Introduction to the Theory and Applications of Functional

Differential Equations

Mathematics and Its Applications

Managing Editor:

M.HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 463

Introduction to the Theory and Applications of Functional Differential Equations

by

V. Kolrnanovskii Moscow State University of Electronics and Mathematics and Space Research Institute (!Kl) of the Russian Academy of Sciences, Moscow, Russia

and

A. Myshkis Moscow State University of Communications (Ml/T), Moscow, Russia

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5148-6 ISBN 978-94-017-1965-0 (eBook) DOI 10.1007/978-94-017-1965-0

Printed on acid-free paper

All Rights Reserved © 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

Dedicated to the memory of the outstanding mathematician,

our friend and teacher Professor Mark Alexandrovich Krasnoselskii

1920 - 1997

Preface

Contents

Part I.

MODELLING BY FUNCTIONAL DIFFERENTIAL EQUATIONS

1

Chapter 1. Theoretical preliminaries 11

1. Functional differential equations (FOEs) 11

1.1 Some cla~ses of FOEs 11 1.2 Solution concept for a FOE 12

1.3 FDE with retardation 14

1.4 A little bit of philosophy 19

Chapter 2. Models 23

1. Viscoelastici t.v 23

2. Aftereffect in mechanics 25

2.1 Motion of a particle in a liquid 25

2.2 Controlled motion of a rigid body 26

2.3 Models of polymer crystallization 28

2.4 Stretchinl!; of a polymer filament 28

3. Hereditary phenomena in physics 30

3.1 Dynamic~ of oscillation 30 3.2 Relativistic dynamics 30

3.3 Nuclear reactors 31

3.4 Distribut~'d networks (long line with tunnel diode) :32 3.5 Heat flow in materials with memory 34 3.6 Models of lasers 35 3. 7 Neural network 35

4. Models with delays in technical problems 36

4.1 Infeed grinding and cutting 36

4.2 Technological delay 38 4.3 Car chasing 39 4.4 Ship course stabilization 39

4.5 Process of combustion in small rockets 39

4.6 Delay- differential equations in engineering applications 40

5. Aftereffect in biology 61

5.1 Evolution equations of a single species 61

viii

5.2 Interaction of two species 65 5.3 Population dynamics model of N interacting species 66 5.4 Coexistence of competitive micro- organisms 67 5.5 Control problems in ecology 67 5.6 Control problems in microbiology ( chemostat models) 68 5. 7 Nicholson blowflies model 70 5.8 Helical movement of tips of growing plants 70 5.9 Grazing system 70

6. Aftereffect in medicine 71

6.1 Mathematical models of the sugar quantity in blood 71 6.2 Model of arterial blood pressure regulation 72 6.3 Cancer chemotherapy 74

6.4 Mathematical models of learning 74

6.5 Mathematical models in immunology and epidemiology 75 6.6 Model of the human immunodeficiency virus (HIV)

epidemic 75 6. 7 Model of survival of red blood cells 78 6.8 Vision process in the compound eye 78 6.9 Human respiratory system 78 6.10 Regulation of glucose- insulin system 79 6.11 A disease transmission model 79

7. Aftereffect in economy and other sciences 80

7.1 Optimal skill with retarded controls 80 7.2 Optimal ildvertising policies 81 7.3 Commodity price fluctuations 82 7.4 Model of the fishing process 82 7.5 River pollution control 83 7.6 Control of financial management 83

Part II.

THEORETICAL BACKGROUND OF FUNCTIONAL DIFFERENTIAL EQUATIONS

Chapter 3. General theory 87

1. Introduction. Method of steps 87

1.1 Notation 87 1.2 Cauchy problem for FDEs 88 1.3 Steps method for FDEs of retarded type (RDEs) 89

ix

1.4 Steps methods for FDEs of neutral type (NDEs) 91 1.5 Problem for a process with aftereffect renewal 92

2. Cauchy problem for RDEs 94

2.1 Basic solvability theorem 94 2.2 Variants 96 2.3 Semigroup relation 98 2.4 Absolutely continuous solutions 100 2.5 RDEs with infinite delay 101 2.6 Features of the Cauchy problem for RDEs 104

3. Cauchy prob!Pm for NDEs 107 3.1 Smooth solutions 107 3.2 NDEs with a functional of integral type 111 3.3 Application of the steps method 114 3.4 Transitio11 to the operator equation 116 3.5 Hale's form of NDEs 119

4. Differential inclusions of retarded type (RDis) 120

4.1 Introduction 120 4.2 Multimaps 121 4.3 Solvability of the Cauchy problem for RDis 122 4.4 Generalized solutions of RDEs and RDis 126

5. General linear FOEs with aftereffect 131 5.1 Cauchy problem for linear RDEs 1:n 5. 2 Generalization 136 5.3 Integral n·presentation for the solution of the

Cauchy problem (variation of constants formula) 139 5.4 Adjoint equation. Periodic solutions 141 5.5 Linear NDEs 142 5.6 Simplest uonautonomous RDEs of the first and

second orrlers 146 6. Linear autonomous FDEs 163

6.1 Exponential solutions of linear autonomous RDEs 163 6.2 Solution of the Cauchy problem 167 6.3 Example of a showering person 170 6.4 Linear autonomous NOEs 174

7. Hopf bifurcation of FDEs 176 7.1 Introduction 176

X

7.2 Example 177 7.3 General r<tse 182

7.4 Variants 186

7.5 Example of an RDE with constant delay:

intraspecific struggle for a common food 187

7.6 Example of an RDE with autoregulative delay:

combustion in the chamber of a turbojet engine 189 7. 7 Example NDE: auto-oscillation in a long

line with tunnel diod 191

8. Stochastic retarded differential equations (SRDEs) 191

8.1 Initial value problem 192 8.2 Existence and uniqueness of solution 194 8.3 Some characteristics of solutions of linear equations 195

Part III.

STABILITY

Chapter 4. Stability of retarded differential equations 199

1. Liapunov's direct method 199

1.1 Stability definitions 199 1.2 Stability theorems for equations with bounded delay 204 1.3 Stability nf equations with unbounded delay 211

1.4 Stability of linear nonautonomous RDEs 216

1.5 Stability of linear periodic RDEs 217 1.6 Application of comparison theorems 222 1.7 Stability in the first approximation 223 1.8 Case of non- positive derivative 224

2. Linear autonomous RDEs 226

2.1 Laplace transformation 226 2.2 Stability conditions 228

3. Stability investigation methods for linear autonomous RDEs 232

3.1 Introduction 232

3.2 Mikhailov criterium 232 3.3 Scalar n- th order equations 233

3.4 Equations with discrete delays 236

4. Razumikhin 's method 247

4.1 Introduction 247

xi

4.2 Guiding functions for systems without delay 248

4.3 Guiding fnnctionals for RDEs 249

4.4 Direct application of Liapunov functions to RDEs 2.51

4.5 The main idea of B.S. Razumikhin 251

4.6 " Impossibility of the first breakdown" 252

4. 7 Precize formulation 253

4.8 Connection between Razumikhin 's method and Liapunov functionals 2.54

4.9 Asymptotic stability 2.54

4.10 Refinement of estimations 2.SEi

4.11 Example 2.55

4.12 Transform at ion of RD Es 2.58

4.13 Other applications of Razumikhin's method 260

Chapter 5. Stability of RDEs with autonomous linear part 263

1. Notations 263

2. L2 - stability 266

3. Estimates for the Green function 270

4. A bound for a region of attraction 274

Chapter 6. Liapunov functionals for concrete FDEs 279

1. Statement of t he problem 279

2. Formal description of the procedure 280

3. Dissipative systems 284

3.1 Stability 284 3.2 Exponential contractivity 291

4. Stability in the first approximation 293

4.1 Exponentially stable linear part 294 4.2 Smooth coefficients 294

5. Scalar RDEs 296

5.1 Scalar equations of n- th order 296 5.2 Scalar eq11ations of second order 299 5.3 Stability of chemostat 304

Chapter 7. Riccati type stability conditions of some linear systems with delay 307

1. Introduction 307

2. Special case 309

xii

2.1 The stability condition for this case

2.2 An application of a form of NDE

2.3 One morE> stability condition

3. Discrete delay -independent stability conditions

4. Delay- dependent stability conditions for equations with

309

310

312

313

discrete delay~ 316

4.1 The stability condition 317

4.2 An application of a form of NDE 318

4 .3 One morf' stability condition 319

5. Distributed delay :l21

5.1 The stability condition 322

5.2 An application of a form of NDE 323

5.3 One more stability condition 324

Chapter 8. Stability of neutral type functional differential equations 329

1. Direct Liapunov's method 329

1.1 Degenerate Liapunov functionals 329

1.2 Stability in a first approximation 335

1.3 The use of functionals depending on derivatives 336

1.4 Instability of NDEs 337

2. Stability of linear NDEs 34::!

2.1 Linear autonomous NDEs 343

2.2 Scalar NDEs 346

2.3 Stability of NDEs with discrete delays 349

2.4 The influence of small delays on stability 351

2.5 Linear inhomogeneous NDEs 3.52

2.6 Boundedness of derivatives for linear NDEs :l.52

2. 7 Boundedness of derivatives for nonlinear NDEs 353

2.8 Linear periodic NDEs 355

Chapter 9. Application of the direct Liapunov method 359

1. Description of the procedure 359

2. Scalar N DEs nf n- th order 362

3. Linear NDEs 367

3.1 The stability condition 367

3.2 Another ~tability condition 368

3.3 The summarizing result

4. Nonlinear NDEs

5. Stability of the second order NDEs

6. An illustratiVP example for dimension n = 3

7. Matrix Riccati equations in stability of NDEs

Chapter 10. Stability of stochastic functional differential

xiii

370

371

375

380

386

equations 387

1. Statement of the problem 387

1.1 Definition s of stability 387

1.2 Ito's formula 389

2. Liapunov's direct method 389

2.1 Asymptotic stability 389

2.2 Examples 390 2.3 Exponential stability 394

2.4 Stability in the first approximation 395

2.5 Stability under persistent disturbances 396

3. Roundedness of moments of solutions 397

3.1 General conditions for boundedness of moments 397

3.2 Scalar SRDE 398

3.3 Second order SRDE 401

4. Construction of Liapunov functionals for SNDEs 402

4.1 Statement of the problem 402

4.2 Description of the procedure 404

4.3 Scalar SN DE 405 4.4 Nonlinear example 407

5. Riccati matrix equations in stability of linear SRDEs 415

Part IV.

BOUNDARY VALUE PROBLEMS AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS

Chapter 11. Boundary value problems for functional differential equations 443

1. Boundary value problems for FDEs of evolutionary type 443

1.1 Introduction 443

1.2 Problems with a finite defect 443

1.3 Halanay 's boundary value problem 446

xiv

1.4 Periodic problem 448

2. Boundary value problems for FOEs of nonevolutionary type 449

2.1 FOEs with unique principal term 450

2.2 FOEs with nonunique principal term 453

Chapter 12. Fredholm alternative for periodic solutions of linear FOEs 4.59

1. Existence of periodic solutions 4.59

1.1 Statement of the problem 459

1.2 Conditions of the Fredholm alternative validity 461

2. Connection bPtween boundedness and periodicity 467

3. Periodic solution of linear difference equations (DcEs) 468

3.1 Statement of the problem 468 3.2 Stationary case. Commensurable shifts of the argument 468

3.3 Arbitrary delays 470

3.4 Variable coefficients , delays divisible by a period 4 71 3.5 Variable coefficients , delays commensurable

with a period

3.6 Variable coefficients, arbitrary constant delays

4. NOEs with small nonlinearities

5. Periodic solutions of autonomous FOEs with small

parameter

Chapter 13. Generalized periodic solutions of Functional Differential Equations

1. Some prerequisites

2. Conditions of existence of periodic solutions

3. Relation between stability and periodicity

3.1 Application of the direct Liapunov method

3.2 Stability of periodic solutions

4. Periodic solutions of concrete classes of equations

4.1 The case of quasilinear deterministic equation

4.2 Linear equations

5. Periodic solutions of the Ito's SFDEs

5.1 Existence of periodic solutions

5.2 Scalar SRDEs

5.3 Method of Liapunov functionals

5.4 Uniqueness of periodic solutions

474

478

479

481

489

489

491

494

494

498

499

499

502

.503

503

506

510

517

Part V.

CONTROL AND ESTIMATION IN HEREDITARY

SYSTEMS

XV

Chapter 14. Problems of control for deterministic f'DEs 523

1. The dynamic programming method for deterministic

RD Es. Bellman's equation 52:3

1.1 Statement of the problem 523

1.2 Optimality conditions .525

2. Linear quadratic problems 526

2.1 Optimal control synthesis 526

2.2 Exact sol11tion 528

2.3 Systems with delays in the control 529

2.4 Effects of delays in regulators 532

2.5 NDE 533

3. Optimal control of bilinear hereditary systems 534

3.1 Optimality conditions 534

3.2 Construction of the optimal control synthesis 535

3.3 Model of optimal feedback control for microbial growth 538

4. Control prob]Pms with phase constraint formula 538

4.1 General optimality conditions 538

4.2 Equations with discrete delays 540

5. Necessary optimality conditions 543

5.1 Systems with state delays 543

5.2 Systems with delays in the control 545

5.3 Systems with distributed delays 546

5.4 Linear systems with discrete and distributed delays 547

5.5 Neutral t_ype systems 549

6. Adaptive control of FDEs .5.50

6.1 Scalar equations .5.50

6.2 Delay identification .5.53

6.3 Multidimensional systems 5.54

xvi

Chapter 15. Optimal control of stochastic delay systems 557

1. Dynamic programming method for controlled

stochastic hen"di tary processes

2. The linear quadratic problem

2.1 Bellman functional and optimal control

2.2 Approximate solution

2.3 Some generalizations

3. Approximate optimal control for equations with

557

.'558

.558

560 564

small parameters .'564

3.1 Formal algorithm 564

3.2 Quasilinear systems with quadratic cost 566

4. Another approach to the problem of

optimal synthesis control 568

4.1 Admissible functionals 568

4.2 Quasilinear quadratic problems 569

Chapter 16. State estimates of stochastic systems with delay 573

1. Filtering of Gaussian processes .'573

1.1 Problem statement 573 1.2 Integral representation for the optimal estimate 574 1.3 The fundamental filtering equation 57.5

1.4 Dual optimal control problem 578

1.5 Particular cases 580 1.6 Dependence of the error of the optimal estimate

on the delay .581 1. 7 Some generalizations 587

2. Filtering of solutions of Ito's equations with delay .'589

2.1 Problem statement .589 2.2 Dual control problem 590

3. Minimax filtering in systems with delay 592

3.1 Statement of the problem .592

3.2 Approximate solution 595

Bibliography 601 Index 643

PREFACE

At the beginning of this century Emil Picard wrote: "Les equations differentielles de la mecanique classique sont telles qu 'il

en resulte que le mouvement est determine par la simple connaissance des positions et des vitesses, c 'est-a-dire par l 'etat a un instant donne et a ['instant infiniment voison.

Les etats anterieurs n'y intervenant pas, l'heredite y est un vain mot. L 'application de ces equations ou le passe ne se distingue pas de l 'avenir, ou les mouvements sont de nature reversible, sont done inapplicables aux etres vivants".

"Nous pouvons rever d'equations fonctionnelles plus compliquees que les equations classiques parce qu 'elles renfermeront en outre des integrates prises entre un temps passe tres eloigne et le temps actuel, qui apporteront la part de l'heredite". (See "La mathematique dans ses rapports avec la physique, Actes du rv• congres international des Mathematiciens, Rome, 1908.)

Many years have passed since this publication. These years have seen substantial progress in many aspects of Functional Differential Equations (FDEs ). A distinguishing feature of the FDEs under consideration is that the evolution rate of the proc{lsses described by such equations depends on the past history. The discipline of FDEs has grown tremendously, and publication of literature has increased perhaps twofold over publication in the previous decade. Several new scientific journals have been introduced to absorb this increased productivity. These journals reflect the broadening interests of scientists, with ever greater attention being paid to applications.

In the middle 1980s, we wrote a book on applications of FDEs enti­tled "Applied theory of functional differential equations", published in the beginning 1992 by Kluwer Academic Publishers.

The first edition of the book have been used numerous time, mostly in teaching FDEs to students of applied mathematics and science, where the purpose was to provide an understanding of the FDEs techniques as well as practice in using them. The book should prove useful for professional applied mathematicians and scientists because of its emphasis on analysis of mathematical models using FDEs and their detailed investigation. Stu­dents, colleagues and other users have provided us with many constructive comments and proposals to make the future edition both better medium for prospective users of FDEs and better for classroom instruction or self­study.

At the time of writing of the first edition, the field of FDEs is making significant breakthroughs in its practice which are no longer only a special­ist's field. The number of FDEs used in the sciences and in applied areas has been increasing tremendously, where they continue to play a crucial

2 PREFACE

role. Applications range from physiology and enzyme kinetics to whaling control and food webs, from neural networks to lazer optics , and from stud­ies of engines to the theory of business cycles. Moreover as our activities get more and more automated, the importance of taking into consideration delay becomes evident as increased numbers of large industrial equipment such as engines, blowers, fans, transformers, and compressors are in use.

The present book is intended as a replacement of the previous one, and is by no means a revision of it . We tried to write a new book for those wishing to learn about FDEs with little or no prior knowledge and lead them from the simplest to the most complicated subjects in simple and understandable way. Yet we never forget the readers with a main interest in applications.

A considerable amount of thought, feedback, and effort have gone in­to the writing and production of the new book which represents a wide­ranging interdisciplinary study of FDEs.

In spite of the general topic remains the same, the overlap between two books has turned out to be very small indeed. But the intended readership is precisely the same, namely postgraduate and advanced undergraduate students and various users of FDEs. Moreover we note that the book will serve as a basic course in FDEs theory for students of applied mathematics, engineering and physical sciences.

Emphasis in the subject of FDEs has changed substantially during last years. New methods in stability theory, optimal control, optimal estimation, periodic solutions were being proposed. Along with this huge variety of new applications has emerged.

Yet, many recent results are inaccessible and have never brought to­gether. For this reason, in this book we have tried to reflect those changes and to bring coverage of literature up to date. So some parts of the old book were dropped and replaced by newer material, to the current situa­tion, that updates the subject relative to the latest developments. Problems of implementation of general methods are considered in much more detail. Great care is done to include topics that can be used in everyday practice and make the book more effective as a textbook.

Examples are interspersed with the text at the appropriate sections. In the main, they are intended to illustrate and to extend the text.

Studying a smaller number of methods in greater depth tends to raise the level of difficulty. In order to avoid this we are giving sometimes non­rigourous accounts where appropriate. If rigour is essential, we have quoted theorems, and supplied proofs especially if they are constructive.

Every efforts were done to explain basic ideas with the minimal math­ematical equipment and to appeal to ones intuition along with ones math­ematical skills.

PREFACE 3

The approach of the book is similar to that of the classical applied mathematics, whose genuine interest in real world problems does not de­tract from his delight in watching mathematics at work.

Keeping in mind a special applied flavour of the book we tried sometimes avoid the strict axiomatic approach, but there has been no dilution of the standard of mathematical argument. Learning to follow and construct a logical sequence of ideas is one of the most important feature of this book and at the same time to convey some notion of the connection between a mathematical model and the real world.

We hope that the book provide a readable account of the newest devel­opment of FD Es theory.

The preparation of the new book presented an opportunity to improve the old one by rewriting many sections, by eliminating errors and misprints, by strengthening the part devoted to theoretical developments of FDEs and their applications.

Tn particular we have included new material on:

-phenomena modelled by FDEs, - proofs of all main theorems, - boundary value problems, - formal procedure for Liapunov functionals construction for different

classes of FDEs, - non autonomous equations of first and second orders, - periodic solutions of FDEs, - periodicity in the theory of stochastic FDEs, - minimax filtering and estimation, - model reference adaptive control, - regularity properties of solutions of stochastic FDEs,

to name a few. Most of the material included in the book is acceptable to any reader

with a reasonable background in real analysis, and some acquaintance with differential equations and probability theory.

There are several sections which require more sophisticated knowledge. In such cases adequate references are provided. A detailed table of con­tents should enable a reader to fairly quickly navigate through the book in searching for desired topics.

Chapters 1,2 are rather elementary, providing an easy introduction to the subject. They contain introductory sections on a number of topics. Al­though some readers would like to jump right over this elementary chapter, they might find it convenient to refer back to it to put later parts'of the book into context. Detailed description of some new phenomena. modelled by FDEs is given here.

4 PREFACE

The main purpose of Ch. 2 is to describe various ways in which FDEs arise as models in various applications. One of the most important tasks in the analysis of systems is mathematical modeling of the systems. Modeling of nonlinear systems is usually quite complex. Therefore one often has the task of determining not only how to accurately describe a system math­ematically, but , more importantly, how to make proper assumptions and approximations, whenever necessary, so that the system may be adequately characterized by a simple mathematical model.

We are sure the topics featured in this chapter will convince the reader that FDEs constitute a very useful tool in contemporary mathematical modeling theory. Moreover, it is shown that sometimes delays do really exist in the laws of Nature (out there waiting to be discovered), and sometimes they are enacted by ourselves to guide us through the jungle of experience. It is becoming increasingly clear that FDEs allow us to improve essentially the clarity of mathematically complex models. As a result they become understandable to those who are not directly involved with modelling, thus enhancing the usefulness of these models.

We also demonstrate that ignoring effects that are adequately repre­sented by functional differential equations is not an alternative because it can lead to wrong (and potentially disastrous) consequences in terms of stability analysis and control design.

For example, engine cycle delays and transport delays are, typically, included in mean- value models of automotive internal combustion engines. These delays are time-variable because they depend on the engine speed that can vary. The implications of these delays for stability analysis and control synthesis of advanced technology internal combustion engines can be described for several problems such as the idle speed control problem and the closed loop air-fuel ratio control problem.

Modern general theory of FDEs is considered in Ch. 3, where all main theorems are equipped with full proofs which are quite instructive and are thus outlined in detail. Various techniques for qualitative investigation have been considered at appropriate places. Theoretical developments and mathematical treatment of the principles involved are included as needed for understanding of the techniques presented.

More intensive treatment of these subjects should be obtained, if de­sired, from many references available.

Stability theory is treated in chapters 4-10. The most important meth­ods and techniques used in applications (e.g., Laplace transformation me­thod, direct Liapunov method, comparison theorems, etc.) are included in these chapters. In particular, formal procedure to construct Liapunov func­tionals for various classes of FDEs is proposed and illustrated by numerous

PREFACE 5

examples as well as applications of differential inequalities for stability in­vestigation. New stability conditions are given for particular FDEs.

Chapters 11-13 are devoted to periodicity of FDEs and boundary value problems. Problems of existence of periodic solutions for retarded, neutral and stochastic FDEs both linear and quasilinear are considered.

Chapters 14-16 are connected with optimal control and estimate prob­lems under incomplete state information. Minimax estimation problems and model reference adaptive control are considered in chapters 18 and 19. In particular, it is shown that purposeful introduction of delay into system can essentially improve its qualitative and quantitative characteristics and many phenomena, such as periodicity, oscillation, instability, etc., can be explained in terms of delay.

Chapters of the book are relatively independent of each other. It means that they can be read in any order. At the same time a beginner is en­countered to first read chapter 1 which is a simplified introduction in the subject.

Finally, the book contains an extensive bibliography of the subject, while it cannot be claimed to be complete. There many references to jour­nal papers, aimed at indicating the origination of the results, or contain supplementary material directly related to the text.

Two numbers are used for theorems and formulas inside a single chapter, and three numbers when referring to other chapters .

Preparation of the book was supported by Russian Fund for Fundamen­tal Researches; INTAS; Institut franco- russe A.M. Liapunov d'informatique et de mathematiques appliquees; International· Soros Science Education Program (ISSEP) .

Readers having comments and suggestions are invited to send these to us. Of course, full acknowledgments will be made.

We are grateful to many our students, associates and colleagues for their quite fruitful discussions and many useful suggestions for improvements, problem statements, consultation, and numerous changes made throughout the text to improve readability and presentation. But our decision was not to name them individually as an indication of how many collaborators we have had.