Control Theory and its Applications

E.O. Roxin The University of Rhode Island

USA

GORDON AND BREACH SCIENCE PUBLISHERS Australia • Canada. China • France • Germany • India. Japan • Luxembourg

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Contents

Introduction to the series xi

Preface xiii

Introduction xv

1 About Mathematical Modelling or What Should We Ask from Nature? . . . . 1

1.1 Historical background 1

1.2 Posing questions and problems 2

1.3 Systems with and without inputs 2 1.3.1 Isolated Systems 2 1.3.2 Systems with inputs 3

1.4 Dynamical Systems and control Systems 4 1.4.1 Dynamical Systems 4 1.4.2 Control Systems 5 1.4.3 Differential inclusions 6 1.4.4 Calculus of variations 7 1.4.5 Modern control theory 8 1.4.6 Differential games 8

2 General Properties of Control Systems 13

2.1 Definitions and examples 13 2.1.1 Problem setting 13 2.1.2 Caratheodory conditions 13

v

d CONTENTS

2.1.3 Admissable controls 14 2.1.4 Examples 15 2.1.5 Linear control Systems 17 2.1.6 The attainable set of the linear control System 18

2.2 The attainability relation 19 2.2.1 Autonomous Systems 19 2.2.2 Definitions 19 2.2.3 Propertiesof the attainability relation 21 2.2.4 Holding and transient sets 22 2.2.5 Equivalence classes 22

2.3 Linear Systems 23 2.3.1 Autonomous linear Systems 23 2.3.2 Examples of linear Systems 24 2.3.3 Systems in canonical form 28

2.4 Controllability 30 2.4.1 Linear Systems with unbounded controls 30 2.4.2 Necessary and sufficient condition 32 2.4.3 The controllability matrix 33

3 Optimal Control and Related Results 37

3.1 Optimal control 37 3.1.1 Bolza, Lagrange and Mayer problems 37 3.1.2 Pontryagin's maximum principle . . . 40 3.1.3 Maximum principle for linear autonomous Systems 41 3.1.4 Examples 42 3.1.5 The maximum principle in the general case 45

3.2 The existenceof optimal controls 47 3.2.1 A counterexample 47 3.2.2 The compactness of the attainable set 47 3.2.3 Cesari's "property Q" 48 3.2.4 Results for linear Systems 49

3.3 Invariant sets 50 3.3.1 Invariance in control Systems 50 3.3.2 Rest points 52

3.4 Stabilityof invariant sets 52 3.4.1 Stability for control Systems 52 3.4.2 Asymptotic stability 53 3.4.3 Finite stability 53

3.5 Attractors and repellers 54 3.5.1 Attractors 54 3.5.2 Repellers 55

CONTENTS vii

3.6 Viability 55

3.6.1 Viability kemel 55

4 Typical Behavior of Control Systems 59

4.1 Restpoints 59 4.1.1 Rest points of control Systems 59 4.1.2 Example 60

4.2 Attracting, repelling and saddle holding sets 63 4.2.1 Attracting holding sets 63 4.2.2 Repelling holding sets 63 4.2.3 Saddle holding sets 63

4.3 Examples 64 4.3.1 Linear Systems 64 4.3.2 Non-linear Systems 67

4.4 Periodic orbits and tubes of periodic orbits 73 4.4.1 Periodic orbits 73 4.4.2 Tubes of periodic orbits 73

4.5 The index for control Systems 73 4.5.1 The index in the plane 73 4.5.2 Index of a circuit with respect to a vector field 75 4.5.3 Index of a point with respect to a vector field 76 4.5.4 Index with respect to a control system 77

5 Conversion of Dynamical Systems into Control Systems 83

5.1 Additive control 83 5.1.1 Adding control to a dynamical system 83 5.1.2 "Spreading" control Systems 84 5.1.3 Random noise 84

5.2 Multiplicative control. Bilinear Systems 85 5.2.1 Bilinear control Systems 85 5.2.2 Solution of (5.9) for bang-bang controls 86 5.2.3 The multiplicative integral 86 5.2.4 The problem of the attainable set 87

5.3 Parameter control . 88 5.3.1 Direct parameter control 88 5.3.2 Examples 88 5.3.3 Indirect parameter control 90

5.4 Systems coupled by control 91 5.4.1 Coupling by control 91 5.4.2 Examples : 92

vüi CONTENTS

5.5 Expansion and fusion of holding sets 96 5.5.1 Natural ordering of control Systems 96 5.5.2 Examples 97

5.6 The controlled Lienard equation 101 5.6.1 The Lienard equation 101 5.6.2 The Cartwright-Littlewood equation 102 5.6.3 The controlled van der Pol equation 103

5.7 Control of chaotic Systems 106 5.7.1 Chaos in dynamical Systems 106 5.7.2 Control of discrete Systems 107 5.7.3 Effect of control on chaos 108

5.8 Control of Systems on manifolds 109 5.8.1 Dynamical Systems on manifolds 109 5.8.2 A control System on the sphere 109 5.8.3 A control System on the cylinder 111 5.8.4 Control Systems on the torus 114 5.8.5 Control Systems on the cone 117

6 Applications 121

6.1 The regulator problem 121 6.1.1 A thermostat problem 121 6.1.2 Chattering controls 124 6.1.3 The general problem ofregulation 125

6.2 Stabilization 126 6.2.1 Stable linear Systems 126 6.2.2 Stabilization by linear feedback 126 6.2.3 Region of stabilization 126

6.3 Population problems 127 6.3.1 Controllable exponential growth 127 6.3.2 The logistic equation 128

6.4 The problem of coexisting species 129 6.4.1 The "chemostat" 129 6.4.2 The mathematical model 129 6.4.3 Some approximations to the model 131 6.4.4 Mainresults 132 6.4.5 Selective growth and survival 133

6.5 Fishery management 133 6.5.1 Management of natural resources 133 6.5.2 The logistic equation in fisheries 134 6.5.3 Fishing effort and the Schaefer model 135 6.5.4 Economic considerations 136 6.5.5 Several species 137

CONTENTS ix

6.6 Predator-prey Systems 138 6.6.1 Predator-prey interaction. Lotka-Volterra model 138 6.6.2 Additive control for the Lotka-Volterra model 139 6.6.3 Parameter control of the Lotkta-Volterra model 140 6.6.4 Systems of related type 141

6.7 A problem of Cancer chemotherapy 142 6.7.1 Control Systems in medical treatment 142 6.7.2 Cancer chemotherapy. The Gompertzian model 142 6.7.3 The dynamics of the Gompertzian model 143 6.7.4 Other optimality considerations 147

Bibliographie References 151

Solutions to Problems and Exercises 157

Index 175