ ker W ⊥ = imW ∗ , we take u(t) = W (t) w and then solve  t1  W (s)W (s) wds = Wτ [0, t1 ]w, v = W (W (·) w) = 0

which gives w = Wτ [0, t1 ]−1 v, and this implies u(t) = W −1 w = W (t) Wτ [0, t1 ]−1 w, where we consider Lb (Rn , L2 (J, Rn )/ ker W ) = ker W ⊥ . Moreover, by (2.65), we derive  t1  t1 2 u(s) ds = W (s) Wτ [0, t1 ]−1 w2 ds 0 0  t1  = W (s) Wτ [0, t1 ]−1 w, W (s) Wτ [0, t1 ]−1 w ds 0  t1  (Wτ [0, t1 ]−1 ) W (s)W (s) Wτ [0, t1 ]−1 w, w ds = 0 

 t1 −1   −1 W (s)W (s) dsWτ [0, t1 ] w, w = (Wτ [0, t1 ] ) 0   = (Wτ [0, t1 ]−1 ) w, w = w, Wτ [0, t1 ]−1 w , which gives M=

 Wτ [0, t1 ]−1 .

(2.66)

Delay systems Chapter | 2

35

We note that (2.65) also implies W  = W ∗  =



Wτ [0, t1 ].

[H2 ]: The function f : J × Rn → Rn is continuous and there exists a constant q > 1 and Lf (·) ∈ Lq (J, R+ ) such that f (t, x1 ) − f (t, x2 ) ≤ Lf (t)x1 − x2 , xi ∈ Rn , i = 1, 2. In view of [H1 ], for arbitrary x(·) ∈ C, consider a control function ux (t) given by

−1 ux (t) = W x1 − eA(t1 +τ ) eτB1 t1 ϕ(−τ )  − −

0

−τ  t1 0

eA(t1 −s) eτB1 (t1 −τ −s) (ϕ  (s) − Aϕ(s))ds  eA(t1 −s) eτB1 (t1 −τ −s) f (s, x(s))ds (t), t ∈ J.

(2.67)

Now we state our main idea to prove our main result via the fixed point method. We firstly show that, using control (2.67), the operator P : C → C defined by  0 A(t+τ ) B1 t eτ ϕ(−τ ) + eA(t−s) eτB1 (t−τ −s) [ϕ  (s) − Aϕ(s)]ds (Px)(t) = e −τ  t eA(t−s) eτB1 (t−τ −s) f (s, x(s))ds + 0  t eA(t−s) eτB1 (t−τ −s) Cux (s)ds + 0

has a fixed point x, which is just a solution of system (2.52). Secondly, we check that (Px)(t1 ) = x1 and (Px)(0) = ϕ(0) = x0 , which means that ux steers system (2.52) from x0 to x1 in finite time t1 . This implies system (2.52) is relatively controllable on J . For each positive number r, define Br = {x ∈ C : xC ≤ r}. Then, for each r, Br is obviously a bounded, closed, and convex set of C. For brevity, we set Rf = sup f (t, 0) and N = A + B1 . t∈J

In what follows, we apply Krasnoselskii’s fixed point theorem (see [82]) to derive the relative controllability result for system (2.52). Theorem 2.13. Suppose that [H1 ] and [H2 ] are satisfied. Then system (2.52) is relatively controllable provided that

 M N t1 (2.68) M2 1 + (e − 1)C < 1, N

36 Stability and Controls Analysis for Delay Systems 1

1 where M2 = [ Np (eNpt1 − 1)] p Lf Lq (J,R+ ) , given by (2.66).

+

1 p

1 q

= 1, p, q > 1, and M is

Proof. To examine the conditions for Krasnoselskii’s fixed point theorem (see [82]), we divide our proof into several steps. Step 1. We prove that there exists a positive number r such that P(Br ) ⊆ Br . In light of [H2 ] and the Hölder inequality, we obtain 



t

e

N (t−s)

Lf (s)ds ≤

0

e

≤ and



1 

t

Np(t−s)

0

e

N (t−s)

ds 0

1 Npt (e Np 

t

t

p

f (s, 0)ds ≤ Rf

0

t

1 q Lf (s)ds

q

1 p − 1) Lf Lq (J,R+ )

eN (t−s) ds ≤

0

Rf N t (e − 1). N

Taking into account (2.67), using [H1 ], [H2 ], and Lemma 2.1, we have

−1 ux (t) ≤ W L(Rn ,L2 (J,Rn )/ ker W ) x1  + eN (t1 +τ ) ϕ(−τ )  +

0

−τ

e

N (t1 −s)





ϕ (s) − Aϕ(s)ds +

t1

e

N (t1 −s)

 f (s, x(s))ds

0 0

 ≤ Mx1  + MeN (t1 +τ ) ϕ(−τ ) + M eN (t1 −s) ϕ  (s) − Aϕ(s)ds −τ  t1  t1 eN (t1 −s) Lf (s)x(s)ds + M eN (t1 −s) f (s, 0)ds +M 0

≤ Mx1  + MeN (t1 +τ ) ϕ(−τ ) + M



0 0

−τ

eN (t1 −s) ϕ  (s) − Aϕ(s)ds

1 p MRf N t1 1 Npt1 (e (e − 1) − 1) Lf Lq (J,R+ ) x + +M Np N ≤ Mx1  + Ma + MM2 r,

where a = eN (t1 +τ ) ϕ +



0 −τ

eN (t1 −s) ϕ  (s) − Aϕ(s)ds +

Rf N t1 (e − 1) N

and M2 is defined above. From [H1 ] and [H2 ] we have  (Px)(t) ≤ e

N (t+τ )

ϕ(−τ ) +

0 −τ

eN (t−s) ϕ  (s) − Aϕ(s)ds

Delay systems Chapter | 2

 +

t

 eN (t−s) f (s, x(s))ds +

0

t

eN (t−s) Cux (s)ds

0

 0 eN (t−s) ϕ  (s) − Aϕ(s)ds ≤ eN (t+τ ) ϕ(−τ ) + −τ  t  t eN (t−s) Lf (s)x(s)ds + eN (t−s) f (s, 0)ds + 0 0  t + eN (t−s) C(Mx1  + Ma + MM2 x)ds 0



≤ eN (t+τ ) ϕ(−τ ) +

0

−τ

eN (t−s) ϕ  (s) − Aϕ(s)ds

1 p Rf N t 1 Npt (e (e − 1) − 1) Lf  q1 x + + L (J,R+ ) Np N  t  t N (t−s) + Cx1 M e ds + MCa eN (t−s) ds 0 0  t N (t−s) e ds + MM2 Cx 0

 M M ≤ a 1 + (eN t1 − 1)C + (eN t1 − 1)Cx1  N N

 M + M2 1 + (eN t1 − 1)C r = r N

for

a 1+ r=

 N t1 − 1)Cx  − 1)C + M 1 N (e

 . N t1 − 1)C 1 − M2 1 + M (e N

M N t1 N (e

Hence, we obtain P(Br ) ⊆ Br for such an r. Now, we divide P into two operators P1 and P2 on Br as  0 eA(t−s) eτB1 (t−τ −s) [ϕ  (s) − Aϕ(s)]ds (P1 x)(t) = eA(t+τ ) eτB1 t ϕ(−τ ) + −τ  t eA(t−s) eτB1 (t−τ −s) Cux (s)ds, + 0  t (P2 x)(t) = eA(t−s) eτB1 (t−τ −s) f (s, x(s))ds, 0

for t ∈ J , respectively. Step 2. We show that P1 is a contraction mapping.

37

38 Stability and Controls Analysis for Delay Systems

Let x, y ∈ Br . In view of [H1 ] and [H2 ], for each t ∈ J , we have  t ux (t) − uy (t) ≤ M eN (t−s) Lf (s)x(s) − y(s)ds 0  t ≤M eN (t−s) Lf (s)dsx − y 0

1 p 1 Npt − 1) Lf Lq (J,R+ ) x − y ≤M (e Np ≤ MM2 x − y.

Using the above, we derive  (P1 x)(t) − (P1 y)(t) ≤

t

eN (t−s) Cux (s) − uy (s)ds  t eN (t−s) dsMM2 x − y ≤ C 0

0

C N t1 (e − 1)MM2 x − y, ≤ N which gives P1 x − P1 y ≤ Lx − y,

L :=

MM2 N t1 (e − 1)C. N

In view of (2.68), we conclude that L < 1, which implies P1 is a contraction. Step 3. We show that P2 is a compact and continuous operator. Let xn ∈ Br with xn → x in Br . Denote Fn (·) = f (·, xn (·)) and F (·) = f (·, x(·)). Using [H2 ], we have Fn → F in C and thus  t eN (t−s) Fn (s) − F (s)ds → 0 as n → ∞ (P2 xn )t − (P2 x)(t) ≤ 0

uniformly for t ∈ J , which implies that P2 is continuous on Br . To check the compactness of P2 , we prove that P2 (Br ) ⊂ C is equicontinuous and bounded. In fact, for any x ∈ Br , t1 ≥ t + h ≥ t > 0, we have (P2 x)(t + h) − (P2 x)(t)  t+h  t = eA(t+h−s) eτB1 (t+h−τ −s) F (s)ds − eA(t−s) eτB1 (t−τ −s) F (s)ds 0

0

= I1 + I2 + I3 , where



t+h

I1 = t

eA(t+h−s) eτB1 (t+h−τ −s) F (s)ds,

Delay systems Chapter | 2

 I2 =

t

A(t+h−s)

e  t 0

I3 =

0

eτB1 (t+h−τ −s)

− eτB1 (t−τ −s)

39

 F (s)ds,

 eA(t+h−s) − eA(t−s) eτB1 (t−τ −s) F (s)ds.

From the above, we derive (P2 x)(t + h) − (P2 x)(t) ≤ I1  + I2  + I3 . Next, we check Ii  → 0 as h → 0, i = 1, 2, 3, uniformly for t. For I1 , using [H2 ],  t+h I1  ≤ eN (t+h−s) F (s)ds 

t



t

t+h

≤

t+h

≤ t

eN (t+h−s) (f (s, x(s)) − f (s, 0) + f (s, 0))ds  eN (t+h−s) Lf (s)x(s)ds + 

≤ xC

t t+h

e

N (t+h−s)

t+h



Lf (s)ds + Rf

t

eN (t+h−s) f (s, 0)ds

t+h

eN (t+h−s) ds

t

1 p Rf N h 1 pN h (e (e − 1) → 0 as h → 0. − 1) Lf Lq (J,R+ ) r + ≤ Np N

One can apply [H2 ] to derive  t eA(t+h−s) eτB1 (t+h−τ −s) − eτB1 (t−τ −s) F (s)ds I2  ≤ 0  t At1 eτB1 (t+h−τ −s) − eτB1 (t−τ −s) Lf (s)dsx ≤e 0  t At1 +e eτB1 (t+h−τ −s) − eτB1 (t−τ −s) f (s, 0)ds 0

≤e



At1

t

eτB1 (t+h−τ −s)

− eτB1 (t−τ −s) p ds

Lf Lq (J,R+ ) r 0  t + eAt1 Rf eτB1 (t+h−τ −s) − eτB1 (t−τ −s) ds 0

≤e

At1



Lf Lq (J,R+ ) r

+ eAt1 Rf



t1 −τ

−τ

t1 −τ

−τ

eτB1 (s+h)

− eτB1 s p ds

 p1

eτB1 (s+h) − eτB1 s ds → 0 as h → 0,

by using Lebesgue’s dominated convergence theorem.

 p1

40 Stability and Controls Analysis for Delay Systems

For I3 , it is easy to get  t eA(t+h−s) − eA(t−s) eB1 (t−s) F (s)ds I3  ≤ 0  t ≤ eA(t+h−s) − eA(t−s) eB1 (t−s) (Lf (s)x + f (s, 0))ds 0  t eAh − I eN (t−s) (Lf (s)r + Rf )ds ≤ 0

 t   t Ah N (t−s) N (t−s) ≤ e − I  r e Lf (s)ds + Rf e ds

≤ eAh − I 

0

1 Npt (e Np

0

1  p Rf N t (e − 1) → 0, − 1) Lf Lq (J,R+ ) r + N

as h → 0. From the above, we immediately obtain (P2 x)(t + h) − (P2 x)(t) → 0, h → 0, uniformly for all t and x ∈ Br . Therefore, P2 (Br ) ⊂ C is equicontinuous. Next, repeating the above computations, we have  t eN (t−s) (Lf (s)r + Rf )ds (P2 x)(t) ≤ 0

≤

1 p Rf N t1 1 Npt1 − 1) Lf Lq (J,R+ ) r + (e (e − 1). Np N

Hence P(Br ) is bounded. By the Arzela–Ascoli theorem, P2 (Br ) ⊂ C is relatively compact in C. Thus, P2 is a compact operator. Hence, using Krasnoselskii’s fixed point theorem (see [82]), P has a fixed point x on Br . Obviously, x is a solution of system (2.52) satisfying x(t1 ) = x1 . The boundary condition x(t) = ϕ(t), −τ ≤ t ≤ 0, holds by Lemma 2.6. The proof is completed.

2.2.1.3 Numerical examples and discussion Firstly, we give some reasonable explanations for system (2.52) from the point of view of practical application. We also note that relative controllability for linear systems of neutral differential equations with a delay is studied in [85] and an example of competing biological systems is used to illustrate the theoretical results. For instance, in a combustion chamber or reactor, we consider the change process of pressure. Based on our knowledge, the movement process of pressure depends not only on the current state value, but also on the past state value and

Delay systems Chapter | 2

41

control. From this point of view, system (2.52) has a practical significance, and f (t, x(t)) of system (2.52) can be considered as an interference which depends on the state. According to the actual needs, people often hope that the pressure of the combustion chamber cannot exceed the expected threshold and it can be controlled. Therefore, we study the controllability of system (2.52), (see also [85, Section 4]). Delay differential equations have been widely used to model the population growth of certain species [86] in the past decades. Now, we present a model of population dynamics with delayed birthrates and delayed logistic terms given by the system ! " ⎧ z˙ 1 (t) = z1 (t) − a1 − b1 z2 (t) − c(z1 (t − τ ) + z2 (t − τ )) ⎪ ⎪ ⎪ ⎨ + d1 z1 (t − τ ) + ϕ1 (t), ! " (2.69) ⎪ z˙ 2 (t) = z2 (t) − a2 + b2 z1 (t) − c(z1 (t − τ ) + z2 (t − τ )) ⎪ ⎪ ⎩ + d2 z2 (t − τ ) + ϕ2 (t), where t > 0, τ > 0, a2 > a1 > 0, b1 , b2 , d1 , d2 > 0, and ϕ1 , ϕ2 are given functions. We put       z1 (t) 0 d1 0 −a1 Z(t) = , B= , , A= 0 −a2 0 d2 z2 (t) and F : R2 × R2 → R2 is defined by F (t, Z(t), Z(t − τ ))   −b1 z1 (t)z2 (t) − cz1 (t)z1 (t − τ ) − cz1 (t)z2 (t − τ ) + ϕ1 (t) = . b2 z1 (t)z2 (t) − cz2 (t)z1 (t − τ ) − cz2 (t)z2 (t − τ ) + ϕ2 (t) ˙ = AZ(t) + BZ(t − τ ) + F (t, Z(t), Z(t − Then, (2.69) can be turned into Z(t) τ )). This may be an explanation from the practical point of view for studying systems with permutable matrices. Concerning the controllability of such systems, we can understand that people find a strategy to attain a target growth of populations by adjusting one of them. Secondly, we give numerical examples to demonstrate the validity of our method and provide some discussion. We use in this section the Euclidean norm. Example 2.3. Set t1 = 1, τ = 0.2, J1 := [0, 1], and u ∈ L2 (J1 , R2 ). Consider the following semilinear delay differential controlled system:  x(t) ˙ = Ax(t) + Bx(t − 0.2) + f (t, x(t)) + Cu(t), x(t) ∈ R2 , t ∈ J1 , ϕ(t) = (0.4, 0.3) , − 0.2 ≤ t ≤ 0. (2.70)

42 Stability and Controls Analysis for Delay Systems

For the sake of simplicity, we set       1 (t + 0.1)x (t) 0.3 0 0.2 0 1 A= , B= , f (t, x(t)) = 10 , C = I. 1 0 0.3 0 0.2 (t + 0.1)x2 (t) 

10



0.06 0 since A, B are diagonal matrices. 0 0.06 By elementary calculation, we have   0.1884 0 −Aτ B= , N = A + B1  = 0.4884, (2.71) B1 = e 0 0.1884 Obviously, AB = BA =

and e

A(1−s)

=

 e0.3(1−s) 0



eτB1 t1

 1.2 0 = . 0 1.2

 0 e0.3(1−s)

,e

A(t1 +τ )

  1.4333 0 = , 0 1.4333

Now we use (2.66) to estimate M. For this purpose, we need to obtain Wτ [0, t1 ] and then derive Wτ [0, t1 ]−1 . Obviously, A = A , B = B  , B1 = B1 , and C = C  = I . Hence, the delay Grammian matrix (2.56) has the following explicit form:  t1 eA(t1 −s) eτB1 (t1 −τ −s) CCeτB1 (t1 −τ −s) eA(t1 −s) ds Wτ [0, t1 ] = 0  t1 eA(t1 −s) eτB1 (t1 −τ −s) eτB1 (t1 −τ −s) eA(t1 −s) ds = 0

 =

0

1

B (0.8−s) B1 (0.8−s) A(1−s) e0.2 e ds

eA(1−s) e0.21

= W1 + W2 + W3 + W4 + W5 , where

(0.6 − s)2 eA(1−s) I + B1 (0.8 − s) + B12 2! 0 2 3 4 (0.4 − s) (0.2 − s) + B14 + B13 eA(1−s) ds, 3! 4!

  0.4 (0.6 − s)2 (0.4 − s)3 2 W2 = eA(1−s) I + B1 (0.8 − s) + B12 + B13 2! 3! 0.2 

W1 =

0.2

× eA(1−s) ds,

Delay systems Chapter | 2

43



(0.6 − s)2 2 A(1−s) eA(1−s) I + B1 (0.8 − s) + B12 e ds, 2! 0.4

2  0.8 A(1−s) I + B1 (0.8 − s) eA(1−s) ds, e W4 = 

W3 =

0.6

0.6 1

 W5 =

eA(1−s) I 2 eA(1−s) ds.

0.8

By computation, one can get     0.4438 0 0.8099 0 , W2 = , W1 = 0 0.4438 0 0.8099     1.1118 0 1.3559 0 W3 = , W4 = , 0 1.1118 0 1.3559   1.3702 0 W5 = . 0 1.3702 Therefore, we obtain 

 5.0916 0 Wτ [0, 1] = , 0 5.0916 and we further derive 

−1

Wτ [0, 1]

 0.1964 0 = . 0 0.1964

(2.72)

Consequently, we obtain  √ M = Wτ [0, 1]−1  = 0.1964 = 0.4432. Hence, W satisfies assumption [H1 ]. Further, it is easy to see that for any x(t), y(t) ∈ R2 and t ∈ J1 ,  1 f (t, x) − f (t, y) = (t + 0.1) (x1 (t) − y1 (t))2 + (x2 (t) − y2 (t))2 10 1 ≤ (t + 0.1)x − y. 10 Hence, f satisfies assumption [H2 ], where we set Lf (·) = Obviously, Lf Lq (J1 ,R+ ) =

1 10 (

1.1q+1 −0.1q+1 q+1

0. Next, C = 1, Lf L2 (J1 ,R+ ) = 0.0661, and

1 q

·+0.1 10

∈ Lq (J1 , R+ ).

) and Rf = sup f (t, 0) = t∈J1

44 Stability and Controls Analysis for Delay Systems 1

1 M2 = [ 2N (e2N − 1)] 2 Lf L2 (J1 ,R+ ) = 0.0942 when we choose p = q = 2. Hence,



 M 0.4432 0.4884 (e γ = M2 1 + (eN t1 − 1)C = 0.0942 1 + − 1) N 0.4884 = 0.1480 < 1,

which guarantees that condition (2.68) holds. Thus all conditions of Theorem 2.13 are satisfied. Hence, system (2.70) is relatively controllable on [0, 1]. Example 2.4. Consider the relative controllability of system (2.70) (with f (·, x) ≡ 0) on J1 , where A, B, B1 , C are defined in Example 2.3. According to Theorem 2.12, we can know that system (2.70) is relatively controllable when f (·, x) = 0. Further, keeping in mind (2.58), one can get 

 −0.6486 , η = x1 − −0.4865

x1 ∈ R2 .

By using the selection form of the control in (2.57), we arrive at B  (t1 −τ −t) A (t1 −t)

u(t) = C  eτ 1

e

B (0.8−t) A(1−t)

= e0.21

e

Wτ [0, t1 ]−1 η

Wτ [0, 1]−1 η

⎧ 2 3 4 ⎪ 2 (0.6 − t) 3 (0.4 − t) 4 (0.2 − t) ⎪ ⎪ I + B + B + B (0.8 − t) + B 1 ⎪ 1 1 1 ⎪ 2! 3! 4! ⎪ ⎪ ⎪ ⎪ A(1−t) −1 ⎪ Wτ [0, 1] η, 0 ≤ t < 0.2, ×e ⎪ ⎪ ⎪ ⎪

⎪ 2 3 ⎪ ⎪ 2 (0.6 − t) 3 (0.4 − t) ⎪ + B1 I + B1 (0.8 − t) + B1 ⎪ ⎪ ⎪ 2! 3! ⎪ ⎪ ⎪ A(1−t) −1 ⎨ Wτ [0, 1] η, 0.2 ≤ t < 0.4, ×e = 2 ⎪ ⎪ 2 (0.6 − t) ⎪ ⎪ I + B (0.8 − t) + B 1 ⎪ 1 ⎪ 2! ⎪ ⎪ ⎪ ⎪ A(1−t) −1 ⎪ Wτ [0, 1] η, 0.4 ≤ t < 0.6, ×e ⎪ ⎪ ⎪ 

⎪ ⎪ ⎪ ⎪ ⎪ I + B1 (0.8 − t) × eA(1−t) Wτ [0, 1]−1 η, 0.6 ≤ t < 0.8, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ I × eA(1−t) Wτ [0, 1]−1 η, 0.8 ≤ t < 1, where B1 is given in (2.71) and Wτ [0, 1]−1 is given in (2.72).

Delay systems Chapter | 2

45

2.2.1.4 Conclusions The purpose of this contribution is to develop a controllability method for linear and semilinear delay controlled systems with linear parts defined by permutable matrices. In order to achieve this purpose, a representation of solutions is used with the help of a delayed matrix exponential. Such an approach leads to new criteria for the relative controllability of our issues by constructing a delay Grammian matrix and applying the fixed point method. The results in this part are motivated from [87].

2.3 Iterative learning control 2.3.1 Iterative learning control for delay differential systems In this section, we discuss ILC for time-delay systems via initial state learning. More precisely, we study the following linear controlled systems with pure delay: ⎧ ⎪ ⎨x˙k (t) = Axk (t) + Bxk (t − τ ) + uk (t), t ∈ [0, T ], xk (t) = ϕ(t), − τ ≤ t ≤ 0, τ > 0, (2.73) ⎪ ⎩ yk (t) = Cxk (t) + Duk (t). Let ϕ ∈ Cτ1 := C 1 ([−τ, 0], Rn ), let A and B be two n × n matrices such that AB = BA, let C and D be two m × n matrices, let k denote the k-th learning iteration, and let the variables xk , uk ∈ Rn , and yk ∈ Rm denote the state, input, and output, respectively. By [1, Corollary 2.2], we derive that the state xk (·) has the form  0 A(t+τ ) B1 t xk (t) = e eτ ϕ(−τ ) + eA(t−τ −s) eτB1 (t−τ −s) eAτ [ϕ  (s) − Aϕ(s)]ds −τ  t eA(t−τ −s) eτB1 (t−τ −s) eAτ uk (s)ds, (2.74) + 0

where eτBt is defined in (2.4) and B1 = e−Aτ B. Let yd be a desired trajectory and set ek = yd − yk , which denotes the output error, and δuk = uk+1 − uk . For system (2.73), we consider the open-loop P-type ILC updating law uk+1 (t) = uk (t) + Po ek (t).

(2.75)

For system (2.73) with D = Θ, we consider the open-loop D-type ILC updating law uk+1 (t) = uk (t) + Do e˙k (t), where Po and Do ∈ Rn×m are learning gain matrices.

(2.76)

46 Stability and Controls Analysis for Delay Systems

The main objective is to use a delayed exponential matrix to generate the control input uk . Then the time-delay system output yk is tracking the iteratively varying reference trajectories yd by adopting P-type ILC and D-type ILC as k → ∞ uniformly on [0, T ] in the λ-norm sense. Here we point out that our method is different from the method given in the previous reference. However, we obtain the same convergence results. Our method relies on a direct formula solution, so it is constructive. Lemma 2.7. (see [88, Chapter 5.6]) For a matrix A ∈ Rn×n and ∀ > 0, there is a norm  ·  on Rn×n such that A ≤ ρ(A) + , where ρ(A) denotes the spectral radius of matrix A. Next, we give an alternative formula to compute the solution of the linear system with pure delay, which is a direct corollary of [1, Corollary 2.2]. Lemma 2.8. Let f : J → Rn be a continuous function. The solution x ∈ C 1 (J, Rn ) of  x(t) ˙ = Ax(t) + Bx(t − τ ) + f (t), t > 0, τ ≥ 0, (2.77) x(t) = ϕ(t), − τ ≤ t ≤ 0, has the form x(t) = eA(t+τ ) eτB1 t ϕ(−τ )  0 j −1 # (t − lτ − s)l Aτ  + eA(t−τ −s) B1l e [ϕ (s) − Aϕ(s)]ds l! −τ l=0  t−j τ j j (t − j τ − s) Aτ  e [ϕ (s) − Aϕ(s)]ds eA(t−τ −s) B1 + j! −τ j −1  t−iτ # (t − iτ − s)i Aτ e f (s)ds, eA(t−τ −s) B1i (2.78) + i! 0 i=0

where B1 = e−Aτ B and (j − 1)τ ≤ t < j τ , j = 1, 2, · · · n. Proof. According to [1, (21)], we know the solution of system (2.77) has the form x(t) = eA(t+τ ) eτB1 t ϕ(−τ )  0 + eA(t−τ −s) eτB1 (t−τ −s) eAτ [ϕ  (s) − Aϕ(s)]ds −τ  t + eA(t−τ −s) eτB1 (t−τ −s) eAτ f (s)ds. 0

(2.79)

Delay systems Chapter | 2

47

Without loss of generality, we consider (j − 1)τ ≤ t < j τ , j = 1, 2, · · · n. Next, we combine the formula of the delayed matrix exponential (2.4) with (2.79) to prove the result. We divide our proof into two steps. Step 1. We prove that 

0

eA(t−τ −s) eτB1 (t−τ −s) eAτ [ϕ  (s) − Aϕ(s)]ds

−τ



=

0

eA(t−τ −s)

−τ

B1l

l=0



+

j −1 #

t−j τ

−τ

(t − lτ − s)l Aτ  e [ϕ (s) − Aϕ(s)]ds l!

eA(t−τ −s) B1

j

(t − j τ − s)j Aτ  e [ϕ (s) − Aϕ(s)]ds. j!

(2.80)

Due to the fact that −τ < s < 0, we obtain t − τ < t − τ − s < t and (j − 2)τ < t − τ < t − τ − s < t < j τ . When −τ < s < t − j τ , we have (j − 1)τ < t −τ −s < t < j τ . When t −j τ < s < 0, we have (j −2)τ < t −τ < t −τ −s < (j − 1)τ . Hence 

0

−τ

eA(t−τ −s) eτB1 (t−τ −s) eAτ [ϕ  (s) − Aϕ(s)]ds

=



t−j τ

e

−τ

A(t−τ −s)

j # l=0



0

+

eA(t−τ −s)

t−j τ

 =

0

j −1 # l=0



t−j τ

−τ

j −1 # l=0

eA(t−τ −s)

−τ

+

B1l

B1l

(t − lτ − s)l Aτ  e [ϕ (s) − Aϕ(s)]ds l!

B1l

(t − lτ − s)l Aτ  e [ϕ (s) − Aϕ(s)]ds l!

(t − lτ − s)l Aτ  e [ϕ (s) − Aϕ(s)]ds l!

eA(t−τ −s) B1

j

(t − j τ − s)j Aτ  e [ϕ (s) − Aϕ(s)]ds. j!

Step 2. We check that  0

t

eA(t−τ −s) eτB1 (t−τ −s) eAτ f (s)ds

=

j −1  # i=0 0

t−iτ

eA(t−τ −s) B1i

(t − iτ − s)i Aτ e f (s)ds. i!

(2.81)

Due to the fact that 0 < s < t, we obtain −τ < t − τ − s < t − τ . When t − (i + 1)τ < s < t − iτ , we have (i − 1)τ < t − τ − s < iτ , i = 0, 1, · · · , j − 2. When

48 Stability and Controls Analysis for Delay Systems

0 < s < t − (j − 1)τ , we have (j − 2)τ < t − τ − s < t − τ < (j − 1)τ . Hence  0

t

eA(t−τ −s) eτB1 (t−τ −s) eAτ f (s)ds

=

j −2  #

t−iτ

i=0 t−(i+1)τ  t−(j −1)τ

+

0

=

j −2  #

t−iτ

eA(t−τ −s) eτB1 (t−τ −s) eAτ f (s)ds eA(t−τ −s) eτB1 (t−τ −s) eAτ f (s)ds (t − lτ − s)l Aτ e f (s)ds l!

B1l

(t − lτ − s)l Aτ e f (s)ds l!

eA(t−τ −s) B1l

(t − lτ − s)l Aτ e f (s)ds l!

eA(t−τ −s) B1l

(t − lτ − s)l Aτ e f (s)ds l!

i=0 t−(i+1)τ

l=0

t−(j −1)τ

j −1 #



+

eA(t−τ −s)

0

=

j −2 # i  t−iτ # t−(i+1)τ

i=0 l=0

+

j −1  t−(j −1)τ # l=0

=

j −2  #

+

0 t−lτ

t−(j −1)τ

l=0

j −1  # l=0

=

j −2  #

0 t−iτ

+

0

j −1  # i=0 0

t−iτ

l=0

eA(t−τ −s) B1l

t−(j −1)τ

(t − lτ − s)l Aτ e f (s)ds l!

eA(t−τ −s) B1l

eA(t−τ −s) B1i

i=0 0  t−(j −1)τ

=

i #

B1l

eA(t−τ −s)

(t − iτ − s)i Aτ e f (s)ds i! j −1 (t

eA(t−τ −s) B1

eA(t−τ −s) B1i

(t − lτ − s)l Aτ e f (s)ds l!

− (j − 1)τ − s)j −1 Aτ e f (s)ds (j − 1)!

(t − iτ − s)i Aτ e f (s)ds. i!

Linking (2.79), (2.80), and (2.81), one can get the result (2.78). The proof is finished.

2.3.1.1 Convergence analysis of P-type In this section, we give the first convergence result of P-type.

Delay systems Chapter | 2

49

Theorem 2.14. Let yd (t), t ∈ [0, T ], be a desired trajectory for system (2.73). If ρ(I − DPo ) < 1, then the P-type ILC law (2.75) guarantees lim yk (t) = yd (t) k→∞

uniformly on [0, T ].

Proof. Without loss of generality, we consider (j − 1)τ ≤ t < j τ , j = 1, 2, · · · , n. Linking (2.73) and (2.74), we have ek+1 (t) − ek (t) = yk (t) − yk+1 (t)  t = −C eA(t−τ −s) eτB1 (t−τ −s) eAτ δuk (s)ds − Dδuk (t). 0

According to (2.75) and Lemma 2.8, we have ek+1 (t) = (I − DPo )ek (t) j −1  t−iτ # (t − iτ − s)i Aτ e δuk (s)ds. −C eA(t−τ −s) B1i i! 0

(2.82)

i=0

Further, by Lemma 2.7 we know that for a given  > 0, there is a norm  ·  on Rn such that I − DPo  ≤ ρ(I − DPo ) + . Using (2.82), we have

 ek+1 (t) ≤ ρ(I − DPo ) +  ek (t) + C

j −1  #

t−iτ

i=0 0

eA(t−τ −s) B1i 

(t − iτ − s)i Aτ e δuk (s)ds. i!

Hence, we obtain

 ek+1 (t) ≤ ρ(I − DPo ) +  ek (t) + C

j −1  #

t−iτ

eA(t−s) B1 i

i=0 0

(t − iτ − s)i δuk (s)ds. i! (2.83)

For fixed i, i = 0, 1, · · · , j − 1, we have 

t−iτ

eA(t−s) B1 i

0

B1 i = i!

 0

t−iτ

(t − iτ − s)i δuk (s)ds i!

eA(t−s) (t − iτ − s)i δuk (s)ds

50 Stability and Controls Analysis for Delay Systems

≤

B1 i At e i!



t−iτ

e(λ−A)s (t − iτ − s)i dsδuk λ .

(2.84)

0

For any λ > A, we apply integration by parts via mathematical induction to derive 

 t−iτ 1 e (t − iτ − s) ds = (t − iτ − s)i de(λ−A)s λ − A 0 0  t−iτ 1 i i =− (t − iτ − s)i−1 de(λ−A)s (t − iτ ) + λ − A (λ − A)2 0 1 i (t − iτ )i − =− (t − iτ )i−1 λ − A (λ − A)2  t−iτ i(i − 1) + (t − iτ − s)i−2 de(λ−A)s (λ − A)3 0 .. . i 1 (t − iτ )i−1 − · · · =− (t − iτ )i − λ − A (λ − A)2

 i! i! (λ−A)(t−iτ ) e (t − iτ ) + −1 − (λ − A)i (λ − A)i+1 t−iτ

=−

(λ−A)s

i # p=0

i

i!(t − iτ )i−p i!e(λ−A)(t−iτ ) + . (i − p)!(λ − A)p+1 (λ − A)i+1

(2.85)

Linking (2.83), (2.84), and (2.85), it is not difficult to get

 j −1 # B1 i At e ek+1 (t) ≤ ρ(I − DPo ) +  ek (t) + C i! i=0

× −

i # p=0

 i!(t − iτ )i−p i!e(λ−A)(t−iτ ) δuk λ . + (i − p)!(λ − A)p+1 (λ − A)i+1 (2.86)

By (2.86) and (2.75), we have ek+1 (t)e

−λt

 j −1 # B1 i (A−λ)t e ≤ ρ(I − DPo ) +  ek (t)e−λt + CPo  i!

× −

i=0

i # p=0

− iτ )i−p

 i!e(λ−A)(t−iτ )

i!(t + (i − p)!(λ − A)p+1 (λ − A)i+1

ek λ ,

Delay systems Chapter | 2

51

and taking the λ-norm, we arrive at



ek+1 λ ≤ ρ(I − DPo ) +  ek λ + CPo  ×

# i p=0

j −1 # B1 i i=0

i!

 i! max{1, T p } i! ek λ . + (i − p)!(λ − A)p+1 (λ − A)i+1

 Since ρ(I − DPo ) < 1, for any  ∈ 0, 1−ρ(I4−DPo ) and λ > A sufficiently large, we have ek+1 λ ≤ (ρ(I − DPo ) + 2)ek λ , which implies lim ek λ = 0, since ρ(I − DPo ) + 2 < 1. In addition, ek  ≤ k→∞

eλT ek λ . Hence, lim ek  = 0. The proof is completed. k→∞

Remark 2.5. We use the delayed matrix exponential method to obtain convergence of the P-type ILC algorithm. Next, we applied the norm  · λ just for a technical reason to get uniform convergence only under condition ρ(I −DPo ) <  1 in the end of the above proof. Moreover, fixing  ∈ 0, 1−ρ(I4−DPo ) , the smallest suitable λ > A is given by an equation CPo 

N −1 # i=0



i B1 i # i! max{1, T p } i! + = , i! (i − p)!(λ − A)p+1 (λ − A)i+1 p=0

which is rather awkward to solve.

2.3.1.2 Convergence analysis of D-type In this section, we discuss the ILC convergence of D-type. Theorem 2.15. Let yd (t), t ∈ [0, T ], be a desired trajectory for system (2.73) with D = Θ. If ρ(I − CDo ) < 1 and ek (0) = 0, k = 1, 2, · · · , then the D-type ILC law (2.76) guarantees lim yk (t) = yd (t) uniformly on [0, T ]. k→∞

Proof. First, we consider (j − 1)τ ≤ t < j τ , j = 2, 3, · · · , N . By (2.74) and (2.76), we have e˙k+1 (t) − e˙k (t) = C[x˙k (t) − x˙k+1 (t)] = CA[xk (t) − xk+1 (t)] + CB[xk (t − τ ) − xk+1 (t − τ )] + C[uk (t) − uk+1 (t)]  t = −CA eA(t−τ −s) eτB1 (t−τ −s) eAτ δuk (s)ds 0

52 Stability and Controls Analysis for Delay Systems

 − CB

t−τ

0

eA(t−2τ −s) eτB1 (t−2τ −s) eAτ δuk (s)ds − CDo e˙k (t).

So we have  t e˙k+1 (t) = (I − CDo )e˙k (t) − CA eA(t−τ −s) eτB1 (t−τ −s) eAτ δuk (s)ds 0  t−τ − CB eA(t−2τ −s) eτB1 (t−2τ −s) eAτ δuk (s)ds. 0

Similar to the proof of Theorem 2.14, one can derive e˙k+1 (t) = (I − CDo )e˙k (t) j  t−(i−1)τ # (t − (i − 1)τ − s)i−1 Aτ e δuk (s)ds eA(t−τ −s) B1i−1 − CA (i − 1)! 0 i=1

− CB

j  #

t−(l−1)τ

0

l=2

eA(t−2τ −s) B1l−2

(t − (l − 1)τ − s)l−2 Aτ e δuk (s)ds. (l − 2)!

Obviously, we have e˙k+1 (t)e

−λt

 j # B1 i−1 (A−λ)t −λt e ≤ ρ(I − CDo ) +  e˙k (t)e + CA (i − 1)! i=1  t−(i−1)τ × eA(λ−s) (t − (i − 1)τ − s)i−1 dsδuk λ 0

+ CB

j # B1 l−2

l=2 t−(l−1)τ

 ×

(l − 2)!

e(A−λ)t

eA(λ−τ −s) (t − (l − 1)τ − s)l−2 dsδuk λ .

0

(2.87) By analogy to the computation in (2.84)–(2.86), inequality (2.87) becomes 



e˙k+1 (t)e−λt ≤ ρ(I − CDo ) +  e˙k (t)e−λt + W1 + W2 Do e˙k λ , (2.88) where W1 = CA

i j # B1 i−1 # i=1

(i − 1)!

p=1

(i − 1)!T i−p (i − 1)! + p (i − p)!(λ − A) (λ − A)i



Delay systems Chapter | 2

53

and W2 = CB

l j # B1 l−2 # l=2

(l − 2)!

q=2

 (l − 2)!T l−q (l − 2)! . + (l − q)!(λ − A)q−1 (λ − A)l−1

If j = 1, which means 0 ≤ t < τ , then by (2.74) and (2.76), we have e˙k+1 (t) − e˙k (t) = C[x˙k (t) − x˙k+1 (t)] = CA[xk (t) − xk+1 (t)] + CB[xk (t − τ ) − xk+1 (t − τ )] + C[uk (t) − uk+1 (t)]  t = −CA eA(t−τ −s) eτB1 (t−τ −s) eAτ δuk (s)ds − CDo e˙k (t). 0

We can repeat the above arguments to arrive at (2.88) with W2 = 0 and W1 = 2CA λ−A . Hence (2.88) holds on the whole [0, T ], which implies 

 e˙k+1 λ ≤ ρ(I − CDo ) +  + W1 + W2 Do  e˙k λ . Note that by ρ(I − CDo ) < 1, we obtain lim e˙k λ = 0. Thus, e˙k  ≤ k→∞

e˙λT e˙k λ . So e˙k  → 0 as k → ∞. Due to the fact that ek (0) = 0, we get ek  ≤ T e˙k , and consequently we find that ek  → 0 as k → ∞. The proof is completed. Remark 2.6. Since ek (0) = yd (0) − yk (0) = yd (0) − Cxk (0) = yd (0) − Cϕ(0), we need yd (0) = Cϕ(0), which gives a compatibility condition for ϕ. Since yd (0) is arbitrary, we need C to be surjective.

2.3.1.3 Numerical examples and discussion In this part, four numerical examples are presented to demonstrate the validity of the designed method. Next, noting the fact that ek  ≤ eλT ek λ , we obtain ek  → 0 as k → ∞, i.e., lim sup ek (t) = 0, which yields lim ek (t) = 0 k→∞ t∈J

k→∞

for ∀t ∈ J . In order to simulate the tracking error trajectory of every moment of each time, we adopt the vector 2-norm in our simulations. Example 2.5. Consider the following system: ⎧ ⎪ ⎨x˙k (t) = xk (t) + xk (t − 0.5) + uk (t), x(t) ∈ R, t ≥ 0, xk (t) = t, − 0.5 ≤ t ≤ 0, ⎪ ⎩ yk (t) = xk (t) + 0.3uk (t).

(2.89)

54 Stability and Controls Analysis for Delay Systems

The P-type ILC is set as uk+1 (t) = uk (t) + ek (t). The original reference trajectory is set as yd (t) = 2 sin(4πt), t ∈ [0, 1], where t ∈ [0, 1], τ = 0.5, ϕ(t) = t. Set n = m = 1, A = 1, B = 1, C = 1, and D = 0.3. It is not difficult to find that B1 = e−Aτ B = e−0.5 and ⎧ −0.5 t, t ∈ [0, 0.5], ⎨1 + e t 2 e0.5 = ⎩1 + e−0.5 t + e−1 (t − 0.5) , t ∈ [0.5, 1]. 2! Next, setting Po = 1, one has ρ(1 − DPo ) = 0.7 < 1. Thus, all conditions of Theorem 2.14 are satisfied, and the output yk (t) uniformly converges to yd (t) for t ∈ [0, 1]. The upper figure of Fig. 2.8 shows system (2.89)’s 10th output yk and the reference trajectory yd . The lower figure of Fig. 2.8 shows the tracking error in each iteration.

FIGURE 2.8

The system’s 10th output and the tracking error for (2.89).

Example 2.6. Consider the following system: ⎧ ⎪ ⎨x˙k (t) = xk (t) + xk (t − 0.5) + uk (t), x(t) ∈ R, t ≥ 0, xk (t) = t, − 0.5 ≤ t ≤ 0, ⎪ ⎩ yk (t) = xk (t).

(2.90)

Delay systems Chapter | 2

55

The D-type ILC is set as uk+1 (t) = uk (t) + 0.3e˙k (t). The original reference trajectory is the same as in Example 2.5. Set t ∈ [0, 1], τ = 0.5, ϕ(t) = t, and n = m = 1, A = 1, B = 1, C = 1, and D = 0. It is not t are the same as in Example 2.5. Next, setting difficult to see that B1 and e0.5 Do = 0.3, it is easy to check ρ(1 − CDo ) = 0.7 < 1. Thus, all conditions of Theorem 2.15 are satisfied. The upper figure of Fig. 2.9 shows system (2.90)’s 10th output yk and the reference trajectory yd . The lower figure of Fig. 2.9 shows the tracking error in each iteration.

FIGURE 2.9 The system’s 10th output and the tracking error for (2.90).

Example 2.7. Consider the following system: ⎧ 2 ⎪ ⎨x˙k (t) = xk (t) + xk (t − 0.5) + uk (t), xk (t), uk (t) ∈ R , t ≥ 0, xk (t) = (et , et ) , − 0.5 ≤ t ≤ 0, ⎪ ⎩ yk (t) = (1, 2)xk (t) + (2, 1)uk (t). The P-type ILC is set as  uk+1 (t) = uk (t) +

 0.5 −0.4

ek (t).

The original reference trajectory is yd (t) = 2 cos(4πt) + 3t 2 , t ∈ [0, 1].

(2.91)

56 Stability and Controls Analysis for Delay Systems

Set t ∈ [0, 1], τ = 0.5, ϕ(t) = (et , et ) , and n = 2, m = 1, A = B = I , C = (1, 2), and D = (2, 1). It is not difficult to see that   0.6065 0 −Aτ B= B1 = e 0 0.6065 and ⎧ I, t ∈ [−0.5, 0], ⎪ ⎪ ⎪ ⎨ I + B1 t, t ∈ [0, 0.5], t e0.5 = ⎪ ⎪ (t − 0.5)2 ⎪ ⎩I + B1 t + B12 , t ∈ [0.5, 1]. 2! Next, setting Po = (0.5, −0.4) , one has ρ(I − DPo ) = 0.8 < 1. Thus, all conditions of Theorem 2.14 are satisfied. Then the output yk (t) uniformly converges to yd (t) for t ∈ [0, 1]. The upper figure of Fig. 2.10 shows system (2.91)’s 10th output yk and the reference trajectory yd . The lower figure of Fig. 2.10 shows the tracking error in each iteration.

FIGURE 2.10

The system’s 10th output and the tracking error for (2.91).

Example 2.8. Consider the following system: ⎧ 2 ⎪ ⎨x˙k (t) = xk (t) + xk (t − 0.5) + uk (t), xk (t), uk (t) ∈ R , t ≥ 0, xk (t) = (t, t) , − 0.5 ≤ t ≤ 0, ⎪ ⎩ yk (t) = (1, 2)xk (t).

(2.92)

Delay systems Chapter | 2

57

The D-type ILC is set as  uk+1 (t) = uk (t) +

 1 −0.4

e˙k (t).

To satisfy the condition ek (0) = 0, the original reference trajectory is chosen as yd (t) = 2 sin(8πt) + 3t 2 , t ∈ [0, 1]. It is easy to check that the conditions of Theorem 2.15 are satisfied. Then the output yk (t) uniformly converges to yd (t) for t ∈ [0, 1]. The upper figure of Fig. 2.11 shows system (2.92)’s 10th output yk and the reference trajectory yd . The lower figure of Fig. 2.11 shows the tracking error in each iteration.

FIGURE 2.11 The system’s 10th output and the tracking error for (2.92).

2.3.1.4 Conclusions We present open-loop P- and D-type asymptotic convergence results for timedelay controlled systems via delayed matrix exponentials in the λ-norm sense by virtue of the spectral matrix radius. Moreover, the simulations show that our ILC strategies are effective. The results in this section are motivated from [15].

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Chapter 3

Oscillating delay systems 3.1 Finite time stability 3.1.1 Finite time stability of oscillating delay systems In this chapter, we study finite time stability of the following second order linear differential equations with pure delay term: 

x(t) ¨ + Ω 2 x(t − τ ) = 0, τ > 0, t ∈ J := [0, T ], x(t) ≡ ϕ(t), x(t) ˙ ≡ ϕ(t), ˙ − τ ≤ t ≤ 0,

(3.1)

where x ∈ Rn , τ is the time delay, ϕ is an arbitrary twice continuously differentiable vector function, T is a prefixed positive number, and Ω is an n × n nonsingular matrix. Note that Khusainov et al. [5] give a new representation of the solution for (3.1) as follows: x(t) = cosτ Ωtϕ(−τ ) + Ω −1 sinτ Ωt ϕ(−τ ˙ )  0 + Ω −1 sinτ Ω(t − τ − s)ϕ(s)ds, ¨ −τ

(3.2)

where cosτ Ωt is called the delayed matrix cosine of polynomial degree 2k ([5, Definition 1]) on the intervals (k − 1)τ ≤ t < kτ formulated by ⎧ ⎪ Θ, − ∞ < t < −τ, ⎪ ⎪ ⎪ ⎪ I, − τ ≤ t < 0, ⎪ ⎪ ⎪ ⎨ 2 t I − Ω 2 2! , 0 ≤ t < τ, cosτ Ωt = ⎪ .. ⎪ ⎪ ... ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ I − Ω 2 t 2 + . . . + (−1)k Ω 2k [t−(k−1)τ ]2k , (k − 1)τ ≤ t < kτ 2! (2k)! (3.3) Stability and Controls Analysis for Delay Systems. https://doi.org/10.1016/B978-0-32-399792-8.00009-8 Copyright © 2023 Elsevier Inc. All rights reserved.

59

60 Stability and Controls Analysis for Delay Systems

and sinτ Ωt is called the delayed matrix sine of polynomial degree 2k + 1 ([5, Definition 2]) on the intervals (k − 1)τ ≤ t < kτ formulated by ⎧ ⎪ Θ, − ∞ < t < −τ, ⎪ ⎪ ⎪ ⎪ Ω(t + τ ), − τ ≤ t < 0, ⎪ ⎪ ⎪ ⎨ 3 Ω(t + τ ) − Ω 3 t3! , 0 ≤ t < τ, sinτ Ωt = ⎪ . . ⎪ ⎪ .. .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Ω(t + τ ) − . . . + (−1)k Ω 2k+1 [t−(k−1)τ ]2k+1 , (k − 1)τ ≤ t < kτ. (2k+1)! (3.4) Obviously, when τ = 0, cosτ Ωt and sinτ Ωt reduce to the matrix cosine function cos Ωt and the matrix sine function sin Ωt, respectively, which are given by the following formal matrix series: cos Ωt = 1 − Ω 2

t2 t 2k + · · · + (−1)k Ω 2k + ··· 2! (2k)!

and sin Ωt = Ω

t t3 t 2k+1 − Ω 3 + · · · + (−1)k Ω 2k+1 + ··· . 1! 3! (2k + 1)!

By Gantmakher [89, p. 123], x(t) = x0 cos Ωt + Ω −1 x˙0 sin Ωt, provided Ω −1 exists, is a solution of the second order differential system x(t) ¨ + Ω 2 x(t) = 0, t ≥ 0, n n x(0) = x0 ∈ R , x(0) ˙ = x˙0 ∈ R . Set J = [0, t1 ], t1 > 0. Denote C 2 (J, Rn ) = {x ∈ C(J, Rn ) : x¨ ∈ C(J, Rn )} endowed with the norm xC 2 (J ) = max{x(t), x(t), ˙ x(t)}. ¨ Let X, Y be t∈J

two Banach spaces and let Lb (X, Y ) be the space of bounded linear operators from X to Y . Now, Lp (J, Y ) denotes the Banach space of functions f : J → Y which are Bochner integrable normed by f Lp (J,Y ) for some 1 < p < ∞. In addition, we let ϕ = max ϕ(s), ϕ ˙ = max ϕ(s), ˙ and ϕ ¨ = max ϕ(s). ¨

s∈[−τ,0]

s∈[−τ,0]

s∈[−τ,0]

We need the following rules of differentiation for the delayed matrix cosine of polynomial degree 2k on the interval [(k − 1)τ, kτ ) and sine of polynomial degree 2k + 1 on the interval [(k − 1)τ, kτ ) defined in (3.3) and (3.4), respectively. Lemma 3.1. ([5, Lemmas 1 and 2]) The following rules of differentiation are true for the matrix functions (3.3) and (3.4): d cosτ Ωt = −Ω sinτ Ω(t − τ ), dt

d sinτ Ωt = Ω cosτ Ωt. dt

Oscillating delay systems Chapter | 3

61

Remark 3.1. For simplification of the next computation, one can divide the 0 term −τ sinτ Ω(t − τ − s)ϕ(s)ds ¨ in (3.2) into the following form according to the subintervals [(k − 1)τ, kτ ):  0 sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ

=



t−kτ −τ

 sinτ Ω(t − τ − s)ϕ(s)ds ¨ +

0

sinτ Ω(t − τ − s)ϕ(s)ds. ¨

t−kτ

Obviously, sinτ Ω(t − τ − s) has different formulas in different subintervals [(k − 1)τ, kτ ) by (3.4). By Remark 3.1, the solution (3.2) of system (3.1) can be expressed in the following form: x(t) = cosτ Ωtϕ(−τ ) + Ω −1 sinτ Ωt ϕ(−τ ˙ )  t−kτ + Ω −1 sinτ Ω(t − τ − s)ϕ(s)ds ¨ + Ω −1

−τ 0



sinτ Ω(t − τ − s)ϕ(s)ds ¨

(3.5)

t−kτ

for (k − 1)τ ≤ t ≤ kτ . Observing the solution (3.2) involves ϕ, ¨ which seems to be a slightly stronger requirement than initial conditions. Remark 3.2. In order to obtain some alternative formulas, one can apply integration by parts via Lemma 3.1 to derive  0 sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ



= sinτ Ω(t − τ )ϕ(0) ˙ − sinτ Ωt ϕ(−τ ˙ )+Ω

0

−τ

cosτ Ω(t − τ − s)ϕ(s)ds. ˙

Then the solution (3.2) can be expressed as x(t) = cosτ Ωtϕ(−τ ) + Ω −1 sinτ Ω(t − τ )ϕ(0) ˙  0 cosτ Ω(t − τ − s)ϕ(s)ds. ˙ + −τ

(3.6)

If we use integration by parts again for the integral part of (3.6), then we have  0 cosτ Ω(t − τ − s)ϕ(s)ds ˙ −τ

= cosτ Ω(t − τ )ϕ(0) − cosτ Ωtϕ(−τ ) − Ω



0

−τ

sinτ Ω(t − 2τ − s)ϕ(s)ds,

62 Stability and Controls Analysis for Delay Systems

which implies that (3.6) can be expressed as ˙ x(t) = cosτ Ω(t − τ )ϕ(0) + Ω −1 sinτ Ω(t − τ )ϕ(0)  0 −Ω sinτ Ω(t − 2τ − s)ϕ(s)ds. −τ

(3.7)

Definition 3.1. (see [24, Definition 2.1]) System (3.1) satisfying initial conditions x(t) ≡ ϕ(t) and x(t) ˙ ≡ ϕ(t) ˙ for −τ ≤ t ≤ 0 is finite time stable with respect to {0, J, δ, , τ } if and only if γ < δ,

(3.8)

which implies x(t) < , ∀ t ∈ J, where γ = max{ϕ, ϕ, ˙ ϕ} ¨ denotes the initial time of observation of the system. In addition, δ,  are real positive numbers. Using the form of cosτ Ωt and sinτ Ωt one can prove the following two lemmas, which will be widely used in the sequel. Lemma 3.2. For any t ∈ [(k − 1)τ, kτ ), k = 0, 1, · · · , n, the following formula is true:  cosτ Ωt ≤ cosh(Ωt). Proof. Using the form of (3.3), one can calculate that t2 (t − τ )4 [t − (k − 1)τ ]2k + Ω4 + · · · + Ω2k 2! 4! (2k)! 2 4 2k t t t ≤ 1 + Ω2 + Ω4 + · · · + Ω2k 2! 4! (2k)! ∞ 2k (Ωt) = cosh(Ωt). ≤ (2k)!

 cosτ Ωt ≤ 1 + Ω2

k=0

The proof is completed. Lemma 3.3. For any t ∈ [(k − 1)τ, kτ ), k = 0, 1, · · · , n, the following formula is true:  sinτ Ωt ≤ sinh[Ω(t + τ )]. Proof. Using the form of (3.4), we get  sinτ Ωt ≤ Ω(t + τ ) + Ω3

t3 [t − (k − 1)τ ]2k+1 + · · · + Ω2k+1 3! (2k + 1)!

Oscillating delay systems Chapter | 3

≤ Ω(t + τ ) + Ω3 ≤

(t + τ )3 (t + τ )2k+1 + · · · + Ω2k+1 3! (2k + 1)!

∞ [Ω(t + τ )]2k+1 k=0

(2k + 1)!

63

= sinh[Ω(t + τ )].

The proof is finished. Remark 3.3. When t ∈ (−∞, −τ ), we can get  cosτ Ωt =  sinτ Ωt = 0 by the form (3.3) and (3.4). As is well known, the exponential forms of hyperbolic functions cosh t and sinh t are defined as follows: cosh t =

et + e−t et − e−t , sinh t = , t ∈ R. 2 2

Then for all t ∈ R, sinh t ≤ cosh t holds and the larger the value of t, the closer cosh t and sinh t. When t → +∞, we get cosh t = sinh t. In addition, both cosh t and sinh t are nonnegative, monotone increasing functions for t ≥ 0. Obviously, the derivatives of cosh(·) and sinh(·) are d d cosh t = sinh t, sinh t = cosh t. dt dt

(3.9)

Next, we give an example to verify the results of Lemmas 3.2 and 3.3 and also show the images of the delayed cosine function and the delayed sine function. Example 3.1. Set τ = 0.3, Ω = 3, Ω ∈ R1 . By (3.3) and (3.4) we derive that cos0.3 3t and sin0.3 3t are as follows: ⎧ 1, t ∈ [−0.3, 0), ⎪ ⎪ ⎪ ⎪ ⎪ 2 t2 ⎪ ⎪ ⎪ 1 − 3 2 , t ∈ [0, 0.3), ⎨ 4 2 , t ∈ [0.3, 0.6), 1 − 32 t2 + 34 (t−0.3) cos0.3 3t = 4! ⎪ 4 6 ⎪ 2 (t−0.3) t ⎪ 2 4 ⎪ 1−3 2 +3 − 36 (t−0.6) , t ∈ [0.6, 0.9], ⎪ 4! 6! ⎪ ⎪ ⎪ .. ⎩ . and ⎧ 3(t + 0.3), t ∈ [−0.3, 0), ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ 3(t + 0.3) − 33 t3! , t ∈ [0, 0.3), ⎪ ⎪ ⎨ 5 3 3(t + 0.3) − 33 t3! + 35 (t−0.3) , t ∈ [0.3, 0.6), sin0.3 3t = 5! ⎪ 5 ⎪ 3 ⎪ 3(t + 0.3) − 33 t + 35 (t−0.3) − 37 (t−0.6)7 , t ∈ [0.6, 0.9]. ⎪ ⎪ 3! 5! 7! ⎪ ⎪ ⎪ .. ⎩ .

64 Stability and Controls Analysis for Delay Systems

FIGURE 3.1

 cos0.3 3t and cosh[3(t + 0.3)].

It follows from Fig. 3.1 and Fig. 3.2 that the inequalities in Lemma 3.2 and Lemma 3.3 hold. The images of the delayed cosine cos0.3 3t and the delayed sine sin0.3 3t are shown in Fig. 3.3 and Fig. 3.4, respectively. Obviously, we can see that the delayed cosine and delayed sine do have some similar properties to the classical cosine and sine functions, such as a wave shape, monotonicity, and periodicity. However, the differences between the delayed cosine and delayed sine and the cosine and sine are that the delayed cosine and delayed sine appear in the interval segment [−τ, 0] and with increasing variables t, the upper and lower bounds of the delayed cosine and delayed sine are also increasing. When the delay τ = 0, by (3.3) and (3.4), the delayed cosine and delayed sine coincide with the cosine and sine functions.

3.1.1.1 Finite time stability results for linear systems In this section, we present some sufficient conditions for finite time stability results for the desired system (3.1) by using three possible formulas of solutions, which do enrich the design methods for practical problems. Theorem 3.1. System (3.1) is finite time stable with respect to {0, J, δ, , τ } if cosh(ΩT )
0, (3.33) x(t) = ϕ(t), x(t) ˙ = ϕ(t), ˙ − τ ≤ t ≤ 0, where f : [0, ∞) → Rn , Ω is an n × n nonsingular matrix, τ is the time delay, and ϕ is an arbitrary twice continuously differentiable vector function. A solution of (3.33) has an explicit representation of the form [5, Theorem 2] x(t) = (cosτ Ωt)ϕ(−τ ) + Ω −1 (sinτ Ωt)ϕ(−τ ˙ )  0 + Ω −1 sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ  t + Ω −1 sinτ Ω(t − τ − s)f (s)ds. 0

(3.34)

Oscillating delay systems Chapter | 3

75

Diblík et al. [91] studied a control problem for a system governed by the following delay oscillating equations:  x(t) ¨ + Ω 2 x(t − τ ) = bu(t), t ∈ [0, t1 ], τ > 0, t1 > 0, (3.35) x(t) = ϕ(t), x(t) ˙ = ϕ(t), ˙ t ∈ [−τ, 0], where b ∈ Rn and u : [0, ∞) → R and they give sufficient and necessary conditions of relative controllability [91, Theorem 3.8] for (3.35) from the point of view of the rank criteria rank (b, Ω 2 b, Ω 4 b, · · · , Ω 2(n−1) b) = n

(3.36)

provided by t1 > (n − 1)τ . In addition, an explicit dependence of the control function related to sinτ Ω and cosτ Ω for (3.36) was given in [91, Theorem 3.9]: u∗ (t) = b (Ω −1 sinτ Ω(t1 − τ − t)) C10 + b (cosτ Ω(t1 − τ − t)) C20 , 0 , · · · , c0 ) are the solutions of the where C10 = (c10 , · · · , cn0 ) and C20 = (cn+1 2n algebraic equation in [91, (3.45)]. In this part, we use a different approach to that in [91] to study controllability of a system governed by the following Cauchy problem:  x(t) ¨ + Ω 2 x(t − τ ) = f (t, x(t)) + Bu(t), τ > 0, t ∈ [0, t1 ], t1 > 0, x(t) = ϕ(t), x(t) ˙ = ϕ(t), ˙ − τ ≤ t ≤ 0, (3.37)

where f : J × Rn → Rn , B is an n × m matrix, and we have an input u : [0, t1 ] → Rm . From (3.34), a solution of system (3.37) can be formulated as x(t) = (cosτ Ωt)ϕ(−τ ) + Ω −1 (sinτ Ωt)ϕ(−τ ˙ )  0 + Ω −1 sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ  t sinτ Ω(t − τ − s)f (s, x(s))ds + Ω −1 0  t + Ω −1 sinτ Ω(t − τ − s)Bu(s)ds.

(3.38)

0

Definition 3.3. System (3.37) is controllable if there exists a control function u∗ : [0, t1 ] → Rm such that x(t) ¨ + Ω 2 x(t − τ ) = f (t, x(t)) + Bu∗ (t) has a solution x = x ∗ : [−τ, t1 ] → Rn satisfying x ∗ (t) = ϕ(t), x˙ ∗ (t) = ϕ(t), ˙ − τ ≤ t ≤ 0,

76 Stability and Controls Analysis for Delay Systems

x ∗ (t1 ) = x1 , x˙ ∗ (t1 ) = x1 , where x1 , x1 ∈ Rn are any finite terminal conditions and t1 is an arbitrary given terminal point.

3.2.1.1 Controllability of linear delay systems In this section, we study the controllability of a system governed by a second order linear delay differential equation:  x(t) ¨ + Ω 2 x(t − τ ) = Bu(t), t ∈ [0, t1 ], τ > 0, (3.39) x(t) = ϕ(t), x(t) ˙ = ϕ(t), ˙ t ∈ [−τ, 0]. We introduce a delay Grammian matrix as follows:  t1 Wτ [0, t1 ] = Ω −1 sinτ Ω(t1 − τ − s)BB  sinτ Ω  (t1 − τ − s)ds. (3.40) 0

We give a new sufficient and necessary condition to guarantee (3.39) is controllable. Theorem 3.5. System (3.39) is controllable if and only if Wτ [0, t1 ] defined in (3.40) is nonsingular. Proof. First, we establish the sufficiency. Since Wτ [0, t1 ] is nonsingular, its inverse Wτ−1 [0, t1 ] is well defined. Thus, for any finite terminal conditions x1 , x1 ∈ Rn , one can construct the corresponding control input u(t) as u(t) = B  sinτ Ω  (t1 − τ − t)Wτ−1 [0, t1 ]ζ,

(3.41)

ζ = x1 − (cosτ Ωt1 )ϕ(−τ ) − Ω −1 (sinτ Ωt1 )ϕ(−τ ˙ )  0 − Ω −1 sinτ Ω(t1 − τ − s)ϕ(s)ds. ¨

(3.42)

where

−τ

From (3.38), the solution x(t1 ) of system (3.39) can be formulated as x(t1 ) = (cosτ Ωt1 )ϕ(−τ ) + Ω −1 (sinτ Ωt1 )ϕ(−τ ˙ )  0 + Ω −1 sinτ Ω(t1 − τ − s)ϕ(s)ds ¨ −τ  t1 sinτ Ω(t1 − τ − s)Bu(s)ds. + Ω −1 0

Putting (3.41) into (3.43), we obtain x(t1 ) = (cosτ Ωt1 )ϕ(−τ ) + Ω −1 (sinτ Ωt1 )ϕ(−τ ˙ )

(3.43)

Oscillating delay systems Chapter | 3

+ Ω −1



0

−τ

sinτ Ω(t1 − τ − s)ϕ(s)ds ¨ + Ω −1



t1

77

sinτ Ω(t1 − τ − s)B

0

× B  sinτ Ω  (t1 − τ − s)dsWτ−1 [0, t1 ]ζ.

(3.44)

Now (3.40), (3.42), and (3.44) give ˙ ) x(t1 ) = (cosτ Ωt1 )ϕ(−τ ) + Ω −1 (sinτ Ωt1 )ϕ(−τ  0 + Ω −1 sinτ Ω(t1 − τ − s)ϕ(s)ds ¨ + ζ = x1 , −τ

and we use Lemmas 3.1 and 3.3 to obtain x(t ˙ 1) =

d {(cosτ Ωt1 )ϕ(−τ ) + Ω −1 (sinτ Ωt1 )ϕ(−τ ˙ ) dt  0 + Ω −1 sinτ Ω(t1 − τ − s)ϕ(s)ds ¨ + ζ}

= x1 .

−τ

Next, we check the initial conditions x(t) = ϕ(t), x(t) ˙ = ϕ(t) ˙ hold when −τ ≤ t ≤ 0. From (3.3) and (3.4), the following relations hold: cosτ Ωt = I, sinτ Ωt = Ω(t + τ ), − τ ≤ t ≤ 0,  sinτ Ω(t − τ − s) =

Θ, t < s ≤ 0, Ω(t − s), − τ ≤ s ≤ t.

Linking (3.38) and the above relations, the solution of (3.39) can be expressed by  t −1 x(t) = ϕ(−τ ) + (t + τ )ϕ(−τ ˙ )+Ω sinτ Ω(t − τ − s)ϕ(s)ds. ¨ (3.45) −τ

Integrating by parts and using Lemma 3.3 yields  t sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ  t sinτ Ω(t − τ − s)d ϕ(s) ˙ = −τ  t t ˙ − ϕ(s)d ˙ sinτ Ω(t − τ − s) = sinτ Ω(t − τ − s)ϕ(s)| −τ −τ

= −(t + τ )Ω ϕ(−τ ˙ ) + Ωϕ(t) − Ωϕ(−τ ).

(3.46)

Putting (3.46) into (3.45), we get x(t) = ϕ(−τ ) + (t + τ )ϕ(−τ ˙ ) + Ω −1 [−Ω(t + τ )ϕ(−τ ˙ ) + Ωϕ(t) − Ωϕ(−τ )]

78 Stability and Controls Analysis for Delay Systems

= ϕ(t). Now x(t) ˙ = ϕ(t) ˙ holds. Thus, (3.39) is controllable according to Definition 3.3. Next we establish the necessity. Assume the delay Grammian matrix Wτ [0, t1 ] is singular. Then Wτ [0, t1 ][Ω −1 ] is singular too. Thus, there exists at least one nonzero state x¯ ∈ Rn such that x¯  Wτ [0, t1 ][Ω −1 ] x¯ = 0. It follows from (3.40) that 0 = x¯  Wτ [0, t1 ][Ω −1 ] x¯  t1 = x¯  Ω −1 sinτ Ω(t1 − τ − s)BB  sinτ Ω  (t1 − τ − s)[Ω −1 ] xds ¯ 0  t1    x¯  Ω −1 sinτ Ω(t1 − τ − s)B x¯  Ω −1 sinτ Ω(t1 − τ − s)B ds = 0  t1

2

 −1

=

x¯ Ω sinτ Ω(t1 − τ − s)B ds. 0

This implies that x¯  Ω −1 sinτ Ω(t1 − τ − s)B = 0 , ∀ s ∈ J.

(3.47)

Since (3.39) is controllable, it can be driven from any continuously differentiable initial vector functions ϕ, ϕ˙ : [−τ, 0] → Rn to an arbitrary state x(t1 ) ∈ Rn . Hence there exists a control u0 (t) that drives the initial state to zero. This means that ˙ ) x(t1 ) = cosτ Ωt1 ϕ(−τ ) + Ω −1 sinτ Ωt1 ϕ(−τ  0 + Ω −1 sinτ Ω(t1 − τ − s)ϕ(s)ds ¨ −τ  t1 sinτ Ω(t1 − τ − s)Bu0 (s)ds = 0. + Ω −1

(3.48)

0

Moreover, there exists a control u(t) ˜ that drives the initial state to the state x, ¯ so x(t1 ) = cosτ Ωt1 ϕ(−τ ) + Ω −1 sinτ Ωt1 ϕ(−τ ˙ )  0 + Ω −1 sinτ Ω(t1 − τ − s)ϕ(s)ds ¨ −τ  t1 sinτ Ω(t1 − τ − s)B u(s)ds ˜ = x. ¯ + Ω −1 0

(3.49)

Oscillating delay systems Chapter | 3

79

Combining (3.48) and (3.49) gives  t1 −1 x¯ = Ω sinτ Ω(t1 − τ − s)B[u(s) ˜ − u0 (s)]ds. 0

Multiplying both sides of the equality by x¯  , we get  t1 x¯  x¯ = x¯  Ω −1 sinτ Ω(t1 − τ − s)B[u(s) ˜ − u0 (s)]ds. 0

Note that with (3.47), we obtain x¯  x¯ = 0. That is, x¯ = 0, which conflicts with x¯ being nonzero. Thus, the delay Grammian matrix Wτ [0, t1 ] is nonsingular.

3.2.1.2 Controllability of nonlinear systems In this section, we apply a fixed point method to establish a sufficient condition of controllability for (3.37). We assume the following: (H1 ) f : J × Rn → Rn is continuous (here J = [0, t1 ]), and there exist Lf ∈ q L (J, R+ ) and q > 1 such that f (t, x1 ) − f (t, x2 ) ≤ Lf (t)x1 − x2 , and set Mf = sup f (t, 0). t∈J

(H2 ) Consider the operator W : L2 (J, Rm ) → Rn given by  t1 W = Ω −1 sinτ Ω(t1 − τ − s)Bu(s)ds. 0

Suppose that W −1 exists and there exists a constant M1 > 0 such that W −1 Lb (Rn ,L2 (J,Rm )/kerW ) ≤ M1 . Next, consider a control function ux of the form  ux (t) = W −1 x1 − (cosτ Ωt1 )ϕ(−τ ) − Ω −1 (sinτ Ωt1 )ϕ(−τ ˙ )  0 − Ω −1 sinτ Ω(t1 − τ − s)ϕ(s)ds ¨ −τ  t1  sinτ Ω(t1 − τ − s)f (s, x(s))ds (t), t ∈ J. − Ω −1 0

We define an operator Γ : C([−τ, t1 ], Rn ) → C([−τ, t1 ], Rn ) as follows: (Γ x)(t) = (cosτ Ωt)ϕ(−τ ) + Ω −1 (sinτ Ωt)ϕ(−τ ˙ )  0 + Ω −1 sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ

(3.50)

80 Stability and Controls Analysis for Delay Systems

+ Ω −1



t

sinτ Ω(t − τ − s)f (s, x(s))ds

0

+ Ω −1



t

sinτ Ω(t − τ − s)Bux (s)ds.

(3.51)

0

For each positive number ξ , let Oξ = {x ∈ C([−τ, t1 ], Rn ) : x = sup x(t) ≤ ξ }. Now Oξ is a bounded, closed, and convex set of

t∈[−τ,t1 ]

C([−τ, t1 ], Rn ). Now we use Krasnoselskii’s fixed point theorem to prove our result. We first prove that the operator Γ has a fixed point x, which is a solution of (3.37). d (Γ x)(t) = ϕ(t) ˙ when −τ ≤ t ≤ 0 and Then we check (Γ x)(t) = ϕ(t), dt d

(Γ x)(t1 ) = x1 , dt (Γ x)(t1 ) = x1 via the control ux defined in (3.50), and this means system (3.37) is controllable. Theorem 3.6. Suppose (H1 ) and (H2 ) are satisfied. Then (3.37) is controllable if   cosh(Ωt1 ) − 1 −1 (3.52) Ω BM1 < 1, M2 1 + Ω where M2 = Ω −1  q > 1.



1 Ωpt1 2p Ωp (e

1 p − 1) Lf Lq (J,R+ ) ,

1 p

+

1 q

= 1, p,

Proof. We divide our proof into three steps to verify the conditions required in Krasnoselskii’s fixed point theorem. Step 1. We show Γ (Oξ ) ⊆ Oξ for some positive number ξ . Consider any positive number ξ and let x ξ ∈ Oξ . Let t ∈ [0, t1 ]. From (H1 ) and the Hölder inequality, we obtain    t sinh Ω(t − s) Lf (s)ds 0

≤

 t  0

 ≤

0

t

p  p1  t  q1 q sinh[Ω(t − s)] ds Lf (s)ds

eΩp(t−s) ds 2p

0

1

p

Lf Lq (J,R+ )



1 p 1 Ωpt = p − 1) Lf Lq (J,R+ ) , (e 2 Ωp −t

where we use the fact that sinh t = e −e ≤ e2 , for ∀ t ∈ R. Next, 2      t  t sinh Ω(t − s) f (s, 0)ds ≤ Mf sinh Ω(t − s) ds t

0

t

0

(3.53)

Oscillating delay systems Chapter | 3

  Mf ≤ cosh(Ωt) − 1 . Ω

81

(3.54)

From (3.50), (H1 ), (H2 ), (3.53), (3.54), and Lemmas 3.2 and 3.3, we obtain (here ϕ = max ϕ(s), ϕ ˙ = max ϕ(s), ˙ and ϕ ¨ = max ϕ(s)) ¨ s∈[−τ,0]

s∈[−τ,0]

s∈[−τ,0]

 ux (t) ≤ W −1 L(Rn ,L2 (J,Rm )/kerW ) x1  +  cosτ Ωtϕ(−τ ) + Ω −1  sinτ Ωtϕ(−τ ˙ )  0 + Ω −1   sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ   t −1 + Ω   sinτ Ω(t − τ − s)f (s, x(s))ds 0

≤ M1 x1  + M1 cosh(Ωt)ϕ + M1 Ω

−1



  sinh Ω(t + τ ) ϕ ˙

   0 + M1 Ω −1 ϕ ¨ sinh Ω(t − s) ds −τ    t sinh Ω(t − s) Lf (s)x(s)ds + M1 Ω −1  0    t + M1 Ω −1  sinh Ω(t − s) f (s, 0)ds 0

  ˙ ≤ M1 x1  + M1 cosh(Ωt)ϕ + M1 Ω −1  sinh Ω(t + τ ) ϕ   M1 Ω −1 ϕ ¨ + cosh[Ω(t + τ )] − cosh(Ωt) Ω  1 p 1 −1 Ωpt (e − 1) Lf Lq (J,R+ ) x + M1 Ω  p 2 Ωp   −1 Mf cosh(Ωt) − 1 + M1 Ω  Ω ≤ M1 x1  + M1 ϑ(t) + M1 M2 ξ ≤ M1 x1  + M1 ϑ(t1 ) + M1 M2 ξ, where

 Mf ϑ(t) = cosh(Ωt)ϕ + Ω  sinh Ω(t + τ ) ϕ ˙ + Ω −1  Ω     ¨ Ω −1 ϕ cosh[Ω(t + τ )] − cosh(Ωt) × cosh(Ωt) − 1 + Ω −1

(here we used the fact that

dt d ϑ(t) > 0,



∀ t ∈ J ).

82 Stability and Controls Analysis for Delay Systems

Now (Γ x ξ )(t) ≤  cosτ Ωtϕ(−τ ) + Ω −1  sinτ Ωtϕ(−τ ˙ )  0 + Ω −1   sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ  t  sinτ Ω(t − τ − s)f (s, x(s))ds + Ω −1  0  t  sinτ Ω(t − τ − s)Bux (s)ds + Ω −1  0   −1 ≤ cosh(Ωt)ϕ + Ω  sinh Ω(t + τ ) ϕ ˙   Ω −1 ϕ ¨ + cosh[Ω(t + τ )] − cosh(Ωt) Ω    t sinh Ω(t − s) Lf (s)x(s)ds + Ω −1  0    t + Ω −1  sinh Ω(t − s) f (s, 0)ds + Ω −1  0      t × sinh Ω(t − s) B M1 x1  + M1 ϑ(t1 ) + M1 M2 ξ ds 0

cosh(Ωt) − 1 Ω −1 BM1 x1  Ω cosh(Ωt) − 1 + Ω −1 BM1 ϑ(t1 ) Ω cosh(Ωt) − 1 + Ω −1 BM1 M2 ξ Ω   cosh(Ωt1 ) − 1 Ω −1 BM1 ≤ ϑ(t1 ) 1 + Ω cosh(Ωt1 ) − 1 Ω −1 BM1 x1  + Ω   cosh(Ωt1 ) − 1 −1 Ω BM1 ξ. + M2 1 + Ω ≤ ϑ(t1 ) + M2 ξ +

Thus for some ξ sufficiently large (which we take for the rest of the proof), from (3.52) we have Γ (x ξ ) ∈ Oξ , and as a result, Γ (Oξ ) ⊆ Oξ . Now we write the operator Γ defined in (3.51) as Γ1 + Γ2 , where (Γ1 x)(t) = (cosτ Ωt)ϕ(−τ ) + Ω −1 (sinτ Ωt)ϕ(−τ ˙ )  0 + Ω −1 sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ

Oscillating delay systems Chapter | 3

 t + Ω −1 sinτ Ω(t − τ − s)Bux (s)ds, 0  t sinτ Ω(t − τ − s)f (s, x(s))ds. (Γ2 x)(t) = Ω −1

83

(3.55) (3.56)

0

Step 2. We show Γ1 : Oξ → C([−τ, t1 ], Rn ) is a contraction. Let t ∈ [0, t1 ]. From (3.50), (3.53), (H1 ), and (H2 ), for ∀ x, y ∈ Oξ , we have ux (t) − uy (t) ≤ M1 Ω −1 



t

 sinτ Ω(t − τ − s)Lf (s)x(s) − y(s)ds    t −1 ≤ M1 Ω x − y sinh Ω(t − s) Lf (s)ds 0

0

≤ M1 M2 x − y. Then from (3.55), we have (Γ1 x)(t) − (Γ1 y)(t)  t ≤ Ω −1   sinτ Ω(t − τ − s)Bux (s) − uy (s)ds 0    t ≤ Ω −1 BM1 M2 x − y sinh Ω(t − s) ds 0

≤  x − y, 1 )−1 where  =: cosh(Ωt Ω −1 BM1 M2 . From (3.52),  < 1 implies Γ1 is Ω a contraction. Step 3. We show that Γ2 : Oξ → C([−τ, t1 ], Rn ) is a continuous compact operator. Let xn ∈ Oξ with xn → x in Oξ , let Fn (·) = f (·, xn (·)) and F (·) = f (·, x(·)), and note     sinh Ω(· − s) Fn (s) → sinh Ω(· − s) F (s), a.e. s ∈ J = [0, t1 ].

From (H1 ), we get     sinh Ω(· − s) Fn (s) − F (s) ≤ 2ξ sinh Ω(· − s) Lf (s) ∈ L1 (J, R+ ). Then using (3.56) and Lebesgue’s dominated convergence theorem, we obtain (Γ2 xn )(t) − (Γ2 x)(t)    t ≤ Ω −1  sinh Ω(t − s) Fn (s) − F (s)ds → 0 as n → ∞. 0

84 Stability and Controls Analysis for Delay Systems

Thus Γ2 : Oξ → C([−τ, t1 ], Rn ) is continuous. Next we show Γ2 (Oξ ) ⊂ C([τ, t1 ], Rn ) is equicontinuous. For x ∈ Oξ and 0 < t ≤ t + h ≤ t1 , from (3.56), we have (Γ2 x)(t + h) − (Γ2 x)(t) = Ω

−1



t+h 0

− Ω −1



t

sinτ Ω(t + h − τ − s)F (s)ds sinτ Ω(t − τ − s)F (s)ds

0

= K1 + K 2 , where K1 = Ω −1



t+h

sinτ Ω(t + h − τ − s)F (s)ds

t

and K2 = Ω

−1

 t

 sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s) F (s)ds.

0

Thus (Γ2 x)(t + h) − (Γ2 x)(t) ≤ K1  + K2 .

(3.57)

Now, we check Ki  → 0 as h → 0, i = 1, 2. For K1 (similar to (3.53)) we obtain    t+h sinh Ω(t + h − s) Lf (s)ds t



1 p1 1 Ωp1 h ≤ p (e − 1) Lf Lq1 (J,R+ ) , 2 1 Ωp1

(3.58)

where p11 + q11 = 1, p1 , q1 > 1. Then using (H1 ), (3.58), and Lemmas 3.2 and 3.3, we get    t+h −1 sinh Ω(t + h − s) F (s)ds K1  ≤ Ω  t

≤ Ω −1 



t+h

  sinh Ω(t + h − s)

t

× (f (s, x(s)) − f (s, 0) + f (s, 0))ds    t+h ≤ Ω −1  sinh Ω(t + h − s) Lf (s)x(s)ds t

+ Mf Ω

−1

  t

t+h



 sinh Ω(t + h − s) ds

Oscillating delay systems Chapter | 3

85



1 p1 1 Ωp1 h ≤ ξ Ω  p (e − 1) Lf Lq1 (J,R+ ) 1 2 Ωp1 cosh(Ωh) − 1 + Mf Ω −1  −→ 0 as h → 0. Ω −1

For K2 , from the Hölder inequality, we have  0

t

 sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s)Lf (s)ds



t

≤

  sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s) ds p2

1 p2

Lf Lq2 (J,R+ ) ,

0

where

1 p2

+

K2  ≤ Ω

1 q2 −1

= 1, p2 , q2 > 1. Then we get 

t



 sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s)F (s)ds

0

≤ ξ Ω −1 



t

 sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s)Lf (s)ds  t −1 + Mf Ω   sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s)ds 0

≤ ξ Ω

−1

 

0 t1



 sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s) ds p2

1 p2

0

× Lf Lq2 (J,R+ )  t1 −1 + Mf Ω   sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s)ds. 0

From (3.4), we know that the delayed matrix function sinτ Ωt is uniformly continuous for ∀ t ∈ J , and thus, we get  sinτ Ω(t + h − τ − s) − sinτ Ω(t − τ − s) → 0 as h → 0. Finally, we get K2  → 0. Now K1  → 0 and K2  → 0 with (3.57) yield (Γ2 x)(t + h) − (Γ2 x)(t) → 0 as h → 0, for all x ∈ Oξ . The other cases are treated similarly. From the Arzela–Ascoli theorem we know Γ2 : Oξ → C([τ, t1 ], Rn ) is compact. From Krasnoselskii’s fixed point theorem (see [82]), Γ has a fixed point x on Oξ . From the definition of operator Γ , x is also the solution of system (3.37). Note x(t1 ) = x1 via the control function ux (t). Also x(t ˙ 1 ) = x1 . Finally, we get the initial conditions x(t) = ϕ(t), x(t) ˙ = ϕ(t) ˙ when −τ ≤ t ≤ 0 using the same procedure as in the proof of (3.39) in Theorem 3.5. Thus, system (3.37) is controllable.

86 Stability and Controls Analysis for Delay Systems

3.2.1.3 Numerical examples and discussion Example 3.3. Consider the controllability of the following linear delay differential controlled system:  x(t) ¨ + Ω 2 x(t − 0.6) = Bu(t), t ∈ [0, 1.2], (3.59) x(t) ≡ ϕ(t), x(t) ˙ ≡ ϕ(t), ˙ t ∈ [−0.6, 0], where

 Ω=

1 2 0 1

 , B=

1 1

 , ϕ(t) =

3t 2t

 , ϕ(t) ˙ =

3 2

.

Noting that B is an n × m matrix, with an input u : [0, t1 ] → Rm , we can see n = 2, m = 1, τ = 0.6, t1 = 1.2. Constructing the corresponding delay Grammian matrix of system (3.59) via (3.40), we obtain  1.2 W0.6 [0, 1.2] = Ω −1 sin0.6 Ω(0.6 − s)BB  sin0.6 Ω  (0.6 − s)ds 0

:= E1 + E2 , where E1 = Ω −1



0.6

sin0.6 Ω(0.6 − s)BB  sin0.6 Ω  (0.6 − s)ds,

0

(0.6 − s) ∈ (0, 0.6),  1.2 −1 E2 = Ω sin0.6 Ω(0.6 − s)BB  sin0.6 Ω  (0.6 − s)ds, 0.6

(0.6 − s) ∈ (−0.6, 0), and

⎧ ⎪ Θ, t ∈ (−∞, −0.6), ⎪ ⎪ ⎪ ⎪ I, t ∈ [−0.6, 0), ⎪ ⎪ ⎪ ⎨ 2 I − Ω 2 t2! , t ∈ [0, 0.6), cos0.6 Ωt = ⎪ ⎪ I − Ω 2 t 2 + Ω 4 (t−0.6)4 , t ∈ [0.6, 1.2), ⎪ ⎪ 2! 4! ⎪ ⎪ ⎪ ⎪ .. ⎩ . ⎧ ⎪ Θ, t ∈ (−∞, −0.6), ⎪ ⎪ ⎪ ⎪ Ω(t + 0.6), t ∈ [−0.6, 0), ⎪ ⎪ ⎪ ⎨ 3 Ω(t + 0.6) − Ω 3 t3! , t ∈ [0, 0.6), sin0.6 Ωt = ⎪ 5 3 ⎪ ⎪ Ω(t + 0.6) − Ω 3 t3! + Ω 5 (t−0.6) , t ∈ [0.6, 1.2). ⎪ 5! ⎪ ⎪ ⎪ ⎪ . ⎩ ..

(3.60)

Oscillating delay systems Chapter | 3

Next, we can calculate that  E1 =

21681 102717 15625 218750 227259 269307 156250 546875



, E2 =

27 125 27 125

87



9 125 9 125

.

Then, we get  W0.6 [0, 1.2] =  −1 [0, 1.2] = W0.6

25056 118467 15625 218750 261009 308682 156250 546875

,

428725000 −411343750 364257 364257 −422931250 406000000 121419 121419

.

Thus, system (3.59) is controllable by Theorem 3.5. In addition, for any

, x ) , finite terminal conditions x(t1 ) = x1 = (x11 , x12 ) , x(t ˙ 1 ) = x1 = (x11 12 it follows (see (3.41)) that one can construct the corresponding control input u(t) ∈ R as −1 u(t) = B  sin0.6 Ω  (0.6 − t)W0.6 [0, 1.2]ζ,

(3.61)

where ζ = x1 − (cos0.6 Ω1.2)ϕ(−0.6) − Ω −1 (sin0.6 Ω1.2)ϕ(−0.6) ˙  x11 − 36132604990374860091 7036874417766400000 = . x12 − 9439319638987191039 3518437208883200000 From (3.38) and (3.61), the solution of system (3.59) has the following form: ˙ x(t) = (cos0.6 Ωt)ϕ(−0.6) + Ω −1 (sin0.6 Ωt)ϕ(−0.6)  t + Ω −1 sin0.6 Ω(t − 0.6 − s)B 0 −1 × B  sin0.6 Ω  (0.6 − s)ds W0.6 [0, 1.2]ζ.

(3.62)

term  t Now we consider the integral  sin  sin Ω(t − 0.6 − s)BB 0.6 0.6 Ω (0.6 − s)ds in (3.62). 0 For 0 < t < 0.6, we can obtain −0.6 < t − 0.6 − s < t − 0.6 < 0 and 0 < 0.6 − t < 0.6 − s < 0.6, so the solution (3.62) can be expressed as the following form via (3.60):   2 3 2t −1 3t x(t) = I − Ω Ω(t + 0.6) − Ω ϕ(−0.6) + Ω ϕ(−0.6) ˙ 2 6    t (0.6 − s)3 + Ω −1 ds [Ω(t − s)] BB  Ω  (1.2 − s) − (Ω  )3 6 0 −1 × W0.6 [0, 1.2]ζ.

88 Stability and Controls Analysis for Delay Systems

For 0.6 < t < 1.2, we get 0 < t − 0.6 − s < t − 0.6 < 0.6 when 0 < s < t − 0.6 and −0.6 < t − 0.6 − s < 0 when t − 0.6 < s < t. We can also obtain 0 < 0.6 − s < 0.6 when 0 < s < 0.6 and −0.6 < 0.6 − t < 0.6 − s < 0 when 0.6 < s < t. Finally, (3.62) can be expressed as the following formula via (3.60):  2 4 2t 4 (t − 0.6) ϕ(−0.6) +Ω x(t) = I − Ω 2 24  3 5 −1 3t 5 (t − 0.6) +Ω Ω(t + 0.6) − Ω +Ω ϕ(−0.6) ˙ 6 120   t−0.6  (t − 0.6 − s)3 Ω(t − s) − Ω 3 B + Ω −1 6 0  3 −1 T   3 (0.6 − s) × B Ω (1.2 − s) − (Ω ) dsW0.6 [0, 1.2]ζ 6   0.6 + Ω −1 [Ω(t − s)] BB  Ω  (1.2 − s) t−0.6

− (Ω  )3  + Ω −1

 (0.6 − s)3 −1 dsW0.6 [0, 1.2]ζ 6   t −1 [0, 1.2]ζ. [Ω(t − s)] BB  Ω  (1.2 − s) dsW0.6

0.6

Example 3.4. In this example, we consider the following nonlinear delay differential controlled system:  x(t) ¨ + Ω 2 x(t − 0.4) = f (t, x(t)) + Bu(t), t ∈ [0, 0.8], (3.63) x(t) ≡ ϕ(t), x(t) ˙ ≡ ϕ(t), ˙ t ∈ [−0.4, 0], where we set 

0 1 −1 0 

Ω=

f (t, x(t)) =

 , B = I2×2 , ϕ(t) =

0.3(t − 0.4) sin[x1 (t)] 0.3(t − 0.4) sin[x2 (t)]

Now, we set u(t) =  x , where  x=

2 

5t + 1





2t 2

 , ϕ(t) ˙ =

5 4t

,

.

 x , en en , where en is the orthonormal

n=1

basis of R2 . From the definition of W in (H2 ), we get  0.8 −1 W =Ω sin0.4 Ω(0.4 − s)Bds  x 0

=Ω

−1



0

0.4

sin0.4 Ω(0.4 − s)ds  x+Ω

−1



0.8

0.4

sin0.4 Ω(0.4 − s)ds  x

Oscillating delay systems Chapter | 3

 =  =



452 1875

0

0

452 1875

602 1875

0

0

602 1875

  x+



89



2 25

0

0

2 25

 x

 x.

Define the inverse W −1 : R2 → L2 (J1 , R2 ) by  (W

−1

 x )(t) :=



1875 602

0

0

1875 602

 x,

where J1 = [0, 0.8]. Then, we get (W





1875 0



602 x  = 3.1146 x ,  x )(t) ≤

 1875



0 602

−1

and thus, we obtain W −1  ≤ 3.1146 := M1 . Hence, W satisfies assumption (H2 ). Next, note that | sin a − sin b| ≤ |a − b|, ∀ a, b ∈ R. We have f (t, x) − f (t, y)  = |0.3(t − 0.4)| (sin[x1 (t)] − sin[y1 (t)])2 + (sin[x2 (t)] − sin[y2 (t)])2  ≤ |0.3(t − 0.4)| [x1 (t) − y1 (t)]2 + [x2 (t) − y2 (t)]2 = |0.3(t − 0.4)|x − y, ∀ t ∈ J1 , x(t), y(t) ∈ R2 . We can set Lf = |0.3(t − 0.4)| ∈ Lq (J1 , R+ ) in (H1 ). When we choose p = q = 2, we get  Lf L2 (J1 ,R+ ) =

0.8

1 [0.3(s − 0.4)] ds 2

2

= 0.0620.

0

Then, we obtain M2 = Ω

−1



1 2 1 1.6Ω (e  3 − 1) Lf L2 (J,R+ ) = 0.0436. 2 Ω

Finally, we calculate that   cosh(0.8Ω) − 1 −1 M2 1 + Ω BM1 = 0.0894 < 1, Ω

90 Stability and Controls Analysis for Delay Systems

which implies that condition (3.52) holds. Now all the conditions required in Theorem 3.6 are satisfied, thus, system (3.63) is controllable.

3.2.1.4 Conclusion We give sufficient and necessary conditions for the controllability for the linear second order delay differential system from the point of view of the delay Grammian matrix. In addition, we construct a specific control function for the controllability problem of transferring an initial function to a prescribed point in the phase space. Then, we construct a specific control function involving a nonlinear term and apply the fixed point theorem to establish a sufficient condition of controllability for the nonlinear system by using properties of the delayed matrix sine and the delayed matrix cosine. The results in this part are motivated from [16].

3.3 Iterative learning control 3.3.1 Iterative learning control for an oscillating system In this part, we discuss the ILC problem via a new approach, that is, the delayed matrix sine and cosine of polynomial degrees methods, for an oscillating system with pure delay of the following form: ⎧ ⎪ ⎨ x¨k (t) + Ω 2 xk (t − τ ) = uk (t), t ∈ [0, T ], (3.64) xk (t) = ϕ(t), − τ ≤ t < 0, τ > 0, ⎪ ⎩ yk (t) = Cxk (t) + Duk (t), where ϕ ∈ C 2 ([−τ, 0], Rn ) and C, D are two m × n matrices. The index k denotes the k-th learning iteration and the variables xk (t), uk (t) ∈ Rn and yk (t) ∈ Rm denote the state, input, and output, respectively. Let yd be a desired trajectory, let the output error be ek (t) = yd (t) − yk (t),

(3.65)

and define uk (t) = uk+1 (t) − uk (t). For system (3.64), we consider the following set of ILC updating laws: (i) open-loop P-type ILC updating law, uk+1 (t) = uk (t) + K1 ek (t);

(3.66)

(ii) closed-loop P-type ILC updating law, uk+1 (t) = uk (t) + K2 ek+1 (t);

(3.67)

Oscillating delay systems Chapter | 3

91

(iii) open-closed-loop P-type ILC updating law, uk+1 (t) = uk (t) + Kρ1 ek (t) + Kρ2 ek+1 (t);

(3.68)

(iv) open-loop D-type ILC updating law, uk+1 (t) = uk (t) + K3 e˙k (t);

(3.69)

(v) closed-loop D-type ILC updating law, uk+1 (t) = uk (t) + K4 e˙k+1 (t);

(3.70)

(vi) open-closed-loop D-type ILC updating law, uk+1 (t) = uk (t) + Kρ3 e˙k (t) + Kρ4 e˙k+1 (t).

(3.71)

Let f : J → Rn be a continuous function. Consider the following delayed system:  x(t) ¨ + Ω 2 x(t − τ ) = f (t), t ≥ 0, τ > 0, (3.72) x(t) = ϕ(t), − τ ≤ t < 0. For a given nonsingular matrix Ω, a solution of system (3.72) is derived by [5, Theorem 1] x(t) = ϕ(−τ ) cosτ Ωt + Ω −1 ϕ(−τ ˙ ) sinτ Ωt  0  t −1 −1 +Ω sinτ Ω(t − τ − s)ϕ(s)ds ¨ +Ω sinτ Ω(t − τ − s)f (s)ds. −τ

0

Lemma 3.4. The solution x(·) of system (3.72) can be expressed by the following formula: ˙ ) sinτ Ωt x(t) = ϕ(−τ ) cosτ Ωt + Ω −1 ϕ(−τ  0 η−1 (t − iτ − s)2i+1 ϕ(s)ds ¨ (−1)i Ω 2i + (2i + 1)! −τ i=0  t−ητ (t − ητ − s)2η+1 ϕ(s)ds ¨ + (−1)η Ω 2η (2η + 1)! −τ η−1  t−lτ (t − lτ − s)2l+1 + f (s)ds, (−1)l Ω 2l (2l + 1)! 0 l=0

where (η − 1)τ ≤ t < ητ , η = 1, 2, . . . , N.

92 Stability and Controls Analysis for Delay Systems

Proof. To achieve our aim, we only need to show that  0 sinτ Ω(t − τ − s)ϕ(s)ds ¨ J1 :=  =

−τ

η−1 0

(−1)i Ω 2i+1

−τ i=0  t−ητ

+ and

 J2 := =

−τ

(t − iτ − s)2i+1 ϕ(s)ds ¨ (2i + 1)!

(−1)η Ω 2η+1

t

sinτ Ω(t 0 η−1  t−lτ

− τ − s)f (s)ds

(−1)l Ω 2l+1

l=0

(t − ητ − s)2η+1 ϕ(s)ds ¨ (2η + 1)!

0

(t − lτ − s)2l+1 f (s)ds. (2l + 1)!

We divide the proof into two parts. (i) For −τ < s < 0, we obtain t − τ < t − τ − s < t, and then (η − 2)τ < t − τ − s < ητ . When −τ < s < t − ητ , we obtain (η − 1)τ < t − τ − s < t, and then (η − 1)τ < t − τ − s < ητ . When t − ητ < s < 0, we obtain t − τ < t − τ − s < (η − 1)τ , and then (η − 2)τ < t − τ − s < (η − 1)τ . Thus,  t−ητ  0 J1 = sinτ Ω(t − τ − s)ϕ(s)ds ¨ + sinτ Ω(t − τ − s)ϕ(s)ds ¨ −τ

 =

t−ητ

t−ητ

−τ

 +  =

η (t − iτ − s)2i+1 ϕ(s)ds ¨ (−1)i Ω 2i+1 (2i + 1)! i=0

0

η−1

t−ητ i=0 η−1 0

(−1)i Ω 2i+1

−τ i=0  t−ητ

+

(−1)i Ω 2i+1

−τ

(t − iτ − s)2i+1 ϕ(s)ds ¨ (2i + 1)!

(t − iτ − s)2i+1 ϕ(s)ds ¨ (2i + 1)!

(−1)η Ω 2η+1

(t − ητ − s)2η+1 ϕ(s)ds. ¨ (2η + 1)!

(ii) For 0 < s < t, we obtain −τ < t − τ − s < t − τ , and then −τ < t − τ − s < (η − 1)τ . When 0 < s < t − (η − 1)τ , we obtain (η − 2)τ < t − τ − s < t − τ , and then (η − 2)τ < t − τ − s < (η − 1)τ . When t − (j + 1)τ < s < t − j τ , we have (j − 1)τ < t − τ − s < j τ , j = 0, 1, . . . , η − 2. Thus,  t−(η−1)τ J2 = sinτ Ω(t − τ − s)f (s)ds 0

Oscillating delay systems Chapter | 3

+  =

η−2 

t−j τ

j =0 t−(j +1)τ η−1 t−(η−1)τ

0

+

η−2  j =0

=

+

l=0



+

l=0

(−1)l Ω 2l+1

(t − lτ − s)2l+1 f (s)ds (2l + 1)!

(−1)l Ω 2l+1

t−(η−1)τ t−lτ

(−1)l Ω 2l+1

0 t−(η−l)τ

0 η−1  t−lτ l=0

(t − lτ − s)2l+1 f (s)ds (2l + 1)!

j (t − lτ − s)2l+1 f (s)ds (−1)l Ω 2l+1 (2l + 1)! t−(j +1)τ

0

η−2  l=0

=

(−1)l Ω 2l+1

η−2  t−lτ l=0

=

sinτ Ω(t − τ − s)f (s)ds

t−j τ

η−1  t−(η−1)τ l=0

93

0

(t − lτ − s)2l+1 f (s)ds (2l + 1)!

(−1)η−1 Ω 2η−1

(−1)l Ω 2l+1

(t − lτ − s)2l+1 f (s)ds (2l + 1)!

[t − (η − 1)τ − s]2η−1 f (s)ds (2η − 1)!

(t − lτ − s)2l+1 f (s)ds. (2l + 1)!

The proof is finished. Remark 3.6. By Lemma 3.4, the solution xk (·) of (3.64) has the following form: xk (t) = ϕ(−τ ) cosτ Ωt + Ω −1 ϕ(−τ ˙ ) sinτ Ωt  0 η−1 (t − iτ − s)2i+1 ϕ(s)ds ¨ (−1)i Ω 2i + (2i + 1)! −τ i=0  t−ητ (t − ητ − s)2p+1 ϕ(s)ds ¨ + (−1)η Ω 2η (2η + 1)! −τ η−1  t−lτ (t − lτ − s)2l+1 + uk (s)ds, (−1)l Ω 2l (2l + 1)! 0

(3.73)

l=0

where (η − 1)τ ≤ t < ητ , η = 1, 2, . . . , n. Remark 3.7. By using (3.73) and [5, Lemmas 1, 2], one can get the following derivative form of the solutions of system (3.72): x(t) ˙ = −Ωϕ(−τ ) sinτ Ω(t − τ ) + ϕ(−τ ˙ ) cosτ Ωt

94 Stability and Controls Analysis for Delay Systems

 +

η−1 0

(−1)i Ω 2i

(t − iτ − s)2i ϕ(s)ds ¨ (2i)!

(−1)η Ω 2η

(t − ητ − s)2η ϕ(s)ds ¨ (2η)!

−τ i=0 t−ητ

 + +

−τ

η−1  l=0

t−lτ

(−1)l Ω 2l

0

(t − lτ − s)2l f (s)ds. (2l)!

(3.74)

Proof. Using the fact that d d cosτ Ωt = −Ω sinτ Ω(t − τ ), sinτ Ωt = Ω cosτ Ωt, dt dt  0 η−1 d (t − iτ − s)2i+1 ϕ(s)ds ¨ (−1)i Ω 2i dt −τ (2i + 1)! i=0

 = d dt



−τ i=0 t−ητ

−τ



= d dt

η−1 0



(−1)η Ω 2η

t−ητ

−τ t−lτ

(−1)i Ω 2i

(t − iτ − s)2i ϕ(s)ds, ¨ (2i)!

(t − ητ − s)2η+1 ϕ(s)ds ¨ (2η + 1)!

(−1)η Ω 2η

(t − ητ − s)2p ϕ(s)ds, ¨ (2η)!

(t − lτ − s)2l+1 f (s)ds (2l + 1)! 0  t−lτ (t − lτ − s)2l f (s)ds, = (−1)l Ω 2l (2l)! 0 (−1)l Ω 2l

one can get (3.74) from (3.73).

3.3.1.1 Convergence analysis of P-type ILC In this section, we give convergence results of P-type. Theorem 3.7. For the given system (3.64) and the open-loop P-type ILC law (3.66), lim yk (t) = yd (t) uniformly on [0, T ] in the λ-norm sense if the k→∞

condition I − DK1  < 1 is met. Proof. Consider (η −1)τ ≤ t < ητ , η = 0, 1, . . . , n. By (3.65), (3.66), and (3.73) we have ek+1 (t) = ek (t) + yk (t) − yk+1 (t) = ek (t)(I − DK1 ) − C

η−1  l=0

0

t−lτ

(−1)l Ω 2l

(t − lτ − s)2l+1 uk (t)ds. (2l + 1)!

Oscillating delay systems Chapter | 3

95

Taking the norm  ·  on Rn via fundamental computations, we have ek+1 (t) ≤ I − DK1 ek (t) η−1  t−lτ (t − lτ − s)2l+1 uk (t)ds Ω 2l  + C (2l + 1)! 0 l=0

≤ I − DK1 ek (t) + CΩ K1 ek λ 2l

η−1  l=0

0

t−lτ

(t − lτ − s)2l+1 λs e ds. (2l + 1)! (3.75)

Next, we show that 

t−lτ

0

1 (t − lτ )2l−j +2 eλ(t−lτ ) (t − lτ − s)2l+1 λs e ds = − + 2l+2 . (3.76) (2l + 1)! λj 2l − j + 2 λ 2l+2 j =1

In fact, integration by parts gives 

t−lτ

0

(t − lτ − s)2l+1 λs 1 (t − lτ )2l+1 1 e ds = − + (2l + 1)! λ (2l + 1)! λ .. .



t−lτ 0

(t − lτ − s)2l λs e ds (2l)!

1 (t − lτ )2l+1 1 (t − lτ )2l − 2 − ... λ (2l + 1)! (2l)! λ  t−lτ 1 1 (t − lτ )1 + 2l+1 − 2l+1 eλs ds 1! λ λ 0

=−

=−

2l+2 j =1

1 (t − lτ )2l−j +2 eλ(t−lτ ) + 2l+2 . λj 2l − j + 2 λ

Combining (3.76) with (3.75), we obtain ek+1 (t) ≤ I − DK1 ek (t) + CΩ 2l K1 ek λ θλ , where θλ :=

η−1 l=0

⎡ ⎣−

2l+2 j =1

⎤ 1 (t − lτ )2l−j +2 eλ(t−lτ ) ⎦ + 2l+2 . λj 2l − j + 2 λ

96 Stability and Controls Analysis for Delay Systems

Taking the λ-norm, we arrive at ek+1 λ ≤ I − DK1 ek λ + CΩ 2l K1 ek λ

η−1

⎡ ⎣−

2l+2 j =1

l=0

⎤ 1 (t − lτ )2l−j +2 e−λlτ ⎦ + 2l+2 . λj 2l − j + 2 λ

 p−1   1 (t−lτ )2l−j +2 e−λlτ Note that η is a finite number and l=0 − 2l+2 j =1 λj 2l−j +2 + λ2l+2 tends to zero when λ is sufficiently large. By the condition I − DK1  < 1, one can choose a λ sufficiently large such that ek+1 λ < ek λ , which implies lim ek λ = 0 and lim yk (t) = yd (t) uniformly on [0, T ] in the k→∞

k→∞

λ-norm sense. Next, we give the second convergence result of closed-loop P-type. Theorem 3.8. For the given system (3.64) and the closed-loop P-type ILC law (3.67), we have lim yk (t) = yd (t) uniformly on [0, T ] in the λ-norm sense k→∞

if the condition I + DK2  > 1 is met. Proof. Without loss of generality, we consider (η − 1)τ ≤ t < ητ , η = 0, 1, . . . , n. By (3.65), (3.67), and (3.73) we have ek+1 (t) = ek (t) + yk (t) − yk+1 (t) = ek (t) − DK2 ek+1 (t) η−1  t−lτ (t − lτ − s)2l+1 uk (t)ds. (−1)l Ω 2l −C (2l + 1)! 0 l=0

Then (I + DK2 )ek+1 (t) = ek (t) − C

η−1  l=0

t−lτ

(−1)l Ω 2l

0

(t − lτ − s)2l+1 uk (t)ds. (2l + 1)!

Taking the norm  ·  on Rn , one has I + DK2 ek+1 (t) ≤ ek (t) + C

η−1  l=0

0

t−lτ

Ω 2l 

(t − lτ − s)2l+1 uk (t)ds (2l + 1)!

≤ ek (t) + CΩ K2 ek+1 λ 2l

η−1  l=0

0

t−lτ

(t − lτ − s)2l+1 λs e ds. (2l + 1)!

Oscillating delay systems Chapter | 3

97

According to (3.76), we obtain I + DK2 ek+1 (t) ≤ ek (t) + CΩ 2l K2 ek+1 λ θλ . Taking the λ-norm, we arrive at ek+1 λ ≤

1 1 ek λ + CΩ 2l K2 ek+1 λ I + DK2  I + DK2  ⎡ ⎤ η−1 2l+2 1 (t − lτ )2l−j +2 e−λlτ ⎣− + 2l+2 ⎦ . × λj 2l − j + 2 λ j =1

l=0

Thus, for some λ sufficiently large and I + DK2  > 1 we obtain ek+1 λ < ek λ , which implies lim ek λ = 0. The proof is completed. k→∞

To end this section, we conclude with the following result of open-closedloop P-type. Since the proof is similar to that of the above two theorems, we omit it here. Theorem 3.9. For the given system (3.64) and the open-closed-loop P-type ILC law (3.68), we have lim yk (t) = yd (t) uniformly on [0, T ] in the λ-norm sense if the condition

k→∞ I −DKρ1  I +DKρ2 

< 1 is met.

3.3.1.2 Convergence analysis of D-type ILC In this section, we give the convergence results of D-type. Theorem 3.10. For the given system (3.64) associated with  t uk (t)ds yk (t) = Cxk (t) + D 0

and the open-loop D-type ILC law (3.69), we have lim yk (t) = yd (t) uniformly k→∞

on [0, T ] in the λ-norm sense if the condition I − DK3  < 1 is met. Proof. Consider (η −1)τ ≤ t < ητ , η = 0, 1, . . . , n. By (3.65), (3.69), and (3.74) we have e˙k+1 (t) − e˙k (t) = y˙k (t) − y˙k+1 (t) = C[x˙k (t) − x˙k+1 (t)] − Duk (t) η−1  t−lτ (t − lτ − s)2l uk (t)ds − DK3 e˙k (t). =C (−1)l+1 Ω 2l (2l)! 0 l=0

Then e˙k+1 (t) = (I − DK3 )e˙k (t) + C

η−1  l=0

0

t−lτ

(−1)l+1 Ω 2l

(t − lτ − s)2l uk (t)ds. (2l)!

98 Stability and Controls Analysis for Delay Systems

Taking the norm  ·  on Rn e˙k+1 (t) ≤ I − DK3 e˙k (t) + C

η−1 

t−lτ

Ω 2l 

l=0 0 2l

(t − lτ − s)2l uk (t)ds (2l)!

≤ I − DK3 e˙k (t) + CΩ K3 e˙k λ η−1  t−lτ (t − lτ − s)2l λs e ds. × (2l)! 0 l=0

Similar to (3.76), we have 

t−lτ

0

1 (t − lτ )2l−j +1 eλ(t−lτ ) (t − lτ − s)2l λs e ds = − + 2l+1 . (2l)! λj 2l − j + 1 λ 2l+1

(3.77)

j =1

Then e˙k+1 (t)e−λt ≤ I − DK3 e˙k (t)e−λt + CΩ 2l K3 e˙k λ ⎡ ⎤ η−1 2l+1 1 (t − lτ )2l−j +1 e−λlτ ⎣−e−λt + 2l+1 ⎦ . × λj 2l − j + 1 λ j =1

l=0

Taking the λ-norm, we arrive at e˙k+1 λ ≤ I − DK3 e˙k λ + CΩ 2l K3 e˙k λ

η−1 l=0

⎡ ⎣−

2l+1 j =1

⎤ 1 (t − lτ )2l−j +1 e−λlτ ⎦ + 2l+1 . λj 2l − j + 1 λ

So for some λ sufficiently large and I − DK3  < 1, we have e˙k+1 λ < e˙k λ . This yields lim e˙k λ = 0. Due to the fact that ek (0) = 0, we get ek λ ≤ k→∞

1 λ e˙k λ ;

consequently, we find lim ek λ = 0. The proof is completed. k→∞

Next, we give the second convergence result of closed-loop D-type. Theorem 3.11. For the given system (3.64) associated with yk (t) = Cxk (t) + t D 0 uk (t)ds and the closed-loop D-type ILC law (3.70), we have lim yk (t) = k→∞

yd (t) uniformly on [0, T ] in the λ-norm sense if the condition I + DK4  > 1 is met.

Oscillating delay systems Chapter | 3

99

Proof. Consider (η −1)τ ≤ t < ητ , η = 0, 1, . . . , n. By (3.65), (3.70), and (3.74) we have (I + DK4 )e˙k+1 (t) = e˙k (t) + C

η−1  l=0

t−lτ

(−1)l+1 Ω 2l

0

(t − lτ − s)2l uk (t)ds. (2l)!

Taking the norm  ·  on Rn , we have e˙k+1 (t) ≤ +

1 e˙k (t) I + DK4 

η−1  t−lτ

1 CΩ 2l K4 e˙k λ I + DK4  l=0

0

(t − lτ − s)2l λs e ds. (2l)!

Taking the λ-norm via (3.77), we have e˙k+1 λ ≤

1 1 e˙k λ + CΩ 2l K4 e˙k λ I + DK4  I + DK4  ⎡ ⎤ η−1 2l+1 1 (t − lτ )2l−j +1 e−λlτ ⎣− + 2l+1 ⎦ . × λj 2l − j + 1 λ l=0

j =1

Thus, for some λ sufficiently large and I + DK4  > 1, we have e˙k+1 λ < e˙k λ , which implies lim e˙k λ = 0. We get lim ek λ = 0. The proof is k→∞

k→∞

completed. To end this section, we conclude with the following result of open-closedloop D-type. Since the proof is similar to that of the above two theorems, we omit it here. Theorem 3.12. For the given system (3.64) associated with yk (t) = Cxk (t) + t D 0 uk (t)ds and the closed-loop D-type ILC law (3.71), we have lim yk (t) = yd (t) uniformly on [0, T ] in the λ-norm sense if the condition met.

k→∞ I −DKρ3  I +DKρ4 

< 1 is

3.3.1.3 Numerical examples and discussion In this section, several numerical examples are presented to demonstrate the validity of the designed method. Example 3.5. Consider second order differential equations with pure delay of the form ⎧ ⎪ ⎨ x¨k (t) + xk (t − 0.5) = uk (t), xk (t), uk (t) ∈ R2 , t ∈ [0, 1], (3.78) xk (t) = (2t, 3t) , t ∈ [−0.5, 0), ⎪ ⎩ yk (t) = (1, 2)xk (t) + (0.5, 0.6)uk (t)

100 Stability and Controls Analysis for Delay Systems

and choose an ILC updating law (3.66) as follows: uk+1 (t) = uk (t) + (1, 0.5) ek (t). The desired continuous trajectory is given as yd (t) = 4 sin(3πt), t ∈ [0, 1]. Obviously, we can see T = 1, τ = 0.5, ϕ(t) = (2t, 3t) , n = 2, m = 1, Ω = I , C = (1, 2), D = (0.5, 0.6), K1 = (1, 0.5) , and uk (t) = (u1k (t), u2k (t)) . Clearly, ⎧ ⎨ t2 − 2 + 1, t ∈ [0, 0.5), cos0.5 t = (3.79) ⎩ (t−0.5)4 t 2 − + 1, t ∈ [0.5, 1], 24 2 and sin0.5 t =

⎧ ⎨ ⎩

3

− t6 + t + 12 , t ∈ [0, 0.5), (t−0.5)5 120

−

t3 6

+ t + 12 , t ∈ [0.5, 1].

(3.80)

Combining (3.79), (3.80), and ϕ(t) with (3.73), we can get the state of the k-th iteration: ⎛ ⎞ t t3 t2 1 − + + t − + (t − s)u (s)ds 1k 0 6 2 2 ⎠ , t ∈ [0, 0.5), xk (t) = ⎝ t t3 3 2 − 6 + 4 t + t − 1 + 0 (t − s)u2k (s)ds ⎛ ⎞ (t−0.5)5 (t−0.5)4 t3 t2 1 − − + + t − 24 6 2 2 ⎜ 120 ⎟ ⎜ ⎟   ⎜ + t (t − s)u1k (s)ds − t−0.5 (t−s−0.5)3 u1k (s)ds ⎟ 0 0 6 ⎟ , t ∈ [0.5, 1]. xk (t) = ⎜ ⎜ (t−0.5)5 (t−0.5)4 t 3 3 2 ⎟ ⎜ ⎟ − − + t + t − 1 16 6 4 ⎝ 120 ⎠  t−0.5 (t−s−0.5)3 t u2k (s)ds + 0 (t − s)u2k (s)ds − 0 6 Moreover, I − DK1  = 0.2. Now all the conditions of Theorem 3.7 are required, yk uniformly converges to yd for all t ∈ [0, 1]. The continuous curve of the upper figure of Fig. 3.5 shows the output yk of the first 10 iterations of system (3.78) and the reference trajectory yd . The lower figure of Fig. 3.5 shows the L2 -norm of the tracking error in each iteration. Clearly, the output of the system yk (·) can fast track on our desired trajectory yd (·), and the 10th iteration error is 3.98 × 10−5 , which is relatively small.

Oscillating delay systems Chapter | 3

101

FIGURE 3.5 The tracking performance of system (3.78) of Example 3.5 and the L2 -norm of the tracking error.

Example 3.6. Conditions for this example are the same as in Example 3.5, except we changed the desired trajectory to a discontinuous trajectory:  2 sin(3πt), t ∈ [0, 0.4), yd (t) = 2 cos(3πt) − 1, t ∈ [0.4, 1]. The image of Fig. 3.6 shows the output yk of the first 10 iterations of system (3.78), the referenced trajectory yd , and the L2 -norm of the tracking error in each iteration. The 10th iteration error is 2.71 × 10−5 , which is small too. Example 3.7. In this example, we simulate different situations for ϕ(·). Again, the conditions in this example are the same as in Example 3.5, except in this example we changed ϕ(t) to a constant vector (3, 1) . Simulation results are shown in Fig. 3.7. Due to the change of ϕ(·), we can get the state xk (t) from Example 3.5 as follows: ⎛ ⎞ t 3 2 − t + 3 + (t − s)u (s)ds 1k 0 ⎠ , t ∈ [0, 0.5), xk (t) = ⎝ 22 t − t2 + 1 + 0 (t − s)u2k (s)ds xk (t) ⎛ =⎝

(t−0.5)4 8 (t−0.5)4 24

− −

t  t−0.5 (t−s−0.5)3 3 2 u1k (s)ds 2 t + 3 + 0 (t − s)u1k (s)ds − 0 6 t  t−0.5 (t−s−0.5)3 t2 u2k (s)ds 2 + 1 + 0 (t − s)u2k (s)ds − 0 6

⎞ ⎠,

102 Stability and Controls Analysis for Delay Systems

FIGURE 3.6

The tracking performance of Example 3.6 and the L2 -norm of the tracking error.

FIGURE 3.7

The tracking performance of Example 3.7 and the L2 -norm of the tracking error.

t ∈ [0.5, 1]. Fig. 3.7 shows the output yk of the first 10 iterations, the referenced trajectory yd , and the L2 -norm of the tracking error in each iteration. The 10th iteration error is 6.82 × 10−5 . The simulation results show that for different continuous functions ϕ(·), our method is also effective.

Oscillating delay systems Chapter | 3

103

3.3.1.4 Conclusion We give P-type and D-type convergence results, and several examples are given to demonstrate the applicability of our main results. The results in this part are motivated from [92].

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Chapter 4

Impulsive delay systems 4.1 Asymptotical stability 4.1.1 Basic estimation and Gronwall-type inequalities Lemma 4.1. (see [7, Lemma 12]) If B ≤ αeατ , α ∈ R+ , then eτ t ∈ R.

B(t−τ )

 ≤ eαt ,

Lemma 4.2. For all t ≥ 0, we have eτBt  ≤ eBt . Proof. From (2.4), without loss of generality, for (k − 1)τ ≤ t < kτ , k = 1, 2, · · · , we have  k 

 eτBt  =   ≤

B

n (t

n=0 ∞ 

 k − (n − 1)τ )n  tn   Bn ≤  n! n! n=0

Bn

n=0

tn n!

= eBt .

The result is proved. Lemma 4.3. (see [93]) Let y(·), b(·) ∈ C([t0 , T ], R+ ) and k ≥ 0, M ≥ 0. Further, ω(·) ∈ C([0, ∞), R+ ) and ω(·) is nondecreasing function. Then the inequality  t y(t) ≤ k + M b(s)ω(y(s))ds, t ∈ [t0 , T ], t0

implies the inequality    t y(t) ≤ W −1 W (k) + M b(s)ds , t ∈ [t0 , t˜], t˜ ≤ T , t0

where

 W (μ) =

μ

μ0

dz , μ0 > 0, μ > 0, ω(z)

and W −1 (·) means the inverse function of W (·). Stability and Controls Analysis for Delay Systems. https://doi.org/10.1016/B978-0-32-399792-8.00010-4 Copyright © 2023 Elsevier Inc. All rights reserved.

105

106 Stability and Controls Analysis for Delay Systems

Denote P C(R+ , Rn ) := {ν : R+ → Rn : ν ∈ C((ti , ti+1 ], Rn )}, there exist and ν(ti+ ) with ν(ti− ) = ν(ti ) for any i = 1, 2, · · · } and P C 1 (R+ , Rn ) := + {ν : R → Rn : ν  ∈ P C(R+ , Rn )}. Denote the space P C(Ω, Rn ) of vectorvalued piecewise continuous functions from Ω → Rn endowed with the norm νP C = sup ν(t), where  ·  is the norm on Rn . In addition, ψP C = ν(ti− )

t∈Ω

sup ψ(t). t∈[−τ,0]

Let Y1 , Y2 be two Banach spaces and let Lb (Y1 , Y2 ) denote the space of all bounded linear operators from Y1 to Y2 . Now Lp (Ω, Y2 ) denotes the Banach space of functions y : Ω → Y2 which are Bochner integrable with norm yLp (Ω,Y2 ) (here 1 < p < ∞).

4.1.2 Linear impulsive delay differential systems In this section, we introduce the impulsive delayed Cauchy matrix (which will be used to seek the formula of solutions) for the following linear impulsive delay differential systems: ⎧  ⎪ ⎨ν (t) = Aν(t) + Bν(t − τ ), t ≥ 0, τ > 0, t = ti , ν(ti ) = Ci ν(ti ), i = 1, 2, · · · , (4.1) ⎪ ⎩ ν(t) = ψ(t), − τ ≤ t ≤ 0, where A, B, Ci are constant n × n matrices, AB = BA, ACi = Ci A, and BCi = Ci B for each i = 1, 2, · · · , ψ ∈ Cτ1 := C 1 ([−τ, 0], Rn ), ν(t) ∈ Rn , time sequences {ti }∞ k=1 satisfy 0 = t0 < t1 < · · · < ti < · · · , impulsive conditions ν(ti ) := ν(ti+ ) − ν(ti− ), and ν(ti+ ) = lim ν(ti + ) and ν(ti− ) = ν(ti ) rep →0+

resent the right and left limits of ν(t) at t = ti and lim ti = ∞, respectively. i→+∞

4.1.2.1 Impulsive delayed Cauchy matrix and its properties In what follows, we introduce the concept of impulsive delay matrix functions, an extension of delay matrix functions for linear delay differential equations, which help us to seek explicit formulas of solutions to impulsive delay differential equations. By using (2.4), we define Y (·, ·) : R × R → Rn×n and Y (t, s) = eA(t−s) X(t, s + τ ), t > s,

(4.2)

where ˇ

X(t, s) = eτB(t−s) +



ˇ B(t−τ −tj )

Cj e τ

X(tj , s), Bˇ = e−Aτ B.

(4.3)

s−τ s, by using Lemma 2.2, we obtain d Y (t, s) dt = AeA(t−s) X(t, s + τ )   ˇ B(t−2τ −tj ) ˇ −s) ˇ τB(t−2τ + Bˇ Cj e τ X(tj , s + τ ) + eA(t−s) Be s 0.

By using the well-known classical Gronwall inequality (see [96, Theorem 1]), we have e−

η2 2 t

ν(t) ≤ T2 eKlt ,

which yields ν(t) ≤ T2 e(Kl+

η2 2 )t

→ 0 as t → ∞

due to (H6 ). The proof is finished.

4.1.2.4 Examples In this section, we give two examples to illustrate our above theoretical results. Here, we use MATLAB software to compute some parameters and draw the figures for the examples. Example 4.1. Consider (4.1) with ϑ = 0.2 and     −3 0 1 0 A= , B= , 0 −2.5 0 0.8

 Cj =

j = 1, 2, · · · ,

2+ 0

1 j

 0 2

, (4.19)

and

 ψ(t) =

 0.25 0.4



 t +1 , i(0, t) = , 2

(4.20)

Impulsive delay systems Chapter | 4

117

FIGURE 4.1 The state response ν(t) of (4.1), (4.19), and (4.20).

where [x] is the biggest integer less than real x. Obviously, AB = BA, ACj = Cj A, BCj = Cj B, j = 1, 2, · · · , α(A) = −2.5, and ρ(C) = 3. Next, eAt  = e−2.5t ≤ Ke(α(A)+ε)t , where K = 1 and ε > 0. By computation, ψ1 = 0.4 = ψ(−ϑ) < δ := 0.41, p˜ = lim i(0,t) t = t→∞

[ t+1 2 ] t→∞ t

lim

= 12 . Next, by using MATLAB software, we obtain

    1.8221 0   ˇ = e−Aϑ B =  B  ≤ αe0.2α , choose α = 1.3821,  0 1.3190   T1 ≥ K e

 (α(A)+α)ϑ

ψ(−ϑ) +

0

−ϑ

e

−(α(A)+α)s





ψ (s) − Aψ(s)ds

= 0.5440 > 0, and η2 = α(A) + α + p ln(ρ(C) + 1) = −0.4248 < 0. Now all the conditions of Theorem 4.4 are satisfied. Thus, ν(t) ≤ T1 e

η2 2 t

= T1 e−0.2124t → 0 as t → ∞,

that is, the trivial solution of (4.1), (4.19), and (4.20) is locally asymptotically stable (see Fig. 4.1).

118 Stability and Controls Analysis for Delay Systems

Example 4.2. Consider (4.16) where ϑ = 0.2,     −3.3 0 0.5 sin ν1 , A= , f (ν(t)) = 0.5 sin ν2 0 −3.3     3 0.5 0.8 0.2 , j = 1, 2, · · · , B= , Cj = 0 2.5 0 0.6

(4.21)

and ψ, i(0, t) are defined in (4.20). By the calculation,     −2.64 −0.66 −9.9 −1.65 AB = = BA, ACj = = Cj A, 0 −1.98 0 −8.25   2.4 0.9 BCj = = Cj B, j = 1, 2, · · · . 0 1.5 Obviously, α(A) = −3.3 and eAt  = e−3.3t ≤ Ke(α(A)+ε)t , where K = 1 and ε > 0. In addition, f (ν) < lν, and we can choose l = 0.5. Next, by using MATLAB software, we obtain     1.5478 03870   ˇ = e−Aϑ B =  B  ≤ αe0.2α , and choose α = 1.4483.  0 1.1609  Moreover, η2 = α(A) + α + p ln(ρ(C) + 1) = −1.1586; thus,  0 η2 η2 T2 = Ke 2 ϑ ψ(−ϑ) + Ke− 2 s ψ  (s) − Aψ(s)ds = 0.6228 > 0. −ϑ

Note that Kl + satisfied. Then,

η2 2

= −0.0793 < 0. Now all the conditions of Theorem 4.5 are

ν(t) ≤ T2 e(Kl+

η2 2 )t

= 0.6228e−0.0793t → 0 as t → ∞,

that is, the trivial solution of (4.16), (4.20), and (4.21) is locally asymptotically stable (see Fig. 4.2).

4.1.2.5 Conclusion We introduce the impulsive delayed matrix function and give its norm estimation. With the help of the impulsive delayed Cauchy matrix and the method of variation of constants, we obtain a representation of solutions to linear impulsive delay differential equations. Moreover, we derive some sufficient conditions to guarantee the trivial solution is locally asymptotically stable. The results in this part are motivated from [97].

Impulsive delay systems Chapter | 4

119

FIGURE 4.2 The state response ν(t) of (4.16), (4.20), and (4.21).

4.2 Finite time stability 4.2.1 Representation of solutions Let T be the set of all impulsive points ti . Consider the linear delay systems  ν  (t) = Aν(t) + Bν(t − τ ), t ∈ Ω := [0, T ], t ∈ / T , τ > 0, (4.22) ν(t) = ψ(t), − τ ≤ t ≤ 0, and the nonlinear delay system  ν  (t) = Aν(t) + Bν(t − τ ) + F (t, ν(t), ν(t − τ )), t ∈ Ω, t ∈ /T, ν(t) = ψ(t), − τ ≤ t ≤ 0,

(4.23)

with the following linear impulsive conditions: (I1 ) ν(ti ) = Ci ν(ti ), ti ∈ T , (I2 ) ν(ti ) = Ci ν(ti ) + di , ti ∈ T , where A, B, Ci ∈ Rn×n , AB = BA, ACi = Ci A, BCi = Ci B for each i ∈ {1, 2, · · · } and di ∈ Rn , ν ∈ C 1 ([−τ, 0], Rn ) ∪ P C 1 (Ω, Rn ), ψ ∈ Cτ1 , F ∈ C(Ω × Rn × Rn , Rn ), and ν(ti± ) = lim →0+ ν(ti ± ) and ν(ti− ) = ν(ti ) represent respectively the right and left limits of ν(t) at t = ti . Set Ω ∩ T = {t1 , t2 , · · · , tr }. Any classical solution ν ∈ C 1 ([−τ, 0], Rn ) ∪ P C 1 (Ω, Rn ) of (4.22) with (I1 ) has the form  0 ν(t) = Y (t, −τ )ψ(−τ ) + Y (t, s)[ψ  (s) − Aψ(s)]ds. (4.24) −τ

120 Stability and Controls Analysis for Delay Systems

By Corollary 4.1 any classical solution ν ∈ C 1 ([−τ, 0], Rn ) ∪ P C 1 (Ω, Rn ) of (4.23) with (I1 ) has the form  0 ν(t) = Y (t, −τ )ψ(−τ ) + Y (t, s)[ψ  (s) − Aψ(s)]ds + +

−τ

i−1  

tγ +1

Y (t, s)F (s, ν(s), ν(s − τ ))ds

γ =0 tγ  t

Y (t, s)F (s, ν(s), ν(s − τ ))ds.

(4.25)

ti

Define H (·, ·) : R × R → Rn×n given by ˇ

H (t, s) = eA(t−s) eτB(t−τ −s) + +





ˇ B(t−τ −tj )

Cj eA(t−tj ) eτ

H (tj , s)

s s.

s s, by using Lemma 2.2, we obtain d H (t, s) dt ˇ

ˇ

ˇ A(t−s) eτB(t−2τ −s) = AeA(t−s) eτB(t−τ −s) + Be  ˇ B(t−τ −tj ) + Cj AeA(t−tj ) eτ H (tj , s) s 0, 0 < t1 < t2 < · · · < tk < T , and the control function u(·) takes values from L2 (Ω, Rn ). Definition 4.4. (see [38, Definition 4]) System (4.58) is called relatively controllable if for an arbitrary initial vector function ψ ∈ C 1 ([−τ, 0], Rn ), the final state of the vector ν1 ∈ Rn at time T , there exists a control u ∈ 2 n L has a solution ν ∈ C 1 ([−τ, 0], Rn ) ∪  (Ω, R )n such kthat 1system (4.58)  P C(Ω, R ) ∩ (∪i=0 C (ti , ti+1 ]) (here t0 = 0 and tk+1 = T ) that satisfies ν(T ) = ν1 .   For any solution ν ∈ C 1 ([−τ, 0], Rn ) ∪ P C(Ω, Rn ) ∩ (∪ki=0 C 1 (ti , ti+1 ]) (here t0 = 0 and tk+1 = T ) of (4.58), from Corollary 4.1, we obtain 

0

ν(t) = Y (t, −τ )ψ(−τ ) + Y (t, s)[ψ  (s) − Aψ(s)]ds −τ  t Y (t, s)[f (s, ν(s)) + Du(s)]ds. +

(4.59)

0

4.3.1.1 Relative controllability of linear systems In this section, we investigate the relative controllability of the linear impulsive delay controlled system ⎧  / T , τ > 0, ⎪ ⎨ν (t) = Aν(t) + Bν(t − τ ) + Du(t), t ∈ Ω, t ∈ ν(ti ) = Ci ν(ti ), ti ∈ T , (4.60) ⎪ ⎩ ν(t) = ψ(t), − τ ≤ t ≤ 0, using the impulsive delay Cauchy matrix introduced. Next we construct a suitable control function for (4.58), which means that we give a condition necessary and sufficient for u ∈ L2 (Ω, Rn ) to lead the solution of (4.58) to ν1 at time T .

Impulsive delay systems Chapter | 4

141

We apply Krasnoselskii’s fixed point theorem (see [82]) to show that (4.58) is also relatively controllable under suitable conditions. The impulsive delay Grammian matrix, an extension of the classical Grammian matrix for linear differential systems, is as follows:  Wτ [0, T ] =

T

Y (T , s)DD  Y  (T , s)ds.

(4.61)

0

Theorem 4.21. System (4.60) is relatively controllable if and only if Wτ [0, T ] is nonsingular. Proof. First we establish sufficiency. Since Wτ [0, T ] is nonsingular, its inverse Wτ [0, T ] is well defined. For any final state ν1 ∈ Rn one can select a control function as follows: ˜ u(t) = D  V  (T , t)Wτ−1 [0, T ]η, where  η˜ = ν1 − Y (T , −τ )ψ(−τ ) −

0 −τ

Y (T , s)[ψ  (s) − Aψ(s)]ds.

Then  ν(T ) = Y (T , −τ )ψ(−τ ) +  +

0

−τ

Y (T , s)[ψ  (s) − Aψ(s)]ds

T

Y (T , s)Du(s)ds 0



= Y (T , −τ )ψ(−τ ) +  + 0

T

0

−τ

Y (T , s)[ψ  (s) − Aψ(s)]ds

Y (T , s)DD  Y  (T , s)Wτ−1 [0, T ]ηds ˜

= ν1 . We argue by contradiction to prove our necessity result. Assume Wτ [0, T ] is singular, i.e., there exists at least one nonzero state ν˜ ∈ Rn such that ν˜  Wτ [0, T ]˜ν = 0. One obtains 



0 = ν˜ Wτ [0, T ]˜ν =

T

ν˜  Y (T , s)DD  Y  (T , s)˜ν ds

0

 =

0

T

ν˜  Y (T , s)D2 ds,

142 Stability and Controls Analysis for Delay Systems

which implies ν˜  Y (T , s)D = 0 , ∀s ∈ Ω. Since system (4.60) is relatively controllable, according to Definition 4.4, there exists a control u1 (t) that drives the initial state to zero at T , i.e.,  ν(T ) = Y (T , −τ )ψ(−τ ) + 

T

+

0

−τ

Y (T , s)[ψ  (s) − Aψ(s)]ds

Y (T , s)Du1 (s)ds = 0.

(4.62)

0

Similarly, there also exists a control u2 (t) that drives the initial state to the (nonzero) state ν˜ at T , i.e.,  ν(T ) = Y (T , −τ )ψ(−τ ) + 

0

−τ

Y (T , s)[ψ  (s) − Aψ(s)]ds

T

+

Y (T , s)Du2 (s)ds 0

= ν˜ .

(4.63)

Then from (4.62) and (4.63), we have 

T

ν˜ =

Y (T , s)D[u2 (s) − u1 (s)]ds.

(4.64)

0

Multiplying both sides of (4.64) by ν˜  , we obtain 



ν˜ ν˜ =

T

ν˜  Y (T , s)D[u2 (s) − u1 (s)]ds = 0.

0

Thus ν˜ = 0, which conflicts with ν˜ being nonzero. Thus, the impulsive delay Grammian matrix Wτ [0, T ] is nonsingular. The proof is finished.

4.3.1.2 Relative controllability of semilinear systems We assume the following: (HW ): The operator W : L2 (Ω, Rn ) → Rn defined by  Wu =

T

Y (T , s)Du(s)ds 0

has an inverse operator W −1 which takes values in L2 (Ω, Rn )/kerW . Then we set M = W −1 Lb (Rn ,L2 (Ω,Rn )/kerW ) .

Impulsive delay systems Chapter | 4

From [87, Remark 3.3], we know  M = Wτ−1 [0, T ].

143

(4.65)

(HF ): The function f : Ω × Rn → Rn is continuous and there exist a constant q > 1 and Lf (·) ∈ Lq (Ω, R+ ) such that f (t, ν) − f (t, μ) ≤ Lf (t)ν − μ, ν, μ ∈ Rn . Theorem 4.22. Suppose that (HW ) and (HF ) are satisfied. Then system (4.58) is relatively controllable provided that   dDM aT (e − 1) < 1, (4.66) c 1+ a   k ˇ ˇ − Bτ where d = ) e−Bτ and j =1 (1 + Cj e c=

 k

 1 1 ˇ ˇ (1 + Cj e−Bτ ) e−Bτ [ (eapT − 1)] p Lf Lq (Ω,R+ ) , ap

j =1

1 1 + = 1, p, q > 1. p q Proof. Let ν1 ∈ Rn be the final state. Using hypothesis (HW ) for arbitrary ν(·) ∈ P C(Ω, Rn ), we define the control function uν (t) by   uν (t) = W −1 ν1 − Y (T , −τ )ψ(−τ ) −  −

T

0 −τ

Y (T , s)[ψ  (s) − Aψ(s)]ds

 Y (T , s)f (s, ν(s))ds (t), t ∈ Ω.

(4.67)

0

We show that, using this control, the operator F : P C(Ω, Rn ) → P C(Ω, Rn ) defined by 

0

Y (t, s)[ψ  (s) − Aψ(s)]ds (Fν)(t) = Y (t, −τ )ψ(−τ ) + −τ  t  t Y (t, s)f (s, ν(s))ds + Y (t, s)Duν (s)ds + 0

0

has a fixed point ν, which is a mild solution of (4.58). We check that (Fν)(T ) = ν1 , which means that uν steers system (4.58) from (Fν)(0) to ν1 in finite time T . This implies system (4.58) is relatively controllable on Ω.

144 Stability and Controls Analysis for Delay Systems

For each positive number r, let Br = {ν ∈ P C(Ω, Rn ) : νP C ≤ r} (a bounded, closed, and convex set of P C(Ω, Rn )). Set N = sup f (t, 0). t∈Ω

We divide the proof into several steps. Step 1. We claim that there exists a positive number r such that F(Br ) ⊆ Br . From (HF ) and Hölder’s inequality, we obtain 



t

e

a(t−s)

Lf (s)ds ≤

0

1 

t

e 

≤

ap(t−s)

0

t

p

ds 0

1 apt (e ap

1 q Lf (s)ds

q

1 p − 1) Lf Lq (Ω,R+ )

and 



t

e

a(t−s)

f (s, 0)ds ≤ N

0

t

ea(t−s) ds =

0

N at (e − 1). a

From (4.67), using (HW ) and (HF ), we have  uν (t) ≤ W −1 Lb (Rn ,L2 (Ω,Rn )/kerW ) ν1  + Y (T , −τ )ψ(−τ )  +

0

−τ

Y (T , s)ψ (s) − Aψ(s)ds +



≤ M ν1  + +

 k

 k

 k 

T

 Y (T , s)f (s, ν(s))ds

0

(1 + Cj e

ˇ −Bτ

 ˇ ) e−Bτ ea(T +τ ) ψ(−τ )

j =1

  ˇ ˇ (1 + Cj e−Bτ ) e−Bτ

j =1

+





(1 + Cj e

ˇ −Bτ

0 −τ

ea(T −s) ψ  (s) − Aψ(s)ds

 ˇ ) e−Bτ

j =1 T

×

ea(T −s) (f (s, ν(s)) − f (s, 0) + f (s, 0))ds



0

   k ˇ ˇ (1 + Cj e−Bτ ) e−Bτ ea(T +τ ) ψ(−τ ) ≤ M ν1  + +

 k j =1

+

 k

j =1

j =1

(1 + Cj e

ˇ −Bτ

  ˇ −Bτ ) e

 ˇ ˇ (1 + Cj e−Bτ ) e−Bτ

0 −τ

ea(T −s) ψ  (s) − Aψ(s)ds

Impulsive delay systems Chapter | 4



T

×

ea(T −s) (Lf (s)ν(s) + f (s, 0))ds

145



0

   k ˇ ˇ ≤ M ν1  + (1 + Cj e−Bτ ) e−Bτ ea(T +τ ) ψ(−τ ) +

 k

j =1

(1 + Cj e

ˇ −Bτ

  ˇ −Bτ ) e

−τ

j =1

+

 k

0

ea(T −s) ψ  (s) − Aψ(s)ds

 1 1 ˇ ˇ (1 + Cj e−Bτ ) e−Bτ [ (eapT − 1)] p ap

j =1

× Lf Lq (Ω,R+ ) νP C    k ˇ ˇ N + (1 + Cj e−Bτ ) e−Bτ (eaT − 1) a j =1

≤ Mν1  + Mb + McνP C , where b=

 k

(1 + Cj e

j =1

+

 k

ˇ −Bτ

 ˇ ) e−Bτ ea(T +τ ) ψ(−τ )

  ˇ ˇ (1 + Cj e−Bτ ) e−Bτ

j =1

+

 k

(1 + Cj e

ˇ −Bτ

j =1

0 −τ

ea(T −s) ψ  (s) − Aψ(s)ds

 ˇ N ) e−Bτ (eaT − 1). a

From (HW ) and (HF ) we have (Fν)(t)



0

≤ Y (t, −τ )ψ(−τ ) + Y (t, s)ψ  (s) − Aψ(s)ds −τ  t  t Y (t, s)f (s, ν(s))ds + Y (t, s)Duν (s)ds + 0

≤

 k

j =1



+

0

 ˇ ˇ −Bτ (1 + Cj e ) e−Bτ ea(t+τ ) ψ(−τ )

0

−τ

 s 0 there exists a constant K := K(x1 , r) such that f (x, y) − f (x, z) ≤ Ky − z for all x ∈ [0, x1 ] and y, z ∈ Br = {y ∈ Rn : y ≤ r}. Theorem 5.6. Assume that [H1 ] holds. Then (5.14) has a unique solution y ∈ C([−τ, xm ), Rn ), where xm ≤ ∞. Further, if xm < ∞, then limx→xm y(x) = ∞. Proof. Consider : C([−τ, xm ), Rn ) → C([−τ, xm ), Rn ) defined in (5.16) again. Step 1. A prior estimate. For arbitrary x ∈ [−τ, 0], we have y(x) = ϕ(x). Choose a subink  Bi−1 iα terval I1 = [0, x1 ]. Let μ(x) = Γ (iα+1) (x − (i − 1)τ ) , where k comes i=1

from set max{k ∈ Λ : kτ ≤ x1 }, r1 = (ϕ + M)Eα (Bx1α ) + 1, and N1 = sup f (x, 0). For any y ∈ C([0, x1 ], Rn ) with {y(x) : x ∈ [0, x1 ]} ≤ r1 , we x∈I1

have ( y)(x)

 0    α B(x−τ −t)α    ≤ EBx ϕ(−τ ) + E ϕ (t)dt τ τ   −τ

x   B(x−τ −t)α  + Eτ,α  f (t, y(t))dt 0

x   B(x−τ −t)α  ≤ (ϕ(−τ ) + M)Eα (Bx α ) + Eτ,α  f (t, y(t)) − f (t, 0)dt 0

x   B(x−τ −t)α  + Eτ,α  f (t, 0)dt 0

x   B(x−τ −t)α  α ≤ (ϕ(−τ ) + M)Eα (Bx1 ) + Ky Eτ,α  dt 0

x   B(x−τ −t)α  + N1 Eτ,α  dt 0

≤ (ϕ + M)Eα (Bx1α ) + (Kr1 + N1 )μ(x).

(5.17)

Fractional delay systems Chapter | 5

169

    Choose l1 = min x1 , x1∗ , where x1∗ comes from set x1∗ : μ(x1∗ ) = Kr11+N1 . It follows from (5.17) that ( y)(x) ≤ r1 , for all x ∈ [0, l1 ]. Step 2. Local existence and uniqueness of solution. For x ∈ [0, l1 ] and y, z ∈ Br , use Lemma 5.5 to derive

x −t)α ( y)(x) − ( z)(x) ≤ EB(x−τ f (x, y(t)) − f (x, z(t))dt τ,α 0  x B(x−τ −t)α ≤K Eτ,α dt y − z 0

≤K

 k i=1

Bi−1 iα (l1 − (i − 1)τ ) y − z, Γ (iα + 1)

where k comes from set max{k ∈ Λ : kτ ≤ l1 }. k  Bi−1 iα Choosing l1 satisfying Γ (iα+1) (l1 − (i − 1)τ ) < i=1

1 2K ,

we can obtain

1  y − z ≤ y − z. 2 This implies that has a fixed point on [0, l1 ], which reduces to a solution. Step 3. Extension of solution. Next, we extend the solution for x ≥ l1 by solving the fixed point problem, z = z, where is given by

0 Bx α −t)α  EB(x−τ ϕ (t)dt ( z)(x) = Eτ ϕ(−τ ) + τ −τ

x −t)α EB(x−τ f (t, z(t))dt, x ∈ [0, l2 ]. + τ,α 0

Set z(x) = y(x) for x ∈ [0, l2 ], where l2 is defined as follows. Take x2 > l1 k  Bi−1 iα and define I2 = [0, x2 ], μ(x) = Γ (iα+1) (x − (i − 1)τ ) , where k comes i=1

from set max{k ∈ Λ : kτ ≤ x2 }, r2 = (ϕ + M)Eα (Bx2α ) + 1, and N2 = sup f (x, 0). x∈I2    Choose l2 = l1 + min x2 − l1 , x2∗ , where x2∗ comes from set x2∗ : μ(x2∗ ) =  1 Kr2 +N2 . For x ∈ [l1 , l2 ], we obtain ( y)(x) ≤ r2 . Repeating this procedure, we get the maximal interval of existence of solution. Thus (5.14) has a unique solution y ∈ C([−τ, xm ), Rn ). Now we verify that lim y(xm ) = ∞ for xm is finite. If not, then there x→xm

exists a sequence {ln } converging to xm and a finite positive number r such that y(ln ) ≤ r for all n. Taking n sufficiently large, so that ln is infinitesimally

170 Stability and Controls Analysis for Delay Systems

close to xm , one can use the previous arguments to extend the solution beyond xm , which is a contradiction. The proof is completed.

5.1.1.3 Finite time stability results for Caputo type In this section, we show finite time stability results by using delayed MittagLeffler-type matrices. Definition 5.9. (see [24]) Eq. (5.1) is called finite time stable with respect to {0, J, τ, δ, κ} if and only if ϕ < δ implies y(x) < κ, ∀ x ∈ J , where ϕ(x), − τ ≤ x ≤ 0, is the initial time of observation, δ, κ are real positive numbers, and δ < κ. 0 Theorem 5.7. Suppose M = −τ ϕ  (s)ds < ∞. If Eα (Bx κ )
0, 0 < θ ≤ 1. The proof is completed. Next, we continue to study the finite time stability of (5.14). We give the basic definition of finite time stable as follows. Definition 5.10. (see [24]) Let y be a solution of (5.14). We say (5.14) is finite time stable with respect to {0, J, τ, δ, η} if and only if ϕ < δ implies y(x) < η, ∀ x ∈ J , where ϕ(x), − τ ≤ x ≤ 0 is the initial time of observation, δ, η are real positive numbers, and δ < η. We impose the following assumptions: [H3 ] There exists an ω(·) ∈ C(J, R+ ) such that f (x, y) ≤ ω(x), for x ∈ J and y ∈ Rn . [H4 ] There exists a ψ(·) ∈ Lq (J, R+ ), q1 = 1 − p1 , p > 1, such that  x 1 f (x, y) ≤ ψ(x) for x ∈ J , y ∈ Rn and Q(x) := 0 ψ(t)q dt q < ∞. [H5 ] There exists a L > 0 such that f (x, y) ≤ Ly, x ∈ J , and y ∈ Rn . Theorem 5.9. Assume that [H1 ]–[H3 ] hold. For a fixed k ∈ Λ, if (δ + M)Eα (Bx ) + ω α

 k i=1

Bi−1 (x − (i − 1)τ )iα Γ (iα + 1)

< η, ∀ x ∈ J, then (5.14) is finite time stable with respect to {0, J, τ, δ, η}.



(5.21)

172 Stability and Controls Analysis for Delay Systems

Proof. By [H1 ], [H2 ], and Theorem 5.5, we know (5.14) has a unique solution y ∈ C([−τ, T ], Rn ). By Lemmas 5.4, 5.5, and 5.6 and the properties of norm  · , via (5.15) we have y(x)





α ≤ EτBx ϕ(−τ ) +  

0

−τ

 

−t)α  EB(x−τ ϕ (t)dt  τ +

0

x

   B(x−τ −t)α  Eτ,α  ω(t)dt

≤ (ϕ(−τ ) + M)Eα (Bx α )

x−(k−1)τ  k (x − (i − 1)τ − t)iα−1 dt Bi−1 + ω Γ ((i − 1)α + α) 0 i=1 ⎛ ⎞ j k x−(j −2)τ  (i−1)α−1  (x − (i − 2)τ − t) ⎝ ⎠ dt + ω Bi−2 Γ ((i − 2)α + α) x−(j −1)τ j =2

i=2

≤ (δ + M)Eα (Bx α ) + ω

k  Bi−1 ((x − (i − 1)τ )iα − ((k − i)τ )iα ) Γ (iα + 1) i=1

+ ω

k  j =2

⎛ j  ×⎝ i=2

⎞   Bi−2 ((j − i + 1)τ )α(i−1) − ((j − i)τ )α(i−1) ⎠ Γ ((i − 1)α + 1)

≤ (δ + M)Eα (Bx α ) + ω

 k i=1

Bi−1 (x − (i − 1)τ )iα < η. Γ (iα + 1)

Due to (5.21), the proof is completed.

Theorem 5.10. Assume that [H1 ], [H2 ], [H4 ], and α > 1 − p1 (p > 1) hold. For a fixed k ∈ Λ, if k 

⎛

iα−1+ p1

i−1 (x − (i − 1)τ ) ⎝ B · (δ + M)Eα (Bx α ) + Q(x) 1 Γ (iα) (piα − p + 1) p i=1

< η, ∀ x ∈ J, then (5.14) is finite time stable with respect to {0, J, τ, δ, η}.

⎞ ⎠ (5.22)

Fractional delay systems Chapter | 5

173

Proof. By Lemmas 5.4, 5.5, and 5.6 and the properties of norm  · , via (5.15) and (5.22) we have y(x)



α

α ≤ ϕ(−τ )EBx τ  + Eα (Bx )



0 −τ

ϕ  (t)dt +



x

0

(x − t)α−1 ψ(t)dt Γ (α)

(x − (k − 1)τ − t)kα−1 + ··· + ψ(t)dt Bk−1 Γ (kα) 0 ≤ (ϕ(−τ ) + M)Eα (Bx α )

k  Bi−1 x−(i−1)τ (x − (i − 1)τ − t)iα−1 ψ(t)dt + Γ (iα) 0 x−(k−1)τ

i=1

≤ (ϕ(−τ ) + M)Eα (Bx α ) 1

k p x−(i−1)τ  Bi−1 p(iα−1) + (x − (i − 1)τ − t) dt Γ (iα) 0 i=1

×

1 q

x−(i−1)τ

q

ψ(t) dt 0

≤ (ϕ(−τ ) + M)Eα (Bx α )  1 

1 k p x−(i−1)τ x  q Bi−1 p(iα−1) q + (x − (i − 1)τ − t) dt ψ(t) dt Γ (iα) 0 0 i=1 ⎛ ⎞ 1 k i−1 (x − (i − 1)τ )iα−1+ p  B ⎝ ⎠ < η. · ≤ (δ + M)Eα (Bx α ) + Q(x) 1 Γ (iα) (piα − p + 1) p i=1

The proof is completed. Theorem 5.11. Assume that [H1 ]–[H3 ] and α ≥ (δ + M)Eα (Bx α ) +

1 2

hold. For a fixed k ∈ Λ, if

ω α x Eα,α (Bx α ) < η, ∀ x ∈ J, α

(5.23)

then (5.14) is finite time stable with respect to {0, J, τ, δ, η}. Proof. By Lemmas 5.4, 5.5, 5.6, and 5.7 and the properties of norm  · , via (5.15) and (5.23) we have y(x)



x

≤ (ϕ(−τ ) + M)Eα (Bx ) + Eα,α (Bx ) α

α

0

(x − t)α−1 ω(t)dt

174 Stability and Controls Analysis for Delay Systems



x

≤ (ϕ(−τ ) + M)Eα (Bx ) + ωEα,α (Bx ) α

≤ (δ + M)Eα (Bx α ) +

α

ω α x Eα,α (Bx α ) < η. α

(x − t)α−1 dt

0

The proof is completed. Theorem 5.12. Assume that [H1 ], [H2 ], [H5 ], and α ≥ k ∈ Λ, if Eα (Bx α )Eα (LΓ (α)Eα,α (Bx α )x α )
0, of two-parameter Mittag-Leffler-type generated by A tion Xτ,α,α and B as follows: ⎧ Θ, − τ  t < 0, ⎪ ⎪ ⎪ ⎪ ⎪ t = 0, ⎨I, A,B Xτ,α,β (t) = ∞ p−1  ⎪ (t − j τ )iα+β−1 ⎪ ⎪ , (p − 1)τ < t  pτ, Q (j τ ) ⎪ i+1 ⎪ ⎩ Γ (iα + β) i=0 j =0

(5.27) where the brief form of Qi+1 (j τ ) is

Fractional delay systems Chapter | 5

177

j

Q1 (j τ )

Q2 (j τ )

Q3 (j τ )

Q4 (j τ )

···

Qp+1 (j τ )

0

I

A

A2

A3

···

Ap

1

Θ

B

AB + BA

A(AB + BA) + BA2

···

···

AB 2 + B(AB + BA)

···

···

2

Θ

Θ

B2

3 ···

Θ ···

Θ ···

Θ ···

B3 ···

··· ···

··· ···

p

Θ

Θ

Θ

Θ

···

Bp

A,B and Xτ,α,β (t) can also be written in (5.28) if A and B are permutable matrices, i.e., if AB = BA, ⎧ Θ, − τ  t < 0, ⎪ ⎪ ⎪ ⎪ ⎪ I, t = 0, ⎪ ⎪ ⎪

 ⎪ ∞ ∞ ⎪ iα+β−1   ⎪ t (t − τ )iα+β−1 ⎪ i i ⎪ ⎨ + + ··· A Ai−1 B A,B Γ (iα + β) (t) = i=0 Γ (iα + β) i=1 1 Xτ,α,β ⎪

 ⎪ ∞ ⎪ iα+β−1  ⎪ i ⎪ i−p+1 p−1 (t − (p − 1)τ ) ⎪ + , B A ⎪ ⎪ ⎪ Γ (iα + β) p−1 ⎪ i=p−1 ⎪ ⎪ ⎪ ⎩ (p − 1)τ < t  pτ. (5.28)

In general, the condition of AB = BA is strong and special for the controlled system since we always avoid putting such restriction to consider controllability problems. The flexible coefficient of matrix B will be used to adjust the control function and deal with more difficulty from the main part A. For example, if the linear system generalized by A is not stable, then we can use a suitable B to make that the perturbation system is stable. In such case, it is impossible to require that AB = BA. A,B in (5.27) does generalize (5.28) in [1] and gives an explicit way Next, Xτ,α,β to compute the delay perturbation of matrix functions in the case of nonpermutable matrices. Based on (5.27), Mahmudov [106] studied the nonhomogeneous fractional order delay system 

α (C D−τ + ν)(t) = Aν(t) + Bν(t − τ ) + g(t), t ∈ J, τ > 0,

ν(t) = ψ(t), − τ  t  0,

(5.29)

A,B and obtained the expression of the solution of (5.29) by using Xτ,α,β (t). We have

A,B ν(t) = Xτ,α,1 (t

+ τ )ψ(−τ ) +

0

−τ

A,B α Xτ,α,α (t − s)[(C D−τ + ψ)(s) − Aψ(s)]ds

178 Stability and Controls Analysis for Delay Systems

+ 0

t

A,B Xτ,α,α (t − s)g(s)ds,

(5.30)

α ν)(·) is the Caputo fractional derivative, α ∈ (0, 1), A, B ∈ Rn×n , where (C D−τ + g ∈ C([−τ, τ1 ], Rn ), J := [0, τ1 ], τ1 > 0. Now, we investigate the relative controllability of the following semilinear fractional delay systems by removing the requirement on the permutable matrices: C α ( D−τ + ν)(t) = Aν(t) + Bν(t − τ ) + f (t, ν(t)) + Cu(t), t ∈ J, (5.31) ν(t) = ψ(t), − τ  t  0,

where α ∈ (0, 1), A, B, C ∈ Rn×n , AB = BA, ν ∈ C 1 ([−τ, τ1 ], Rn ), ψ ∈ Cτ1 := C 1 ([−τ, 0], Rn ), f ∈ C(J × Rn , Rn ), J = [0, τ1 ], τ1 > 0, and the control function u(·) takes values from L2 (J, Rn ). From (5.30), we know the solution of (5.31) can be expressed in the following form:

0 A,B A,B α ν(t) = Xτ,α,1 (t + τ )ψ(−τ ) + Xτ,α,α (t − s)[(C D−τ + ψ)(s) − Aψ(s)]ds −τ

t A,B Xτ,α,α (t − s)[f (s, ν(s)) + Cu(s)]ds. (5.32) + 0

By [107, p.1861] and [107, (8)], − 1].

t

α−1 E α α,α (λ(t −s) )ds 0 (t −s)

= λ1 [Eα (λt α )

A,B Lemma 5.8. The function Xτ,α,β (·) is continuous on (0, +∞).

Proof. For (p − 1)τ < t∗ < pτ , p = 1, 2, · · · , we have A,B (t) lim Xτ,α,β

t→t∗

 ∞  t iα+β−1 (t − τ )iα+β−1 i + + ··· = lim A Ai−1 B t→t∗ Γ (iα + β) Γ (iα + β) 1 i=0 i=1

 ∞ iα+β−1  i i−p+1 p−1 (t − (p − 1)τ ) + B A Γ (iα + β) p−1 i=p−1

 ∞ ∞   t iα+β−1 (t − τ )iα+β−1 i i + + ··· = A lim Ai−1 B lim t→t∗ Γ (iα + β) t→t∗ Γ (iα + β) 1 i=0 i=1

 ∞  (t − (p − 1)τ )iα+β−1 i + Ai−p+1 B p−1 lim t→t∗ Γ (iα + β) p−1 i=p−1

 ∞ ∞ iα+β−1   (t∗ − τ )iα+β−1 i i t∗ + + ··· = A Ai−1 B Γ (iα + β) Γ (iα + β) 1  ∞

i=0

i

i=1

Fractional delay systems Chapter | 5

+

∞ 

 i p−1

i=p−1

Ai−p+1 B p−1

179

(t∗ − (p − 1)τ )iα+β−1 Γ (iα + β)

A,B = Xτ,α,β (t∗ ).

For t∗ = pτ , p = 1, 2, · · · , we have A,B (t) lim Xτ,α,β

t→t∗−

 ∞  t iα+β−1 (t − τ )iα+β−1 i + + ··· A = lim Ai−1 B Γ (iα + β) Γ (iα + β) 1 t→pτ − i=0 i=1

 ∞  (t − (p − 1)τ )iα+β−1 i + Ai−p+1 B p−1 Γ (iα + β) p−1 i=p−1

 ∞ ∞  pτ iα+β−1  i (pτ − τ )iα+β−1 + + ··· = Ai Ai−1 B Γ (iα + β) Γ (iα + β) 1 i=0 i=1

 ∞  (pτ − (p − 1)τ )iα+β−1 i + Ai−p+1 B p−1 Γ (iα + β) p−1  ∞

i

i=p−1

A,B A,B = Xτ,α,β (pτ ) = Xτ,α,β (t∗ ),

A,B (t) lim Xτ,α,β

t→t∗+

 ∞ iα+β−1  t (t − τ )iα+β−1 i + + ··· Ai = lim Ai−1 B Γ (iα + β) Γ (iα + β) 1 t→pτ + i=0 i=1

 ∞  (t − (p − 1)τ )iα+β−1 i + Ai−p+1 B p−1 Γ (iα + β) p−1  ∞

i=p−1 ∞  



(t − pτ )iα+β−1 Γ (iα + β) i=p

 ∞ ∞   t iα+β−1 (t − τ )iα+β−1 i i = A lim Ai−1 B lim + + ··· 1 t→pτ + Γ (iα + β) t→pτ + Γ (iα + β) i=0 i=1

 ∞  (t − (p − 1)τ )iα+β−1 i + Ai−p+1 B p−1 lim Γ (iα + β) p−1 t→pτ + +

+

i=p−1 ∞   i=p

i p

i p

Ai−p B p

Ai−p B p lim

t→pτ +

(t − pτ )iα+β−1 Γ (iα + β)

180 Stability and Controls Analysis for Delay Systems

 ∞ iα+β−1  pτ (pτ − τ )iα+β−1 i + + ··· = Ai Ai−1 B Γ (iα + β) Γ (iα + β) 1 i=0 i=1

 ∞  (pτ − (p − 1)τ )iα+β−1 i + Ai−p+1 B p−1 Γ (iα + β) p−1 ∞ 

+

i=p−1 ∞   i=p

i p

Ai−p B p lim

t→pτ +

A,B = Xτ,α,β (pτ ) +

(t − pτ )iα+β−1 Γ (iα + β)

∞   (t − pτ )iα+β−1 i Ai−p B p lim p Γ (iα + β) t→pτ + i=p

A,B A,B = Xτ,α,β (pτ ) = Xτ,α,β (t∗ ). A,B To sum up, Xτ,α,β (·) is continuous on (0, +∞).

Lemma 5.9. For all t ≥ 0, 0 < α < 1, 0 < β  1 satisfying α + β  1, we have A,B Xτ,α,β (t)  t β−1 Eα,β ((A + B)t α ).

Proof. By (5.27), without loss of generality, for (p − 1)τ < t  pτ , p = 1, 2, · · · , we have A,B Xτ,α,β (t)   ∞ ∞  iα+β−1    (t − j τ )    Qi+1 (j τ ) Γ (iα + β)   i=0 j =0 

 ∞ ∞   t iα+β−1 (t − τ )iα+β−1 i  + + ··· Ai Ai−1 B Γ (iα + β) Γ (iα + β) 1 i=0 i=1

 ∞  (t − (p − 1)τ )iα+β−1 i + Ai−p+1 Bp−1 Γ (iα + β) p−1 i=p−1

 ∞ ∞   t iα+β−1 t iα+β−1 i + + ···  Ai Ai−1 B Γ (iα + β) Γ (iα + β) 1 i=0 i=1

 ∞  t iα+β−1 i + Ai−p+1 Bp−1 Γ (iα + β) p−1 i=p−1

=

t β−1

t α+β−1 + (A + B) Γ (β) Γ (α + β)

 " 2α+β−1 ! t 2 2 2 AB + B + A + Γ (2α + β) 1

Fractional delay systems Chapter | 5

! + A + 3

 3 1

A B + 2

 3 2

" AB + B 2

3

181

t 3α+β−1 Γ (3α + β)

+ ···

 " ! t (p−1)α+β−1 p−1 p−1 + A + Ap−2 B + · · · + Bp−1 Γ ((p − 1)α + β) 1



 ! " ∞  i i i i−1 i−p+1 p−1 A B + · · · + A B + A + 1 p−1 i=p

×

t iα+β−1 Γ (iα + β)

t β−1 t α+β−1 + (A + B)1 Γ (β) Γ (α + β) 2α+β−1 t + (A + B)2 Γ (2α + β) ∞  t (p−1)α+β−1 t iα+β−1 + (A + B)i + · · · + (A + B)p−1 Γ ((p − 1)α + β) Γ (iα + β)

 (A + B)0

i=p

=

∞ 

(A + B)i

i=0 β−1

=t

t iα+β−1 Γ (iα + β)

Eα,β ((A + B)t α ).

The result is proved. A,B (t)  Eα,1 ((A + B)t α ) = Remark 5.1. Obviously, for β = 1, Xτ,α,1 A,B (t)  t −0.5 E Eα ((A + B)t α ). If 0 < α = β < 1, then Xτ,α,α α,α ((A + B)t α ) holds for α ≥ 0.5.

Definition 5.11. (see [38, Definition 4])System (5.31) is called relatively controllable if for an arbitrary initial vector function ψ ∈ C 1 ([−τ, 0], Rn ), the final state of the vector ντ1 ∈ Rn , and time τ1 , there exists a control u ∈ L2 (J, Rn ) such that system (5.31) has a solution ν ∈ C 1 ([−τ, 0] ∪ J, Rn ) that satisfies the initial condition ψ and ν(τ1 ) = ντ1 .

5.1.2.1 Relative controllability results for linear systems For f (t, ν(t)) = 0, t ∈ J , system (5.31) reduces to the following linear fractional delay controlled system: C α ( D−τ + ν)(t) = Aν(t) + Bν(t − τ ) + Cu(t), t ∈ J, τ > 0, (5.33) ν(t) = ψ(t), − τ  t  0,

182 Stability and Controls Analysis for Delay Systems

with the solution of the form

0 A,B A,B α (t + τ )ψ(−τ ) + Xτ,α,α (t − s)[(C D−τ ν(t) = Xτ,α,1 + ψ)(s) − Aψ(s)]ds −τ

t A,B Xτ,α,α (t − s)Cu(s)ds. + 0

Next, we introduce the following representation of the fractional order delay Grammian matrix, which is an extension of the classical Grammian matrix of linear differential systems:

τ1 A,B A ,B Wτ,α [0, τ1 ] = Xτ,α,α (τ1 − s)CC Xτ,α,α (τ1 − s)ds. 0

Theorem 5.13. System (5.33) is relatively controllable if and only if Wτ,α [0, τ1 ] is nonsingular. Proof. If Wτ,α [0, τ1 ] is a nonsingular matrix, its inverse Wτ,α [0, τ1 ] is well defined. One can select a control function as follows:



A ,B −1 u(t) = C Xτ,α,α (τ1 − t)Wτ,α [0, τ1 ]η,

where

A,B (τ1 +τ )ψ(−τ )− η = ντ1 −Xτ,α,1

0

−τ

A,B α Xτ,α,α (τ1 −s)[(C D−τ + ψ)(s)−Aψ(s)]ds.

Then

0 A,B A,B α (τ1 + τ )ψ(−τ ) + Xτ,α,α (τ1 − s)[(C D−τ ν(τ1 ) = Xτ,α,1 + ψ)(s) − Aψ(s)]ds −τ

τ1 A,B + Xτ,α,α (τ1 − s)Cu(s)ds 0

0 A,B A,B α (τ1 + τ )ψ(−τ ) + Xτ,α,α (τ1 − s)[(C D−τ = Xτ,α,1 + ψ)(s) − Aψ(s)]ds −τ

τ1 A,B A ,B −1 + Xτ,α,α (τ1 − s)CC Xτ,α,α (τ1 − s)Wτ,α [0, τ1 ]ηds 0

= ντ1 . Next, we prove the necessity result by contradiction. Suppose that Wτ,α [0, τ1 ] is singular, i.e., there exists at least one nonzero state ν˜ ∈ Rn such that ν˜ Wτ,α [0, τ1 ]˜ν = 0. Further, one obtains



0 = ν˜ Wτ,α [0, τ1 ]˜ν = 0

τ1





A,B A ,B ν˜ Xτ,α,α (τ1 − s)CC Xτ,α,α (τ1 − s)˜ν ds

Fractional delay systems Chapter | 5



τ1

= 0

183

A,B ν˜ Xτ,α,α (τ1 − s)C2 ds,

which implies that A,B ν˜ Xτ,α,α (τ1 − s)C = 0 , ∀s ∈ J.

Since system (5.33) is relatively controllable, according to Definition 5.11, there exists a control u1 (t) that drives the initial state to zero state at τ1 , namely,

A,B (τ1 ν(τ1 ) = Xτ,α,1



+

τ1

0

+ τ )ψ(−τ ) +

A,B Xτ,α,α (τ1

0

−τ

A,B α Xτ,α,α (τ1 − s)[(C D−τ + ψ)(s) − Aψ(s)]ds

− s)Cu1 (s)ds = 0.

(5.34)

Similarly, there also exists a control u2 (t) that drives the initial state to state ν˜ at τ1 , i.e.,

A,B ν(τ1 ) = Xτ,α,1 (τ1



+ 0

τ1

+ τ )ψ(−τ ) +

0

−τ

A,B α Xτ,α,α (τ1 − s)[(C D−τ + ψ)(s) − Aψ(s)]ds

A,B Xτ,α,α (τ1 − s)Cu2 (s)ds = ν˜ .

Then by (5.34) and (5.35), we have

τ1 A,B Xτ,α,α (τ1 − s)C[u2 (s) − u1 (s)]ds. ν˜ =

(5.35)

(5.36)

0

Multiplying both sides of (5.36) by ν˜ , we obtain

τ1 A,B ν˜ ν˜ = ν˜ Xτ,α,α (τ1 − s)C[u2 (s) − u1 (s)]ds = 0, 0

and sequentially, ν˜ = 0, which conflicts with ν˜ being nonzero. Thus, the fractional delay Grammian matrix Wτ,α [0, τ1 ] is nonsingular. The proof is finished.

5.1.2.2 Relative controllability results for semilinear systems For f (t, ν(t)) = 0, t ∈ J , system (5.31) is a semilinear fractional delay controlled system. We assume the following hypotheses: (H1 ): The operator W : L2 (J, Rn ) → Rn defined by

τ1 A,B Xτ,α,α (τ1 − s)Cu(s)ds Wu = 0

184 Stability and Controls Analysis for Delay Systems

has an inverse operator W −1 which takes values in L2 (J, Rn )/kerW . Then we set M = W −1 Lb (Rn ,L2 (J,Rn )/kerW ) . From [87, Remark 3.3], we know # −1 M = Wτ,α [0, τ1 ].

(5.37)

(H2 ): The function f : J × Rn → Rn is continuous and Lf (·) ∈ L∞ (J, R+ ) such that f (t, ν) − f (t, μ)  Lf (t)ν − μ, t ∈ J, ν, μ ∈ Rn . Theorem 5.14. Suppose that 1 > α  0.5, (H1 ), and (H2 ) are satisfied. Then system (5.31) is relatively controllable provided that  1 M2 1 + [Eα (λτ1α ) − 1]CM < 1, (5.38) λ where M2 = λ1 [Eα (λτ1α ) − 1]Lf L∞ (J,R+ ) and λ = A + B. Proof. Using hypothesis (H1 ) for an arbitrary function ν(·) ∈ C, it is suitable to define the following control function uν (t): 

0 A,B A,B α ντ1 − Xτ,α,1 (τ1 + τ )ψ(−τ ) − Xτ,α,α (τ1 − s)[(C D−τ uν (t) = W + ψ)(s) −τ

τ1 A,B − Aψ(s)]ds − Xτ,α,α (τ1 − s)f (s, ν(s))ds (t), t ∈ J. −1

0

We show that, using this control, the operator P : C → C defined by

A,B (t (Fν)(t) = Xτ,α,1

0

A,B α Xτ,α,α (t − s)[(C D−τ + ψ)(s) − Aψ(s)]ds

t

t A,B A,B Xτ,α,α (t − s)f (s, ν(s))ds + Xτ,α,α (t − s)Cuν (s)ds + 0

+ τ )ψ(−τ ) +

−τ

0

has a fixed point ν, which is a mild solution of (5.31). We check that (Fν)(τ1 ) = ντ1 and (Fν)(0) = ν0 , which means that uν steers system (5.31) from ν0 to ντ1 in finite time τ1 . This implies system (5.31) is relatively controllable on J . For each positive number r, let Br = {ν ∈ C : ν  r}. Then, for each r, Br is obviously a bound, closed, and convex set of C. Set Rf = max f (t, 0). t∈J

In order to make the following process clear we divide it into several steps.

Fractional delay systems Chapter | 5

185

Step 1. We claim that there exists a positive number r such that F(Br ) ⊆ Br . In light of (H2 ) and the Hölder inequality, we obtain

t

(t − s)α−1 Eα,α (λ(t − s)α )Lf (s)ds 0

t (t − s)α−1 Eα,α (λ(t − s)α )dsLf L∞ (J,R+ )  0

1  [Eα (λt α ) − 1]Lf L∞ (J,R+ ) λ and

t

0

(t − s)α−1 Eα,α (λ(t − s)α )f (s, 0)ds

t Rf ≤ Rf [Eα (λt α ) − 1]. (t − s)α−1 Eα,α (λ(t − s)α )ds = λ 0

Taking into account Lemma 5.9, using (H1 ) and (H2 ), we have uν (t) ≤ W

−1

+ +

 A,B Lb (Rn ,L2 (J,Rn )/kerW ) ντ1  + Xτ,α,1 (τ1 + τ )ψ(−τ )

0

−τ

τ1

A,B α Xτ,α,α (τ1 − s)(C D−τ + ψ)(s) − Aψ(s)ds A,B Xτ,α,α (τ1 − s)f (s, ν(s))ds

!0  M ντ1  + Eα (λ(τ1 + τ )α )ψ(−τ )

+

0

−τ τ1

+

α (τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(C D−τ + ψ)(s) − Aψ(s)ds

"

(τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(f (s, ν(s)) − f (s, 0) + f (s, 0))ds

!0  M ντ1  + Eα (λ(τ1 + τ )α )ψ(−τ )

+ +

0

α (τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(C D−τ + ψ)(s) − Aψ(s)ds

−τ

τ1

(τ1 − s) Eα,α (λ(τ1 − s) )(Lf (s)ν(s) + f (s, 0))ds !0  M ντ1  + Eα (λ(τ1 + τ )α )ψ(−τ ) α−1

α

"

186 Stability and Controls Analysis for Delay Systems

+ +

0

α (τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(C D−τ + ψ)(s) − Aψ(s)ds

−τ

τ1

(τ1 − s)α−1 Eα,α (λ(τ1 − s)α )Lf (s)ν(s)ds "

τ1 α−1 α + (τ1 − s) Eα,α (λ(τ1 − s) )f (s, 0)ds !0  M ντ1  + Eα (λ(τ1 + τ )α )ψ(−τ ) 0

+

0

α (τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(C D−τ + ψ)(s) − Aψ(s)ds " Rf 1 + [Eα (λτ1α ) − 1]Lf L∞ (J,R+ ) ν + [Eα (λτ1α ) − 1] λ λ  Mντ1  + MM1 + MM2 ν, −τ

where Rf [Eα (λτ1α ) − 1] M1 = Eα (λ(τ1 + τ )α )ψ(−τ ) + λ

0 α (τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(C D−τ + + ψ)(s) − Aψ(s)ds, −τ

and M2 is defined in the above. Applying [108, Lemma 2.6], combining conditions (H1 ) and (H2 ), we obtain (Fν)(t) A,B  Xτ,α,1 (t + τ )ψ(−τ )

0 A,B α + Xτ,α,α (t − s)(C D−τ + ψ)(s) − Aψ(s)ds −τ

t

t A,B A,B Xτ,α,α (t − s)f (s, ν(s))ds + Xτ,α,α (t − s)Cuν (s)ds + 0

0

 Eα (λ(t + τ ) )ψ(−τ )

0 α (t − s)α−1 Eα,α (λ(t − s)α )(C D−τ + + ψ)(s) − Aψ(s)ds −τ

t + (t − s)α−1 Eα,α (λ(t − s)α )f (s, ν(s))ds 0

t (t − s)α−1 Eα,α (λ(t − s)α )Cuν (s)ds + α

0

 Eα (λ(τ1 + τ )α )ψ(−τ )

Fractional delay systems Chapter | 5

+ +

0

187

α (τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(C D−τ + ψ)(s) − Aψ(s)ds

−τ

τ1

(τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(Lf (s)ν(s) + f (s, 0))ds

0



τ1

+ 0

(τ1 − s)α−1 Eα,α (λ(τ1 − s)α )C[Mντ1  + MM1 + MM2 ν]ds

1  M1 + M2 νC + [Eα (λτ1α ) − 1]C[Mντ1  + MM1 + MM2 ν] λ  1 1 α = 1 + [Eα (λτ1 ) − 1]CM M1 + [Eα (λτ1α ) − 1]CMντ1  λ λ  1 + M2 1 + [Eα (λτ1α ) − 1]CM ν λ  1 1 α  1 + [Eα (λτ1 ) − 1]CM M1 + [Eα (λτ1α ) − 1]CMντ1  λ λ  1 + M2 1 + [Eα (λτ1α ) − 1]CM r = r, λ for  r=

1 + λ1 [Eα (λτ1α ) − 1]CM M1 + λ1 [Eα (λτ1α ) − 1]CMντ1   . 1 − M2 1 + λ1 [Eα (λτ1α ) − 1]CM

Hence, we obtain F(Br ) ⊆ Br for such an r. Now, we define operators F1 and F2 on Br as

0 A,B A,B α (F1 ν)(t) = Xτ,α,1 (t + τ )ψ(−τ ) + Xτ,α,α (t − s)[(C D−τ + ψ)(s) − Aψ(s)]ds −τ

t A,B Xτ,α,α (t − s)Cuν (s)ds + 0

and

t

(F2 ν)(t) = 0

A,B Xτ,α,α (t − s)f (s, ν(s))ds,

for t ∈ J , respectively. Step 2. We claim that F1 is a contraction mapping. Let ν, γ ∈ Br . In view of (H1 ) and (H2 ), for each t ∈ J , we have

τ1 A,B uν (t) − uγ (t)  M Xτ,α,α (τ1 − s)f (s, ν(s)) − f (s, γ (s))ds 0

188 Stability and Controls Analysis for Delay Systems

M

τ1

(τ1 − s)α−1 Eα,α (λ(τ1 − s)α )Lf (s)(ν(s) − γ (s))ds

0

M

τ1

(τ1 − s)α−1 Eα,α (λ(τ1 − s)α )Lf (s)dsν − γ 

0

M  [Eα (λτ1α ) − 1]Lf L∞ (J,R+ ) ν − γ  λ  MM2 ν − γ . Thus, (F1 ν)(t) − (F1 γ )(t)

t A,B  Xτ,α,α (t − s)Cuν (s) − uγ (s)ds 0

t (t − s)α−1 Eα,α (λ(t − s)α )dsCMM2 ν − γ   0

1 = [Eα (λt α ) − 1]CMM2 ν − γ  λ CMM2  [Eα (λτ1α ) − 1]ν − γ . λ So we obtain F1 ν − F1 γ   P ν − γ , 2 where P = CMM [Eα (λτ1α ) − 1]. λ In view of (5.38), we conclude that P < 1, which implies F1 is a contraction. Step 3. We claim that F2 is a compact and continuous operator. Let νn ∈ Br with νn → ν in Br . Using (H2 ), we have f (s, νn (s)) → f (s, ν(s)) in C and thus, using the dominated convergence theorem,

(F2 νn )(t) − (F2 ν)(t)

t A,B  Xτ,α,α (t − s)f (s, νn (s)) − f (s, ν(s))ds 0

t  (t − s)α−1 Eα,α (λ(t − s)α )f (s, νn (s)) − f (s, ν(s))ds 0

→ 0 as n → ∞, which implies that F2 is continuous on Br . To check the compactness of F2 , we prove that F2 (Br ) ∈ C is equicontinuous and uniformly bounded. In fact, for any ν ∈ Br , 0 < t < t + h  τ1 , (F2 ν)(t + h) − (F2 ν)(t)

Fractional delay systems Chapter | 5





t+h

=

A,B Xτ,α,α (t + h − s)f (s, ν(s))ds −

0



t+h

= t



+

t

0

t

0

189

A,B Xτ,α,α (t − s)f (s, ν(s))ds

A,B Xτ,α,α (t + h − s)f (s, ν(s))ds A,B A,B (Xτ,α,α (t + h − s) − Xτ,α,α (t − s))f (s, ν(s))ds.

Denote

t+h

W1 =

t t

W2 = 0

A,B Xτ,α,α (t + h − s)f (s, ν(s))ds,

A,B A,B (Xτ,α,α (t + h − s) − Xτ,α,α (t − s))f (s, ν(s))ds.

From the above, we derive (F2 ν)(t + h) − (F2 ν)(t)  W1  + W2 . Now, we only need to check Wi  → 0 as h → 0, i = 1, 2. Clearly,

W1  

t+h

A,B Xτ,α,α (t + h − s)f (s, ν(s))ds

t



t+h

(t + h − s)α−1 Eα,α (λ(t + h − s)α )(Lf (s)ν(s) + f (s, 0))ds

t



t+h

t

(t + h − s)α−1 Eα,α (λ(t + h − s)α )dsLf L∞ (J,R+ ) ν

+ Rf

t+h

(t + h − s)α−1 Eα,α (λ(t + h − s)α )ds

t

Rf 1 [Eα (λhα ) − 1]  [Eα (λhα ) − 1]Lf L∞ (J,R+ ) ν + λ λ → 0 as h → 0 and

t

W2  

0 t



A,B A,B Xτ,α,α (t + h − s) − Xτ,α,α (t − s)f (s, ν(s))ds

A,B A,B Xτ,α,α (t + h − s) − Xτ,α,α (t − s)Lf (s)ν(s)ds

t A,B A,B Xτ,α,α (t + h − s) − Xτ,α,α (t − s)f (s, 0)ds + 0

t A,B A,B  Xτ,α,α (t + h − s) − Xτ,α,α (t − s)dsLf L∞ (J,R+ ) ν 0

0

190 Stability and Controls Analysis for Delay Systems



t

+ Rf 0

A,B A,B Xτ,α,α (t + h − s) − Xτ,α,α (t − s)ds

→ 0 as h → 0. As a result, we immediately obtain (F2 ν)(t + h) − (F2 ν)(t) → 0 as h → 0, for all ν ∈ Br . Therefore, F2 (Br ) is equicontinuous in C. Next, repeating the above computations, we have

t A,B Xτ,α,α (t − s)f (s, ν(s))ds (F2 ν)(t)  0

t  (t − s)α−1 Eα,α (λ(t − s)α )f (s, ν(s))ds 0

τ1  (τ1 − s)α−1 Eα,α (λ(τ1 − s)α )(Lf (s)ν(s) + f (s, 0))ds 0

Rf 1 [Eα (λτ1α ) − 1].  [Eα (λτ1α ) − 1]Lf L∞ (J,R+ ) r + λ λ Hence F2 (Br ) is bounded. By the Arzela–Ascoli theorem, F2 (Br ) ⊂ C is relatively compact in C. Thus, F2 is a compact and continuous operator. Using Krasnoselskii’s fixed point theorem (see [82]), F has a fixed point ν on Br . Obviously, ν is a solution of system (5.31) satisfying ν(τ1 ) = ντ1 . The boundary condition ν(t) = ψ(t), −τ  t  0 holds by (5.32). The proof is completed.

5.1.2.3 Numerical examples and discussion Example 5.3. Set α = 0.5. Consider the following semilinear fractional delay differential controlled system: ⎧ α ⎪ (C D−0.5 + ν)(t) = Aν(t) + Bν(t − 0.5) + f (t, ν(t)) + Cu(t), t ∈ [0, 1.5], ⎪ ⎨

 0.4 ⎪ ⎪ ⎩ν(t) = 0.5 , − 0.5  t  0, (5.39) where we set

A=

C=



 0.18 0 , B= , 0 0.2 

 0.1 0.2tν1 (t) , f (t, ν(t)) = . 0.3 0.15tν2 (t)

0.2 0 0 0.2 0.4 0.2

Clearly λ = A + B = 0.4, C = 0.5.

Fractional delay systems Chapter | 5

191

Now we use (5.37) to estimate M. For this purpose, we need to obtain Wτ,α [0, τ1 ] and then Wτ,α [0, τ1 ]−1 . The fractional delay Grammian matrix has the following explicit form:

τ1 A,B A ,B Wτ,α [0, τ1 ] = Xτ,α,α (τ1 − s)CC Xτ,α,α (τ1 − s)ds 0

=

1.5

0





A,B A ,B Xτ,α,α (1.5 − s)CC Xτ,α,α (1.5 − s)ds

= S1 + S2 + S3 , where

S1 =

∞ 0.5 ! 

0

+

i=0

∞ 

iα+α−1 i (1.5 − s)

A

Γ (iα + α)



i 2

Ai−2 B 2

+

∞ 

 i 1

i=1

Ai−1 B

(1 − s)iα+α−1 Γ (iα + α)

" (0.5 − s)iα+α−1

Γ (iα + α)

 ! ∞ ∞ iα+α−1  (1 − s)iα+α−1 i i (1.5 − s) + (A ) × CC (A )i−1 B Γ (iα + α) Γ (iα + α) 1 i=0 i=1

 " ∞  i (0.5 − s)iα+α−1 ds, + (A )i−2 (B )2 Γ (iα + α) 2 i=2

 "

1 ! ∞ ∞ iα+α−1  (1 − s)iα+α−1 i i (1.5 − s) S2 = A Ai−1 B + Γ (iα + α) Γ (iα + α) 1 0.5 i=0 i=1 ! ∞ (1.5 − s)iα+α−1 × C (A )i Γ (iα + α) i=0

 ∞ iα+α−1 "  i i−1 (1 − s) ds, + (A ) B Γ (iα + α) 1 i=2

i=1

and ∞ 1.5 ! 

S3 = 1

i=0

iα+α−1 " i (1.5 − s)

A

Γ (iα + α)

! ∞ iα+α−1 " i (1.5 − s) CC ds. (A ) Γ (iα + α)

i=0

By calculation,

S1 =

S3 =

 0.0959 0.0641 0.0641 0.0783 0.5270 0.3410 0.3410 0.4030

, S2 =

 .

 0.1089 0.0720 0.0720 0.0870

,

192 Stability and Controls Analysis for Delay Systems

Therefore, we obtain

Wτ,α [0, 1.5] =

−1 Wτ,α [0, 1.5] =

and

# M=

 0.7318 0.4771 0.4771 0.5683

,

3.0187 −2.5343 −2.5343 3.8872

−1 Wτ,α [0, 1.5] =

 ,

√ 1.3529 = 1.1631.

Further, for any ν, μ ∈ Rn , f (t, ν) − f (t, μ) = max{0.2t|ν1 − μ1 |, 0.15t|ν2 − μ2 |} ≤ 0.2t max{|ν1 − μ1 |, |ν2 − μ2 |} = 0.2tν − μ. Note Lf (t) = 0.2t and Lf L∞ (J,R+ ) = 0.2 × 1.5 = 0.3, Eα (λτ1α ) = 1.5056. As a result 1 1 × (1.5056 − 1) × 0.3 = 0.3792 M2 = [Eα (λτ1α ) − 1]Lf L∞ (J,R+ ) = λ 0.4 and 

1 α M2 1 + [Eα (λτ1 ) − 1]CM λ 1 × (1.5056 − 1) × 0.5 × 1.1631) = 0.6579 < 1. = 0.3792 × (1 + 0.4 Thus all the conditions of Theorem 5.14 are satisfied, so (5.39) is relatively controllable on [0, 1.5]. We can find the control function u = (4t, 5t) , which makes the solution of system (5.39) satisfy ν(1.5) = (3.3, 4.8) . Example 5.4. Consider the relative controllability of system (5.39) (with f (t, ν(t)) ≡ 0) on J , where A, B, and C are defined in Example 5.3. Letting α = 0.7, by calculation,



 0.1378 0.0912 0.1232 0.0807 S1 = , S2 = , 0.0912 0.1101 0.0807 0.0964

 0.2132 0.1379 S3 = . 0.1379 0.1630

Fractional delay systems Chapter | 5

Then

W0.5,0.7 [0, 1.5] = S1 + S2 + S3 =

193

 0.4742 0.3098 0.3098 0.2228

,

W0.5,0.7 [0, 1.5] is a nonsingular matrix, and

 23.0267 −32.0183 −1 W0.5,0.7 [0, 1.5] = . −32.0183 49.0092 According to Theorem 5.13, we can know that system (5.39) is relative controllability when f (·, ν(·)) = 0. We can get A,B η = ντ1 − Xτ,α,1 (1.5 + 0.5)ψ(−0.5)

0 A,B α − Xτ,α,α (1.5 − s)[(C D−τ + ψ)(s) − Aψ(s)]ds −0.5

 0.6802 = ντ1 − . 0.8713

By using the form of the control u(t), we get



A ,B −1 (τ1 − t)Wτ,α [0, τ1 ]η u(t) = C Xτ,α,α



−1 A ,B (1.5 − t)W0.5,0.7 [0, 1.5]η = C Xτ,α,α

 ⎧ ! ∞ ∞ 0.7i−0.3  ⎪ (1.5 − t)0.7i−0.3  i ⎪ i i−1 (1 − t) ⎪ C + (A ) ) B (A ⎪ ⎪ Γ (0.7i + 0.7) Γ (0.7i + 0.7) 1 ⎪ ⎪ i=0 i=1 ⎪ ⎪

 ⎪ ⎪ ∞ 0.7i−0.3 " ⎪  ⎪ i −1 i−2 2 (0.5 − t) ⎪ ⎪ W0.5,0.7 ) (B ) [0, 1.5]η, + (A ⎪ ⎪ Γ (0.7i + 0.7) 2 ⎪ ⎪ i=2 ⎪ ⎪ ⎪ ⎪ 0  t < 0.5, ⎪ ⎪ ⎨ !

 " ∞ ∞  = (1.5 − t)0.7i−0.3  i (1 − t)0.7i−0.3 i i−1 ⎪ + C (A ) (A ) B ⎪ ⎪ Γ (0.7i + 0.7) Γ (0.7i + 0.7) 1 ⎪ ⎪ i=0 i=1 ⎪ ⎪ ⎪ ⎪ × W −1 [0, 1.5]η, 0.5  t < 1, ⎪ ⎪ ⎪ ⎪ ! 0.5,0.7 ⎪ ∞ ⎪ 0.7i−0.3 "  ⎪ ⎪ i (1.5 − t) −1 ⎪ Wτ,α (A ) [0, 1.5]η, 1  t < 1.5, C ⎪ ⎪ ⎪ Γ (0.7i + 0.7) ⎪ i=0 ⎪ ⎪ ⎪ ⎩ −1 C W0.5,0.7 [0, 1.5]η, t = 1.5.

5.1.2.4 Conclusion Based on the fractional delay Grammian matrix defined by a two-parameter Mittag-Leffler-type delay matrix function, we show sufficient and necessary

194 Stability and Controls Analysis for Delay Systems

FIGURE 5.1

A,B The trajectories of Xϑ,α,β (t) for A = 0.8, B = 0.9, and ϑ = 0.2.

A,B FIGURE 5.2 The trajectories of Xϑ,α,β (t) for A = 0.8, B = 0.9, ϑ = 0.2, α = 0.6, and β = 0.6. The right is an enlarged view of the left, and ◦ is the coordinate (0, 1), that is to say, the value of 0.8,0.9 X0.2,0.6,0.6 (t) at 0 is 1.

conditions to guarantee a linear controlled system is relatively controllable. Next, we present sufficient conditions for controllability results by Krasnoselskii’s fixed point theorem. Finally, we use two numerical examples to illustrate our theories (see Figs. 5.1 and 5.2). The results in this part are motivated from [109].

Fractional delay systems Chapter | 5

195

5.2 Finite time stability and controllability for Riemann–Liouville type 5.2.1 Finite time stability for Riemann–Liouville type In this section, we study the finite time stability of linear fractional delay differential systems given by ⎧ RL α n×n ⎪ ⎪ ⎨ ( D−τ + y)(x) = By(x − τ ) + f (x), B ∈ R , x ∈ (0, T ], τ > 0, n y(x) = ω(x), ω(x) ∈ R , − τ ≤ x ≤ 0, ⎪ ⎪ ⎩ (I1−α y)(−τ + ) = ω(−τ ), ω(−τ ) ∈ Rn , −τ +

(5.40) α y denotes the Riemann–Liouville derivative of order α ∈ (0, 1) where RL D−τ + 1−α (see Definition 5.3), I−τ + y denotes the Riemann–Liouville fractional integral of order 1 − α (see Definition 5.2), f ∈ C([−τ, T ], Rn ), and ω is an arbitrary α ω exists. Riemann–Liouville differentiable vector function, i.e., RL D−τ +

5.2.1.1 Representation of solutions for linear systems Lemma 5.10. Letting (k − 1)τ < x ≤ kτ , −τ ≤ s ≤ t, and k ∈ N+ being a fixed number, we have

x

(x s

−s)α − t)−α EB(t−τ dt τ,α

=

k



x

i=0 iτ +s

(x − t)−α B i

(t − iτ − s)(i+1)α−1 dt. Γ (iα + α)

Proof. The proof is similar to that of Lemma 5.3, so we omit it here. Firstly, we use (5.4) to construct the explicit formula of solutions of ⎧ RL α n ⎪ ⎪ ⎨ ( D−τ + y)(x) = By(x − τ ), y(x) ∈ R , x ∈ (0, T ], τ > 0, (5.41) y(x) = ω(x), ω(x) ∈ Rn , − τ ≤ x ≤ 0, ⎪ ⎪ 1−α ⎩ (I y)(−τ + ) = ω(−τ ), ω(−τ ) ∈ Rn . −τ +

α

n×n , Theorem 5.15. For the delayed Mittag-Leffler-type matrix EB· τ,α : R → R B(x−τ )α

α

α EBt )(x) = BE one has (RL D−τ τ,α + τ,α α y)(x) = By(x (RL D−τ +

−τ ≤ x ≤ 0.

α

, i.e., EBx τ,α

is a solution of

(τ +x) − τ ) that satisfies initial conditions EBx τ,α = I Γ (α) , α

α−1

Proof. For x ∈ ((k − 1)τ, kτ ] and k ∈ N+ , we adopt mathematical induction to prove our result. (i) For k = 1, 0 < x ≤ τ , we have α

y(x) = EBx τ,α = I

x 2α−1 (τ + x)α−1 +B . Γ (α) Γ (α + α)

(5.42)

196 Stability and Controls Analysis for Delay Systems α

Using Definition 5.3 for EB· τ,α , via (5.42) and Lemma 5.1 we obtain α

α Bt (RL D−τ + Eτ,α )(x)

 0

x d 1 (x − t)−α y(t)dt + (x − t)−α y(t)dt Γ (1 − α) dx −τ 0 1 = Γ (1 − α)  x

x d (τ + t)α−1 t 2α−1 × dt + dt (x − t)−α I (x − t)−α B dx Γ (α) Γ (α + α) −τ 0  d B[1 − α, α] B 1 x α−1 I + x α B[1 − α, 2α] = B . = Γ (1 − α) dx Γ (α) Γ (2α) Γ (α) =

(ii) For k = 2, τ < x ≤ 2τ , we have α

y(x) = EBx τ,α = I

x 2α−1 (x − τ )3α−1 (τ + x)α−1 +B + B2 . Γ (α) Γ (α + α) Γ (2α + α)

(5.43)

α

Using Definition 5.3 for EB· τ,α , via (5.43) and Lemma 5.1 we obtain α

α Bt (RL D−τ + Eτ,α )(x)

=

d 1 Γ (1 − α) dx



τ −τ

(x − t)−α y(t)dt +



x

(x − t)−α y(t)dt



τ

1 d (t − τ )3α−1 dt (x − t)−α B 2 Γ (α) Γ (1 − α) dx τ Γ (2α + α) x α−1 (x − τ )2α−1 + B2 . =B Γ (α) Γ (2α) x α−1

=B

+

x

(iii) Supposing k = n, (n − 1)τ < x ≤ nτ and n ∈ N+ , the following relation holds: α

α Bt (RL D−τ + Eτ,α )(x) = B

(x − τ )2α−1 (x − (n − 1)τ )nα−1 x α−1 + B2 + · · · + Bn . Γ (α) Γ (2α) Γ (nα)

Next, for k = n + 1, nτ < x ≤ (n + 1)τ , we have α

y(x) = EBx τ,α =I

x 2α−1 (x − nτ )(n+2)α−1 (τ + x)α−1 +B + · · · + B n+1 . (5.44) Γ (α) Γ (α + α) Γ ((n + 1)α + α) α

By using Definition 5.3 for EB· τ,α , via (5.44) and Lemma 5.1 we obtain

Fractional delay systems Chapter | 5

197

α

α Bt (RL D−τ + Eτ,α )(x)

1 d = Γ (1 − α) dx



0

−τ

−α

(x − t)

y(t)dt + · · · +

x

−α

(x − t)

y(t)dt

nτ

(x − τ )2α−1 (x − (n − 1)τ )nα−1 x α−1 + B2 + · · · + Bn Γ (α) Γ (2α) Γ (nα) 

x d B n+1 1 + (x − t)−α (t − nτ )(n+2)α−1 dt Γ (1 − α) dx Γ ((n + 1)α + α) nτ

=B

=B

(x − τ )2α−1 (x − nτ )(n+1)α−1 x α−1 + B2 + · · · + B n+1 . Γ (α) Γ (2α) Γ ((n + 1)α)

Therefore, for any (k − 1)τ < x ≤ kτ and k ∈ N+ , by mathematical induction, we have α

α Bt (RL D−τ + Eτ,α )(x)  α−1 kα−1 x (x − τ )2α−1 k−1 (x − (k − 1)τ ) =B +B + ··· + B Γ (α) Γ (2α) Γ (kα) α

) . = BEB(x−τ τ,α

The proof is completed. Theorem 5.16. Let (n − 1)τ < x ≤ nτ for all n ∈ {0, 1, 2, · · · , k ∗ }. A solution y ∈ C(X, Rn ) of (5.41) can be expressed by the following formula:

α

y(x) = EBx τ,α ω(−τ ) +

0

−τ

−s) RL α EB(x−τ ( D−τ + ω)(s)ds, τ,α α

where either X = ((n − 1)τ, nτ ] for 0 < α < 1/(n + 1) or X = [(n − 1)τ, nτ ] for α ≥ 1/(n + 1). α

Proof. Let matrix Y0 (x) = EBx τ,α satisfy Theorem 5.15. We should search any solution of (5.41) satisfying initial conditions y(x) = ω(x), −τ ≤ x ≤ 0, in the form

0 −s)α y(x) = Y0 (x)C + EB(x−τ z(s)ds, (5.45) τ,α −τ

where C is an unknown vector constant and z(·) is an unknown Riemann– Liouville differentiable vector function. According to the matrix Y0 (x) is a y)(−τ + ) = ω(−τ ). solution of (5.41). Therefore, we choose C satisfying (I1−α −τ + B(−2τ −s)α

Let us assume x = −τ . Using (5.4), we obtain Eτ,α s ≤ 0. For −τ < x ≤ 0, we have 1−α 1−α + lim (I−τ ω(−τ ) = (I−τ + y)(−τ ) = + y)(x) x→−τ +

= Θ with −τ ≤

198 Stability and Controls Analysis for Delay Systems



x 1 −α = lim (x − t) Y0 (t)Cdt x→−τ + Γ (1 − α) −τ  x C −α α−1 = lim (x − t) (τ + t) dt = lim C = C, x→−τ + Γ (1 − α) x→−τ + −τ 

which implies that formula (5.45) takes the form α y(x) = EBx τ,α ω(−τ ) +



0

−τ

−s) EB(x−τ z(s)ds. τ,α α

Since −τ ≤ x ≤ 0, one should divide the interval into two subintervals: (i) For −τ ≤ s ≤ x and −τ ≤ x − τ − s ≤ x, the delayed Mittag-Leffler-type α−1 B(x−τ −s)α matrix Eτ,α = I (x−s) Γ (α) . (ii) For x ≤ s ≤ 0 and x − τ ≤ x − τ − s ≤ −τ , the delayed Mittag-LefflerB(x−τ −s)α = Θ. Thus on the interval −τ ≤ x ≤ 0, we have type matrix Eτ,α ω(x) = I

(τ + x)α−1 ω(−τ ) + Γ (α)



x

−τ

I

(x − s)α−1 z(s)ds. Γ (α)

(5.46)

By Riemann–Liouville fractional differentiation on both sides of (5.46), we obtain α (RL D−τ + ω)(x) 1 = Γ (1 − α) 

x

x (τ + t)α−1 d (t − s)α−1 −α I × (x − t) I ω(−τ ) + z(s)ds dt dx −τ Γ (α) Γ (α) −τ  x

x d z(s) I (x − t)−α (t − s)α−1 dt ds = Γ (1 − α) dx −τ Γ (α) s

x d z(s)ds = z(x). = dx −τ

The proof is completed. Now we are ready to derive a special solution of (5.40) with zero initial conditions. Theorem 5.17. A solution  y ∈ C([−τ, T ], Rn ) of (5.40) satisfying initial conditions y(x) = 0, x ∈ [−τ, 0] has the form

 y (x) =

x

−τ

−t) EB(x−τ f (t)dt, x ∈ [0, T ]. τ,α α

Fractional delay systems Chapter | 5

199

Proof. Using the method of variation of constants, we will search the solution of the nonhomogeneous system  y (x) in the form

 y (x) =

x

−τ

−t) EB(x−τ c(t)dt, τ,α α

(5.47)

where c(·), −τ ≤ t ≤ x, is an unknown vector function and  y (0) = 0. Taking the Riemann–Liouville fractional differentiate for formula (5.47), we obtain: (i) For k = 1 and 0 < x ≤ τ , according to (5.40), we have

RL

(

α D−τ y )(x) = B y (x +

=B

− τ ) + f (x) = B

x−τ

−τ

x−τ

−τ

−t) EB(x−2τ c(t)dt + f (x) τ,α α

(x − τ − t)α−1 c(t)dt + f (x). I Γ (α)

However, according to Definition 5.3 and Lemma 5.10, we have α (RL D−τ y )(x) +

 t

x d 1 −α B(t−τ −s)α = (x − t) Eτ,α c(s)ds dt Γ (1 − α) dx −τ −τ  x

x d 1 −α B(t−τ −s)α = c(s) (x − t) Eτ,α dt ds Γ (1 − α) dx −τ s 

x

x d (t − s)α−1 1 dt c(s) (x − t)−α I = Γ (1 − α) dx −τ Γ (α) s

x (t − τ − s)2α−1 + dt ds (x − t)−α B Γ (2α) τ +s

x d = c(s)ds dx −τ  x

x−τ 2α−1 d 1 −α (t − τ − s) dt ds c(s) (x − t) B + Γ (1 − α) dx −τ Γ (2α) τ +s

x−τ B d = c(x) + c(s)(x − τ − s)α ds Γ (α + 1) dx −τ

x−τ (x − τ − s)α−1 c(s)ds. = c(x) + B Γ (α) −τ

Hence, we obtain c(x) = f (x). (ii) For (k − 1)τ < x ≤ kτ and k ∈ N+ , according to (5.40), we have

RL

(

α D−τ y )(x) = B y (x +

− τ ) + f (x) = B

x−τ

−τ

−t) EB(x−2τ c(t)dt + f (x) τ,α α

200 Stability and Controls Analysis for Delay Systems k



=

x−iτ

i=1 −τ

(x − iτ − t)iα−1 c(t)dt + f (x). Γ (iα)

Bi

However, according to Definition 5.3, we have  t

x d 1 −s)α (x − t)−α EB(t−τ c(s)ds dt τ,α Γ (1 − α) dx −τ −τ  x

x d 1 −s)α = c(s) (x − t)−α EB(t−τ dt ds. τ,α Γ (1 − α) dx −τ s

α (RL D−τ y )(x) = +

According to Lemma 5.1 and (5.40), we obtain α (RL D−τ y )(x) +

x k  1 d c(s)ds + = dx −τ Γ (1 − α) i=1



x−iτ x (i+1)α−1 d −α i (t − iτ − s) c(s) (x − t) B × dt ds dx −τ Γ (iα + α) iτ +s

x−iτ k  d Bi c(s)(x − iτ − s)iα ds = c(x) + Γ (iα + 1) dx −τ i=1

= c(x) +

k



x−iτ

i=1 −τ

Bi

(x − iτ − t)iα−1 c(t)dt. Γ (iα)

Hence, we obtain c(x) = f (x). The proof is completed. Linking Theorems 5.16 and 5.17 via the superposition principle, we have the following result. Theorem 5.18. A solution y ∈ C(X, Rn ) ∩ C([−τ, T ], Rn ) of (5.40) satisfying initial conditions y(x) = ω(x), −τ ≤ x ≤ 0, has the form α y(x) = EBx τ,α ω(−τ ) +

+

x

−τ



0

−τ

−s) RL α EB(x−τ ( D−τ + ω)(s)ds τ,α α

−t) EB(x−τ f (t)dt. τ,α α

Remark 5.2. A discontinuity effect is displayed in (5.40) with x = 0. α y)(x) = Ay(x − τ ) + f (x) I. Since the variable of the equation (RL D−τ + takes value in the interval (0, ∞), the time point x = 0 is a discontinuity point of α y denotes a Riemann–Liouville fractional derivathe first kind, where RL D−τ + tive.

Fractional delay systems Chapter | 5

201

α y)(x) = Ay(x − τ ) + f (x), x > 0, at point x = 0 II. The equation (RL D−τ + is continuous if and only if lim y(x) = lim ω(x) = ω(0), where x→0+

x→0−

lim y(x)

x→0+

τ α−1 =I ω(−τ ) + Γ (α) τ α−1 ω(−τ ) + =I Γ (α)



0

−τ 0



−τ

−s)α RL α EA(−τ ( D−τ + ω)(s)ds τ,α

+

(−s)α−1 RL α ( D−τ + ω)(s)ds + Γ (α)



0

−t) EA(−τ f (t)dt τ,α α

−τ 0 (−t)α−1

−τ

Γ (α)

f (t)dt.

α

n×n in (5.4) is nonRemark 5.3. The Mittag-Leffler-type matrix EA· τ,α : R → R α negative. However, the monotonicity of the Mittag-Leffler-type matrix EA· τ,α : R → Rn×n depends on constants 0 < α < 1 and n ∈ N.

5.2.1.2 Finite time stability for linear systems In this section, we study the finite time stability of ⎧ RL α n×n ⎪ ⎪ ⎪ ( D−τ + y)(x) = Ay(x − τ ) + f (x), 0 < α < 1, A ∈ R , ⎪ ⎨ x ∈ (0, T ], τ > 0, ⎪ y(x) = μ(x), μ(x) ∈ Rn , − τ < x ≤ 0, ⎪ ⎪ ⎪ ⎩ (I1−α y)(−τ + ) = b, b ∈ Rn .

(5.48)

−τ +

 For 0 < γ < 1, we denote Cγ ([a, b], Rn ) = y(x) ∈ C((a, b], Rn ): (x −  γ n a) y(x) ∈ C([a, b], R ) . Then, Cγ ([a, b], Rn ) is a Banach space endowed with yCγ = max (x − a)γ y(x). a≤x≤b

Definition 5.12. (see [24]) Eq. (5.48) is finite time stable with respect to {0, J¯, τ, δ, η} if and only if μC < δ and b < δ implies a solution y of (5.48) satisfying yCγ < η and δ < η, δ, η > 0, where J¯ := ((n − 1)τ, nτ ]. Lemma 5.11. For any x ∈ ((n − 1)τ, nτ ], n ∈ {1, 2, · · · , k ∗ } and 0 < γ < 1, we obtain: 1 , we have (I) Letting 0 < α < n+1 α+γ −1 (x − (n − 1)τ )γ EAx Eα,α (A(x − (n − 1)τ )α ). τ,α  ≤ (x − (n − 1)τ ) α

(II) Letting

1 n+1

≤ α ≤ 12 , we have α

γ (x − (n − 1)τ )γ EAx τ,α  ≤ (x − (n − 1)τ )

n  j =0

Aj

(x − (j − 1)τ )(j +1)α−1 . Γ (j α + α)

202 Stability and Controls Analysis for Delay Systems

(III) Letting α > 12 , we have α+γ −1 (x − (n − 1)τ )γ EAx Eα,α (Ax α ). τ,α  ≤ x α

α

Proof. According to the definition of EA· τ,α , one has the following three cases: 1 (I) Letting x ∈ ((n − 1)τ, nτ ] and 0 < α < n+1 , we have α

(x − (n − 1)τ )γ EAx τ,α   (τ + x)α−1 (x − τ )3α−1 x 2α−1 ≤ (x − (n − 1)τ )γ + A + A2 Γ (α) Γ (α + α) Γ (2α + α) (n+1)α−1 (x − (n − 1)τ ) + · · · + An Γ (nα + α)  (x − (n − 1)τ )α−1 (x − (n − 1)τ )2α−1 + A ≤ (x − (n − 1)τ )γ Γ (α) Γ (α + α) 3α−1 (x − (n − 1)τ ) (x − (n − 1)τ )(n+1)α−1 + · · · + An + A2 Γ (2α + α) Γ (nα + α) n (j +1)α−1  (x − (n − 1)τ ) ≤ (x − (n − 1)τ )γ Aj Γ (j α + α) j =0

α+γ −1

≤ (x − (n − 1)τ )

Eα,α (A(x − (n − 1)τ )α ).

(II) Letting x ∈ ((n − 1)τ, nτ ] and

1 n+1

≤ α ≤ 12 , we have

α

(x − (n − 1)τ )γ EAx τ,α   α−1 x 2α−1 (x − τ )3α−1 γ (τ + x) + A + A2 ≤ (x − (n − 1)τ ) Γ (α) Γ (α + α) Γ (2α + α) (n+1)α−1 (x − (n − 1)τ ) + · · · + An Γ (nα + α) n  (x − (j − 1)τ )(j +1)α−1 . ≤ (x − (n − 1)τ )γ Aj Γ (j α + α) j =0

(III) Letting x ∈ ((n − 1)τ, nτ ] and α > 12 , we have α

(x − (n − 1)τ )γ EAx τ,α   (τ + x)α−1 (x − τ )3α−1 x 2α−1 ≤ xγ + A + A2 Γ (α) Γ (α + α) Γ (2α + α) (n+1)α−1 (x − (n − 1)τ ) + · · · + An Γ (nα + α)

Fractional delay systems Chapter | 5

 ≤ xγ

203

x 2α−1 x α−1 x 3α−1 x (n+1)α−1 + A + A2 + · · · + An Γ (α) Γ (α + α) Γ (2α + α) Γ (nα + α)

≤ x α+γ −1 Eα,α (Ax α ). This proof is finished. Lemma 5.12. For any x ∈ ((n − 1)τ, nτ ], n ∈ {1, 2, · · · , k ∗ }, we have

0 −τ

−s) EA(x−τ ds ≤ τ,α α

n  j =0

−

Aj (x − (j − 1)τ )(j +1)α Γ ((j + 1)α + 1)

n  Aj −1 (x − (j − 1)τ )j α . Γ (j α + 1) j =1

α

Proof. According to the definition of EA· τ,α , we obtain

0

−τ

−s) EA(x−τ ds τ,α α



=

x−nτ

−τ x−nτ

−s)α EA(x−τ ds τ,α



+

0 x−nτ

−s) EA(x−τ ds τ,α α

(x − s)α−1 (x − τ − s)2α−1 (x − 2τ − s)3α−1 + A + A2 ≤ Γ (α) Γ (α + α) Γ (2α + α) −τ nα−1 (x − (n − 1)τ − s) ds + · · · + An−1 Γ ((n − 1)α + α)

x−nτ (x − nτ − s)(n+1)α−1 + An ds Γ (nα + α) −τ

0  (x − s)α−1 (x − τ − s)2α−1 (x − 2τ − s)3α−1 + A + A2 + Γ (α) Γ (α + α) Γ (2α + α) x−nτ nα−1 (x − (n − 1)τ − s) ds + · · · + An−1 Γ ((n − 1)α + α)

x−nτ (x − nτ − s)(n+1)α−1 ds ≤ An Γ (nα + α) −τ 

0 (x − s)α−1 (x − τ − s)2α−1 + + A Γ (α) Γ (α + α) −τ 3α−1 (x − 2τ − s) (x − (n − 1)τ − s)nα−1 ds + A2 + · · · + An−1 Γ (2α + α) Γ ((n − 1)α + α)

x−nτ (x − nτ − s)(n+1)α−1 ds ≤ An Γ (nα + α) −τ



204 Stability and Controls Analysis for Delay Systems

+

n  Aj −1 Γ (j α)

j =1

≤

≤

0

−τ

(x − (j − 1)τ − s)j α−1 ds

An

(x − (n − 1)τ )(n+1)α Γ ((n + 1)α + 1)  n  Aj −1 (x − (j − 2)τ )j α − (x − (j − 1)τ )j α + Γ (j α + 1)

j =1 n  j =0

−

Aj (x − (j − 1)τ )(j +1)α Γ ((j + 1)α + 1)

n  Aj −1 (x − (j − 1)τ )j α . Γ (j α + 1) j =1

This proof is completed. Lemma 5.13. For any α > 1 − {1, 2, · · · , k ∗ }, we have

0

−τ

1 p

(p > 1), x ∈ ((n − 1)τ, nτ ] and n ∈

−s) α EA(x−τ (RL D−τ + μ)(s)ds τ,α α

≤

n   j =0



×

(j +1)α−1+ p1

Aj (x − (j − 1)τ ) Γ (j α + α) (p(j + 1)α − p + 1) p1 0

−τ

(

RL

α q D−τ + μ)(s) ds



q1 .

α

Proof. According to the definition of EA· τ,α , using Hölder’s inequality, we derive

0 −τ

−s) α EA(x−τ (RL D−τ + μ)(s)ds τ,α α



=

x−nτ

−τ



+

≤

−s) α EA(x−τ (RL D−τ + μ)(s)ds τ,α α

0

−s) α EA(x−τ (RL D−τ + μ)(s)ds τ,α α

x−nτ x−nτ

(x − nτ − s)(n+1)α−1 RL α ( D−τ + μ)(s)ds Γ (nα + α) −τ

n  Aj −1 0 α + (x − (j − 1)τ − s)j α−1 (RL D−τ + μ)(s)ds Γ (j α) −τ An

j =1

Fractional delay systems Chapter | 5

205

 x−nτ 1 p An p((n+1)α−1) ≤ (x − nτ − s) ds Γ (nα + α) −τ  x−nτ q1 RL α q × ( D−τ + μ)(s) ds +

−τ n−1  j =0



× ≤

j =0



where

1 p

+

1 q

0

−τ

n  

×

Aj Γ (j α + α) (

RL



0

−τ

(x − j τ − s)p((j +1)α−1) ds

α q D−τ + μ)(s) ds

(j +1)α−1+ p1

−τ

(

RL

α q D−τ + μ)(s) ds

p

q1

Aj (x − (j − 1)τ ) Γ (j α + α) (p(j + 1)α − p + 1) p1 0

1



q1 ,

= 1 for every p > 1. This proof is completed.

We now try to give finite time stable results. We impose the following assumptions: α μ)(s) < ∞; [A1 ] 0 < M = sup−τ ≤s≤0 (RL D−τ +  0 RL α [A2 ] 0 < M1 = −τ ( D−τ + μ)(s)ds < ∞; 1  0 α μ)(s)q ds q < ∞, 1 = 1 − 1 , p > 1; [A3 ] 0 < M2 = −τ (RL D−τ + q p [A4 ] for every constant 0 < γ < 1 and 0 < α < 1, we have α + γ − 1 ≥ 0; [A5 ] we suppose f (·) ∈ C([−τ, T ], Rn ) and f  = max f (x) < ∞; x∈[−τ,T ]

[A6 ] ∃ ν(·) ∈ Lq ([−τ, T ], R+ ), q1 = 1 − p1 , p > 1, such that f (x) ≤ ν(x)  x 1 for x ∈ [−τ, T ] and φ(x) := −τ ν(t)q dt q < ∞. For every (n − 1)τ < x ≤ nτ and n ∈ {1, 2, · · · , k ∗ }, define ⎧ ⎪ (x − (n − 1)τ )α+γ −1 Eα,α (A(x − (n − 1)τ )α ), 0 < α < ⎪ ⎪ ⎪ ⎨ n  )(j +1)α−1 1 Φ1 (x) = Aj (x−(jΓ−1)τ , n+1 ≤ α ≤ 12 , (x − (n − 1)τ )γ (j α+α) ⎪ ⎪ j =0 ⎪ ⎪ ⎩ α+γ −1 Eα,α (Ax α ), α > 12 , x and Φ2 (x) =

n  j =0

Aj (x − (j − 1)τ )(j +1)α Γ ((j + 1)α + 1)

1 n+1 ,

206 Stability and Controls Analysis for Delay Systems n  Aj −1 − (x − (j − 1)τ )j α , Γ (j α + 1) j =1

Φ3 (x) =

n  j =0

Φ4 (x) =

(x − (j − 1)τ )(j +1)α Aj , Γ (j α + α) (j + 1)α

n   j =0

(j +1)α−1+ p1

Aj (x − (j − 1)τ ) Γ (j α + α) (p(j + 1)α − p + 1) p1

.

Theorem 5.19. Assume that [A1 ], [A4 ], and [A5 ] hold. Eq. (5.48) is finite time stable with respect to {0, J¯, τ, δ, η} provided that δΦ1 (x) + (x − (n − 1)τ )γ (MΦ2 (x) + f Φ3 (x)) < η, ∀ x ∈ J¯.

(5.49)

Proof. By Theorem 5.18, the solution of (5.48) can be given by α y(x) = EAx τ,α b +



0

−τ

−s)α RL α EA(x−τ ( D−τ + μ)(s)ds τ,α

+

x

−τ

−t) EA(x−τ f (t)dt. τ,α α

Using Lemmas 5.11 and 5.12, via (5.49) we obtain (x − (n − 1)τ )γ y(x) α

≤ (x − (n − 1)τ )γ EAx τ,α b

0 −s)α RL α + (x − (n − 1)τ )γ EA(x−τ ( D−τ + μ)(s)ds τ,α −τ

x −t)α (x − (n − 1)τ )γ EA(x−τ f (t)dt + τ,α −τ

 0 −s)α EA(x−τ ds ≤ δΦ1 (x) + (x − (n − 1)τ ) M τ,α −τ

x −t)α + f  EA(x−τ dt τ,α −τ  γ ≤ δΦ1 (x) + (x − (n − 1)τ ) MΦ2 (x) γ

+ f 

n  j =0

Aj Γ (j α + α)



x−j τ

−τ

(x − j τ − t)(j +1)α−1 dt



≤ δΦ1 (x) + (x − (n − 1)τ )γ (MΦ2 (x) + f Φ3 (x)) < η, ∀ x ∈ J¯. This proof is completed.

Fractional delay systems Chapter | 5

207

Theorem 5.20. Assume that [A2 ], [A4 ], and [A5 ] hold. For all α > 12 , (5.48) is finite time stable with respect to {0, J¯, τ, δ, η} provided that δΦ1 (x) + x α+γ −1 Eα,α (Ax α )M1 + x γ f Φ3 (x) < η, ∀ x ∈ J¯.

(5.50)

Proof. By using Lemma 5.11 via (5.50), we obtain (x − (n − 1)τ )γ y(x) α

≤ (x − (n − 1)τ )γ EAx τ,α b

0 −s)α RL α + (x − (n − 1)τ )γ EA(x−τ ( D−τ + μ)(s)ds τ,α −τ

x −t)α (x − (n − 1)τ )γ EA(x−τ f (t)dt + τ,α −τ

 0 −s)α RL α EA(x−τ ( D−τ + μ)(s)ds ≤ δΦ1 (x) + (x − (n − 1)τ )γ τ,α −τ

x −t)α + f  EA(x−τ dt τ,α −τ  ≤ δΦ1 (x) + x γ x α−1 Eα,α (Ax α )M1 + f Φ3 (x) ≤ δΦ1 (x) + x α+γ −1 Eα,α (Ax α )M1 + x γ f Φ3 (x) < η, ∀ x ∈ J¯. This proof is completed. Theorem 5.21. Assume that [A3 ], [A4 ], [A6 ], and α > 1 − p1 (p > 1) hold. Eq. (5.48) is finite time stable with respect to {0, J¯, τ, δ, η} provided that  δΦ1 (x) + (x − (n − 1)τ )γ Φ4 (x) M2 + φ(x) < η, ∀ x ∈ J¯. (5.51) Proof. Using Lemmas 5.11 and 5.12, via (5.51) we obtain (x − (n − 1)τ )γ y(x)



0

−s) RL α ≤ δΦ1 (x) + (x − (n − 1)τ ) EA(x−τ ( D−τ + μ)(s)ds τ,α −τ

x −t)α + EA(x−τ f (t)dt τ,α −τ  γ ≤ δΦ1 (x) + (x − (n − 1)τ ) M2 Φ4 (x) γ

+

n  j =0

Aj Γ (j α + α)



x−j τ

−τ

α

(x − j τ − t)(j +1)α−1 ν(t)dt



208 Stability and Controls Analysis for Delay Systems

 ≤ δΦ1 (x) + (x − (n − 1)τ )

γ



×

M2 Φ4 (x) +

n  j =0

x−j τ

−τ

p((j +1)α−1)

(x − j τ − t)

1 

p

dt

Aj Γ (j α + α) 1

x −τ

q

q

ν(t) dt

 γ ≤ δΦ1 (x) + (x − (n − 1)τ ) Φ4 (x) M2 + φ(x) < η, ∀ x ∈ J¯. This proof is completed. Theorem 5.22. Assume that [A2 ], [A4 ], and [A6 ] hold. For all α > max{ 12 , 1 − 1 ¯ p } (p > 1), (5.48) is finite time stable with respect to {0, J , τ, δ, η} provided that  α−1+ p1 (x + τ ) Eα,α (Ax α ) δΦ1 (x) + x α+γ −1 M1 + x γ φ(x) 1 (p(α − 1) + 1) p ¯ < η, ∀ x ∈ J . (5.52) Proof. Using Lemma 5.11, via (5.52) we obtain (x − (n − 1)τ )γ y(x)

 0 −s)α RL α ≤ δΦ1 (x) + (x − (n − 1)τ )γ EA(x−τ ( D−τ + μ)(s)ds τ,α −τ

x −t)α + EA(x−τ f (t)dt τ,α −τ 

x −t)α EA(x−τ f (t)dt ≤ δΦ1 (x) + x γ x α−1 Eα,α (Ax α )M1 + τ,α ≤ δΦ1 (x) + x + xγ

n  j =0

−τ

α+γ −1

α

Eα,α (Ax )M1

x−j τ (x − j τ − t)(j +1)α−1 ν(t)dt Γ (j α + α) −τ Aj

≤ δΦ1 (x) + x α+γ −1 Eα,α (Ax α )M1

x n  (Ax α )j + xγ (x − t)α−1 ν(t)dt Γ (j α + α) −τ j =0



≤ δΦ1 (x) + x

α+γ −1

< η, ∀ x ∈ J¯. This proof is completed.

M1 + x φ(x) γ



α−1+ p1

(x + τ )

1

(p(α − 1) + 1) p

Eα,α (Ax α )

Fractional delay systems Chapter | 5

209

5.2.1.3 Numerical example and discussion Example 5.5. Set α = 0.6, τ = 0.2, k ∗ = 3, T = 0.6. We consider ⎧ ⎪ ⎪ ⎨

RL D 0.6 y(x) = Ay(x −0.2+

− 0.2) + f (x),

x ∈ (0, 0.6],

+ 0.2)2 ) ,

−0.2 < x ≤ 0,

μ(x) = (x + 0.2, 2(x ⎪ ⎪ ⎩ (I0.4 y)(−0.2+ ) = b = (0.05, 0.05) , −0.2+

(5.53)

where

 y1 (x) y2 (x)

y(x) =

, A=

 0.2 0.1 0.3 0.5

, f (x) =

x



x2

.

By Theorem 5.18, for x ∈ (0.2(n − 1), 0.2n], n ∈ {1, 2, 3}, any solution of (5.53) has the following form: 0.6 y(x) = EAx 0.2,0.6 b +

+

x −0.2



0 −0.2

A(x−0.2−s)0.6 RL

E0.2,0.6

(

0.6 D−0.2 + μ)(s)ds

A(x−0.2−t)0.6 E0.2,0.6 f (t)dt,

where ⎧ −0.4 0.2 ⎪ I (x+0.2) + A Γx(1.2) , ⎪ ⎪ Γ (0.6) ⎪ ⎪ ⎪ ⎨ I (x+0.2)−0.4 + A x 0.2 + A2 (x−0.2)0.8 , Ax 0.6 Γ (0.6) Γ (1.2) Γ (1.8) E0.2,0.6 = −0.4 0.8 0.2 ⎪ (x−0.2) (x+0.2) x 2 ⎪ I Γ (0.6) + A Γ (1.2) + A Γ (1.8) ⎪ ⎪ ⎪ ⎪ 1.4 ⎩ +A3 (x−0.4) Γ (2.4) ,

0

A(x−0.2−s)0.6 RL

E0.2,0.6

0 < x ≤ 0.2, 0.2 < x ≤ 0.4,

0.4 < x ≤ 0.6,

0.6 D−0.2 + μ)(s)ds −0.2 ⎛ ⎞

0 1.4 0.4 0.6 B[2, 0.4](s + 0.2) A(x−0.2−s) ⎝ Γ (0.4) ⎠ ds, = E0.2,0.6 4.8 1.4 −0.2 B[3, 0.4](s + 0.2) Γ (0.4)

and

x

−0.2

A(x−0.2−t)0.6

E0.2,0.6

f (t)dt

(

210 Stability and Controls Analysis for Delay Systems

⎧



 ⎪   t t −0.4 0.2 ⎪ x x−0.2 (x−t) (x−0.2−t) ⎪ ⎪ dt + −0.2 A Γ (1.2) dt, 0 < x ≤ 0.2, ⎪ −0.2 I Γ (0.6) ⎪ t2 t2 ⎪ ⎪ ⎪



 ⎪ ⎪ ⎪   t t −0.4 0.2 ⎪ x x−0.2 (x−t) (x−0.2−t) ⎪ ⎪ dt + −0.2 A Γ (1.2) dt ⎪ −0.2 I Γ (0.6) ⎪ t2 t2 ⎪ ⎪ ⎪

 ⎪ ⎪ ⎪ t ⎨  x−0.4 2 (x−0.4−t)0.8 + −0.2 A dt, 0.2 < x ≤ 0.4, Γ (1.8) = t2 ⎪ ⎪



 ⎪ ⎪ ⎪ x  x−0.2 (x−0.2−t)0.2 t t ⎪ (x−t)−0.4 ⎪ ⎪ dt + −0.2 A Γ (1.2) dt ⎪ −0.2 I Γ (0.6) ⎪ t2 t2 ⎪ ⎪ ⎪



 ⎪ ⎪ ⎪  x−0.6 3 (x−0.6−t)1.4 t t ⎪  x−0.4 2 (x−0.4−t)0.8 ⎪ ⎪ + −0.2 A dt + −0.2 A dt, ⎪ Γ (1.8) Γ (2.4) ⎪ t2 t2 ⎪ ⎪ ⎩ 0.4 < x ≤ 0.6. Now put n = 3, p = 2, q = 2, and γ = 0.5. By calculation, one has μ = 0.28, f  = 0.96, M = 0.9304, M1 = 0.1128, M2 = 0.2713, φ(0.6) = 0.1621.  0.6i × Next, Φ1 (0.6) = 0.60.1 E0.6,0.6 (0.60.7 ) = 2.2993, Φ2 (0.6) = 3i=0 Γ (0.6i+1.6) 3 0.6i−1 (i+1)0.6 0.6i (0.8 − 0.2i) − i=1 Γ (0.6i+1) (0.8 − 0.2i) = 0.2998, Φ3 (0.6) = 3 0.6i (0.8−0.2i)(i+1)0.6 3 0.6i (0.8−0.2i)0.6i+0.1 1 = i=0 0.6(i+1)Γ (0.6(i+1)) = 1.3167, and Φ4 (0.6) = i=0 Γ (0.6(i+1))(1.2i+0.2) 2

1.9317. By Definition 5.12, we seek a suitable η making xCγ of (5.48) not exceed η on J¯. On the one hand, we can use the explicit formula of solution to (5.53) via numerical simulation to find a corresponding η = 0.875 for a fixed T = 0.6. On the other hand, by checking conditions in Theorems 5.19, 5.20, 5.21, and 5.22 for [−0.2, 0.6], one can choose the better value η = 1.10.

5.2.1.4 Conclusions We construct the explicit formulas of solutions for Caputo- and Riemann– Liouville-type fractional systems and give sufficient conditions to guarantee finite time stability and controllability for these fractional systems. The results in this part are motivated from [110,111]. 5.2.2 Relative controllability for Riemann–Liouville type In this section, we study the relative controllability of ⎧ RL α ⎪ ⎪ ⎨ ( D−τ + y)(x) = Ay(x − τ ) + Bu(x), x ∈ (0, x1 ], τ > 0, y(x) = μ(x), μ(x) ∈ Rn , − τ < x ≤ 0, ⎪ ⎪ ⎩ (I1−α y)(−τ + ) = b, b ∈ Rn , −τ +

(5.54)

Fractional delay systems Chapter | 5

211

where y : J = (−τ, x1 ] → Rn is Riemann–Liouville differentiable on (−τ, x1 ] with (n − 1)τ < x1 ≤ nτ , A, B ∈ Rn×n , and the control function u ∈ Lp (J, Rn ), p ∈ (1, ∞). Definition 5.13. Eq. (5.54) is called relatively controllable if for ∀x1 ∈ Rn , ∃ u∗ ∈ Lp (J, Rn ) such that ⎧ RL α ∗ ⎪ ⎪ ⎨ ( D−τ + y)(x) = Ay(x − τ ) + Bu (x), x ∈ (0, x1 ], x1 > 0, τ > 0, n y(x) = μ(x), μ(x) ∈ R , − τ < x ≤ 0, ⎪ ⎪ ⎩ (I1−α y)(−τ + ) = b, b ∈ Rn , −τ +

(5.55) has a solution y(x, u∗ ) := y ∗ (x) satisfying y ∗ (x) = μ(x), −τ < x ≤ 0, via y ∗ (x1 ) = y1 . Lemma 5.14. A solution y ∈ C(Ω, Rn )∩C([−τ, T ], Rn ) of (5.54) can be given by α y(x) = EAx τ,α b +



0

−τ

−s)α RL α EA(x−τ ( D−τ + μ)(s)ds τ,α

+

x

−τ

−t) EA(x−τ Bu(t)dt. τ,α α

Consider a linear mapping Ψτ (u(·)) : Lp (J, Rn ) → Rn given by

x −t)α Ψτ (u(x)) = EA(x−τ Bu(t)dt. τ,α −τ

Now we prove that Ψτ (u(·)) is bounded. Lemma 5.15. Let x ∈ ((n − 1)τ, nτ ] and n ∈ {1, 2, · · · , k ∗ }. If α > max{ 12 , p1 } with 1 < p < ∞, then Ψτ (u(·)) is bounded. Proof. Letting x ∈ ((n − 1)τ, nτ ] and n ∈ {1, 2, · · · , k ∗ }, we have

x −t)α EA(x−τ Bu(t)dt τ,α −τ



= B

x−nτ

−τ



+ ··· +

x−τ x−nτ



≤ B

x

−τ

−t)α EA(x−τ u(t)dt τ,α A(x−τ −t)α

Eτ,α 

+



x−(n−1)τ

x−nτ

−t) EA(x−τ u(t)dt τ,α α

u(t)dt

(x − t)α−1 (x − τ − t)2α−1 + A Γ (α) Γ (α + α)

(x − 2τ − t)3α−1 (x − (n − 1)τ − t)nα−1 + · · · + An−1 Γ (2α + α) Γ ((n − 1)α + α) (n+1)α−1 (x − nτ − t) u(t)dt + An Γ (nα + α) + A2

212 Stability and Controls Analysis for Delay Systems

+

x−(n−1)τ

x−nτ



(x − t)α−1 (x − τ − t)2α−1 + A Γ (α) Γ (α + α)

− 2τ − t)3α−1 (x − (n − 1)τ − t)nα−1 + A + · · · + An−1 Γ (2α + α) Γ ((n − 1)α + α) × u(t)dt + · · ·

x−τ  (x − t)α−1 (x − τ − t)2α−1 + + A u(t)dt Γ (α) Γ (α + α) x−2τ

x (x − t)α−1 u(t)dt + Γ (α) x−τ 

x−j τ n Aj (j +1)α−1 (x − j τ − t) u(t)dt ≤ B Γ (j α + α) −τ 2 (x

j =0

≤ B

 n j =0

(Ax α )j Γ (iα + α)

≤ BEα,α (Ax α )



x −τ



(x − t)

α−1

α− p1

(x + τ )

(1 + q(α − 1))

1 q

u(t)dt

uLp ,

1 1 + = 1. p q

This proof is completed. Lemma 5.16. (see [112, Lemma 3.2])Let α > max{ 12 , p1 } with 1 < p < ∞. If a pair (A, B) of (5.54) is controllable, i.e., rank Sk = k, where Sk = α {B, AB, A2 B, · · · , Ak−1 B}, then the matrix function ωτ,α (x) = EAx τ,α B consists of elements that are row linearly independent on the interval −τ < x ≤ x1 with x1 > (k − 1)τ , i.e., there exists no nonzero vector b¯ = (b1 , b2 , · · · , bn ) such that b¯ ωτ,α (x) = 0.

5.2.2.1 Relative controllability for linear systems We study the relative controllability of (5.54) by imposing α > max{ 12 , p1 } with 1 < p < ∞. Consider

x1 α α 1 −τ −t) BB EA (x1 −τ −t) dt. Wτ,α [−τ, x1 ] = EA(x (5.56) τ,α τ,α −τ

Now we will give the delayed fractional Grammian matrix criterion result. Theorem 5.23. Wτ,α [−τ, x1 ] defined in (5.56) is nonsingular if and only if system (5.54) is relatively controllable. Proof. Note that Wτ,α [−τ, x1 ] is a nonsingular matrix, which guarantees −1 [−τ, x ] exists. For any y ∈ Rn , one can choose u(·) ∈ Lp (J, Rn ) such Wτ,α 1 1 that

(x1 −τ −t) −1 u(t) = B EA Wτ,α [−τ, x1 ]ζ, τ,α α

(5.57)

Fractional delay systems Chapter | 5

213

where ζ

Ax α = y1 − Eτ,α1 b −



0

−τ

1 −τ −s) (RL D α EA(x τ,α −τ + μ)(s)ds. α

(5.58)

By using Lemma 5.14 and (5.57), we have

0 α Ax α 1 −τ −s) (RL D α EA(x y(x1 ) = Eτ,α1 b + τ,α −τ + μ)(s)ds −τ

x1 α α 1 −τ −t) BB EA (x1 −τ −t) dtW −1 [−τ, x ]ζ. EA(x + 1 τ,α τ,α τ,α −τ

(5.59)

Linking (5.56) and (5.59), via (5.58) we have Ax α y(x1 ) = Eτ,α1 b +



0

−τ

1 −τ −s) (RL D α EA(x τ,α −τ + μ)(s)ds + ζ = y1 . α

By Definition 5.13 and Lemma 5.14, y(x) = μ(x) with −τ < x ≤ 0 holds. Therefore, (5.54) is relatively controllable. Suppose Wτ,α [−τ, x1 ] is a singular matrix and one has at least one nonzero state  y ∈ Rn satisfying  y Wτ,α [−τ, x1 ] y = 0. Therefore, we have 0 = y Wτ,α [−τ, x1 ] y

x1 α α 1 −τ −t) BB EA (x1 −τ −t)  =  y EA(x y dt τ,α τ,α −τ x1

=

−τ

   A(x −τ −t)α 2 1   dt, y E B τ,α  

which implies 1 −τ −t) B = 0 , ∀ t ∈ J.  y EA(x τ,α α

(5.60)

Note that (5.54) is relatively controllable. Therefore, ∃ u(·) ¯ such that y(x1 ) = 0, i.e.,

0 α Ax α 1 −τ −s) (RL D α y(x1 ) = Eτ,α1 b + EA(x τ,α −τ + μ)(s)ds −τ

x1 α 1 −τ −t) B u(t)dt EA(x ¯ = 0. + τ,α −τ

Analogically, ∃  u(·) such that y(x1 ) =  y , i.e., Ax α y(x1 ) = Eτ,α1 b +



0

−τ

1 −τ −s) (RL D α EA(x τ,α −τ + μ)(s)ds α

(5.61)

214 Stability and Controls Analysis for Delay Systems

+

x1

−τ

1 −τ −t) B EA(x u(t)dt =  y. τ,α α

(5.62)

Then, by (5.61) and (5.62), we have

x1 α 1 −τ −t) Bu(t)dt, u =  EA(x u − u. ¯  y= τ,α

(5.63)

−τ

According to (5.63), we have

x1 α 1 −τ −t) Bu(t)dt.  y  y=  y EA(x τ,α −τ

Note that by (5.60), one can get  y  y = 0. This contradicts the hypothesis that  y = 0. Therefore, Wτ,α [−τ, x1 ] is nonsingular. Next, we will give the rank criterion result. Theorem 5.24. System (5.54) is relatively controllable if and only if x1 > (k − 1)τ and rank Sk = k. Proof. Assume that (5.54) is relatively controllable. Then, ∃ u∗ (·) such that (5.55) has a solution y ∗ (·) satisfying y ∗ (x1 ) = y1 . By Lemma 5.14, a solution y ∈ C(Ω, Rn ) ∩ C([−τ, T ], Rn ) of (5.54) can be given by α



y(x) = EAx τ,α b +

0 −τ

−s) RL α EA(x−τ ( D−τ + μ)(s)ds + τ,α α



x −τ

−t) EA(x−τ Bu∗ (t)dt. τ,α α

By Definition 5.13, we have Ax α y(x1 ) = Eτ,α1 b +

+

x1

−τ



0

−τ

1 −τ −s) (RL D α EA(x τ,α −τ + μ)(s)ds α

1 −τ −t) Bu∗ (t)dt. EA(x τ,α α

(5.64)

Consider an arbitrary vector ξ satisfying the follow equation: Ax α

y1 − Eτ,α1 b −



0 −τ

1 −τ −s) (RL D α EA(x τ,α −τ + μ)(s)ds = ξ. α

(5.65)

Without loss of generality, let x1 ∈ ((n − 1)τ, nτ ] and note that with the A(x −τ −s)α in (5.4), we get representation of Eτ,α 1

x1 α 1 −τ −s) Bu∗ (t)dt EA(x τ,α −τ

Fractional delay systems Chapter | 5



215

x1 −τ (x1 − t)α−1 ∗ (x1 − τ − t)2α−1 ∗ u (t)dt + AB u (t)dt + · · · + Γ (α) Γ (2α) −τ −τ

x1 −(n−1)τ (x1 − (n − 1)τ − t)nα−1 ∗ (n−1) +A B u (t)dt. Γ (nα) −τ

=B

x1

Denote

ϕ1 (x1 ) =

x1

−τ x1 −τ

ϕ2 (x1 ) =

(x1 − τ − t)2α−1 ∗ u (t)dt, · · · , Γ (2α)

−τ x1 −(n−1)τ

ϕn (x1 ) =

(x1 − t)α−1 ∗ u (t)dt, Γ (α)

(x1 − (n − 1)τ − t)nα−1 ∗ u (t)dt. Γ (nα)

−τ

Note that by (5.64) and (5.65), we have Bϕ1 (x1 ) + ABϕ2 (x1 ) + · · · + A(n−1) Bϕn (x1 ) = ξ.

(5.66)

By the fact that (5.54) is relatively controllable, (5.66) has a solution for ∀ξ . Note that for n ≥ k, (5.54) is relatively controllable such that x1 > (n − 1)τ ≥ (k − 1)τ . The matrix A can be expressed by a linear combination of I, A, A2 , · · · , Ak−1 for ∀Ai , i ≥ k [89,113]. For n ≥ k Eq. (5.66) can be replaced by B ϕ¯1 (x1 ) + AB ϕ¯2 (x1 ) + · · · + A(k−1) B ϕ¯k (x1 ) = ξ,

(5.67)

where ϕ¯ j (x1 ) with j = 1, 2, · · · , k are some functions at x1 . If (5.67) has the solution ϕ¯ j (x1 ) for arbitrary ξ , then rankSk = k. We prove our results by contradiction. Suppose that (5.54) is not relatively controllable and rankSk = k. By Theorem 5.23, the matrix Wτ,α [−τ, x1 ] is singular. Therefore, there exists at least one nonzero state y¯ ∈ Rn such that

x1 α α 1 −τ −t) BB EA (x1 −τ −t) ydt 0 = y¯ Wτ,α [−τ, x1 ]y¯ = y¯ EA(x ¯ τ,α τ,α

=

x1

−τ



−τ

1 −τ −t) B y¯ EA(x τ,α α

 α 1 −τ −t) B y¯ EA(x dt, τ,α

which implies 1 −τ −t) B = 0 , y¯ EA(x τ,α α

− τ < t ≤ x1 .

Changing variables x1 − τ − t = x, we have y¯ EAx τ,α B = 0 , α

− τ < x ≤ x1 .

This contradicts Lemma 5.16. This proof is completed.

216 Stability and Controls Analysis for Delay Systems

5.2.2.2 Numerical example and discussion Example 5.6. Set α = 0.6, p = 2, τ = 0.4, x1 = 0.8, and k = 2. We study ⎧ RL D 0.6 y(x) = Ay(x − 0.4) + Bu(x), x ∈ (0, x ], ⎪ ⎪ 1 ⎨ −0.4+ 2 ) , − 0.4 < x ≤ 0, (5.68) μ(x) = (x, 2x ⎪ ⎪ ⎩ (I0.4 y)(−0.4+ ) = b = (−0.4, 0.32) , −0.4+

where u ∈ L2 ([−0.4, x1 ], R2 ) and





 y1 (x) 1 0.2 1 0.5 y(x) = , A= , B= . 0.2 0.3 0.2 1 y2 (x) One computes that the fractional delayed Grammian matrix of system (5.68) via (5.56) can be achieved:

0.8 0.6 (0.4−t)0.6 EA(0.4−t) BB EA dt = W1 + W2 + W3 , W0.4,0.6 [−0.4, 0.8] = 0.4,0.6 0.4,0.6 −0.4

where 

0.8 (0.8 − t)−0.4 (0.4 − t)0.2 2 (−t) I +A +A BB W1 = Γ (0.6) Γ (1.2) Γ (1.8) −0.4  (0.4 − t)0.2 (−t)0.8 (0.8 − t)−0.4 × I + A + (A )2 dt, Γ (0.6) Γ (1.2) Γ (1.8) 

0.4  (0.4 − t)0.2 (0.8 − t)−0.4 (0.8 − t)−0.4 +A BB I I W2 = Γ (0.6) Γ (1.2) Γ (0.6) 0 0.2 (0.4 − t) dt, + A Γ (1.2)  

0.8 (0.8 − t)−0.4 (0.8 − t)−0.4 W3 = BB I dt. I Γ (0.6) Γ (0.6) 0.4



0

By a direct computation, one can get





 2.47 1.04 1.48 0.7 2.35 1.31 W1 = , W2 = , W3 = . 1.04 0.67 0.7 0.607 1.31 1.95 Therefore, we have

W0.4,0.6 [−0.4, 0.8] =

−1 W0.4,0.6 [−0.4, 0.8] =

 6.3 3.05 3.05 3.22 0.28 − 0.29 −0.29 0.56

,  .

Fractional delay systems Chapter | 5

217

Set y(x1 ) = (y1 , y2 ). By (5.57), one can construct u ∈ L2 ([−0.4, x1 ], R2 ) as A (0.4−t)0.6

−1 u(t) = B Z0.4,0.6 W0.4,0.6 [−0.4, 0.8]ζ ⎧  ⎪ I (0.8−t)−0.4 + A (0.4−t)0.2 + (A )2 (−t)0.8 W −1 ⎪ ⎪ B ⎪ 0.4,0.6 [−0.4, 0.8]ζ, Γ (0.6) Γ (1.2) Γ (1.8) ⎪ ⎪ ⎪ ⎪ ⎪ t ∈ [−0.4, 0), ⎨  = −0.4 0.2 −1 (0.4−t) ⎪ W0.4,0.6 B I (0.8−t) [−0.4, 0.8]ζ, t ∈ [0, 0.4), ⎪ Γ (0.6) + A Γ (1.2) ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ −0.4 ⎪ −1 ⎩ B I (0.8−t) W0.4,0.6 [−0.4, 0.8]ζ, t ∈ [0.4, 0.8), Γ (0.6)

(5.69) where 0.6



ζ = x(x1 ) − EA0.8 0.4,0.6 b −

 y1 + 0.449 = , y2 − 0.092

0

A(0.4−s)0.6 RL

−0.4

E0.4,0.6

(

0.6 D−0.4 + μ)(s)ds

where 0.6 (RL D−0.4 + μ)(s)



s t d 1 −0.6 (s − t) = dt Γ (0.4) ds −0.4 2t 2

25 0.4 + 1 (25s − 4)(s + 0.4)−0.6 1 14 (s + 0.4) 35 = 5 Γ (0.4) (50s − 4)(s + 0.4)0.4 + 1 (25s 2 − 4s + 42

21

Consider

Sk = {B, AB} =

 28 25 )(s

+ 0.4)−0.6

 1 0.5 1.04 0.7 0.2 1 0.26 0.4

.

Obviously, rankSk = 2. From the above, system (5.68) is relatively controllable via Theorems 5.23 and 5.24. From Lemma 5.14 and (5.69), the solution of system (5.68) has the following form:

0 0.6 Ax 0.6 0.6 y(x) = E0.4,0.6 b + EA(x−0.4−s) (RL D−0.4 + μ)(s)ds 0.4,0.6 −0.4

x A(x−0.4−t)0.6 E0.4,0.6 Bu(t)dt. + −0.4

.

218 Stability and Controls Analysis for Delay Systems

x A(x−0.4−t)0.6 Now we consider the integral term −0.4 Z0.4,0.6 Bu(t)dt in (5.69). Letting −0.4 ≤ t ≤ x and 0 < x ≤ 0.4, we can obtain −0.4 < x − 0.4 − t ≤ x − 0.4 and x − 0.4 < x − 0.4 − t ≤ x. Therefore, the integral term x A(x−0.4−t)0.6 Bu(t)dt in (5.69) can be given by −0.4 E0.4,0.6

x

A(x−0.4−t)0.6

−0.4

E0.4,0.6



=

x

−0.4 x

+

−0.4 0

+

(x − t)−0.4 (0.8 − t)−0.4 BB dt Γ (0.6) Γ (0.6) (x − t)−0.4 (0.4 − t)0.2 BB A dt Γ (0.6) Γ (1.2) (x − t)−0.4 (−t)0.8 BB (A )2 dt Γ (0.6) Γ (1.8)

−0.4 x−0.4

+

−0.4

Bu(t)dt

A

(x − 0.4 − t)0.2 BB Γ (1.2)

0.2 0.8 (0.8 − t)−0.4 (0.4 − t) 2 (−t) +A + (A ) dt × I Γ (0.6) Γ (1.2) Γ (1.8)



−1 × W0.4,0.6 [−0.4, 0.8]ζ.

Letting −0.4 ≤ t ≤ x and 0.4 < x ≤ 0.8, we can obtain −0.4 < x − 0.4 − t ≤ x − 0.8, x − 0.8 < x − 0.4 − t ≤ x − 0.4, and x − 0.4 < x − 0.4 − t ≤ x. Therex x A(x−0.4−t)0.6 A(x−0.4−t)0.6 fore, the integral term −0.4 E0.4,0.6 Bu(t)dt in −0.4 E0.4,0.6 Bu(t)dt in (5.69) can be given by

x

A(x−0.4−t)0.6

−0.4

E0.4,0.6



=

x

−0.4 0.4

+



−0.4 0



−0.4 0

+

Bu(t)dt

(x − t)−0.4 (0.8 − t)−0.4 BB dt Γ (0.6) Γ (0.6) (x − t)−0.4 (0.4 − t)0.2 BB A dt Γ (0.6) Γ (1.2) (x − t)−0.4 (−t)0.8 BB (A )2 dt Γ (0.6) Γ (1.8)

(−t)0.8 (x − 0.4 − t)0.2 BB (A )2 dt Γ (1.2) Γ (1.8) −0.4 

x−0.4 (0.8 − t)−0.4 (0.4 − t)0.2 (x − 0.4 − t)0.2 dt A + BB I + A Γ (1.2) Γ (0.6) Γ (1.2) −0.4 

x−0.8 0.8 (0.8 − t)−0.4 (0.4 − t)0.2 2 (x − 0.8 − t) + BB I + A A Γ (1.8) Γ (0.6) Γ (1.2) −0.4

+

A

Fractional delay systems Chapter | 5

+ (A )2

219

(−t)0.8 −1 [−0.4, 0.8]ζ. dt W0.4,0.6 Γ (1.8)

5.2.2.3 Conclusions We give the delayed Grammian matrix and rank criterion to examine whether a linear delay system is relatively controllable. The results in this part are motivated from [111].

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Chapter 6

Difference delay systems 6.1 Controllability 6.1.1 Controllability for linear discrete delay systems In this part, we find other control functions for the problem from [4]. Moreover, we state an equivalent condition for the relative controllability assuming the final point to be from a specific linear subspace, i.e., for the restricted relative q controllability. We introduce the notation Zp := {p, p + 1, . . . , q − 1, q}, where p, q ∈ Z, Z := {0, ±1, ±2, . . .} with p ≤ q. B· was given in [2,3]. The definition of discrete delayed matrix exponential em Definition 6.1. For an n × n constant matrix B, we define the discrete matrix B· : R → Rn×n by the following discrete matrix function: delayed exponential em ⎧ −m−1 ⎪ Θ, if k ∈ Z−∞ , ⎪ ⎪ ⎪ ⎪ ⎨ I, if k ∈ Z0 , −m Bk em := (6.1) k! 2 (k−m)! s (k−(s−1)m)! ⎪ ⎪ ⎪ I + B 1!(k−1)! + B 2!(k−m−2)! + . . . + B s!(k−(s−1)m−s)! , ⎪ ⎪ ⎩ if k ∈ Zs(m+1) (s−1)(m+1)+1 , s = 1, 2, . . . , where m ≥ 1. Bk proved in [3, Theorem 2.1] is The main property of em Bk B(k+1) Bt B(k−m) em := em − em = Bem ,

k ∈ Z∞ −m .

We shall always consider the forward difference x(k) := x(k + 1) − x(k) and investigate the relative controllability in the sense of the following definition. Definition 6.2. Let m ∈ N, b ∈ Rn and let B be an n × n matrix. System x(k) = Bx(k − m) + bu(k),

k ∈ Z∞ 0 ,

(6.2)

is relatively controllable if for any initial function ϕ : Z0−m → Rn , any finite terminal state x ∗ ∈ Rn , and any finite terminal point k1 ≥ k ∗ ∈ N with a fixed k ∗ there exists a discrete function u∗ : Z0k1 −1 → R such that the system x(k) = Bx(k − m) + bu∗ (k),

k ∈ Z0k1 −1 ,

Stability and Controls Analysis for Delay Systems. https://doi.org/10.1016/B978-0-32-399792-8.00012-8 Copyright © 2023 Elsevier Inc. All rights reserved.

221

222 Stability and Controls Analysis for Delay Systems 1 has a solution x : Zk−m → Rn such that x(k1 ) = x ∗ and

x(k) = ϕ(k),

k ∈ Z0−m .

(6.3)

n We recall that a solution x : Z∞ −m → R of Cauchy problem (6.2)–(6.3) can be represented in the form [3]

Bk x(k) = em ϕ(−m) +

0 

B(k−m−j ) em ϕ(j

− 1) +

j =−m+1

k 

B(k−m−j )

em

bu(j − 1).

j =1

(6.4) Now, we recall two results on relative controllability from [4] which we shall extend in this section. Theorem 6.1. Problem x(k) = Bx(k − m) + bu(k), x(k) = ϕ(k), x(k1 ) = x

∗

k

k ∈ Z0k1 −1 ,

∈ Z0−m ,

(6.5) (6.6) (6.7)

is relatively controllable if and only if assumptions rank(b, Bb, B 2 b, . . . , B n−1 b) = n

(6.8)

k1 ≥ k ∗ := (n − 1)(m + 1) + 1

(6.9)

and

hold simultaneously. Theorem 6.2. Let conditions (6.8) and (6.9) be valid. Then a control function u = u∗ for the problem (6.5)–(6.7) can be expressed in the form B(k1 −m−k−1)  −1 u∗ (k) = b (em ) G ξ

(6.10)

for k ∈ Z0k1 −1 , where G=

k1 

B(k1 −m−j )

em

B  (k1 −m−j )

bb em

,

(6.11)

ϕ(j − 1).

(6.12)

j =1 Bk1 ϕ(−m) − ξ = x ∗ − em

0 

B(k1 −m−j )

em

j =−m+1

Note that the authors of [4] followed a classical construction of a control function (6.10) for linear difference equations (see, e.g., [114]), although they

Difference delay systems Chapter | 6

223

used the discrete delayed matrix exponential and obtained results for delay difference equations. In this section, we generalize this approach by constructing new control functions. Throughout this section, [u, v] shall denote the linear span of vectors u, v and θp shall denote a p-dimensional zero vector (we omit the index if p = n). We present a simple corollary for the relative controllability of weakly nonlinear systems. n n n 1 Corollary 6.1. Let f : Z∞ 0 × R × R × R × R → R be a given C -smooth function in all arguments and let ε ∈ R be close to 0. The problem consisting of equation

x(k) = Bx(k − m) + bu(k) + εf (k, x(k), x(k − m), u(k), ε)

(6.13)

for k ∈ Z0k1 −1 and conditions (6.6)–(6.7) is relatively controllable if conditions (6.8)–(6.9) hold simultaneously. Proof. Consider the Banach spaces X = Rk1 , Y = R, Z = Rn with usual maximum norms. Problem (6.13), (6.6), (6.7) is relatively controllable if there is a solution u of equation F(u, ε) = θ with F : X × Y → Z given by F(u, ε) :=

k1 

B(k1 −m−j )

em

(bu(j − 1)

j =1

+ εf (j − 1, x(j − 1), x(j − m − 1), u(j − 1), ε)) − ξ for ξ of (6.12) and u = (u(0), u(1), . . . , u(k1 − 1)). By Theorem 6.1 we get the existence of uˆ ∈ Rk1 satisfying F(u, ˆ 0) = θ , and moreover, we find that the linear mapping K = Du F(u, ˆ 0) : Rk1 → Rn : v →

k1 

B(k1 −m−j )

em

bv(j − 1)

(6.14)

j =1

is surjective. So, the surjective function theorem can be applied, and one obtains the existence of neighborhoods U1 × U2 ⊂ Rk1 of uˆ in Rk1 for U1 ⊂ ker K with dim ker K = k1 − n and U2 ⊂ X2 with a complement X2 of ker K in Rk1 , so dim X2 = n, and V ⊂ R of 0 in R, and a unique C 1 -function u2 (u1 , ε) such that F(u, ε) = θ for (u, ε) ∈ U1 × U2 × V if and only if u = (u1 , u2 (u1 , ε)). Moreover, u2 (u1 , 0) = uˆ 2 for uˆ = (uˆ 1 , uˆ 2 ) ∈ U1 × U2 . The proof is complete. Remark 6.1. We note that neighborhoods U1 , U2 , V and the smallness of ε depend on the inputs in (6.6)–(6.7). On the other hand, when f is globally Lipschitz continuous in the second, third, and fourth variables with a constant L, then writing F(u, ε) = Ku + εG(u, ε) − ξ,

224 Stability and Controls Analysis for Delay Systems

we see that G : Rk1 × R → Rn is also globally Lipschitz continuous with a constant LG , where, of course, LG can be computed by L but this formula is awkward so we omit it. Since K : Rk1 → Rn is surjective, it has a right inverse, one is given by the Moore–Penrose pseudoinverse K  (KK  )−1 from Theorem 6.2 determined by (6.10) and (6.11). To see this, we take the matrix representation of linear mapping K given by (6.14),  B(k1 −m−1) Ku = em b B(k1 −m−j )

Denoting sj := em K we get

B(k1 −m−2)

em

b

B(−m)

. . . em

 b u.

(6.15)

b = (sj 1 , sj 2 , . . . , sj n ) for j ∈ Zk11 the columns of ⎛

s1

⎞

⎜ ⎟ ⎜s ⎟ ⎜ 2⎟ KK  = (s1 , s2 , . . . , sk1 ) ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ sk1 = (s1 s11 + s2 s21 + · · · + sk1 sk1 1 , s1 s12 + s2 s22 + · · · + sk1 sk1 2 , . . . , s1 s1n + s2 s2n + · · · + sk1 sk1 n ) = (s1 s11 , s1 s12 , . . . , s1 s1n ) + (s2 s21 , s2 s22 , . . . , s2 s2n ) + · · · + (sk1 sk1 1 , sk1 sk1 2 , . . . , sk1 sk1 n ) = s1 s1 + s2 s2 + · · · + sk1 sk1 =

k1 

sj sj ,

j =1

which is exactly matrix G of (6.11). Moreover, (K  u)j = sj u ∀j ∈ Zk11 for the j -th coordinate of the vector K  u, and formula (6.10) follows. Clearly, any right inverse K −1 : Rn → Rk1 gives X2 and vice versa. Then we split u = u1 + u2 , u1 ∈ ker K, u2 ∈ X2 , and F(u, ε) = θ is equivalent to u2 = K −1 (ξ − εG(u1 + u2 , ε)) .

(6.16)

Certainly, the map X2 u2 → K −1 (ξ − εG(u1 + u2 , ε)) ∈ X2 is globally Lipschitz with a constant |ε| K −1 LG , so if |ε| K −1 LG < 1, then the Banach fixed point theorem gives a unique solution u2 = u2 (ξ, u1 , ε) of (6.16) for any ξ ∈ Rn , u1 ∈ ker K, and u2 (ξ, u1 , ε) is globally Lipschitz in ξ and u1 .

Difference delay systems Chapter | 6

225

6.1.1.1 New control functions In this section, we provide control functions for the boundary problem (6.5)–(6.7) which are, in general, different from u∗ given by (6.10). Later, we state conditions for relative controllability restricted to a special subspace of Rn . It will be shown that if the right-hand condition x ∗ in (6.7) lies in this subspace, it can be achieved sooner by the choice of u∗ , i.e., k1 can be lower than in (6.9). First, we state the results on the control functions. Theorem 6.3. Let v : Zk11 → Rn be such that the matrix W :=

k1 

B(k1 −m−j )

em

bv  (j )

(6.17)

j =1

is nonsingular and the conditions for relative controllability (6.8)–(6.9) are valid. Then a control function u = u∗ for the problem (6.5)–(6.7) can be expressed in the form u∗ (k) = v  (k + 1)W −1 ξ,

k ∈ Z0k1 −1 .

(6.18)

Proof. We look for the control function in the form u(j − 1) = v  (j )D with constant vector D = (D1 , . . . , Dn ) ∈ Rn . Since u = u∗ has to satisfy (6.4) at k1 and (6.7) at once, we get Bk1 x ∗ = em ϕ(−m) +

0 

B(k1 −m−1) em ϕ(j − 1) +

j =−m+1

k1 

B(k1 −m−j )

em

bu(j − 1)

j =1

or, in the notation (6.12), k1 

B(k1 −m−j )

em

bv  (j )D = ξ.

j =1

Using the property of matrix W we obtain D = W −1 ξ . Hence u(j − 1) = v  (j )W −1 ξ . B(k −m−k)

Remark 6.2. When one sets v(k) = em 1 b in the latter theorem, one gets ⎡ ⎤−1 k1   B(k −m−j )  B (k1 −m−j ) ⎦ B  (k1 −m−k−1) ⎣ u∗ (k) = b em em 1 bb em ξ. j =1

Thus, in this case, the theorem coincides with Theorem 6.2. We note that in [4, Lemma 3.4] it is also proved that the matrix G of (6.11) is nonsingular. The next lemma provides a necessary condition for function v(j ) to satisfy the assumption of Theorem 6.3.

.

226 Stability and Controls Analysis for Delay Systems

Lemma 6.1. If matrix  := W

k1 

w(j )v  (j )

(6.19)

j =1

with some w : Zk11 → Rn is nonsingular, then the elements (v(j ))i , i = 1, . . . , n, are linearly independent on Zk11 , i.e., there is no nontrivial constant vector μ = (μ1 , . . . , μn ) such that μ v(j ) = 0 for all j ∈ Zk11 . Proof. Let there exist μ = (μ1 , . . . , μn ) = θ such that μ v(j ) = 0 on Zk11 . Then k1 k1   ( w(j )v  (j ))μ = w(j )(v  (j )μ) = 0, j =1

j =1

. which is a contradiction with the nonsingularity of the matrix W The next example compares Theorem 6.2 and Theorem 6.3. Example 6.1. Consider the following boundary value problem: x(k) = x(k − 2) + 2y(k − 2) + u(k),

k ∈ Z30 ,

y(k) = x(k − 2),

k ∈ Z30 ,

x(k) = y(k) = 0,

k ∈ Z0−2 ,

(x(4), y(4)) = (ξ1 , ξ2 ) ∈ R2 . It is easy to see that n = m = 2, k1 = 4, ϕ(k) = (0, 0) for k ∈ Z0−2 , B = 1 2   ∗ 1 0 , and b = (1, 0) . Hence Bb = (1, 1) , k = 4, and conditions (6.8)–(6.9) are immediately satisfied. Theorem 6.1 yields the existence of u : Z30 → R such that there is a solution of this problem. From the definition of the discrete delayed matrix exponential we get  B(k −m−j ) em 1 b

=

(2, 1) ,

j = 1,

(1, 0) ,

j = 2, 3, 4.

(6.20)

Following Theorem 6.2, one constructs     1 1 −2 7 2 −1 G= ,G = , ξ = (x(k ∗ ), y(k ∗ )) = (ξ1 , ξ2 ) 3 −2 7 2 1 and control function u = u∗ , which is concluded in Table 6.1 together with the values of solution (x, y) given by (6.4). On the other side, using Theorem 6.3

Difference delay systems Chapter | 6

227

TABLE 6.1 Control function and the corresponding solution of Example 6.1. k

−2

−1

0

1

2

3

ξ2

ξ1 −2ξ2 3

ξ1 −2ξ2 3 2ξ1 −ξ2 3

ξ1

0

ξ2

x(k)

0

0

0

ξ2

ξ1 −2ξ2 3 ξ1 +ξ2 3

y(k)

0

0

0

0

0

u∗ (k)

with

⎧  ⎪ ⎪ ⎨(1, 0) , v(j ) = (0, 1) , ⎪ ⎪ ⎩(0, 0) ,

4

j = 1, j = 2, j = 3, 4

(which does not violate the necessary condition of Lemma 6.1) leads to           2 1 1  2  0 1 −1 , W = W= 0 1 = 1 0 + 1 0 0 1 1 −2 by (6.17) and, consequently, to Table 6.2 by (6.18) and (6.4). TABLE 6.2 Another control function and the corresponding solution of Example 6.1. k u∗ (k) x(k) y(k)

−2 0 0

−1

0

1

2

3

4

0 0

ξ2 0 0

ξ1 − 2ξ2 ξ2 0

0 ξ1 − ξ2 0

0 ξ1 − ξ2 0

ξ1 ξ2

From the previous example, one can see how to find a nonzero control function implying a nonzero solution that satisfies the zero initial condition and even the zero final condition x(k1 ) = θ . It suffices to set u∗ = u1 − u2 , where u1 , u2 are two different control functions of one problem with a nonzero final condition. Such a nonzero control function u∗ leading to a nonzero solution satisfying zero initial and final conditions can be found neither by Theorem 6.2 nor by Theorem 6.3, due to multiplication by ξ in (6.10) and (6.18). For instance, subtracting control functions of Table 6.1 and Table 6.2 leads 2 to a family of control functions {u∗ζ }ζ ∈R parametrized by ζ = ξ1 −2ξ ∈ R, and 3 ∗ each uζ results in the zero final condition of Example 6.2 (see Table 6.3). Another way is to use the next result providing a complete characterization of control functions for problem (6.5)–(6.7). In what follows, we denote by |Z the restriction onto a linear space Z and by (ker K)⊥ an orthogonal complement to a null space of a linear operator K.

228 Stability and Controls Analysis for Delay Systems

TABLE 6.3 Control function and the corresponding solution of Example 6.1 with ξ = (0, 0). k

−2

−1

u∗ζ (k) xζ (k) yζ (k)

0 0

0 0

0

1

2

3

0

−2ζ

ζ

ζ

0 0

0 0

−2ζ 0

−ζ 0

4 0 0

Theorem 6.4. Let system (6.5)–(6.7) be relatively controllable. Then each control function u∗ : Z0k1 −1 → Rn fulfills u∗ = K|−1 ξ +U (ker K)⊥ for some U ∈ ker K, where u∗ = (u∗ (0), . . . , u∗ (k1 − 1)) is a vector representation of function u∗ and K : Rk1 → Rn is a linear operator defined by (6.14) and represented by the n × k1 matrix (6.15). Proof. Let u∗ be a control function for (6.5)–(6.7). Then by (6.4) and (6.7), u∗ satisfies k1 

B(k1 −m−j )

em

bu∗ (j − 1) = ξ.

j =1

Equivalently, we can use matrix notation and write Ku∗ = ξ . Simultaneously, there is a unique decomposition u∗ = u∗1 + u∗2 , u∗1 ∈ (ker K)⊥ , u∗2 ∈ ker K. Therefore, u∗1 = K|−1 ξ and u∗2 = U . (ker K)⊥ the Moore–Penrose Remark 6.3. Of course we can take instead of K|−1 (ker K)⊥ pseudoinverse K T (KK T )−1 .

Now, we apply the above theorem on Example 6.1 to find all control functions leading to zero final conditions, i.e., we look for the controls for Example 6.1 with ξ = (0, 0) . By (6.20) one gets K = ( 21 10 10 10 ), ker K = [(0, 1, −1, 0) , (0, 1, 0, −1) ], (ker K)⊥ = [(1, 0, 0, 0) , (0, 1, 1, 1) ]. So in fact, there is a family of two-parameter control functions {u∗ζ1 ,ζ2 }ζ1 ,ζ2 ∈R such that the corresponding vector representation is u∗ζ1 ,ζ2 = (0, ζ1 + ζ2 , −ζ1 , −ζ2 ) . One can see that the control u∗ζ of Table 6.3 is just a special case when ζ1 = ζ2 = −ζ .

6.1.1.2 Relative controllability Theorem 6.5. Let M ⊂ Rn be B an invariant linear subspace of dimension p, x ∗ ∈ M, ϕ(k) ∈ M for each k ∈ Z0−m , and b ∈ M. Problem (6.5)–(6.7) is relatively controllable if and only if assumptions rank(b, Bb, B 2 b, . . . , B p−1 b) = p

(6.21)

Difference delay systems Chapter | 6

229

and k1 ≥ kp∗ := (p − 1)(m + 1) + 1

(6.22)

hold simultaneously. Proof. Let {v1 , v2 , . . . , vp } be a basis of M and let {v1 , v2 , . . . , vp , wp+1 , wp+2 , . . . , wn } be a basis of Rn . From [115, Section 39] we know that the matrix representation of the linear transformation B in coordinate system {v1 , v2 , . . . , vp , wp+1 , wp+2 , . . . , wn }  B C1  has the form ( Θn−p,p C2 ), where B is the p × p matrix of B considered as linear transformation on the space M with respect to the coordinate system {v1 , v2 , . . . , vp }, Θn−p,p is an (n − p) × p zero matrix, and C1 , C2 are some arrays of scalars of dimensions p × (n − p), (n − p) × (n − p), respectively. In  B C1 other words, S −1 BS = ( Θn−p,p C2 ) with a regular matrix

S := (v1 , v2 , . . . , vp , wp+1 , wp+2 , . . . , wn ).

(6.23)

Since S −1 S = E, we have S −1 vi = ei for each i = 1, . . . , p, where ei = (0, . . . , 0, 1, 0, . . . , 0)       n−i

i−1

are canonical vectors in Rn . Thus if a = S −1 a =

p  i=1

ai S −1 vi =

p 

p

i=1 ai vi

∈ M, then

ai ei = (a1 , . . . , ap , 0, . . . , 0) ,

i=1 

b ). We introduce the i.e., S −1 M = Rp × {0}n−p , in particular S −1 b =: ( θn−p change of coordinates x = Sy. Accordingly,

y(k) = S −1 x(k) = S −1 x(k) = S −1 Bx(k − m) + S −1 bu(k) = S −1 BSy(k − m) + S −1 bu(k). So we have the next problem for y = ( yy12 ) ∈ Rp × Rn−p :  1 (k − m) + C1 y2 (k − m) +  bu(k), y1 (k) = By

k ∈ Z0k1 −1 ,

y2 (k) = C2 y2 (k − m),    ϕ1 (k) y(k) =  ϕ (k) := ,  ϕ2 (k)

k ∈ Z0k1 −1 , k ∈ Z0−m ,

230 Stability and Controls Analysis for Delay Systems



 ∗ y y(k1 ) = y ∗ = 1∗ , y2 with  ϕ := S −1 ϕ, y ∗ := S −1 x ∗ . Note that by the assumptions of the theorem, ∗ y2 = θn−p ,  ϕ2 (k) = θn−p for each k ∈ Z0−m . Hence we get immediately the solu1 tion y2 (k) = θn−p ∀k ∈ Zk−m . Now, it is easy to see that the problem (6.5)–(6.7) is relatively controllable if and only if the problem  1 (k − m) +  y1 (k) = By bu(k), ϕ1 (k), y1 (k) = 

k ∈ Z0k1 −1 ,

(6.24)

∈ Z0−m ,

(6.25)

k

y1 (k1 ) = y1∗

(6.26)

is relatively controllable. If problem (6.24)–(6.26) is relatively controllable, then (by Definition 6.2) there is a kp∗ ∈ N such that for each  ϕ1 : Z0−m → Rp , y1∗ ∈ Rp , k1 ≥ kp∗ there

exists a control function u∗ : Z0k1 −1 → R such that there is a solution y1 of the system (6.24)–(6.26) with u = u∗ . By (6.4), at the point k1 we have 

Bk1 y1∗ = y1 (k1 ) = em  ϕ1 (−m) +

0 

 1 −m−j ) B(k

em

 ϕ1 (j − 1)

j =−m+1

+

k1 

 1 −m−j ) ∗ B(k 

em

bu (j − 1),

j =1

i.e., k1 

 1 −m−j ) ∗ B(k 

em

bu (j − 1) = ξ1

(6.27)

j =1

with 

Bk1 ξ1 := y1∗ − em  ϕ1 (−m) −

0 

 1 −m−j ) B(k

em

 ϕ1 (j − 1).

j =−m+1

Definition 6.1 of discrete delayed matrix exponential yields that the left-hand  q  side of (6.27) is a linear combination of q + 1 vectors  b, B b, . . . , B b, where    ! k1 k1 − 1 k1 − m − 1 = −1= q := m+1 m+1 m+1 

and · and · are the ceiling function and floor function, respectively. By the p can be written as a linear combination of Caley–Hamilton theorem, matrix B

Difference delay systems Chapter | 6

 ...,B p−1 . Thus, we can consider matrices E, B, " ! # k1 − 1 q = min ,p − 1 . m+1

231

(6.28)

So Eq. (6.27) has the form of a linear equation, q 

j  B bαj = ξ1 ,

j =0

with constants α0 , α1 , . . . , αq depending on u∗ (0), u∗ (1), . . . , u∗ (k1 − 1). A solution (α0 , α1 , . . . , αq ) of this system exists for any right-hand side ξ1 ∈ Rp if and only if two conditions are fulfilled simultaneously: 1. the number of equations is not greater than the number of variables,  p−1 b) has full rank. 2. matrix ( b, B b, . . . , B The first condition means that q ≥ p − 1, i.e., ! k1 − 1 k1 − 1 ≥ ≥ p − 1, m+1 m+1 k1 ≥ (p − 1)(m + 1) + 1. By (6.28), it is equivalent to q = p − 1. Now, the second condition is equivalent  p−1 b) = p. Clearly, the rank of a matrix will not change if to rank( b, B b, . . . , B we add n − p zero rows or multiply this matrix by a nonsingular matrix. Hence,         p−1 B b b B b p−1    rank( b, B b, . . . , B , ,..., ) b) = rank( θn−p θn−p θn−p = rank(S −1 b, S −1 Bb, . . . , S −1 B p−1 b) = rank(b, Bb, . . . , B p−1 b). So the second condition is equivalent to (6.21). Now, assume the conditions (6.21) and (6.22) are valid. We show that the problem (6.24)–(6.26) is relatively controllable. Denote by ϕ1 = ψ} Qψ := {y1 (k1 ) | y1 satisfies (6.24)–(6.25) with  the domain of reachability for a fixed initial function ψ with the values in ϕ1 does not vanish on Z0−m , one can split problem (6.24)–(6.26) to a Rp . If  homogeneous problem with nonhomogeneous initial function and a nonhomogeneous problem with homogeneous initial function. More precisely, we write y1 = u ϕ1 + v, where u ϕ1 solves   u ϕ1 (k) = Bu ϕ1 (k − m), u ϕ1 (k), ϕ1 (k) = 

k ∈ Z0k1 −1 , k ∈ Z0−m ,

232 Stability and Controls Analysis for Delay Systems

and v is a solution of (6.24)–(6.26) with  ϕ1 (k) = θp ∀k ∈ Z0−m . Obviously, (6.24)–(6.26) is relatively controllable if and only if the corresponding problem for v is relatively controllable. Therefore, it suffices to investigate only the case  ϕ1 (k) = θp ∀k ∈ Z0−m with the corresponding domain of reachability Q0 . First, we prove that dim Q0 = p, i.e., there are p linearly independent final vectors y1∗ ∈ Q0 ⊂ Rp that can be reached at k1 by the choice of a convenient control u∗ . Let us suppose, on the contrary, that there is a nontrivial constant vector z ∈ Rp such that for any control function u : Z0k1 −1 → R and the corresponding solution y1 of (6.24)–(6.25) we have z y1 (k1 ) = 0. Then, by (6.4) this equation becomes z

k1 

 1 −m−j ) B(k 

bu(j − 1) = 0

em

j =1  −m−j ) B(k  b = 0 ∀j ∈ Zk11 . On setfor any control function u. Accordingly, z em 1 b = 0. Next, if j = k1 − (m + 1), then ting j = k1 , we get z  1 −m−j ) B(k 



B.1   b = z  em b = z b + z B b = z B b = 0.

z  em

Subsequently, we obtain  p−1 b = z B b = · · · = z B b=0 z (in the last term j = k1 − (p − 1)(m + 1) ≥ 1 by (6.22)) which is a contradiction with nonzero z due to condition (6.21). Clearly, if Q0 contains point y1 (k1 ), then also −y1 (k1 ) ∈ Q0 (one simply takes −u and applies formula (6.4)). Similarly, it can be shown that Q0 is linear, i.e., Q0 = Rp . This says that for the zero initial condition (and also for any ϕ ≡ θp on Z0−m ) and any final point y1∗ there exists a control u∗ and the corresponding solution of (6.24)–(6.26). That is exactly the relative controllability of this system. The proof is finished. The following statement for weakly nonlinear systems immediately results from the latter theorem. Corollary 6.2. Let M, x ∗ , ϕ, b be as in Theorem 6.5, let f : R × Rn × Rn × R × R → M be a given function C 1 -smooth in all arguments, and let ε ∈ R be close to 0. The problem (6.13), (6.6), (6.7) is relatively controllable if conditions (6.21)–(6.22) hold simultaneously. Proof. In the notation of the proof of Theorem 6.5, we write x = Sy to get the following problem for y = ( yy12 ) ∈ Rp × Rn−p :  1 (k − m) + C1 y2 (k − m) +  bu(k) y1 (k) = By + ε f(k, y(k), y(k − m), u(k), ε),

k ∈ Z0k1 −1 ,

Difference delay systems Chapter | 6

233

k ∈ Z0k1 −1 ,

y2 (k) = C2 y2 (k − m),    ϕ1 (k) y(k) =  ϕ (k) := ,  ϕ2 (k)   y∗ y(k1 ) = 1∗ , y2

k ∈ Z0−m ,

where



  f (k, y(k), y(k − m), u(k), ε) . S −1 f (k, Sy(k), Sy(k − m), u(k), ε) = θn−p

ϕ2 (k) = θn−p ∀k ∈ Z0−m , and the problem As in the mentioned proof y2∗ = θn−p ,  (6.13), (6.6), (6.7) is relatively controllable if and only if the problem consisting of equation y (k) y (k−m)  1 (k − m) +  y1 (k) = By bu(k) + ε f(k, ( θ1n−p ), ( 1θn−p ), u(k), ε)

for k ∈ Z0k1 −1 and conditions (6.25), (6.26) is relatively controllable. Here one applies Corollary 6.1 with p instead of n. The statement is proved. Remark 6.1 can be extended to Corollary 6.2. Motivated by Theorem 6.3, we state the next result on the control function for the restricted relative controllability from Theorem 6.5. Theorem 6.6. Let M, x ∗ , ϕ, b be as in Theorem 6.5, let v : Zk11 → Rn be such that rankW = p for W :=

k1 

B(k1 −m−j )

em

bv  (j ),

(6.29)

j =1

and let conditions (6.21)–(6.22) be valid. Then a control function u = u∗ for the problem (6.5)–(6.7) can be expressed in the form u∗ (k) = v  (k + 1)(W |−1 ξ + F ), (ker W )⊥

k ∈ Z0k1 −1 ,

(6.30)

for an arbitrary fixed F ∈ ker W . Proof. As in the proof of Theorem 6.3, we look for a control function in the form u(j − 1) = v T (j )D with a constant vector D ∈ Rn . In this case, we write D = D1 + D2 , where D1 ∈ (ker W )⊥ , D2 ∈ ker W . Since u = u∗ satisfies (6.4) and (6.7), we get W D1 = W D =

k1  j =1

B(k1 −m−j )

em

bv  (j )D = ξ

234 Stability and Controls Analysis for Delay Systems

for ξ given by (6.12). Note that the left-hand side of the last equality is a fiBk b ∈ M ∀k ∈ Z∞ , i.e., nite linear combination of vectors from M since em −m n W R ⊂ M. Clearly also ξ ∈ M by the assumptions on x ∗ and ϕ. The assumption on rankW is equivalent to dim(ker W )⊥ = p, which yields that W |(ker W )⊥ : (ker W )⊥ → M is bijective. Therefore, D1 = W |−1 ξ and the (ker W )⊥ proof is complete. Of course, Remark 6.3 can be also applied (see Corollary 6.3 below). Now, we generalize Lemma 6.1 to nonsingular matrices to obtain a necessary condition for rankW = p.  = p for W  of (6.19), then there exist at most n − p Lemma 6.2. If rank W linearly independent constant vectors μ1 , μ2 , . . . , μn−p ∈ Rn such that μ i v(j ) = 0,

n−p

∀i ∈ Z1

, j ∈ Zk11 .

Proof. Suppose that μ1 , μ2 , . . . , μn−p+1 ∈ Rn are linearly independent constant vectors such that μ i v(j ) = 0,

n−p+1

∀i ∈ Z1

, j ∈ Zk11 .

Then k1 

 μi = 0, w(j )v  (j )μi = W

n−p+1

∀i ∈ Z1

,

j =1

 ≥ n − p + 1. Equivalently, dim(ker W  )⊥ ≤ p − 1, which is i.e., dim ker W  taken as vectors in a contradiction with p linearly independent rows of W ⊥  (ker W ) . Remark 6.4. All n − p vectors from the previous lemma do not have to exist. For instance, set w(1) = w(2) = (1, 2) , v(1) = (1, 0) , v(2) = (0, 1) . Then W = ( 12 12 ) has rank 1, but only the trivial vector μ1 = (0, 0) satisfies μ 1 v(j ) = 0 for each j = 1, 2. The following statement shows that the set of control functions for problem (6.5)–(6.7) remains nonempty if conditions (6.8)–(6.9) are replaced with the assumptions of Theorem 6.5 and conditions (6.21)–(6.22), as it constructs a concrete u∗ given more generally by (6.30). Corollary 6.3. Let M, x ∗ , ϕ, b be as in Theorem 6.5 and let conditions (6.21)–(6.22) be valid. A control function u = u∗ for the problem (6.5)–(6.7) can be expressed in the form $−1 $ k1    B(k1 −m−j )  B (k1 −m−j ) $$ ∗  B (k1 −m−k−1) u (k) = b em ( em bb em )$ ξ $ j =1 M

for k ∈ Z0k1 −1 .

Difference delay systems Chapter | 6

235

Proof. Denote  := G

k1 

B(k1 −m−j )

em

B  (k1 −m−j )

bb em

(6.31)

.

j =1

 = p and ker G  = M ⊥ , we will be able to apply TheoIf we prove that rank G B(k1 −m−k) rem 6.6 with v(k) = em b.  has the form of (6.29), so GR  n ⊂ M as discussed in the proof of Clearly, G  ≤ dim M = p. In particular, GM  ⊂ M. Theorem 6.6, i.e., rank G Now, take the matrix S of (6.23). From the proof of Theorem 6.5 we know  B C1 that S −1 BS = ( Θn−p,p C2 ). Consequently, ⎛

⎞

 Bk em

C(k)⎠

Bk Bk = S(S −1 em S)S −1 = S ⎝ em Θn−p,p

C2 k em

S −1

for some p × (n − p) matrix function C(k). Analogically, Bk Bk b = S(S −1 em S)(S −1 b) em ⎛ ⎞     Bk   Bk  e C(k) b b e m ⎠ =S⎝ =S m C2 k θn−p θn−p Θn−p,p em

and =S G

k1  j =1

=S

k1 





 1 −m−j ) B(k 

b

em 

θn−p

  (k −m−j ) B  b  em 1

 −m−j )  B  (k −m−j ) B(k  em 1 b b em 1

Θn−p,p

j =1

 θn−p



S 

Θp,n−p S  . Θn−p,n−p

 = M. Then there exists a vector ζ ∈ M such that Gζ  = θ. Suppose that GM Accordingly, 





0 = ζ Gζ = (S ζ )

k1  j =1

= ζ1

k1  j =1



 −m−j )  B  (k −m−j ) B(k  em 1 b b em 1

Θn−p,p

 1 −m−j )  B  (k −m−j ) B(k  ζ1 , b b em 1

em

 Θp,n−p S  ζ Θn−p,n−p (6.32)

236 Stability and Controls Analysis for Delay Systems

where S  ζ = (ζ1 , ζ2 ) ∈ Rp × Rn−p . Note that ζ1 = θp whenever ζ ∈ M.  p−1 Clearly, vectors  b, B b, . . . , B b are linearly independent if and only if vectors 

 b θn−p



 =S

−1

b,

    p−1 B b B b −1 = S Bb, . . . , = S −1 B p−1 b θn−p θn−p

 p−1 are linearly independent. Hence, rank( b, B b, . . . , B b) = p, and we can apply [4, Lemma 3.4] (see also Remark 6.2) saying that the matrix

H :=

k1 

 1 −m−j )  B  (k −m−j ) B(k  b b em 1

em

j =1

is nonsingular for each k1 ≥ (p − 1)(m − 1) + 1. More precisely, in the proof of that lemma it is shown that H is positive definite. So the right-hand side of (6.32)  = M, i.e., rank G  ≥ p. is positive and a contradiction follows. Therefore, GM    = In conclusion, rank G = p = dim M implying M = imG = imG ⊥  since G  is symmetric. (ker G)  = M ⊥ we have Moreover, for any F ∈ ker G 

Bk B k F  em b = b  em F = 0, ∀k ∈ Z∞ −m . B(k −m−k)

Thus by setting v(k) = em 1 b, the control function of (6.30) is independent of F , and the statement of the corollary coincides with the statement of Theorem 6.6.

6.1.1.3 Numerical example and discussion Example 6.2. Consider the following boundary value problem for x = (x1 , x2 , x3 , x4 ) ∈ R4 : x1 (k) = x1 (k − 2) + u(k), k ∈ Z30 , x2 (k) = x2 (k − 2) + u(k), k ∈ Z30 , x3 (k) = x1 (k − 2) − x2 (k − 2) + x3 (k − 2), k ∈ Z30 , x4 (k) = x1 (k − 2) + x4 (k − 2), k ∈ Z30 , x(−2) = (0, 0, 0, 0) , x(k) = (0, 0, 0, 1) , k = −1, 0, x(4) = (1, 1, 0, −1) .

Difference delay systems Chapter | 6

237

In this case n = 4, m = 2, k1 = 4,  ϕ(k) =

⎛ 1 0 0 ⎜ k = −2, ⎜0 1 0 B =⎜ ⎝1 −1 1 k = −1, 0, 1 0 0

(0, 0, 0, 0) , (0, 0, 0, 1) ,

⎞ 0 ⎟ 0⎟ ⎟, 0⎠ 1

and b = (1, 1, 0, 0) . It is easy to verify that M = [(1, 1, 0, 0) , (0, 0, 0, 1) ] = [b, Bb] is a B-invariant linear space of dimension p = 2, containing x ∗ , ϕ(k) ∀k ∈ Z0−2 , and b. Thus conditions (6.21) and (6.22) are immediately verified. From the definition of the discrete delayed matrix exponential one gets  B(k −m−j ) b em 1

=

(2, 2, 0, 1) , j = 1, (1, 1, 0, 0) , j = 2, 3, 4.

 defined by (6.31) and ξ of (6.12) have the form Matrix G ⎛ 7 ⎜ ⎜  = ⎜7 G ⎝0 2

7 7 0 2

0 0 0 0

⎞ ⎛ ⎞ 2 1 ⎟ ⎜ ⎟ 2⎟ ⎜1⎟ ⎟, ξ = ⎜ ⎟, ⎝0⎠ 0⎠ 1 −5

 −1 ξ , we solve equation respectively. To find G| M ⎛ ⎞ ⎛ ⎞ 0 1 ⎜ ⎟ ⎜ ⎟ 1 ⎟ ⎜ ⎜  ⎜0⎟  ⎜ ⎟ + βG αG ⎟=ξ ⎝0⎠ ⎝0⎠ 1 0 for α, β ∈ R. This is the same as solving the system 14α + 2β = 1, 4α + β = −5. 37 11 11 −74   −1 Hence, α = 11 6 , β = − 3 , and G|M ξ = ( 6 , 6 , 0, 6 ) . Now, we apply ∗ Corollary 6.3 to derive the control function u and, consequently, the solution of Example 6.2. These values are shown in Table 6.4.

238 Stability and Controls Analysis for Delay Systems

TABLE 6.4 Control function and the corresponding solution of Example 6.2 obtained by Corollary 6.3. k

−2

−1

0

1

2

3

⎛ ⎞ 0 ⎜ ⎟ ⎜0⎟ ⎜ ⎟ ⎝0⎠

⎛ ⎞ 0 ⎜ ⎟ ⎜0⎟ ⎜ ⎟ ⎝0⎠

−5 ⎛ ⎞ 0 ⎜ ⎟ ⎜0⎟ ⎜ ⎟ ⎝0⎠

11 3

11 3

11 3

0

1

1

u∗ (k)

x(k)

⎛

⎞

⎛ ⎞

⎟ ⎜ ⎜ 4⎟ ⎜− 3 ⎟ ⎟ ⎜ ⎝ 0 ⎠ 2

⎜3⎟ ⎜7⎟ ⎜3⎟ ⎜ ⎟ ⎝0⎠

⎛

⎞ −5 ⎜ ⎟ ⎜−5⎟ ⎜ ⎟ ⎝0⎠ 1

− 43

Now, for instance let us set ⎧ ⎪ (0, 0, 0, 1) , ⎪ ⎪ ⎪ ⎨(1, 0, 1, 0) , v(k) = ⎪ (0, 1, 1, 0) , ⎪ ⎪ ⎪ ⎩ (0, 0, 0, 0) , Then the matrix W of (6.29) has the form ⎛ 1 1 ⎜ ⎜1 1 W =⎜ ⎝0 0 0 0

2 2 0 0

k = 1, k = 2, k = 3, k = 4.

⎞ 2 ⎟ 2⎟ ⎟ 0⎠ 1

and it is easy to check that ker W = [(1, −1, 0, 0) , (−2, 0, 1, 0) ], (ker W )⊥ = [(1, 1, 2, 2) , (0, 0, 0, 1) ]. We find W |−1 ξ using equation (ker W )⊥ ⎛ ⎞ ⎛ ⎞ 1 0 ⎜ ⎟ ⎜ ⎟ ⎜1⎟ ⎜0⎟ αW ⎜ ⎟ + βW ⎜ ⎟ = ξ ⎝2⎠ ⎝0⎠ 2 1 for α, β ∈ R. This is equivalent to 10α + 2β = 1, 2α + β = −5,

7

3

4 ⎛

⎞ 1 ⎜ ⎟ ⎜1⎟ ⎜ ⎟ ⎝0⎠ −1

239

Difference delay systems Chapter | 6

resulting in α =

11 6 ,

−1 11 11 11  β = − 26 3 . Hence W |(ker W )⊥ ξ = ( 6 , 6 , 3 , −5) . Choos-

ing a general vector F = (f1 − 2f2 , −f1 , f2 , 0) ∈ ker W , f1,2 ∈ R, and Theorem 6.6, we obtain control function u∗ and the corresponding solution of Example 6.2 shown in Table 6.5. TABLE 6.5 Control function and the corresponding solution of Example 6.2 obtained by Theorem 6.6. k

−2

−1

u∗ (k)

x(k)

⎛ ⎞ 0 ⎜ ⎟ ⎜0⎟ ⎜ ⎟ ⎝0⎠ 0

⎛ ⎞ 0 ⎜ ⎟ ⎜0⎟ ⎜ ⎟ ⎝0⎠ 1

0 −5 ⎛ ⎞ 0 ⎜ ⎟ ⎜0⎟ ⎜ ⎟ ⎝0⎠ 1

1

2

11 + f − f 1 2 2

11 − f + f 1 2 2

⎛ ⎞ −5 ⎜ ⎟ ⎜−5⎟ ⎜ ⎟ ⎝0⎠ 1

⎛

3 ⎞

1 +f −f 1 2⎟ ⎜2 ⎜1 ⎟ + f − f ⎜2 1 2⎟ ⎜ ⎟

⎝

0 2

⎠

0 ⎛ ⎞ 6 ⎜ ⎟ ⎜6⎟ ⎜ ⎟ ⎝0⎠ 3

4 ⎛

⎞ 1 ⎜ ⎟ ⎜1⎟ ⎜ ⎟ ⎝0⎠ −1

6.1.1.4 Conclusions New control functions are derived for linear difference equations with delay, which can be used to construct a nontrivial solution of a boundary value problem with zero boundary conditions. A special case of invariant linear subspace is considered and corresponding control functions are constructed. Results for weakly nonlinear problems are also discussed. The results in this part are motivated from [9].

6.2 Iterative learning control for fixed trial lengths 6.2.1 Iterative learning control for linear systems In this section, we consider the ILC problem for the following discrete controlled systems with single delay: ⎧ T ⎪ ⎨ xk (t + 1) = Axk (t) + A1 xk (t − m) + Buk (t), t ∈ Z0 , xk (t) = ϕ(t), t ∈ Z0−m , ⎪ ⎩ yk (t) = Cxk (t) + Duk (t),

(6.33)

where m ≥ 1 is a prefixed integer and A, A1 ∈ Rn×n satisfying AA1 = A1 A with rank(A) = n, which implies A−1 exists. The index k = 1, 2, . . . denotes the k-th iteration and T is a prefixed positive integer. The variable xk : ZT−m → Rn denotes the state, uk : ZT0 → Rr denotes the dominant input, and yk : ZT0 → Rm∗ denote the output. In addition, B, C, and D denote n × r, m∗ × n, and m∗ × r constant matrices, respectively. Recently, Diblík and Khusainov [2] investigated the representation of solutions of the following Cauchy problem of linear discrete systems with single

240 Stability and Controls Analysis for Delay Systems

delay: 

x(t + 1) = Ax(t) + A1 x(t − m) + f (t), t ∈ Z∞ 0 , x(t) = ϕ(t), t ∈ Z0−m ,

(6.34)

∞ n n where A, A1 , and m are defined in (6.33), x : Z∞ −m → R , f : Z0 → R , and 0 n ϕ : Z−m → R . With the aid of the discrete matrix delayed exponential function, [2] developed the classical idea in ODEs to derive the representation of solutions of linear discrete systems. Here we collect a matrix-formed solution of (6.34), which is useful to investigate ILC problems and asymptotical behavior of solutions.

Lemma 6.3. (see [2, Theorem 3.5]) The solution x(t), t ∈ Z∞ −m , of (6.34) has the form 0 

B1 t −m x(t) = At em A ϕ(−m) +

B (t−m−j )

A(t−j ) em1

[ϕ(j ) − Aϕ(j − 1)]

j =−m+1

+

t 

B (t−m−j )

A(t−j ) em1

f (j − 1),

(6.35)

j =1 B1 · is defined in (6.1). where B1 = A−1 A1 A−m provided that A−1 exists and em

With the help of (6.1) and (6.35), we apply the representation of solution in (6.35) to study the ILC problem for discrete controlled systems with single delay (6.33). By (6.35), one can see that the state xk (t) of (6.33) has the following form: 0 

B1 t −m xk (t) = At em A ϕ(−m) +

B (t−m−j )

A(t−j ) em1

[ϕ(j ) − Aϕ(j − 1)]

j =−m+1

+

t 

B (t−m−j )

A(t−j ) em1

Buk (j − 1).

(6.36)

j =1

The main novelty of this part is that we do not turn our original ILC problem for (6.33) to a Roesser model to derive the convergence result, which is very different from the existing literature. In other words, we fully use formulation (6.35) for the solution to system (6.33) via two given learning laws to generate the control input uk for the output yk of the system to track the desired reference trajectory as accurately as possible with k tending to infinity uniformly on a finite time interval. For a constant matrix Q ∈ Rn×n , we define Q = max Qz generated

z =1

by · , where z denotes the norm for z ∈ Rn . For a discrete vector function x : ZT0 → Rn , we define the λ-norm x λ = sup {λt x(t) }, 0 < λ < 1. t∈ZT0

Difference delay systems Chapter | 6

241

Lemma 6.4. For all t ∈ Z, Gt

em

≤ e G (t+m) .

(6.37)

s(m+1) Proof. For any t ∈ Z∞ 1 , that is, t ∈ Z(s−1)(m+1)+1 , where s = 1, 2, . . ., taking the matrix norm for (6.1), we get

(t − m)! t! + G 2 + ... 1!(t − 1)! 2!(t − m − 2)! (t − (s − 1)m)! + G s s!(t − (s − 1)m − s)! t t2 ts ≤ 1 + G + G 2 + . . . + G s 1! 2! s! ∞ n  ( G t) = e G t . ≤ n!

Gt

em

≤ 1 + G

n=0

Gt = 0 for t ∈ Z−m−1 and eGt = 1 for t ∈ Z0 . Next, (6.1) implies also em −∞ −m m Summarizing, (6.37) holds for all t ∈ Z.

Lemma 6.5. Let G ∈ Rn×n , whose eigenvalues λ1 , λ2 , · · · , λn are different from each other. The following relation holds: Gt λ1 t λ2 t λn t = P diag(em , em , · · · , em )P −1 , t ∈ Z, em

where P := (α1 , α2 , · · · , αn ) ∈ Rn×n is invertible, P −1 denotes the invertible matrix of P , and αi ∈ Rn denotes the corresponding eigenvector of λi , i = 1, 2, · · · , n. Proof. Note that λi αi = Gαi . Then, GP = (Gα1 , Gα2 , · · · , Gαn ) = (λ1 α1 , λ2 α2 , · · · , λn αn ) = (α1 , α2 , · · · , αn )diag(λ1 , λ2 , · · · , λn ) = P diag(λ1 , λ2 , · · · , λn ).

(6.38)

Since P is invertible, P −1 exists. Multiplying by P −1 on both sides of (6.38), one obtains G = P diag(λ1 , λ2 , · · · , λn )P −1 . Further, one can derive the following facts: G2 = P diag(λ1 , λ2 , · · · , λn )P −1 P diag(λ1 , λ2 , · · · , λn )P −1 = P diag(λ21 , λ22 , · · · , λ2n )P −1 ,

242 Stability and Controls Analysis for Delay Systems

.. . Gn = P diag(λn1 , λn2 , · · · , λnn )P −1 . s(m+1)

From the above facts, by (6.1), for any t ∈ Z(s−1)(m+1)+1 , s = 1, 2, · · · , we get (t − m)! (t − (s − 1)m)! t! Gt em + G2 + · · · + Gs =I +G 1!(t − 1)! 2!(t − m − 2)! s!(t − (s − 1)m − s)! ⎛ ⎞ (t−(s−1)m)! 1 + · · · + λs1 s!(t−(s−1)m−s)! ⎜ ⎟ ⎜ ⎟ .. =P ⎜ ⎟ . ⎝ ⎠ (t−(s−1)m)! 1 + · · · + λsn s!(t−(s−1)m−s)! × P −1 λ1 t λ2 t λn t = P diag(em , em , · · · , em )P −1 .

The proof is completed.

6.2.1.1 ILC design and convergence analysis Let yd be a desired reference trajectory and let the k-th iteration error be ek (t) := yd (t) − yk (t).

(6.39)

Denote xk (t) := xk+1 (t) − xk (t) and uk (t) := uk+1 (t) − uk (t). For (6.33), we set uk (t) = L1 ek (t).

(6.40)

For (6.33) with D = Θ, we set uk (t) = L2 ek (t + 1),

(6.41)

where L1 and L2 are r × m∗ learning gain parameter matrices that need to be determined in (6.43) and (6.50), respectively, below. For example, one can choose L1 = σ D −1 , σ ∈ [0, 1) and L1 = σ (CB)−1 , σ ∈ [0, 1). By (6.36), one has xk (t) =

t 

B (t−m−j )

A(t−j ) em1

Buk (j − 1).

(6.42)

j =1

Remark 6.5. In practical applications, we can stop our iteration if there exists a k ∈ Z∞ 1 such that ek (t) < , where  > 0 is a prefixed parameter that depends on the specific requirements.

Difference delay systems Chapter | 6

243

Now consider (6.33) associated with (6.40) and (6.41). We are ready to provide the convergence analysis for error ek λ . Theorem 6.7. Consider (6.33) associated with (6.40). For arbitrary initial input u1 (t), we have lim ek λ = 0 on ZT0 provided that k→∞

ρ(I − DL1 ) < 1.

(6.43)

Proof. For (6.33) with t ∈ ZT0 , by (6.39), one has ek+1 (t) − ek (t) = yk (t) − yk+1 (t) = −Cxk (t) − Duk (t). According to (6.40), we have ek+1 (t) = (I − DL1 )ek (t) − Cxk (t).

(6.44)

Taking the norm · on Rn for (6.44), by Lemma 2.7, one gets

ek+1 (t) ≤ [ρ(I − DL1 ) + ] ek (t) + C xk (t) ,

(6.45)

where  is an arbitrary positive number. Obviously, xk (0) becomes an n-dimensional zero vector. According to (6.43) and (6.45), it is easy to get lim ek (0) = 0. k→∞

When t ∈ ZT1 , multiplying both sides of (6.45) by λt and then taking the λ-norm, we have

ek+1 λ ≤ [ρ(I − DL1 ) + ] ek λ + C

xk λ .

(6.46)

Now we estimate the value of λt xk (t) . According to (6.37), (6.40), and (6.42), we obtain λt xk (t) ≤ λt

t 

B (t−m−j )

A(t−j ) em1

B uk (j − 1)

j =1

≤ λt A0T −1 e B1 T B L1

t 

ek (j − 1)

j =1

≤λ

t

A0T −1 e B1 T B

L1

t 

λ−(j −1) λj −1 ek (j − 1)

j =1

≤ A0T −1 e B1 T B L1 ek λ

t 

λt−(j −1)

j =1

≤ λT A0T −1 e B1 T B

L1 ek λ ,

(6.47)

244 Stability and Controls Analysis for Delay Systems

where Adc := sup Ai . i∈Zdc

Taking the supremum norm for both sides of (6.47), one can obtain

xk λ = sup {λt xk (t) } ≤ λT A0T −1 e B1 T B L1 ek λ .

(6.48)

t∈ZT0

Now linking (6.46) and (6.48), we have

ek+1 λ ≤ [ρ(I − DL1 ) +  + μλ ] ek λ , where μλ := λT A0T −1 e B1 T B C % L1 . Finally, due to (6.43), when λ ∈ 0,

1−ρ(I −DL1 )− T AT0 −1 e B1 T B C L1

(6.49) &

'

(0, 1),

one can obtain

ek+1 λ < ek λ , which implies lim ek λ = 0. The proof is finished. k→∞

Theorem 6.8. Consider (6.33) with D = Θ and (6.41). For arbitrary initial input u1 (t), we have lim ek λ = 0 on ZT1 provided that k→∞

ρ(I − CBL2 ) < 1.

(6.50)

Proof. For (6.33) with D = 0 and t ∈ ZT1 , we can get the relation between the k-th error and the (k + 1)-th error via (6.39): ek+1 (t) − ek (t) = yk (t) − yk+1 (t) = −Cxk (t).

(6.51)

By (6.51) via (6.42), we obtain ek+1 (t) = ek (t) − C

t 

B (t−m−j )

A(t−j ) em1

Buk (j − 1)

j =1 B1 (−m) = ek (t) − Cem Buk (t − 1)

−C

t−1 

B (t−m−j )

A(t−j ) em1

Buk (j − 1).

j =1

Due to (6.1) and (6.41), we have ek+1 (t) = (I − CBL2 )ek (t) − C

t−1  j =1

B (t−m−j )

A(t−j ) em1

BL2 ek (j ).

(6.52)

Difference delay systems Chapter | 6

245

Taking the norm · for (6.52), by Lemma 2.7, we get

ek+1 (1) ≤ [ρ(I − CBL2 ) + ] ek (1) for t = 1. By (6.50), it is easy to obtain lim ek (1) = 0. k→∞

When t ∈ ZT2 , we arrive at

ek+1 (t) ≤ [ρ(I − CBL2 ) + ] ek (t) + C

t−1 

B (t−m−j )

A(t−j ) em1

B L2 ek (j )

j =1

≤ [ρ(I − CBL2 ) + ] ek (t) + A1T −1 e B1 T B C L2 ek λ

t−1 

λ−j .

j =1

Then by taking the λ-norm, we obtain

ek+1 λ ≤ [ρ(I − CBL2 ) + ] ek λ + A1T −1 e B1 T B C L2 ek λ

t−1 

λt−j

j =1

≤ [ρ(I − CBL2 ) +  + νλ ] ek λ ,

(6.53)

where 0 < λ < 1 and νλ := λ(T − 1)AT1 −1 e B1 T B C L2 . Similar to Theorem 6.7, choosing a λ from the set   ( 1 − ρ(I − CBL2 ) −  (0, 1), λ ∈ 0, T −1 B1 T (T − 1)A1 e

B C L2 by (6.50),

ek+1 λ < ek λ , which implies lim ek λ = 0. Thus, the proof is completed. k→∞

Remark 6.6. In Theorem 6.8, if yd (0) is in the range of C, then taking Cϕ(0) = yd (0), we have yk (0) = Cxk (0) = Cϕ(0) = yd (0), which implies that ek (0) = 0 for any k = 1, 2, . . .. As a result, lim ek λ = 0 on ZT0 . k→∞

Remark 6.7. Obviously, the smaller spectral radius ρ(·) in Theorems 6.7 and 6.8, the better the convergence performance. In addition, by analyzing formulas (6.49) and (6.53), when the selected λ is determined, the smaller the value of the variable t, the better the convergence performance. These conclusions will also be verified in the next section.

246 Stability and Controls Analysis for Delay Systems

6.2.1.2 Numerical examples and discussion Example 6.3. Set n = 2, r = m∗ = 1, T = 12. Consider the following discrete control system with single delay: ⎧ 12 ⎪ ⎨ xk (t + 1) = Axk (t) + A1 xk (t − 5) + Buk (t), t ∈ Z0 , xk (t) = ϕ(t) = (2, 1) , t ∈ Z0−5 , ⎪ ⎩ yk (t) = Cxk (t) + Duk (t),

(6.54)

where 

     x1,k (t) 0 1 1 0 xk (t) = , , A= , A1 = −2 3 0 1 x2,k (t)     1 B= , C = 0.2 0.3 , D = 1. 2 We set the learning law (6.40) as uk+1 (t) = uk (t) + L1 · ek (t), where we choose L1 := 0.7D −1 = 0.7 and give the desired discrete reference trajectory as yd (t) = 5t cos(0.5t), t ∈ Z12 0 . Obviously, we can see the state xk ∈ R2 , input uk ∈ R, and output yk ∈ R. Moreover, AA1 = A1 A is satisfied and by (6.36), the state of (6.54) has the following form:

xk (t) = e5A1 t ϕ(−5) +

t 

A (t−5−j )

e5 1

Buk (j − 1).

(6.55)

j =1

According to Lemma 6.5, one can obtain  e5A1 t

= P diag(e5t , e52t )P −1

=

2 · e5t − e52t

e52t − e5t

2 · e5t − 2 · e52t

2 · e52t − e5t

 ,

Difference delay systems Chapter | 6

 where P =

 1 1 1 2

 , p −1 =

2 −1 −1 1

247

 , and

⎧ ⎪ 0, if t ∈ Z−6 ⎪ −∞ , ⎪ ⎪ ⎪ ⎪ 1, if t ∈ Z0−5 , ⎪ ⎪ ⎨ t! , if t ∈ Z61 , 1 + 1!(t−1)! e5t = ⎪ ⎪ (t−5)! t! ⎪ 1 + 1!(t−1)! + 2!(t−5−2)! , if t ∈ Z12 ⎪ 7 , ⎪ ⎪ ⎪ ⎪ . ⎩ . . ⎧ −6 ⎪ ⎪ ⎪ 0, if t ∈ Z−∞ , ⎪ ⎪ ⎪ 1, if t ∈ Z0−5 , ⎪ ⎪ ⎨ t! , if t ∈ Z61 , 1 + 2 1!(t−1)! e52t = ⎪ ⎪ (t−5)! t! ⎪ 1 + 2 1!(t−1)! + 4 2!(t−5−2)! , if t ∈ Z12 ⎪ 7 . ⎪ ⎪ ⎪ ⎪ . ⎩ . .

(6.56)

Thus, (6.55) becomes    2 · e5t − e52t 2 e52t − e5t xk (t) = 1 2 · e5t − 2 · e52t 2 · e52t − e5t ⎛ ⎞ t t−5−j 2(t−5−j ) 2(t−5−j ) t−5−j  2 · e − e e − e 5 5 5 5 ⎝ ⎠ + t−5−j 2(t−5−j ) 2(t−5−j ) t−5−j 2 · e − 2 · e 2 · e − e j =1 5 5 5 5   1 × uk (j − 1) 2 ⎛ ⎞ t  2(t−5−j ) 3 · e5t − e52t + e5 uk (j − 1) ⎜ ⎟ ⎜ ⎟ j =1 ⎟. =⎜ ⎜ ⎟ t  ⎝ ⎠ 2(t−5−j ) t 2t 3 · e5 − 2 · e5 + 2 · e5 uk (j − 1) j =1

Furthermore, ρ(1 − 1 · 0.7) = 0.3 < 1. All the conditions of Theorem 6.7 are satisfied; thus, lim ek λ = 0 uniformly on Z12 0 . k→∞

2 We set the first input u1 (t) = 0, t ∈ Z12 0 . Define the l -norm of error ek l 2 =  2 1/2 (

ek (t) ) . t∈ZT0

The upper image of Fig. 6.1 shows the reference trajectory yd (t) (red (light gray in print version) plus sign) and the output (blue (dark gray in print version) fold line) of system (6.54). The lower image of Fig. 6.1 shows the l 2 -norm of

248 Stability and Controls Analysis for Delay Systems

FIGURE 6.1

The tracking performance of system (6.54) and the l 2 -norm of the tracking error.

FIGURE 6.2

The reference trajectory yd (t) and output y20 (t) of system (6.54).

the tracking error in each iteration. The reference trajectory (red (light gray in print version) plus sign) and the output y20 (t) (blue (dark gray in print version) box) are shown in Fig. 6.2. Obviously, the output yk (t) can track the reference trajectory yd (t) effectively. In specific applications, when the accuracy of the error reaches the requirements, we can stop iterating.

Difference delay systems Chapter | 6

249

Example 6.4. Set n = r = m∗ = 2, T = 12. In this example, we discuss the system of the following form: ⎧ 12 ⎪ ⎨ xk (t + 1) = Axk (t) + A1 xk (t − 5) + Buk (t), t ∈ Z0 , 0 xk (t) = ϕ(t) = (0, 0) , t ∈ Z−5 , ⎪ ⎩ yk (t) = Cxk (t),

(6.57)

where  xk (t) =

 A=



 x1,k (t) x2,k (t)

, uk (t) =





1 0 1 1

 1 0 6 1

, A1 =



 u1,k (t) u2,k (t) 

, B=

 y1,k (t) y2,k (t)

, yk (t) =

1 −0.8 0 1



,

 , C=

 1 1 0 1

.

We set the learning law (6.41) for (6.57) as uk+1 (t) = uk (t) + L2 ek (t + 1), where we choose  −1

L2 = 0.8(CB)

= 0.8

1 −0.2 0 1



and give the reference trajectory as  yd (t) =



3t cos(0.5t) + 1.5t

, t ∈ Z12 0 ,

0.5 sin(t)(e0.25t − 1)

where we mark  yd (t) =



 y1,d (t) y2,d (t)

, ek (t) =

 e1,k (t) e2,k (t)

.

Clearly, the state xk ∈ R2 , input uk ∈ R2 , and output yk ∈ R2 . Next, AA1 = A1 A. By (6.36), the state of system (6.57) has the form xk (t) = A5

t  j =1

A−1 A1 A−5 (t−5−j )

At−5−j e5

Buk (j − 1)

250 Stability and Controls Analysis for Delay Systems

The tracking performance of system (6.57) and the l 2 -norm of the tracking error.

FIGURE 6.3

=

t 

1 t −j

j =1



× ⎛

⎛

 0 1

u1,k (j − 1) u2,k (j − 1)

⎜ ⎜ =⎜ ⎜  ⎝ t j =1

t 



⎞

t−5−j

⎝ e5

0 t−5−j

0

⎠

e5

1 −0.8 0 1



* t−5−j ) u1,k (j − 1) − 0.8u2,k (j − 1) e5

⎞

⎟ ⎟ ⎟, * ⎟ ⎠ t−5−j ) (t − j )u1,k (j − 1) + (1 − 0.8t + 0.8j )u2,k (j − 1) e5 j =1

where e5t is shown in (6.56). Next, one can see that  ρ (I − CBL2 ) = ρ

 0.2 0 0 0.2

= 0.2 < 1.

Similar to Theorem 6.7, we also set the first input u1 (t) = 0, t ∈ Z12 0 . Thus, 12 Theorem 6.8 guarantees lim ek λ = 0 uniformly on Z1 for (6.57). k→∞

Figs. 6.3 and 6.5 show the tracking performance of system (6.57) and the l 2 -norm of the tracking error. Figs. 6.4 and 6.6 show the results of yd (t) and y15 (t). Simulation results show that the iterative output yk (t) can converge to the desired reference trajectory.

Difference delay systems Chapter | 6

251

FIGURE 6.4 The reference trajectory y1,d (t) and output y1,15 (t) of system (6.57).

FIGURE 6.5 The tracking performance of system (6.57) and the l 2 -norm of the tracking error.

6.2.1.3 Conclusion We adopt a new framework to establish the same convergence theorem for the ILC problem of linear discrete delayed systems by virtue of the representation of solutions via discrete matrix delayed exponential functions. The results in this part are motivated from [116].

252 Stability and Controls Analysis for Delay Systems

FIGURE 6.6

The reference trajectory y2,d (t) and output y2,15 (t) of system (6.57).

6.3 Iterative learning control for varying trial lengths 6.3.1 Iterative learning control for linear discrete delay systems Let E{X} be the expectation of a stochastic variable X and let P[Y ] be the occurrence probability of event Y . We note that ZT0 a means the shortest running T interval and Z0 d means the desired running interval. Clearly, Td > Ta > 0.

6.3.1.1 Discrete matrix delayed exponential function When we consider the nonhomogeneous initial Cauchy problem (6.34), we note that the condition on commutative matrices is acceptable for various matrices, for example, normal matrices and diagonal matrices (see [117, Section 30, Lemma 4] and [118, Corollary 9.8, p. 152]). The main motivation for AA1 = A1 A is that the solution formula (6.35) can be found by adopting a similar idea to linear difference systems without delay. In B1 t fact, X(t) := At em is an explicit form of the fundamental matrix for x(t + 1) = 0 t Ax(t) + A1 x(t − m), t ∈ Z∞ 0 , with x(t) = A , t ∈ Z−m . The simple form allows us to generate a sequence of outputs to track the desired reference. We also note that a model of population dynamics with delayed birthrates and delayed logistic terms can be described by delay differential or difference equations with permutable matrices [7,119].

Difference delay systems Chapter | 6

253

6.3.1.2 Randomly varying trial lengths A very important difficulty that needs to be dealt with is that the trial lengths are randomly varying for each iteration. Now, we analyze the randomness of trial lengths. In fact, two situations need to be considered in the iterative process; one is Tk ≥ Td and the other is Ta ≤ Tk < Td , where Td is the desired operation length, Tk is the actual operation length, and Ta is the possible smallest operation length. T T In the following, we mark the running interval of the k-th iteration as Z0 k (Z0 d Td Ta and Z0 can be defined similarly). For the former case, only the data in Z0 can be used to update the input signal. Therefore, without losing generality, we view this case as Tk = Td . For the latter case, the output does not exist in ZTTdk+1 , i.e., T

only the date in Z0 k can be used to update the input signal. In order to characterize this process, we introduce a stochastic variable ηk (t), t ∈ ZT0 d , satisfying a Bernoulli distribution and taking values 0 and 1 in the kth iteration. Here ηk (t) = 1 indicates that the controlled systems can run up to time instant t with a probability of p(t), 0 < p(t) ≤ 1. On the contrary, ηk (t) = 0 indicates that the controlled systems cannot run up to time instant t with probability 1 − p(t). Since the stochastic variable ηk (t) obeys the Bernoulli distribution, we obtain the expectation E{ηk (t)} = 0·(1−p(t))+1·p(t) = p(t). Hypothesis: Assume the probability P[MTk ] = pTk , in which we mark event MTk as running terminal at Tk in the k-th iteration. Obviously, controlled systems can run for the interval ZT0 a , which implies that p(t) = 1 for t ∈ ZT0 a . For t ∈ ZTTda+1 , we can calculate Td Td Td +   p(t) = P[ Mi ] = P[Mi ] = pi . i=t

i=t

i=t

In addition,

P[

Td +

i=Ta

Mi ] =

Td 

P[Mi ] =

i=Ta

Td 

pi = 1.

i=Ta

Thus, we conclude possibility p(t) is ⎧ T ⎪ ⎨ 1, t ∈ Z0 a , p(t) = Td  ⎪ ⎩ pi , t ∈ ZTTda+1 . i=t

254 Stability and Controls Analysis for Delay Systems

6.3.1.3 ILC design and convergence analysis In this part, we consider the ILC problem for the following linear discrete delay controlled system: ⎧ T ⎪ ⎨ xk (t + 1) = Axk (t) + A1 xk (t − σ ) + Buk (t), t ∈ {ηk (i) · i, i ∈ Z0 d }, x (t) = ϕ(t), t ∈ Z0−σ , ⎪ ⎩ k yk (t) = Cxk (t) + Duk (t), (6.58) where A, A1 ∈ Rn×n satisfying AA1 = A1 A and A−1 exist, B ∈ Rn×r , C ∈ Rm×n , D ∈ Rm×r , the delay σ ≥ 1 is a prefixed integer, and ϕ : Z0−σ → Rn denotes the initial state. The index k = 1, 2, . . . denotes the k-th iteration and the variables xk (t) ∈ Rn , uk (t) ∈ Rr , and yk (t) ∈ Rm denote the state, input, and output, respectively. In addition, considering the randomness of the trial lengths, t ∈ {ηk (i) · i, i ∈ ZT0 d } can accurately depict the running interval of the T k-th iteration by the stochastic variable ηk (i), i ∈ Z0 d . ∞ By Lemma 6.3, the state xk (t), t ∈ Z−σ , of (6.58) can be represented in the form xk (t) = At eσB1 t A−σ ϕ(−σ ) +

0 

A(t−j ) eσB1 (t−σ −j ) [ϕ(j ) − Aϕ(j − 1)]

j =−σ +1

+

t 

A(t−j ) eσB1 (t−σ −j ) Buk (j − 1),

(6.59)

j =1

where B1 = A−1 A1 A−σ and eσB1 · is defined similar to (6.1). Assuming that system (6.58) is stable, controllable, and observable, for arbitrary bounded reference yd (t) ∈ Rm , there exists a unique control input ud (t) ∈ Rr such that ⎧ T ⎪ ⎨ xd (t + 1) = Axd (t) + A1 xd (t − m) + Bud (t), t ∈ Z0 d , x (t) = ϕ(t), t ∈ Z0−σ , ⎪ ⎩ d yd (t) = Cxd (t) + Dud (t), where xd (t) ∈ Rn is the corresponding referential state. In ILC, the tracking error of controlled systems is used to update the input, where the tracking error is defined as ek (t) := yd (t) − yk (t). However, the error T of interval ZTdk+1 does not exist for the k-th iteration in our case where Tk denotes the running terminal of the k-th iteration. Thus, we mark the tracking error for

Difference delay systems Chapter | 6

255

t ∈ ZTTdk+1 as zero, and for t ∈ ZT0 d , we define a corrected tracking error as ⎧ ⎨ e (t), t ∈ ZTk , k 0 e˜k (t) := ηk (t)ek (t) = ⎩ 0, t ∈ ZTd . Tk+1

T

Now we can observe the running interval of each iteration as Z0 d by using the stochastic variable ηk (t). Consider the following ILC updating laws: T

uk+1 (t) = uk (t) + L1 e˜k (t), t ∈ Z0 d ,

(6.60)

uk+1 (t) = uk (t) + L2 e˜k (t + 1), t ∈ ZT0 d .

(6.61)

and

Note t + 1 = Td + 1 when t = Td in law (6.61), so we consider the desired running interval as Z0Td +1 to ensure that the error can be used to update input. Now we are ready to present the first main result in this subsection. Theorem 6.9. Consider system (6.58) associated with updating law (6.60). For arbitrary initial input u1 (·), we have lim E{ e˜k λ } = 0

k→∞

on ZT0 d if the Euclidean matrix norm · related to parameter matrix L1 satisfies

I − L1 D < 1.

(6.62)

Proof. Denote δxk (t) := xd (t) − xk (t) as the state error and δuk (t) := ud (t) − uk (t) as the input error. We divide the proof into two parts. Part 1: We show that lim E{ δuk λ } = 0, for ∀ t ∈ ZT0 d . k→∞

By updating law (6.60), for t ∈ ZT0 d , we have δuk+1 (t) = δuk (t) − L1 e˜k (t) = δuk (t) − ηk (t)L1 ek (t).

(6.63)

According to (6.58), one can compute ⎧ T ⎪ ⎨ δxk (t + 1) = Aδxk (t) + A1 δxk (t − σ ) + Bδuk (t), t ∈ {ηk (i) · i, i ∈ Z0 d }, δx (t) = 0, t ∈ Z0−σ , ⎪ ⎩ k ek (t) = Cδxk (t) + Dδuk (t). (6.64) Inserting ek (t) of (6.64) into (6.63), we get δuk+1 (t) = [I − ηk (t)L1 D]δuk (t) − ηk (t)L1 Cδxk (t).

(6.65)

256 Stability and Controls Analysis for Delay Systems

Taking the Euclidean norm for (6.65), we have

δuk+1 (t) ≤ I − ηk (t)L1 D δuk (t) + ηk (t) L1 C δxk (t) . (6.66) For t = 0, noting that ηk (0) = 1 and δxk (0) = 0, we obtain

δuk+1 (0) ≤ I − L1 D δuk (0) . By condition (6.62), we get δuk+1 (0) < δuk (0) and lim δuk (0) = 0.

(6.67)

k→∞

For t ∈ ZT1 d , multiplying both sides of (6.66) by e−λt , we get e−λt δuk+1 (t) ≤ I − ηk (t)L1 D e−λt δuk (t) + ηk (t) L1 C e−λt δxk (t) .

(6.68)

By (6.59), one can derive δxk (t) in (6.64) as follows: δxk (t) =

t 

A(t−j ) eσB1 (t−σ −j ) Bδuk (j − 1), t ∈ {ηk (i) · i, i ∈ Z0 d }. T

(6.69)

j =1

Note that we consider the convergence of errors in the sense of expectation. Without loss of generality, we can assume that (6.69) well holds on the full inT T terval Z0 d . Again note that ηk (i), i ∈ Z0 d , takes the value 0 on the part (not T T running) interval Z0 d \ {ηk (i) · i, i ∈ Z0 d }. Thus, the main contributions to computing the convergence of error in the sense of expectation are the accumulation effects on the running interval {ηk (i) · i, i ∈ ZT0 d }. Now, taking the λ-norm for both sides of (6.69) and using Lemma 6.4, one can obtain sup e−λt δxk (t) ≤ sup e−λt T t∈Z1 d

T t∈Z1 d

t 

A(t−j ) eσB1 (t−σ −j ) B δuk (j − 1)

j =1

≤ sup e−λt AT0 d −1 e B1 Td B T t∈Z1 d

t  j =1

≤ sup A0Td −1 e B1 Td B δuk λ T t∈Z1 d

≤e

B1 Td −λ

where Acb := maxc Aj . j ∈Zb

eλ(j −1) e−λ(j −1) δuk (j − 1)

t 

eλ(j −1−t)

j =1

Td A0Td −1 B

δuk λ ,

(6.70)

Difference delay systems Chapter | 6

257

Taking the λ-norm for (6.68) via (6.70), we arrive at

δuk+1 λ ≤ I − ηk (t)L1 D δuk λ + ηk (t)e B1 Td −λ Td A0Td −1 L1 C B δuk λ .

(6.71)

Applying the operator E{·} on both sides of (6.71), one can obtain E{ δuk+1 λ } ≤ E{ I − ηk (t)L1 D }E{ δuk λ } + E{ηk (t)}e B1 Td −λ Td A0Td −1 L1 C B E{ δuk λ }.

(6.72)

Noting that E{ηk (t)} = p(t) and E{ I − ηk (t)L1 D } = I − 0 · L1 D · (1 − p(t)) + I − 1 · L1 D · p(t) = 1 − p(t) + p(t) I − L1 D (6.73) = 1 + p(t)( I − L1 D − 1), because 0 < p(·) ≤ 1, according to (6.62), we have ϑ := max (1 + p(t)( I − L1 D − 1)) < 1. T

(6.74)

t∈Z1 d

Thus, noting (6.73) and (6.74), (6.72) becomes ˜ E{ δuk+1 λ } ≤ ϑE{ δu k λ },

(6.75)

where ϑ˜ := ϑ + e B1 Td −λ Td A0Td −1 L1 C B . Next, due to (6.74) we can choose a number ,  - . 1−ϑ λ > max B1 Td − ln ,0 T −1 Td A0 d L1 C B such that ϑ˜ < 1. Therefore, linking (6.67) and (6.75), we obtain lim E{ δuk λ } = 0, for ∀ t ∈ ZT0 d .

k→∞

(6.76)

Part 2: We prove that lim E{ e˜k λ } = 0, for ∀ t ∈ ZT0 d . k→∞

Taking the λ-norm for ek (·) in (6.64) and applying E{·}, we get E{ ek λ } ≤ C E{ δxk λ } + D E{ δuk λ }, ∀ t ∈ {ηk (i) · i, i ∈ ZT0 d }.

(6.77)

258 Stability and Controls Analysis for Delay Systems

By (6.70), one can obtain T −1

E{ δxk λ } ≤ e B1 Td −λ Td A0 d

B E{ δuk λ }.

According to (6.76), for a fixed number λ, we have lim E{ δxk λ } = 0.

k→∞

(6.78)

Combining (6.76) with (6.78), one can obtain lim E{ ek λ } = 0 via (6.77). k→∞

T

Thus, lim E{ e˜k λ } = 0 for ∀ t ∈ Z0 d . The proof is completed. k→∞

Next we present the second main result in this subsection. Theorem 6.10. Consider system (6.58) with D = Θ and updating law (6.61). For arbitrary initial input u1 (·), lim E{ e˜k λ } = 0

k→∞ T

on Z1 d if the Euclidean matrix norm · related to parameter matrix L2 satisfies

I − L2 CB < 1.

(6.79)

Proof. Considering D = Θ of system (6.58), we have ⎧ T ⎪ ⎨ δxk (t + 1) = Aδxk (t) + A1 δxk (t − σ ) + Bδuk (t), t ∈ {ηk (i) · i, i ∈ Z1 d }, δx (t) = 0, t ∈ Z0−σ , ⎪ ⎩ k ek (t) = Cδxk (t). (6.80) As mentioned, it is sufficient to consider the error of the accumulation effect T on the running interval {ηk (i) · i, i ∈ Z1 d } when we consider the convergence of error in the sense of expectation. Without loss of generality, for t ∈ ZT1 d , by updating law (6.61) and (6.80), one can obtain δuk+1 (t) = δuk (t) − L2 e˜k (t + 1) = δuk (t) − ηk (t + 1)L2 Cδxk (t + 1) = [I − ηk (t + 1)L2 CB]δuk (t) − ηk (t + 1)L2 CAδxk (t) − ηk (t + 1)L2 CA1 δxk (t − σ ).

(6.81)

Taking the λ-norm for (6.81), we have

δuk+1 λ ≤ I − ηk (t + 1)L2 CB δuk λ + ηk (t + 1) L2 C A J1 + ηk (t + 1) L2 C A1 J2 , (6.82)

Difference delay systems Chapter | 6

259

where J1 := sup e−λt δxk (t) and J2 := sup e−λt δxk (t − σ ) . T t∈Z1 d

(6.83)

T t∈Z1 d

Now, we estimate the values of J1 and J2 in (6.83). Similar to (6.70), we get T −1

J1 ≤ e B1 Td −λ Td A0 d

T

B δuk λ , ∀ t ∈ Z1 d .

(6.84)

For J2 , when t ∈ Zσ1 , we get δxk (t − σ ) = 0 by (6.80). When t ∈ ZTσd+1 , by (6.59) and Lemma 6.4, one can obtain sup e−λt δxk (t − σ ) T

t∈Zσd+1

≤ sup e−λt T t∈Zσd+1

t−σ 

A(t−σ −j ) eσB1 (t−2σ −j ) B δuk (j − 1)

j =1

≤ sup e−λt A0Td −σ −1 e B1 (Td −σ ) B T t∈Zσd+1

t−σ  j =1

T −σ −1 B1 (Td −σ )

≤ sup A0 d

e

B δuk λ

T t∈Zσd+1

≤e

eλ(j −1) e−λ(j −1) δuk (j − 1)

t−σ 

eλ(j −1−t)

j =1

B1 (Td −σ )−(σ +1)λ

(Td − σ )A0Td −σ −1 B

δuk λ ,

where Acb := maxc Aj . j ∈Zb

From the above, one can conclude that J2 ≤ e B1 Td −λ Td A0Td −1 B δuk λ , ∀ t ∈ ZT1 d .

(6.85)

Substituting (6.84) and (6.85) into (6.82), we have

δuk+1 λ ≤ I − ηk (t + 1)L2 CB δuk λ + ηk (t + 1) L2 C ( A + A1 ) × e B1 Td −λ Td AT0 d −1 B δuk λ . Then applying the expectation operator E{·} to both sides, we obtain E{ δuk+1 λ } ≤ E{ I − ηk (t + 1)L2 CB } E{ δuk λ } + E{ηk (t + 1)} × e B1 Td −λ Td AT0 d −1 L2 C ( A + A1 ) B E{ δuk λ }.

(6.86)

260 Stability and Controls Analysis for Delay Systems

Similar to (6.73), we get E{ I − ηk (t + 1)L2 CB } = 1 + p(t + 1)( I − L2 CB − 1).

(6.87)

According to E{ηk (t + 1)} = p(t + 1) and (6.87), (6.86) becomes E{ δuk+1 λ } ≤ [1 + p(t + 1)( I − L2 CB − 1)]E{ δuk λ } + p(t + 1) × e B1 Td −λ Td A0Td −1 L2 C ( A + A1 ) B E{ δuk λ }.

(6.88)

Due to 0 < p(t + 1) ≤ 1 for t ∈ ZT1 d and (6.79), we get ϑ := max {1 + p(t + 1)( I − L2 CB − 1)} < 1. T

(6.89)

t∈Z1 d

Next, (6.88) becomes ˜ E{ δuk+1 λ } ≤ ϑE{ δu k λ },

(6.90)

where ϑ˜ := ϑ + e B1 Td −λ Td AT0 d −1 L2 C ( A + A1 ) B . Then, according to (6.89) we can choose a number  , - . 1−ϑ λ > max B1 Td − ln ,0 T −1 Td A0 d L2 C ( A + A1 ) B such that ϑ˜ < 1. Therefore, (6.90) implies T

lim E{ δuk λ } = 0, for ∀ t ∈ Z1 d .

k→∞

Applying E{·} to (6.84), one can obtain E{ δxk λ } ≤ e B1 Td −λ Td A0Td −1 B E{ δuk λ }, ∀ t ∈ ZT1 d . For a fixed λ, we have lim E{ δxk λ } = 0, for ∀ t ∈ ZT1 d . k→∞

Taking the λ-norm for ek (·) in (6.80), we obtain E{ ek λ } ≤ C E{ δxk λ }. Finally, lim E{ ek λ } = 0. Thus, lim E{ e˜k λ } = 0 for ∀ t ∈ ZT1 d . The k→∞

proof is finished.

k→∞

Difference delay systems Chapter | 6

261

6.3.1.4 Numerical examples and discussion Example 6.5. Consider the following discrete delay system: ⎧ 20 ⎪ ⎨ xk (t + 1) = Axk (t) + A1 xk (t − 5) + Buk (t), t ∈ {ηk (i) · i, i ∈ Z0 }, xk (t) = ϕ(t) = (2, 1) , t ∈ Z0−5 , ⎪ ⎩ yk (t) = Cxk (t) + Duk (t), (6.91) where 

     x1,k (t) 0 1 1 0 xk (t) = , , A= , A1 = −2 3 0 1 x2,k (t)     1 B= , C = 0.2 0.3 , D = 1. 2 Running terminal Tk of each iteration is randomly selected between 17 and 22. Without loss of generality, set u0 (t) = 0. We select the learning law (6.60) as uk+1 (t) = uk (t) + 0.45e˜k (t),

(6.92)

and the desired reference trajectory is yd (t) = 3t sin(2t), t ∈ Z20 0 .

(6.93)

Clearly, AA1 = A1 A holds. According to (6.59), the state of (6.91) has the following form: xk (t) = e5A1 t ϕ(−5) +

t 

A (t−5−j )

e5 1

Buk (j − 1).

j =1

Note that e5A1 t = P diag(e52t , e5t )P −1 ,  where P =



 1 1 1 2

,

P −1 

e5A1 t

=

=

2 −1 −1 1

 . Thus,

2 · e5t − e52t

e52t − e5t

2 · e5t − 2 · e52t

2 · e52t − e5t

 .

262 Stability and Controls Analysis for Delay Systems

Thanks to Lemma 6.3, one can obtain the following explicit formula of solution: ⎛ ⎞ t  2(t−5−j ) 3 · e5t − e52t + e5 uk (j − 1) ⎜ ⎟ ⎜ ⎟ j =1 ⎟, (6.94) xk (t) = ⎜ ⎜ ⎟ t  ⎝ ⎠ 2(t−5−j ) t 2t 3 · e5 − 2 · e5 + 2 · e5 uk (j − 1) j =1

where e5t and e52t have the following forms: ⎧ ⎪ 0, if t ∈ Z−6 ⎪ −∞ , ⎪ ⎪ ⎪ ⎪ 1, if t ∈ Z0−5 , ⎪ ⎪ ⎪ ⎪ t! ⎪ ⎪ , if t ∈ Z61 , 1 + (t−1)! ⎪ ⎪ ⎨ (t−5)! t! 1 + (t−1)! + 2!(t−7)! , if t ∈ Z12 e5t = 7 , ⎪ ⎪ (t−5)! (t−10)! t! 18 ⎪ 1+ ⎪ ⎪ (t−1)! + 2!(t−7)! + 3!(t−13)! , if t ∈ Z13 , ⎪ ⎪ ⎪ (t−5)! (t−10)! (t−15)! t! ⎪ 1 + (t−1)! + 2!(t−7)! + 3!(t−13)! + 4!(t−19)! , if t ∈ Z24 ⎪ 19 , ⎪ ⎪ ⎪ ⎪ ⎩ .. .

(6.95)

and ⎧ ⎪ 0, if t ∈ Z−6 ⎪ −∞ , ⎪ ⎪ ⎪ ⎪ 1, if t ∈ Z0−5 , ⎪ ⎪ ⎪ ⎪ t! ⎪ ⎪ , if t ∈ Z61 , 1 + 2 (t−1)! ⎪ ⎪ ⎨ (t−5)! t! 1 + 2 (t−1)! + 4 2!(t−7)! , if t ∈ Z12 e52t = 7 , ⎪ ⎪ (t−5)! (t−10)! t! ⎪ 1 + 2 (t−1)! + 4 2!(t−7)! + 8 3!(t−13)! , if t ∈ Z18 ⎪ 13 , ⎪ ⎪ ⎪ ⎪ (t−5)! (t−10)! (t−15)! t! ⎪ 1 + 2 (t−1)! + 4 2!(t−7)! + 8 3!(t−13)! + 16 4!(t−19)! , if t ∈ Z24 ⎪ 19 . ⎪ ⎪ ⎪ ⎪ ⎩ .. . (6.96) With the help of solution (6.94) and discrete matrix delayed exponential functions (6.95) and (6.96), we can present the structure of state xk very clearly, which also makes the output yk fully observed on each running time interval associated with the number of time delay. This advantage may avoid overuse of data each time on the whole running time interval. In this case, one can save computation resources and improve the running efficiency in some sense. Next, noting L1 = 0.45 in learning law (6.92), one can check that I − L1 D = 0.55 < 1. Now all the conditions of Theorem 6.9 are satisfied; thus, lim E{ e˜k λ } = 0 uniformly on Z20 0 .

k→∞

Difference delay systems Chapter | 6

263

FIGURE 6.7 yd , y5 , and y6 of Example 6.5.

FIGURE 6.8 Tracking error e˜k l 2 of Example 6.5.

Fig. 6.7 shows the reference trajectory yd and output trajectory yk . When the output does not exist, we let yk = yd to make sure e˜k (·) = ηk (·)ek (·). Fig. 6.8 shows the Euclidean distance e˜k l 2 of error e˜k in each iteration. The error of the 20th iteration is 7.1014 × 10−6 and the running interval of the 20th iteration is Z20 0 . Figs. 6.9 and 6.10 show the reference trajectory yd , output trajectory yk , and error Euclidean distance ek l 2 for the same trial lengths (Z20 0 ) in each iteration. −6 The error of the 20th iteration is 2.1839 × 10 .

264 Stability and Controls Analysis for Delay Systems

FIGURE 6.9

FIGURE 6.10

yd , y5 , and y6 of Example 6.5 for the same trial lengths.

Tracking error ek l 2 of Example 6.5 for the same trial lengths.

Example 6.6. Consider ⎧ 15 ⎪ ⎨ xk (t + 1) = xk (t) + 2xk (t − 5) + 2uk (t), t ∈ {ηk (i) · i, i ∈ Z0 }, 0 xk (t) = 0, t ∈ Z−5 , ⎪ ⎩ yk (t) = 0.5xk (t).

(6.97)

Difference delay systems Chapter | 6

265

FIGURE 6.11 yd and yk (k = 4, 5, 8, 10) of Example 6.6.

FIGURE 6.12 Tracking error e˜k l 2 of Example 6.6.

Set uk (0) = 0 and let Tk randomly take values from Z16 13 . Select the desired reference trajectory as (6.93) and the learning law (6.61) as uk+1 (t) = uk (t) + 0.4e˜k (t + 1). We can calculate that E − L2 CB = |1 − 0.4 × 0.5 × 2| = 0.6 < 1. Theorem 6.10 guarantees lim E{ e˜k λ } = 0 on Z15 1 . k→∞

266 Stability and Controls Analysis for Delay Systems

FIGURE 6.13

yd and yk (k = 4, 5, 8, 10) of Example 6.6 for the same trial lengths.

FIGURE 6.14

Tracking error ek l 2 of Example 6.6 for the same trial lengths.

Fig. 6.11 shows the reference trajectory yd and output yk of (6.97). Fig. 6.12 shows the error e˜k l 2 for each iteration. The trial length of the 50th iteration −4 is Z13 0 and the error of the 50th iteration is 3.3265 × 10 . Figs. 6.13 and 6.14 show the tracking performance of a delay system with the same trial lengths, and the error of the 50th iteration is 6.5889 × 10−4 . In Figs. 6.11 and 6.13 the lines denote output trajectories of the system; the more iterations, the closer the trajectories are to the desired trajectory.

Difference delay systems Chapter | 6

267

6.3.1.5 Conclusion We adopt a new framework to establish the same convergence theorem for ILC problems of linear discrete delayed systems by virtue of the representation of the solution via a discrete matrix delayed exponential function. The results in this part are motivated from [120].

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Chapter 7

Stochastic delay systems 7.1 Controllability for first order systems 7.1.1 Null controllability stochastic delay systems In this part, we study null controllability of the following stochastic systems via delayed perturbation of matrices: ⎧ t ⎪ ⎨ x  (t) = Ax(t) + Bx(t − τ ) + Cu(t) + 0 (s, x(s))dW (s), t ∈ J = [0, T ], τ ≥ 0, ⎪ ⎩ x(t) = ψ(t), t ∈ [−τ, 0],

(7.1)

where A, B ∈ Rn×n , by assuming that A and B are permutation matrices and C ∈ Rn×m , state x(t) ∈ Rn , control u(t) ∈ Rm , and initial function ψ(t) : [−τ, 0] −→ Rn and defining the delayed matrix exponential expτ (Bt) corresponding to delay τ and matrix B and the semilinear function  ∈ C(J × Rn , Rn×n ). The set of all n-dimensional Wiener process W = (W1 (t), W2 (t), · · · , Wn (t)) . Let (Ω, F, P) be a probability space with probability measure P on Ω and let {Ft |t ∈ J } be a filtration generated by {W (s) : s ∈ [0, t]} and F = FT . Let L2 (Ω, FT , Rn ) be the Hilbert space of all FT -measurable square inten grable variables with values in Rn and let LF 2 (J, R ) be the Hilbert space of all square integrable and Ft -measurable processes with values in Rn . Let A := C([−τ, T ], L2 (Ω, Ft , P, Rn )) be the Banach space of all square integrable and Ft -adapted processes x(t) with norm x2C = sup Ex(t)2 ≤ r < ∞. t∈[−τ,T ]

We consider the matrix norm A = sup Ax for the matrix A : Rn → Rn for initial values

ψ2

C

x=1 Eψ(s)2

= sup s∈[−τ,0]

m and Uad = LF 2 (J, R ) denotes the

set of all admissible controls. Lemma 7.1. ([121]) Let  : J × Ω → L02 be a strongly measurable mapping such that 

T 0

p L2

E(t) 0 dt < ∞.

Stability and Controls Analysis for Delay Systems. https://doi.org/10.1016/B978-0-32-399792-8.00013-X Copyright © 2023 Elsevier Inc. All rights reserved.

269

270 Stability and Controls Analysis for Delay Systems

Then p  t  t   p  ≤ L (s)dW (s) E(s) 0 ds, E   0

L2

0

for all t ∈ J and p ≥ 2, where L is the constant in p and T (for more details see [122]). We now introduce the following operators and matrices for further discussion. The linear bounded operator LT0 ∈ L(Uad , L2 (Ω, FT , Rn )) is defined as  LT0 u =

0

T

eA(T −s) eτB1 (T −τ −s) Cu(s)ds.

Clearly (LT0 ) : L2 (Ω, FT , Rn ) −→ Uad defined by B  (T −τ −·) A (T −·)

(LT0 ) y(·) = C  eτ 1

e



E y F· .

The linear controllability operator Γτ [0, T ] ∈ L(L2 (Ω, FT , Rn ), L2 (Ω, FT , Rn )) which is associated with the operator LT0 is defined by Γτ [0, T ] = LT0 (LT0 ) {·}  T  B  (T −τ −s) A (T −s)

= eA(T −s) eτB1 (T −τ −s) CC  eτ 1 e E · Fs ds, 0

and we have the following controllability Grammian matrix Wτ [0, T ] ∈ L(Rn , Rn ):  T B  (T −τ −s) A (T −s) Wτ [0, T ] = eA(T −s) eτB1 (T −τ −s) CC  eτ 1 e ds. (7.2) 0

Definition 7.1. [123, Definition 2.4] System (7.1) is said to be null controllable on J if for every initial function ψ ∈ C 1 ([−τ, 0], Rn ), there exists a control u ∈ Uad such that system (7.1) has a solution x ∈ A that satisfies the initial condition x(t) = ψ(t), t ∈ [−τ, 0] and x(T ) = 0 at finial time T . Lemma 7.2. [123, Lemma 4] If a triplet (A, B1 , C) is controllable, i.e., rank Sn = n, where Sn = C|AC|A2 C| · · · |An−1 C|B1 C|AB1 C|A2 B1 C|

· · · |An−1 B1 C| · · · |B1n−1 C| · · · |An−1 B1n−1 C ,

271

Stochastic delay systems Chapter | 7

then the matrix function ωτ (t) = eAt eτB1 t consists of elements that are linearly independent on the interval −τ ≤ t ≤ T − τ where T > (n − 1)τ , i.e., there exists no nonzero vector a  = (a1 , a2 , · · · , an ) such that a  ωτ (t) = 0 .

7.1.1.1 Null controllability for linear systems Consider the following linear stochastic delay system:  t x  (t) = Ax(t) + Bx(t − τ ) + Cu(t) + 0 (s)dW (s), t ∈ J, τ ≥ 0, x(t) = ψ(t), t ∈ [−τ, 0], (7.3) where the linear stochastic function  ∈ C(J, Rn×n ) is continuous. By [1, Theorem 1.2], the solution of (7.3) can be expressed in the following form:  0 eA(t−s) eτB1 (t−τ −s) [ψ  (s) − Aψ(s)]ds x(t) = eA(t+τ ) eτB1 t ψ(−τ ) + −τ  t eA(t−s) eτB1 (t−τ −s) Cu(s)ds + 0   s  t A(t−s) B1 (t−τ −s) + e eτ (η)dW (η) ds. (7.4) 0

0

In the following Theorem 7.1, we investigate the necessary and sufficient conditions for (7.3) to be null controllable by using controllability matrices. Theorem 7.1. System (7.3) is null controllable on J if and only if Wτ [0, T ] is nonsingular. Proof. Sufficiency: Since Wτ [0, T ] is nonsingular, its inverse is well defined. For all ψ(t), one can define a control function u(t) as follows:   B  (T −τ −t) A (T −t)

u(t) = C  eτ 1  − 

0 −τ T

− 0

e

E Wτ−1 [0, T ] − eA(T +τ ) eτB1 T ψ(−τ )

eA(T −s) eτB1 (T −τ −s) [ψ  (s) − Aψ(s)]ds eA(T −s) eτB1 (T −τ −s)



s 0

 



(η)dW (η) ds Ft .

(7.5)

Substituting t = T in (7.4) and inserting (7.5) in (7.4), we get  0 x(T ) = eA(T +τ ) eτB1 T ψ(−τ ) + eA(T −s) eτB1 (T −τ −s) [ψ  (s) − Aψ(s)]ds  + 0

−τ

T

B  (T −τ −s) A (T −s)

eA(T −s) eτB1 (T −τ −s) CC  eτ 1

e

272 Stability and Controls Analysis for Delay Systems





× E Wτ−1 [0, T ] − eA(T +τ ) eτB1 T ψ(−τ )  −

0

−τ

 −

T

0

 +

T

0

eA(T −s) eτB1 (T −τ −s) [ψ  (s) − Aψ(s)]ds  



(η)dW (η) ds Fs ds

0  s  eA(T −s) eτB1 (T −τ −s) (η)dW (η) ds. eA(T −s) eτB1 (T −τ −s)



s

0

Hence x(T ) = 0. Further, the initial condition x(t) = ψ(t), t ∈ [−τ, 0], holds by Eqs. (7.3) and (7.4). Thus (7.3) is null controllable on J . Necessity: We prove the necessary part by contradiction. Assume that (7.3) is null controllable. We must prove that Wτ [0, T ] is nonsingular. On the other hand, if it is singular, then there exists at least one nonzero vector a˜ ∈ Rn such that 0 = a˜  Wτ [0, T ]a˜  T B  (T −τ −s) A (T −s) = a˜  eA(T −s) eτB1 (T −τ −s) CC  eτ 1 e ads ˜ 0

 =

T

0

 =

T

0





 a˜  eA(T −s) eτB1 (T −τ −s) C a˜  eA(T −s) eτB1 (T −τ −s) C ds

    A(T −s) B1 (T −τ −s) 2 eτ C  ds, a˜ e

which implies that a˜  eA(T −s) eτB1 (T −τ −s) C = 0 , ∀ s ∈ J.

(7.6)

Since system (7.3) is null controllable, there exist control functions u1 (t) and u2 (t) that drive the initial state to zero at final time T , that is,  0 A(T +τ ) B1 T e eτ ψ1 (−τ ) + eA(T −s) eτB1 (T −τ −s) [ψ1 (s) − Aψ1 (s)]ds  +

−τ

T

0

 +

0

T

eA(T −s) eτB1 (T −τ −s) Cu1 (s)ds  s  eA(T −s) eτB1 (T −τ −s) (η)dW (η) ds = 0 0

and eA(T +τ ) eτB1 T ψ1 (−τ ) +



0

−τ

eA(T −s) eτB1 (T −τ −s) [ψ1 (s) − Aψ1 (s)]ds

(7.7)

Stochastic delay systems Chapter | 7



T

+

eA(T −s) eτB1 (T −τ −s) Cu2 (s)ds  s  eA(T −s) eτB1 (T −τ −s) (η)dW (η) ds = 0.

0



T

+ 0

273

(7.8)

0

From Eqs. (7.7)–(7.8), one can get 



eA(T −s) eτB1 (T −τ −s) C u2 (s) − u1 (s) ds = a, ˜

T

0

(7.9)

where

  a˜ = eA(T +τ ) eτB1 T ψ1 (−τ ) − ψ2 (−τ )   0    A(T −s) B1 (T −τ −s)   ψ1 (s) − ψ2 (s) + A ψ2 (s) − ψ1 (s) ds. e eτ + −τ

Multiplying by a˜  on both sides of (7.9), one can get a˜  a˜ =

 0

T



a˜  eA(T −s) eτB1 (T −τ −s) C u2 (s) − u1 (s) ds = 0.

By Eq. (7.6), we get a˜  a˜ = 0, i.e., a˜ = 0, which is contracted to a˜ = 0. Thus, the Grammian matrix Wτ [0, T ] is nonsingular. The proof is completed. Also, we prove the null controllability results in the following Theorem 7.2 by using rank correlation of the Cayley–Hamilton (C-H) theorem. Theorem 7.2. System (7.3) is null controllable on J if and only if T ≥ (n − 1)τ and det Sn = 0, where Sn = C|AC|A2 C| · · · |An−1 C|B1 C|AB1 C|A2 B1 C|

· · · |An−1 B1 C| · · · |B1n−1 C| · · · |An−1 B1n−1 C = n. Proof. Necessity: Assume that system (7.3) is null controllable, i.e., the solution of system (7.3) satisfies the conditions x(T ) = 0 and x(t) = ψ(t), t ∈ [−τ, 0]. As in Eq. (7.4), the solution of (7.3) at time t = T is  x(T ) = e (T A

 +

+ τ )eτB1 T ψ(−τ ) +

T

0

 +

0

T

0

−τ

eA(T −s) eτB1 (T −τ −s) [ψ  (s) − Aψ(s)]ds

eA(T −s) eτB1 (T −τ −s) Cu(s)ds  s  eA(T −s) eτB1 (T −τ −s) (η)dW (η) ds. 0

(7.10)

274 Stability and Controls Analysis for Delay Systems

Denote an arbitrary vector ν, which satisfies the following equations:  ν = e (T A



T

+  x(T ) − ν = 0

0 T

+ τ )eτB1 T ψ(−τ ) + eA(T −s) eτB1 (T −τ −s)

0

−τ



eA(T −s) eτB1 (T −τ −s) [ψ  (s) − Aψ(s)]ds s

 (η)dW (η) ds,

(7.11)

0

eA(T −s) eτB1 (T −τ −s) Cu(s)ds,

using the representation of delayed exponential matrix function eA(T −s) × B (T −τ −s) in Definition 2.4. By change of variables, T − τ − s = μ, we obeτ 1 tain  0

T

eA(T −s) eτB1 (T −τ −s) Cu(s)ds 

=

T −τ

−τ 0

eA(τ +μ) eτB1 μ Cu((T − τ − μ))dμ

 k−1  (τ + μ) k−1 (τ + μ) I +A + ··· + A I Cu(T − τ − μ)dμ = 1! (k − 1)! −τ   τ (τ + μ) (τ + μ)k−1  μ + · · · + Ak−1 I + B1 I +A + 1! (k − 1)! 1! 0 × Cu(T − τ − μ)dμ   2τ  (τ + μ) (τ + μ)k−1 + I +A + · · · + Ak−1 1! (k − 1)! τ   2 μ (μ − τ ) × I + B1 + B12 Cu(T − τ − μ)dμ 1! 2! + .. .  T −τ  k−1  (τ + μ) k−1 (τ + μ) + ··· + A I +A + 1! (k − 1)! (k−2)τ   2 μ (μ − τ ) (μ − (k − 2)τ )k−1 + · · · + B1k−1 × I + B1 + B12 1! 2! (k − 1)! × Cu(T − τ − μ)dμ   0 (τ + μ) (τ + μ)k−1 = I +A + · · · + Ak−1 Cu(T − τ − μ)dμ 1! (k − 1)! −τ  τ (τ + μ) (τ + μ)k−1 + · · · + Ak−1 I +A + 1! (k − 1)! 0 

Stochastic delay systems Chapter | 7

275

μ μ (τ + μ) μ (τ + μ)k−1  + AB1 + · · · + Ak−1 B1 1! 1! 1! 1! (k − 1)! × Cu(T − τ − μ)dμ  2τ  (τ + μ)2 (τ + μ)k−1 (τ + μ) + A2 + · · · + Ak−1 + I +A 1! 2! (k − 1)! τ k−1 μ μ (τ + μ) μ (τ + μ) (μ − τ )2 + · · · + Ak−1 B1 + B12 + B1 + AB1 1! 1! 1! 1! (k − 1)! 2!  2 2 k−1 (μ − τ ) (τ + μ) (μ − τ ) (τ + μ) + · · · + Ak−1 B12 + AB12 2! 1! 2! (k − 1)! × Cu(T − τ − μ)dμ

+ B1

+ .. . +



T −τ



I +A

(k−2)τ

(τ + μ) (τ + μ)2 (τ + μ)k−1 + A2 + · · · + Ak−1 1! 2! (k − 1)!

μ μ (τ + μ) μ (τ + μ)k−1 (μ − τ )2 + AB1 + · · · + Ak−1 B1 + B12 1! 1! 1! 1! (k − 1)! 2! 2 (τ + μ) 2 (τ + μ)k−1 (μ − τ ) (μ − τ ) + · · · + Ak−1 B12 + AB12 2! 1! 2! (k − 1)! + ··· + B1

(μ − (k − 2)τ )k−1 (μ − (k − 2)τ )k−1 (τ + μ) + AB1k−1 + ... (k − 1)! (k − 1)! 1! (μ − (k − 2)τ )k−1 (τ + μ)k−1  + Ak−1 B1k−1 Cu(T − τ − μ)dμ. (k − 1)! (k − 1)!

+ B1k−1

Denote  Φ11 (T ) = Φ12 (T ) =

T −τ

−τ  T −τ −τ T −τ

 Φ13 (T ) =

−τ T −τ

 Φ1k (T ) =

−τ T −τ

 Φ21 (T ) =

0

u(T − τ − μ)dμ, (τ + μ) u(T − τ − μ)dμ, 1! (τ + μ)2 u(T − τ − μ)dμ, · · · , 2! (τ + μ)k−1 u(T − τ − μ)dμ, (k − 1)! μ u(T − τ − μ)dμ, 1!

276 Stability and Controls Analysis for Delay Systems

 Φ22 (T ) =

T −τ

0

 Φ23 (T ) =

T −τ

0

 Φ2k (T ) =

T −τ

0

 Φ31 (T ) =

T −τ

τ

 Φ32 (T ) =

T −τ

τ

 Φ33 (T ) =

T −τ

τ

 Φ3k (T ) =

T −τ

τ

···  Φm1 (T ) =

T −τ

(k−2)τ T −τ

 Φm2 (T ) =

(k−2)τ T −τ

 Φm3 (T ) =

(k−2)τ T −τ

 Φmk (T ) =

(k−2)τ

μ (τ + μ) u(T − τ − μ)dμ, 1! 1! μ (τ + μ)2 u(T − τ − μ)dμ, · · · , 1! 2! μ (τ + μ)k−1 u(T − τ − μ)dμ, 1! (k − 1)! (μ − τ )2 u(T − τ − μ)dμ, 2! (μ − τ )2 (τ + μ) u(T − τ − μ)dμ, 2! 1! (μ − τ )2 (τ + μ)2 u(T − τ − μ)dμ, · · · , 2! 2! (μ − τ )2 (τ + μ)k−1 u(T − τ − μ)dμ, 2! (k − 1)! (μ − (k − 2)τ )k−1 u(T − τ − μ)dμ, (k − 1)! (μ − (k − 2)τ )k−1 (τ + μ) u(T − τ − μ)dμ, (k − 1)! 1! (μ − (k − 2)τ )k−1 (τ + μ)2 u(T − τ − μ)dμ, · · · , (k − 1)! 2! (μ − (k − 2)τ )k−1 (τ + μ)k−1 u(T − τ − μ)dμ. (k − 1)! (k − 1)!

Note that by (7.10) and (7.11), we have CΦ11 (T ) + ACΦ12 (T ) + · · · + Ak−1 CΦ1k (T ) + B1 CΦ21 (T ) + AB1 CΦ22 (T ) + · · · + Ak−1 B1 CΦ1k (T ) + · · · + B1k−1 CΦm1 (T ) + AB1k−1 CΦm2 (T ) + · · · + Ak−1 B1k−1 CΦmk (T ) = x(T ) − ν.

(7.12)

Since the system is null controllable, (7.12) has a solution for an arbitrary vector x(T ) − ν. If k < n, then the system is overdetermined and not always has a solution. Therefore, for the system to be null controllable, it is necessary that T > (k − 1)τ ≥ (n − 1)τ . It follows from the C-H theorem that an arbitrary power Ai B1i , i ≥ n, of the matrices A, B1 can be expressed as a linear combination of matrices (see [89]):

A|A2 | · · · |An−1 |B1 |AB1 |A2 B1 | · · · |An−1 B1 | · · · |B1n−1 | · · · |An−1 B1n−1 .

Stochastic delay systems Chapter | 7

277

Therefore, for k ≥ n Eq. (7.12) can be replaced by 11 (T ) + AC Φ 12 (T ) + · · · + An−1 C Φ 1n (T ) + B1 C Φ 21 (T ) CΦ 22 (T ) + · · · + An−1 B1 C Φ 1n (T ) + · · · + B n−1 C Φ m1 (T ) + AB1 C Φ 1 m2 (T ) + · · · + An−1 B n−1 C Φ mn (T ) + AB1n−1 C Φ 1 = x(T ) − ν

(7.13)

j i (T ), j = 1, 2, · · · , m, i = 1, 2, · · · , n are some functions of T . where Φ j i (T ), j = 1, 2, · · · , m, i = 1, 2, · · · , n, for arbitrary If (7.13) has the solution Φ ν, then rank Sn = C|AC|A2 C| · · · |An−1 C|B1 C|AB1 C|A2 B1 C| · · · |An−1 B1 C|

· · · |B1n−1 C| · · · |An−1 B1n−1 C = n. Sufficiency: We prove our result by contradiction. Assume that system (7.3) is not null controllable and rank Sn = n. By Theorem 7.1, Wτ [0, T ] is singular. Namely, there exists at least one nonzero vector a˜ ∈ Rn such that 0 = a˜  Wτ [0, T ]a˜  T B  (T −τ −s) A (T −s) = a˜  eA(T −s) eτB1 (T −τ −s) CC  eτ 1 e ads ˜ 0

 =

0

T





 a˜  eA(T −s) eτB1 (T −τ −s) C a˜  eA(T −s) eτB1 (T −τ −s) C ds,

which implies that a˜  eA(T −s) eτB1 (T −τ −s) C = 0 , s ∈ J. Changing variables, T − τ − s = μ, we have a˜  eA(τ +μ) eτB1 μ C = 0 , − τ ≤ t ≤ T − τ, which contradicts the statement of Lemma 7.2 that the matrix function ωτ (t) = eAt eτB1 t consists of elements that are linearly independent on the interval −τ ≤ t ≤ T − τ with T > (n − 1)τ . The proof is complete.

7.1.1.2 Null controllability for semilinear systems We derive the sufficient conditions for null controllability of the stochastic semilinear system (7.1). For that we set the following hypotheses. [H1]. Controllability operator Γτ [0, T ] has its inverse Γτ−1 [0, T ] ∈ Uad / ker Γτ [0, T ]. Then we take k2 = EΓτ−1 [0, T ]2L(L2 (Ω,FT ,Rn ),Uad / ker Γ ) .  From Wang et al. [87], one can write k2 = Γτ−1 [0, T ]2 .

278 Stability and Controls Analysis for Delay Systems

[H2].  ∈ C(J × Rn , Rn ), ∃ q > 1 and M (t) ∈ Lq (J, R+ ) such that (t, x) − (t, y)2 ≤ M (t)x − y2 , x, y ∈ Rn .   [H3]. Set 4T 2 L M2 1 + 3k1 k2 < 1. For brevity, we set k1 = Wτ [0, T ]2 , R = supt∈J E(t, 0)2 , L = 2k1 k2 T 2 × L M2 + 2T 2 L M2 , ϑ = A2 + B1 2 , a = eϑ(T +τ ) Eψ(−τ )2 + ϑpT 1 0 ϑT p × τ −τ eϑ(T −s) Eψ  (s) − Aψ(s)2 ds + T 2 L R e ϑ−1 , M2 = e ϑp−1 M Lq (J,R+ ) . The solution of (7.1) can be expressed as  0 x(t) = eA(t+τ ) eτB1 t ψ(−τ ) + eA(t−s) eτB1 (t−τ −s) [ψ  (s) − Aψ(s)]ds −τ  t eA(t−s) eτB1 (t−τ −s) Cu(s)ds + 0  s   t A(t−s) B1 (t−τ −s) e eτ (η, x(η))dW (η) ds, + 0

0

and its control function u(t) is defined as follows:    (T −t)

u(t) = C  eA 

B  (T −τ −t)

eτ 1

E Γτ−1 [0, T ] − eA(T +τ ) eτB1 T ψ(−τ )

0

eA(T −s) eτB1 (T −τ −s) [ψ  (s) − Aψ(s)]ds − −τ  s  



× (η, x(η))dW (η) ds Ft .

0 −

 0

T

eA(T −s) eτB1 (T −τ −s)

We focus on the theoretical results of null controllability of stochastic delay systems via the fixed point technical route. Define the operator T :A→A by  0 eA(t−s) eτB1 (t−τ −s) [ψ  (s) − Aψ(s)]ds (Ax)(t) = eA(t+τ ) eτB1 t ψ(−τ ) + −τ  t A(t−s) B1 (t−τ −s) e eτ Cu(s)ds + 0  s   t + eA(t−s) eτB1 (t−τ −s) (η, x(η))dW (η) ds, 0

0

Stochastic delay systems Chapter | 7

279

which has a fixed point x, which is (7.1). Next, we check that system (7.1) is null controllable on J . Now, we present our theoretical results for semilinear systems by the following Theorem 7.3. Theorem 7.3. Suppose that hypotheses [H1]–[H3] hold. Then, (7.1) is null controllable on J provided that L 0, ⎨ y(t) y(t) = ψ(t), ⎪ ⎩ ˙ y(t) ˙ = ψ(t), t ∈ J1 := [−τ, 0], (7.16) where A ∈ Rn×n is a nonsingular matrix, B ∈ Rn×m , state vector y(t) ∈ Rn , control function u : J −→ Rm , H ∈ ( 12 , 1), and  : J × Rn −→ T (Rn ), where T (Rn ) denotes the Thorin class which is the smallest class of distributions on Rn that contains all Gamma distributions and is closed under convolution and weak convergence. The initial function ψ(t) : J1 −→ Rn is an F0 -measurable Rn -valued stochastic process independent of the Rosenblatt process ZH with finite second moment.

Stochastic delay systems Chapter | 7

289

The Hermite process of order one is the so-called fractional Brownian motion and that of order two is called the Rosenblatt process. Initially, Rosenblatt introduced the following distribution for t ≥ 0:    t −(1+R) −(1+R) ZR (t) = D(R) (ν − x1 )+ 2 (ν − x2 )+ 2 dν dW (x1 )dW (x2 ), R2

0

where R ∈ (0, 12 ) and {W (x), x ∈ R} is a standard Brownian motion. The process of ZR (1) is called the “1 non-Gaussian limiting distribution” (Rosenblatt distribution); for more details see [131]. By employing pathwise Malliavin calculus and the Itô–Wiener multiple integral, the Rosenblatt process was investigated in [121]. In [132] the Rosenblatt process for every t1 , t2 , · · · , td ≥ 0 (ZR (t1 ), ZR (t2 ), · · · · · · , ZR (td ))(d)  ∞ ∞ $ $ 2 2 μn (t1 )(ζn − 1), · · · · · · , μn (td )(ζn − 1) n=1

n=1

was considered, where {ζn } are independent and identically distributed N (0, 1) random variables. Let A := C([−τ, T ], Lp (Ω, FT , P, Rn )) be the Banach space and Ft  p 2 adapted processes ϕ(t) with norm  · , where ϕp = sup Eϕ(t)2 . Also, t∈J

we introduce the Banach space C 2 (J, Lp (Ω, FT , P, Rn ))  = y ∈ C(J, Lp (Ω, FT , P, Rn )) : y¨ ∈ C(J, Lp (Ω, FT , P, Rn )) with norm

 ˙ Ey(t) ¨ . yC 2 (J ) = sup Ey(t), Ey(t), t∈J

We consider the matrix norm A = sup Ay for the matrix A : Rn → Rn 

p

y=1

for initial values ψC = sup Eψ(s)2 

2 ¨ ¨ ψ C = sup Eψ(s) p

s∈J1

p 2

 p 2 2 , and ˙ ˙ p = sup Eψ(s) , ψ C

p 2

s∈J1

.

s∈J1

p

Let h be a function such that E[h(ζo )] = 0 and E[h(ζo )] < ∞. Suppose the function % h has Hermite rank n, i.e., if h allows the Hermite polynomials h(x) = i≥0 Ci Hi (x), where Ci = i!1 E[h(ζ0 Hi (ζ0 ))], then n = min{i; Ci = 0}. Since E[h(ζo )] = 0, we have n ≥ 1. Then by the noncentral limit theorem, [kt] 1 $ h(ζi ) kH i=1

290 Stability and Controls Analysis for Delay Systems

converges as k −→ ∞ in the sense of the Rn -distribution to the process    t & n −( 12 + 1−H ) n n (s − yi )+ ZH (t) = C(H, n) dsdW (y1 ), · · · · · · , dW (yn ), Rn

0

i=1

(7.17) where x+ = max(x, 0) and the above integral is a multiple Wiener–Itô stochastic integral with respect to a Brownian motion W (y)y∈R . The constant C(H, n) > 0 n (1)2 = 1]. The process (Z n (t) and it will be taken such that E[ZH t≥0 ) is called H the Hermite process and it is H -self-similar in the sense that for any C > 0, n (t))(d)(C H Z n (t)), where (d) means equivalence of all Rn -distributions (ZH H and it has stationary increments. • If n = 1, then (7.17) is called fractional Brownian motion with • If n ≥ 2, then (7.17) is called a non-Gaussian process. • If n = 2, then (7.17) is called a Rosenblatt process.

1 2

< H < 1.

Consider the linear stochastic control systems driven by the Rosenblatt process of the form ⎧ ⎪ ¨ = −A2 y(t − τ ) + Bu(t) + (t)dZH (t), t ∈ J, τ > 0, ⎨ y(t) (7.18) y(t) = ψ(t), ⎪ ⎩ ˙ y(t) ˙ = ψ(t), t ∈ J1 , where  ∈ C(J, T (Rn )). Definition 7.2. System (7.16) is said to be controllable if there exists a control function u : J −→ Rm such that y(t) ¨ = −A2 y(t − τ ) + Bu (t) + (t, y(t))dZH (t), t ∈ J, τ > 0, has a solution y = y  : [−τ, T ] −→ Rn satisfying ˙ t ∈ J1 , y  (T ) = y1 , y  (t) = ψ(t), y˙  (t) = ψ(t), where y1 is a finite terminal state which belongs to Rn . From [5, Theorems 1 and 2], the solution of (7.18) can be expressed in the following form: ˙ ) y(t) = (cosτ At)ψ(−τ ) + A−1 (sinτ At)ψ(−τ  0  t ¨ + A−1 sinτ A(t − τ − s)ψ(s)ds + A−1 sinτ A(t − τ − s)Bu(s)ds −τ 0  t sinτ A(t − τ − s)(s)dZH (s), (7.19) + A−1 0

Stochastic delay systems Chapter | 7

291

where cosτ A : R → Rn×n (see (3.3)) and sinτ A : R → Rn×n (see (3.4)). Let L(X, Y ) denote the space of linear bounded operators from a Banach m n space X to a Banach space Y . The operator LT0 ∈ L(LF p (J, R ), Lp (Ω, FT , R )) is defined by LT0 u = A−1



T

sinτ A(T − τ − s)Bu(s)ds

0

and its adjoint operator 

(LT0 ) x = B  sinτ A (T − τ − t)E x Ft m is mapping from Lp (Ω, FT , Rn ) to LF p (J, R ). We consider the linear controllability operator

(Γτ )T0 {·} = A−1



T 0



sinτ A(T − τ − s)BB  sinτ A (T − τ − s)E · Fs ds

∈ L(Lp (Ω, Ft , Rn ), Lp (Ω, Ft , Rn )) and the delayed controllability Grammian matrix (Wτ )T0 ∈ L(Rn , Rn ): (Wτ )T0 = A−1



T

sinτ A(T − τ − s)BB  sinτ A (T − τ − s)ds.

(7.20)

0

Here  denotes the transpose.

7.2.1.1 Controllability of stochastic linear oscillating delay systems Theorem 7.4. The linear system (7.18) is controllable if and only if the delayed controllability Grammian matrix (7.20) is nonsingular. Proof. Sufficiency: Since (Wτ )T0 is positive definite, i.e., it is nonsingular, and ' (−1 therefore its inverse (Wτ )T0 is well defined. Define a control function

−1 y1 − (cosτ AT )ψ(−τ ) u(t) = B  sinτ A (T − τ − t) (Wτ )T0  0 ˙ ¨ − A−1 (sinτ AT )ψ(−τ ) − A−1 sinτ A(T − τ − s)ψ(s)ds − A−1



−τ

T

sinτ A(T − τ − s)(s)dZH (s) .

0

Substituting t = T and inserting (7.21) in (7.19), we get ˙ ) y(T ) = (cosτ AT )ψ(−τ ) + A−1 (sinτ AT )ψ(−τ

(7.21)

292 Stability and Controls Analysis for Delay Systems

+ A−1



0

−τ  T

¨ sinτ A(T − τ − s)ψ(s)ds

−1 sinτ A(T − τ − s)BB  sinτ A (T − τ − s) (Wτ )T0 + A−1 0 ˙ × y1 − (cosτ AT )ψ(−τ ) − A−1 (sinτ AT )ψ(−τ )  0 ¨ − A−1 sinτ A(T − τ − s)ψ(s)ds − A−1 + A−1

 

−τ T

sinτ A(T − τ − s)(s)dZH (s) (s)ds

0 T

sinτ A(T − τ − s)(s)dZH (s).

0

Now, using the delayed controllability Grammian matrix (7.20) we obtain ˙ ) y(T ) = (cosτ AT )ψ(−τ ) + A−1 (sinτ AT )ψ(−τ  0



−1 ¨ + A−1 sinτ A(T − τ − s)ψ(s)ds + (Wτ )T0 (Wτ )T0 −τ ˙ × y1 − (cosτ AT )ψ(−τ ) − A−1 (sinτ AT )ψ(−τ )  0 ¨ − A−1 sinτ A(T − τ − s)ψ(s)ds − A−1

−τ T



sinτ A(T − τ − s)(s)dZH (s)

0

+ A−1



T

sinτ A(T − τ − s)(s)dZH (s)

0

= y1 . ˙ Next, we verify the boundary conditions y(t) = ψ(t) and y(t) ˙ = ψ(t), t ∈ J1 . From (3.4) and (3.3), we have sinτ At = A(t + τ ), cosτ At = I , t ∈ J1 , and we define  Θ, s ∈ (t, 0], sinτ A(t − τ − s) = A(t − s), s ∈ [−τ, t]. From the above relation, the solution (7.19) becomes ˙ ) + A−1 y(t) = I ψ(−τ ) + A−1 A(t + τ )ψ(−τ



t

−τ

¨ sinτ A(t − τ − s)ψ(s)ds. (7.22)

Stochastic delay systems Chapter | 7

293

For simplicity, 

t

¨ sinτ A(t − τ − s)ψ(s)ds  t ˙ sinτ A(t − τ − s)d ψ(s) =

−τ

−τ

 t t ˙ ˙ = sinτ A(t − τ − s)ψ(s)| − d(sinτ A(t − τ − s))ψ(s)ds −τ −τ  t ˙ ˙ ψ(s)ds = −A(t + τ )ψ(−τ )+A −τ

˙ = −A(t + τ )ψ(−τ ) + Aψ(t) − Aψ(−τ ).

(7.23)

˙ Substituting (7.23) in (7.22), we get y(t) = ψ(t). Now, y(t) ˙ = ψ(t), t ∈ J1 , holds. Thus, the linear system (7.18) is controllable according to Definition 7.2. Necessity: On the other hand, if the delayed controllability Grammian matrix (7.20) is singular, then (Wτ )T0 [A−1 ] is also singular, i.e., there exists at least one nonzero state vector x such that 0 = x  (Wτ )T0 [A−1 ] x  T = x  A−1 sinτ A(T − τ − s)BB  sinτ A (T − τ − s)[A−1 ] xds 0



T

= 0



T

= 0



x  A−1 sinτ A(T − τ − s)B



 x  A−1 sinτ A(T − τ − s)B ds

 2   −1  x A sinτ A(T − τ − s)B  ds,

which implies that x  A−1 sinτ A(T − τ − η)B = 0 , ∀ η ∈ J.

(7.24)

Since the linear system (7.18) is controllable, according to Definition 7.2, there exists a control u1 (t) that steers the initial state to zero at time T , i.e., ˙ ) y(T ) = (cosτ AT )ψ(−τ ) + A−1 (sinτ AT )ψ(−τ  0 ¨ + A−1 sinτ A(T − τ − s)ψ(s)ds + A−1 + A−1

 

−τ T

sinτ A(T − τ − s)Bu1 (s)ds

0 T 0

sinτ A(T − τ − s)(s)dZH (s) = 0,

(7.25)

294 Stability and Controls Analysis for Delay Systems

where 0 denotes the n-dimensional zero vector. Similarly, there exists a control u2 (t) that steers the initial state to x at time T , i.e., ˙ y(T ) = (cosτ AT )ψ(−τ ) + A−1 (sinτ AT )ψ(−τ )  0 ¨ + A−1 sinτ A(T − τ − s)ψ(s)ds + A−1

−τ T



sinτ A(T − τ − s)Bu2 (s)ds

0

+ A−1



T

sinτ A(T − τ − s)(s)dZH (s) = x.

(7.26)

0

Then, by (7.25) and (7.26), we have  T −1 x=A sinτ A(T − τ − s)B[u2 (s) − u1 (s)]ds. 0

Multiplying by x  on both sides of the above equation, we get  T  x x= x  A−1 sinτ A(T − τ − s)B[u2 (s) − u1 (s)]ds. 0

Note (7.24). One can obtain x  x = 0, i.e., x = 0, which is a contradiction. Thus, the delayed controllability Grammian matrix (7.20) is nonsingular. The proof is finished.

7.2.1.2 Controllability of stochastic nonlinear oscillating delay systems Consider the following hypotheses: [H1]. The linear stochastic delay system (7.18) is controllable on J . [H2]. Let p ∈ (1, ∞). The function  ∈ C(J × Rn , T (Rn )) is continuous, and ∃M ∈ Lq (J, R + ) and q > 1 such that (t, y1 ) − (t, y2 ) ≤ M (t)y1 − y2 , yi ∈ Rn , i = 1, 2. Let R = sup (t, 0)p . 0≤t≤T

If hypothesis [H1] holds, then for some γ > 0 we have E (Γτ )T0 x, x ≥ γ Exp , ∀ x ∈ Lp (Ω, Ft , Rn ) (see [133, Lemma 2]). Also, [(Γτ )T0 ]−1  ≤ 1 T p γ := M1 (see [134]) and we set M = max{(Wτ )s  : s ∈ J }. The solution of (7.16) can be expressed in the form ˙ ) y(t) = (cosτ At)ψ(−τ ) + A−1 (sinτ At)ψ(−τ  t + A−1 sinτ A(t − τ − s)Buy (s)ds 0

Stochastic delay systems Chapter | 7

+ A−1 + A−1



0

−τ  t

295

¨ sinτ A(t − τ − s)ψ(s)ds sinτ A(t − τ − s)(s, y(s))dZH (s)

0

and its control function uy (t) is defined as uy (t) = B  sinτ A (T − τ − t)E [(Γτ )T0 ]−1 y1 − (cosτ AT )ψ(−τ )  0 ˙ ¨ − A−1 (sinτ AT )ψ(−τ ) − A−1 sinτ A(T − τ − s)ψ(s)ds − A−1

 0

−τ

T



sinτ A(T − τ − s)(s, y(s))dZH (s) Ft .

(7.27)

Define the operator Q : A → A by ˙ (Qy)(t) = (cosτ At)ψ(−τ ) + A−1 (sinτ At)ψ(−τ )  0 ¨ + A−1 sinτ A(t − τ − s)ψ(s)ds −τ  t sinτ A(t − τ − s)Buy (s)ds + A−1 0  t + A−1 sinτ A(t − τ − s)(s, y(s))dZH (s), 0 d ˙ and suppose it has a fixed point y. Note (Qy)(t) = ψ(t), dt (Qy)(t) = ψ(t), t∈ J1 , (Qy)(T ) = y1 , from the control uy , and this means that (7.16) is controllable on J . Define the set    p p 2 2 Br = y ∈ A : EyC = sup Ey(t) ≤r , t∈[−τ,T ]

for each positive number r. Now Br is a closed, bounded, and convex set of A for each r. Next, we derive sufficient conditions for controllability of the nonlinear system. Theorem 7.5. Suppose that hypotheses [H1] and [H2] hold. Then, (7.16) is controllable on J if M2 [1 + 5p−1 MM1 ] < 1,

(7.28)

where  M2 : = 5p−1 A−1 p CH T (p−1)H

eA p1 T − 1 2p1 p1 Ap p



1 p1

296 Stability and Controls Analysis for Delay Systems

× M Lq (J,R + ) , p1 ∈ (1, ∞). Proof. We verify the conditions of Krasnoselskii’s fixed point theorem (see [82]) as follows. Lemma 7.3. Under hypotheses [H1]–[H2], there exists an r > 0 such that Q(Br ) ⊆ Br . Proof. Note E(Qy r )(t)p ˙ ≤ 5p−1  cosτ Atp Eψ(−τ )p + A−1 p  sinτ Atp Eψ(−τ )p  0 p   −1 p  ¨ + A  E  sinτ A(t − τ − s)ψ(s)ds   −τ  t p   −1 p  + A  E  sinτ A(t − τ − s)(s, y(s))dZH (s)  0

 p  5 $  −1 t   = + E A sinτ A(t − τ − s)Buy (s)ds  In .  0

(7.29)

n=1

From Lemmas 3.2 and 3.3, we have p

I1 = 5p−1  cosτ Atp Eψ(−τ )p ≤ 5p−1 cosh(Ap t)EψC , ˙ I2 = 5p−1 A−1 p  sinτ Atp Eψ(−τ )p ˙ p, ≤ 5p−1 A−1 p sinh(Ap (t + τ ))Eψ C  0 p   p−1 −1 p  ¨ I3 = 5 A  E  sinτ A(t − τ − s)ψ(s)ds   −τ

¨ ≤ 5p−1 A−1 p τ p−1 Eψ C p

−1 p p−1 Eψ ¨ p C p−1 A  τ ≤5 Ap



0 −τ



 sinτ A(t − τ − s)p ds

    cosh Ap (t + τ ) − cosh Ap t .

By employing Lemmas 7.1, 3.2, and 3.3 and hypothesis [H2] we have  t p   p−1 −1 p  I4 = 5 A  E  sinτ A(t − τ − s)(s, y(s))dZH (s)  0  t ≤ 5p−1 A−1 p CH T (p−1)H E sinτ A(t − τ − s)(s, y(s))p ds 0  t p−1 −1 p (p−1)H A  CH T sinh[Ap (t − s)]E (s, y(s))p ds ≤5 0  t p−1 −1 p (p−1)H ≤5 A  CH T sinh[Ap (t − s)]M (s)E y(s)p ds 0

Stochastic delay systems Chapter | 7



297

sinh[Ap (t − s)]E (s, 0)p ds 0  t sinh[Ap (t − s)]M (s)ds ≤ 5p−1 A−1 p CH T (p−1)H r 0  t

+ R sinh[Ap (t − s)]ds . t

+

0

From Hölder’s inequality, we have 

t

sinh[A(t − s)]M (s)ds

0

 ≤

t

(sinh[A(t − s)])

0

+

where

1 p1

I4 ≤ 5

p−1

1 q

 × 0

t

 p1  ds

1

0

t

q M (s)ds

 q1 ,

= 1 and q is as in [H2]. Thus, we have

−1 p

A

p1

 CH T

q

M (s)ds

(p−1)H

 q1

 p1  t ) *p1 1 p r sinh[A (t − s)] ds 

+ R

0 t

sinh[Ap (t − s)]ds

0

 ≤ 5p−1 A−1 p CH T (p−1)H r 0

t

exp(Ap p1 (t − s)) ds 2p1

 p1

1

M Lq (J,R + )

R p cosh(A T ) − 1 Ap   1 exp(Ap p1 T ) − 1 p1 p−1 −1 p (p−1)H r A  CH T M Lq (J,R + ) ≤5 2p1 p1 Ap  R + cosh(Ap T ) − 1 . Ap +

From (7.27), Lemmas 7.1, 3.2, and 3.3, [H2], I1 to I4 , and Hölder’s inequality, we obtain  p   −1 t   A sin A(t − τ − s)Bu (s)ds I5 = 5p−1 E  τ y   0 ≤ 5p−1 (Wτ )T0 p [(Γτ )T0 ]−1 p 5p−1 Ey1 p ˙ +  cosτ AT p Eψ(−τ )p + A−1 p  sinτ AT p Eψ(−τ )p  0 p    ¨ + A−1 p E  sinτ A(T − τ − s)ψ(s)ds   −τ

298 Stability and Controls Analysis for Delay Systems

 T p     + A−1 p E  sin A(T − τ − s)(s, y(s))dZ (s) τ H   0

≤ 52(p−1) MM1 Ey1 p + χ(T ) + M2 r , where p ˙ p χ(T ) : = cosh(Ap T )EψC + A−1 p sinh(Ap (T + τ ))Eψ C     ¨ p A−1 p τ p−1 Eψ C p p + × cosh A (T + τ ) − cosh A T Ap R cosh(Ap T ) − 1. + A−1 p CH T (p−1)H Ap

From I1 to I5 , (7.29) becomes E(Qy r )(t)p p ˙ p ≤ 5p−1 cosh(Ap t)EψC + A−1 p sinh(Ap (t + τ ))Eψ C

¨ p A−1 p τ p−1 Eψ C p p cosh(A (t + τ )) − cosh(A t) Ap   + 5p−1 MM1 Ey1 p + χ(T ) + M2 r + A−1 p CH

+

  t  p1  t  q1 *p ) 1 q × T (p−1)H r M (s)ds sinh[Ap (t − s)] 1 ds 0 0   t p + R sinh[A (t − s)]ds 0 p ˙ p ≤ 5p−1 cosh(Ap t)EψC + A−1 p sinh(Ap (t + τ ))Eψ C

¨ p A−1 p τ p−1 Eψ C cosh(Ap (t + τ )) − cosh(Ap t) p A   p−1 MM1 Ey1 p + χ(T ) + M2 r +5

+

  1 exp(Ap p1 t) − 1 p1 r + A  CH T M Lq (J,R + ) 2p1 p1 Ap  R + cosh(Ap t) − 1 Ap ≤ 5p−1 χ(T ) + 5p−1 MM1 Ey1 p  + 5p−1 MM1 χ(T ) + 5p−1 MM1 M2 r + M2 r −1 p

(p−1)H

Stochastic delay systems Chapter | 7

299

  ≤ 5p−1 χ(T ) 1 + 5p−1 MM1 + 5p−1 MM1 Ey1 p   + M2 r 1 + 5p−1 MM1 . Thus, for some r sufficiently large, from (7.28) we have (Qy r )(t) ∈ Br , so as a result Q(Br ) ⊆ Br . Next, we write the operator Q as Q1 + Q2 , where ˙ ) (Q1 y)(t) = (cosτ At)ψ(−τ ) + A−1 (sinτ At)ψ(−τ  0 ¨ sinτ A(t − τ − s)ψ(s)ds + A−1 −τ  t sinτ A(t − τ − s)Buy (s)ds + A−1 0

and −1



(Q2 y)(t) = A

t

sinτ A(t − τ − s)(s, y(s))dZH (s).

(7.30)

0

Lemma 7.4. Assume that hypotheses [H1] and [H2] hold. Then, the operator Q1 is a contraction mapping. Proof. Let t ∈ J . For all y, z ∈ Br , we have E(Q1 y)(t) − (Q1 z)(t)p  p   −1 t   A = E sin A(t − τ − s)B[u (s) − u (s)]ds τ y z   0

≤ (Wτ )T0 p [(Γτ )T0 ]−1 p A−1 p  T p    × E sinτ A(T − τ − s)[(s, z(s)) − (s, y(s))]dZH (s)  0

MM1 M2 p p ≤ E y − zC ≤ δE y − zC , 5p−1 1 M2 where δ = MM . From (7.28), note that δ < 1, which implies that Q1 is a 5p−1 contraction.

Lemma 7.5. The operator Q2 : Br → A is continuous and compact. Proof. Let yn ∈ Br with yn −→ y ∈ Br . For convenience, let n (·) = (·, yn (·)) and (·) = (·, y(·)), and note



sinh Ap (· − s) n (s) −→ sinh Ap (· − s) (s), s ∈ J.

300 Stability and Controls Analysis for Delay Systems

From [H2], we get



sinh Ap (· − s) En (s) − (s)p ≤ 2r sinh Ap (· − s) M (s) ∈ L1 (J, R + ). Then using (7.30) and Lebesgue’s dominated convergence theorem, one can obtain E(Q2 yn )(t) − (Q2 y)(t)p  t

≤ A−1 p CH T (p−1)H sinh Ap (t − s) En (s) − (s)p ds → 0 0

as n → ∞. Thus, Q2 : Br −→ A is continuous. Next, we show that Q2 (Br ) ⊂ A is equicontinuous. For that y ∈ Br , 0 < t ≤ t + h ≤ T , and from (7.30), we obtain  t+h −1 (Q2 y)(t + h) − (Q2 y)(t) = A sinτ A(t + h − τ − s)(s)dZH (s) 0  t sinτ A(t − τ − s)(s)dZH (s) − A−1 0

= K1 + K2 , where K1 = A−1



t+h

sinτ A(t + h − τ − s)(s)dZH (s)

t

and K2 = A−1



t

[sinτ A(t + h − τ − s) − sinτ A(t − τ − s)](s)dZH (s).

0

Therefore,   E(Q2 y)(t + h) − (Q2 y)(t)p ≤ 2p−1 EK1 p + EK2 p . Now, we need to check EKi p −→ 0 when h −→ 0, where i = 1, 2. For K1 , q , we have let q be as in [H2]. With p1 = q−1 

t+h

t



sinh Ap (t + h − s) M (s)ds

 ≤

exp(Ap hp1 ) − 1 2p1 p1 Ap



1 p1

M Lq (J,R + ) .

Stochastic delay systems Chapter | 7

301

Using the above inequality, we have  t+h

p −1 p (p−1)H EK1  ≤ A  CH h sinh Ap (t + h − s) E(s)p ds t

≤ A−1 p CH h(p−1)H



t+h



sinh Ap (t + h − s)

t

× E(s, y(s)) − (s, 0) + (s, 0)p ds  t+h

≤ 2p−1 A−1 p CH h(p−1)H sinh Ap (t + h − s) t   × E(s, y(s)) − (s, 0)p + E(s, 0)p ds ≤ 2p−1 A−1 p CH h(p−1)H  t+h

× sinh Ap (t + h − s) M (s)Ey(s)p ds t

+ 2p−1 A−1 p CH h(p−1)H



t+h



sinh Ap (t + h − s) R (s)ds

t

1 exp(Ap hp1 ) − 1 p1 A  CH h r M Lq (J,R + ) ≤2 2p1 p1 Ap

R p cosh(A + 2p−1 A−1 p CH h(p−1)H h) − 1 → 0 as h → 0. Ap p−1

−1 p



(p−1)H

q For K2 , let q be as in [H2] and as before p1 = q−1 . Applying Hölder’s inequality we have  t E[sinτ A(t + h − τ − s) EK2 p ≤ A−1 p CH T (p−1)H 0

− sinτ A(t − τ − s)](s, y(s))p ds   t p−1 −1 p (p−1)H r ≤2 A  CH T ( sinτ A(t + h − τ − s)  − sinτ A(t − τ − s) ) ds p p1

0 1 p1



× M Lq (J,R + ) + 2p−1 A−1 p CH T (p−1)H R  t ×  sinτ A(t + h − τ − s) − sinτ A(t − τ − s)p ds. 0

From (3.4) and (3.3), we know that sinτ At is uniformly continuous for all t ∈ J , and thus, we get  sinτ A(t + h − τ − s) − sinτ A(t − τ − s)p −→ 0 as h −→ 0. Finally, E(Q2 y)(t + h) − (Q2 y)(t)p −→ 0 as h −→ 0 for all y ∈ Br . The other parts are treated similarly. From the Arzela–Ascoli theorem, the operator Q : Br −→ A is compact.

302 Stability and Controls Analysis for Delay Systems

Therefore, from Krasnoselskii’s fixed point theorem (see [82]), Q has a fixed point y which is the solution of (7.16). Furthermore, y(T ) = y1 through the control function uy (t). Thus, the nonlinear stochastic delay system (7.16) is controllable on J .

7.2.1.3 Numerical examples and discussion Consider the following nonlinear stochastic delay system: y(t) ¨ = −A2 y(t − 0.75) + Bu(t) + (t, y(t))dZH (t), t ∈ [0, 1.5], ˙ y(t) = ψ(t), y(t) ˙ = ψ(t), t ∈ [−0.75, 0], where  A=

1 −1 0 2



 , B=

⎛

 1 1

e(t+0.1) 10 y1 (t) (t+0.1) e 10 y2 (t)

, (t, y(t)) = ⎝

(7.31)

⎞ ⎠.

The corresponding linear part is ⎛ (t) = ⎝

e(t+0.1) 10 e(t+0.1) 10

and  ψ(t) =

t 2t



⎞ ⎠

 ˙ , ψ(t) =

 1 2

.

Here B is an n × m matrix with n = 2, m = 1, control function u : J −→ Rm , terminal time T = 1.5, τ = 0.75. The corresponding delayed controllability Grammian matrix of system (7.31) is  1.5 −1 [W0.75 ]1.5 = A sin0.75 A(0.75 − s)BB  sin0.75 A (0.75 − s)ds 0 0

=: W1 + W2 , where W1 = A−1



0.75

sin0.75 A(0.75 − s)B

0

× B  sin0.75 A (0.75 − s)ds, (0.75 − s) ∈ (0, 0.75),  1.5 W2 = A−1 sin0.75 A(0.75 − s)B 0.75

× B  sin0.75 A (0.75 − s)ds, (0.75 − s) ∈ (−0.75, 0),

Stochastic delay systems Chapter | 7

and

sin0.75 (At) =

⎧ ⎪ Θ, ⎪ ⎪ ⎪ ⎪ ⎨A(t + 0.75),

−∞ < t < −0.75, −0.75 ≤ t < 0, 3

⎪ A(t + 0.75) − A3 t3! , ⎪ ⎪ ⎪ ⎪ ⎩A(t + 0.75) − A3 t 3 + A5 (t−0.75)5 , 3! 5!

cos0.75 (At) =

303

⎧ ⎪ Θ, ⎪ ⎪ ⎪ ⎪ ⎨I,

0 ≤ t < 0.75, 0.75 ≤ t < 1.5,

−∞ < t < −0.75, −0.75 ≤ t < 0, 2

⎪ I − A2 t2! , ⎪ ⎪ ⎪ ⎪ ⎩I − A2 t 2 + A4 (t−0.75)4 , 2! 4!

0 ≤ t < 0.75, 0.75 ≤ t < 1.5.

By simple computations, we obtain the delay Grammian matrix   3.2495 4.6042 , [W0.75 ]1.5 0 = 0.8491 2.9060 where



 0.1131 1.8891 0.0941 1.7009

W1 = Its inverse is

 −1 1.5 [W0.75 ]0

=

 , W2 =

 3.1364 2.7151 0.7550 1.2051

0.5252 −0.8320 −0.1534 0.5872

.

 .

Thus, the corresponding linear system of (7.31) is controllable on [0,  1.5].More   y11 y11 over, for any final states y(T ) = y1 = , and , y(T ˙ ) = y1 =  y12 y12 uy (t) ∈ R, −1 1.5 ]0 χ1 , uy (t) = B  sin0.75 A (0.75 − t)[W0.75

where ˙ χ1 = y1 − (cos0.75 A1.5)ψ(−0.75) − A−1 (sin0.75 A1.5)ψ(−0.75)    1.5−0.75 (1.5 − 0.75 − s)3 − A−1 (s)dZH (s) A(1.5 − s) − A3 6 0  0.75  1.5  + [A(1.5 − s)] (s)dZH (s) + [A(1.5 − s)] (s)dZH (s) 1.5−0.75

0.75

304 Stability and Controls Analysis for Delay Systems

 =

y11 − 1.5987 y12 − 6.4079

 .

The solution of the corresponding linear systems of (7.31) is ˙ y(t) = (cos0.75 At)ψ(−0.75) + A−1 (sin0.75 At)ψ(−0.75)  t + A−1 sin0.75 A(t − 0.75 − s)BB  sin0.75 A (0.75 − s)ds 0  t −1 1.5 × [W0.75 ]0 χ1 + A−1 sin0.75 A(t − 0.75)(s)dZH (s). 0

Consider the first integral term from the above equation,  t sin0.75 A(t − 0.75 − s)BB  sin0.75 A (0.75 − s)ds. 0

For the interval t ∈ (0, 0.75), we get −0.75 < t − 0.75 − s < t − 0.75 < 0 and 0 < 0.75 − t < 0.75 − s < 0.75. Then the solution   2 3 2t −1 3t ˙ y(t) = I − A A(t + 0.75) − A ψ(−0.75) + A ψ(−0.75) 2 6    t (0.75 − s)3 + A−1 ds [A(t − s)] BB  A (1.5 − s) − (A )3 6 0  t −1 1.5 ]0 χ1 + A−1 A(t − s)(s)dZH (s). (7.32) × [W0.75 0

Also for the interval t ∈ (0.75, 1.5), we get 0 < t − 0.75 − s < t − 0.75 < 0.75 when 0 < s < t − 0.75 and −0.75 < t − 0.75 − s < 0 when t − 0.75 < s < t. Furthermore, we obtain 0 < 0.75 − s < 0.75 when 0 < s < 0.75 and −0.75 < 0.75 − t < 0.75 − s < 0 when s ∈ (0.75, t). Finally, solution (7.32) can be represented in the following form:    t2 (t − 0.75)4 t3 y(t) = I − A2 + A4 ψ(−0.75) + A−1 A(t + 0.75) − A3 2 24 6   t−0.75  5 (t − 0.75) ˙ ψ(−0.75) + A−1 A(t − s) + A5 120 0    (t − 0.75 − s)3 (0.75 − s)3 − A3 BB  A (1.5 − s) − (A )3 ds 6 6   0.75 −1 1.5 × [W0.75 ]0 χ1 + A−1 [A(t − s)] BB  A (1.5 − s) t−0.75

3 −1 1.5  3 (0.75 − s) ds[W0.75 ]0 χ1 − (A ) 6

Stochastic delay systems Chapter | 7



305



−1 1.5 ]0 χ1 [A(t − s)] BB  A (1.5 − s) ds[W0.75 0.75   t−0.75  (t − 0.75 − s)3 A(t − s) − A3 + A−1 (s)dZH (s) 6 0  0.75 + A−1 [A(t − s)] (s)dZH (s) t−0.75  t −1 +A [A(t − s)] (s)dZH (s). + A−1

t

0.75

Let p = 2 = p1 and q = 2. For any x(t), y(t) ∈ R 2 , t ∈ [0, 1.5],

et+0.1 (x1 (t) − y1 (t))2 + (x2 (t) − y2 (t))2 10

t+0.1



e

x − y2 . ≤

10

(t, x) − (t, y)2 =

Hence,  satisfies [H2], where by set M (·) = sup (t, 0)2 = 0. Now

e·+0.1 10

∈ Lq ([0, 1.5], R + ), R =

t∈[0,1.5]

 M Lq ([0,1.5],R + ) =

0

1.5  es+0.1 2

10

 12 ds

= 0.6203

and M1 = 1.3115. Furthermore, M2 : = 5

p−1

−1 p

A

 CH T

 (p−1)H

M Lq (J,R + )

exp(Ap p1 T ) − 1 2p1 p1 Ap



1 p1

= 0.0812. Hence we obtain



M2 1 + 5p−1 MM1 = 0.7561 < 1,

which guarantees that (7.28) holds. Thus, the hypotheses of Theorem 7.5 are satisfied. Hence, system (7.31) is controllable on [0, 1.5]. By using MATLAB, the controlled trajectories of the states and steering control are computed and depicted in Figs. 7.2, 7.3, and 7.4 in various time intervals and whenever one extends the time interval one can see the different stochastic natures.

7.2.1.4 Conclusions This part contributes some meaningful results on controllability for second order stochastic nonlinear delay systems driven by the Rosenblatt process with per-

306 Stability and Controls Analysis for Delay Systems

FIGURE 7.2

The trajectory of system (7.31) steers from the initial state.

FIGURE 7.3

The trajectory of system (7.31) steers from the initial state.

FIGURE 7.4

The trajectory of system (7.31) steers from the initial state.

turbation matrices in finite-dimensional settings. Several sufficient conditions are established for controllability of the stochastic nonlinear system using fixed point theory, delayed Grammian matrices, and appropriate local conditions on the nonlinear term. The results in this part are motivated from [135].

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Index

A Admissible controls, 269 Asymptotical stability, 3, 105, 113

B Banach space, 6, 60, 106, 155, 201, 269, 289, 291 Brownian motion, 290 fractional, 289, 290 standard, 289

C Caputo fractional derivative, 178 differentiated expression, 162, 163 differentiation, 165 Cauchy matrix impulsive delayed, 106, 107, 118, 154 problem, 2, 22, 74, 75, 222, 239 problem nonhomogeneous initial, 252 Classical Grammian matrix, 31 Combustion chamber, 23, 40, 41 Commutative matrices, 252 Constant matrix, 155, 239 Continuous compact operator, 83 curve, 100 differentiable, 30 function, 46, 91, 102 linear/discrete delay systems, 1 operator, 38, 147, 150, 188, 190

Continuous/discrete delayed exponential matrices, 1 Control function, 2, 29, 31, 75, 90, 140, 141, 177, 182, 221, 222, 225, 227, 228, 230, 233, 272, 290, 291 input, 46, 76, 87, 240, 254 methods for delay differential systems, 2 problems, 1, 75 systems, 1, 246 Controllability, 1, 2, 28, 41, 74, 76, 79, 86, 140, 195, 210, 221, 269, 295 for Caputo type, 155 impulsive delay systems, 3 linear discrete delay systems, 221 oscillating systems, 288 Grammian matrix, 270, 285, 291, 292, 302 matrices, 271 method, 45 operator, 277 problem, 29, 90, 177 results, 194, 283, 288 Controlled trajectories, 288 Convergence analysis, 48, 51, 94, 97, 242, 243, 254

D Delay Cauchy matrix impulsive, 140 controlled system impulsive, 154 linear, 31, 154 313

314 Index

linear discrete, 254 linear impulsive, 140 difference equations, 223 differential controlled systems, 2 equations, 1, 24, 26, 31, 41, 69, 74, 252 systems, 2, 3, 24, 33, 90, 139 differential/discrete equations, 1 Grammian matrix, 31, 33, 42, 45, 76, 78, 79, 86, 90, 151, 182, 303 matrix functions, 106, 193 oscillating equations, 75 oscillating systems controllability, 74 perturbation, 176, 177 system, 1–3, 5, 23, 69, 177, 266 impulsive, 1, 139 linear impulsive, 115 linear stochastic, 271, 294 term, 5 time, 155 Delayed birthrates, 31, 41, 252 cosine, 64 cosine function, 63 exponential matrix, 5, 6, 15, 17, 46, 156 approach, 28 function, 3, 274 fractional Grammian matrix criterion, 212 Grammian matrix, 216, 219, 306 logistic, 31, 41, 252 matrix, 74, 90 cosine, 59, 60, 90 exponential, 2, 16, 29, 45, 47, 51, 176, 269 function, 85 sine, 60, 90 Mittag-Leffler-type matrix, 164, 170 matrix function, 156 perturbation of matrices, 269, 283 sine, 64 sine function, 63 system, 91 Deterministic semilinear systems, 283 Diagonal matrices, 252 Difference delay systems, 3

Differential controlled systems, 3 Differential equations, 17, 71, 99 delay, 1, 24, 26, 31, 41, 69, 74 fractional delay, 155, 164 impulsive delay, 106 linear delay, 76, 106 Discontinuous trajectory, 101 Discrete controlled systems, 239, 240 delay system, 1, 261 delayed matrix exponential, 230 matrix delayed exponential, 1, 221 matrix delayed exponential function, 240, 252, 262 matrix function, 221 Dynamical systems, 1, 2

E Error bounds, 7 Euclidean distance, 263 input, 255 state, 255 Euclidean distance, 263 distance error, 263 matrix norm, 255, 258 norm, 6, 24, 41, 256 scalar product, 33 vector norm, 155 Expectation operator, 259 Exponential stability, 2

F Finite stability time, 14, 15 Finite time interval, 1, 2, 175 stability, 1–3, 5, 16, 21, 23, 25, 59, 68, 71, 119, 139, 155, 171, 175, 176, 195, 201, 210 analysis, 17 concept, 1 for Caputo type, 155 for linear delay differential, 5 for linear systems, 201 for semilinear delay differential, 16

Index 315

problems for delay systems, 5 results, 5, 7, 12, 16, 17, 20, 64, 74, 124, 170, 176 results for Caputo type, 170 results for linear systems, 64 stable, 2, 6, 7, 9, 10, 12, 13, 16, 17, 19–22, 25, 27, 62, 64, 67–69, 124–136, 170–174, 201, 205–208 Fixed trial lengths, 239 Formal matrix series, 60 Fractional Brownian motion, 289, 290 delay differential equations, 155, 164 Grammian matrix, 183, 191, 193 systems, 1, 3, 162, 178 differential systems, 3, 23 integral, 155 systems, 2, 3, 210 Function continuous, 46, 91, 102 control, 2, 29, 31, 75, 90, 140, 141, 177, 182, 221, 222, 225, 227, 228, 230, 233, 272, 290, 291 delayed exponential matrix, 3 vector, 59, 74, 155, 164 Fundamental delay matrix, 16 matrix, 1, 2, 6, 17, 30, 107, 120, 252 matrix methods, 2

G G-Brownian motion, 283 Grammian matrix, 29, 30, 151, 273 classical, 31 controllability, 285, 292, 302 criterion, 29 delay, 31, 33, 42, 45, 76, 78, 79, 86, 90, 151, 182, 303 delayed, 216, 219 fractional delay, 191, 193 impulsive delay, 141, 154 method, 30

Gronwall inequality, 5, 15, 17, 18, 70, 74, 105, 116, 127, 134, 139

I Impulsive condition, 125, 131 condition linear, 119 delay Cauchy matrix, 140 controlled systems, 154 differential equations, 106 differential systems controllability, 140 fundamental matrices, 3 Grammian matrix, 141, 142, 154 matrix functions, 106 solutions, 138 systems, 1, 139 delayed Cauchy matrix, 106, 107, 118, 154 matrix exponential, 139 matrix function, 118 points, 113, 119 semilinear delay differential systems, 140 Input error, 255 Integral equation, 167 fractional, 155 part, 61 Riemann–Liouville fractional, 195 term, 87, 218 Interval finite time, 1, 2, 175 Inverse operator, 33, 142, 184 Iteration error, 100–102, 242 Iterative learning control (ILC), 1, 3, 45, 46, 54, 55, 90, 239, 252 algorithm, 51 convergence, 51 design, 3, 242, 254 for delay differential systems, 45 for linear discrete delay systems, 252 for linear systems, 239 law, 49, 51, 97 problem, 3, 90, 239, 240, 251, 254 strategies, 57 theory, 3 updating law, 45, 90, 100, 255

316 Index

L Learning gain matrices, 45 parameter matrices, 242 Learning law, 240, 246, 249, 261, 262, 265 Linear bounded operators, 270, 291 combination, 230, 234 continuous delay systems, 33 continuous systems, 15 controllability operator, 270, 291 controlled systems, 29, 45, 194 delay continuous systems, 15, 16, 28 controlled system, 31, 154 differential controlled system, 86 differential equations, 76, 106 differential/discrete equations, 1 systems, 14, 119, 138, 219 systems controllability, 76 difference equations, 222, 239 difference systems, 252 differential delay systems, 1 equation, 2, 59, 74 systems, 2, 29, 31, 141, 182 differential/difference controlled systems, 2 discrete delay controlled system, 254 delay systems, 33 delayed systems, 251 pure delay systems, 2 systems, 2, 240 equation, 231 fractional delay controlled system, 181 delay differential systems, 195 ODEs, 162, 164 growth conditions, 28 homogeneous delay differential system, 5 impulsive conditions, 119 delay controlled system, 140 delay differential equations, 118

delay differential systems, 106 delay system, 115 homogeneous delay systems, 110 nonhomogeneous delay differential equations, 110 mapping, 223, 224 ODEs solution, 1 operator, 227, 228 parts, 2, 29, 139, 302 space, 227, 237 span, 223 stochastic control systems, 290 delay system, 271, 294 function, 271 subspace, 221, 228, 239 systems, 5, 33, 46, 151, 177, 291, 293, 303, 304 relative controllability, 140 stable manifolds, 2 transformation, 229 Linear matrix inequality (LMI), 2, 5, 16 Linearly independent constant vectors, 234 elements, 212, 226, 271, 277 final vectors, 232 rows, 234 vectors, 236 Lozinski matrix norm, 7 Lyapunov function, 16 function methods, 5 functionals, 2 method, 1

M Mathematical induction, 50, 108, 109, 113, 124, 158, 162, 195, 197 principle, 110 Matrix commutative, 252 constant, 155, 239 controllability, 271 cosine function, 60 delayed, 74, 90 delayed exponential functions, 251 diagonal, 252

Index 317

exponential, 2, 57, 139, 221, 223, 226, 237 functions, 60, 176, 177, 212, 235, 271, 277 functions delay, 106 linear combination, 276 logarithmic norm, 6 measure, 6, 7, 16 norm, 6, 16, 155, 241, 269, 289 normal, 252 notation, 228 permutable, 31, 41, 45, 139, 154, 177, 178, 252 permutation, 17, 269 perturbation, 283, 288 regular, 229 representation, 229 sine function, 60 Mild solution, 143, 184 Mittag-Leffler delay matrix function, 193 functions, 156 matrix, 156, 162, 195, 198, 201 matrix function, 156 Modern data-driven control, 3 Multiple control functions, 2

N Neutral differential equations, 40 Nonhomogeneous initial Cauchy problem, 252 initial function, 231 linear delay systems, 30 linear difference equations, 2 Nonlinear case, 115 delay differential controlled system, 88 differential equations finite time stability, 28 system, 25–28, 119 dynamical systems, 2 fractional delay differential equations, 167 stochastic delay system, 302 systems, 90, 151, 295

controllability, 79 finite time stability, 2 term, 23, 26, 28, 69, 90, 115, 125 Nonlinearity, 16, 24, 26, 74, 139 Nonpermutable matrices, 177 Nonsingular matrix, 59, 74, 91, 182, 193, 212, 231, 234, 288 Nonsingularity, 226 Normal matrices, 252 Null controllability, 269, 277, 278, 283, 288 for linear systems, 271 semilinear systems, 277 stochastic linear, 288 results, 273 stochastic delay systems, 269 Numerical examples, 13, 16, 23, 24, 40, 41, 53, 71, 86, 99, 138, 151, 174, 176, 190, 194, 209, 216, 236, 246, 261, 283, 302

O Operator continuous, 38, 147, 150, 188, 190 linear, 228 Optimal control, 3 Ordinary differential equation (ODE), 1, 7, 30, 162, 164, 240 Oscillating delay systems, 1, 3 delay systems finite time stability, 59 systems, 90

P Pairwise permutable matrices, 2, 29, 31 Partial derivative, 70 Permutable matrices, 31, 41, 45, 139, 154, 177, 178, 252 Permutation matrices, 17, 269 Perturbation matrices, 288 Positive symmetric constant matrix, 7 Prefixed integer, 239, 254 iteration, 5 Process control, 3

318 Index

R Regular matrix, 229 Relative controllability, 2, 28–30, 33, 44, 45, 75, 140, 153, 154, 178, 192, 193, 210, 212, 221–223, 225, 228, 232, 282 for delay differential, 28 linear delay differential systems, 31 linear discrete pure delay systems, 3 linear systems, 40, 212 semilinear, 33 results for Caputo type, 176 linear systems, 151, 181 semilinear systems, 183 system, 35 Representation matrix, 229 solution, 1–3, 5, 29, 45, 59, 110, 118, 119, 239, 240, 251 Riemann–Liouville derivative, 156, 195 differentiable vector function, 197 fractional derivative, 200 differentiate, 199 differentiation, 198 integral, 195 systems, 210 Rosenblatt distribution, 288, 289 process, 288–290, 305

S Semilinear delay controlled systems, 45 differential controlled system, 41 differential systems, 28, 33 systems, 288 fractional delay controlled system, 183 differential controlled system, 190 functions, 269, 288 impulsive delay differential controlled system, 151 stochastic systems, 283

systems, 279 systems relative controllability, 142 Single delay, 239, 240, 246 Solution for nonlinear system, 167 representation, 1–3, 5, 29, 45, 59, 110, 118, 119, 239, 240, 251 representation for linear systems, 162, 195 stability, 3, 68 Spectral matrix radius, 57 Stability analysis, 1 finite time, 1–3, 5, 16, 21, 23, 25, 59, 68, 71, 119, 139, 155, 171, 175, 176, 195, 201, 210 result, 114, 115 solution, 3, 68 time interval, 25–28 Stable finite time, 2, 6, 7, 9, 10, 12, 13, 16, 17, 19–22, 25, 27, 62, 64, 67–69, 124–136, 170–174, 201, 205–208 system, 1, 138, 177, 254 Standard Brownian motion, 289 State error, 255 feedback control, 23 interference, 23 system, 2, 28 vector, 288, 293 Steering control, 288, 305 Stochastic delay systems, 1, 3, 269, 278 linear oscillating delay systems controllability, 291 nonlinear controllability oscillating delay systems, 294 delay differential, 288 system, 306 oscillating delay systems controllability, 288 semilinear system, 277, 283, 288 variable, 252–255 Symmetric positive definite matrix, 7

Index 319

System state, 2, 28 trajectories boundedness, 1

T Terminal conditions, 76, 87 state, 221, 290 Tracking error, 54–57, 100, 101, 248, 250, 254, 255, 263–266 error trajectory, 53 performance, 101, 102, 248, 250, 251, 266

Transition matrix, 5, 15 Trial lengths, 239, 253, 254, 263, 266

U Ulam’s stability, 2 Uniform trial lengths, 3 Updating law, 255, 258

V Variable delay, 283 Vector function, 59, 74, 155, 164 representation, 228 state, 293

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