Control Theory and its Applications
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1 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 Malaysia The Netherlands. Russia. Singapore. Switzerland. Thailand United Kingdom
2 Contents Introduction to the series Preface Introduction xi xiii xv 1 About Mathematical Modelling or What Should We Ask from Nature? Historical background Posing questions and problems Systems with and without inputs Isolated Systems Systems with inputs Dynamical Systems and control Systems Dynamical Systems Control Systems Differential inclusions Calculus of variations Modern control theory Differential games 8 2 General Properties of Control Systems Definitions and examples Problem setting Caratheodory conditions 13 v
3 d CONTENTS Admissable controls Examples Linear control Systems The attainable set of the linear control System The attainability relation Autonomous Systems Definitions Propertiesof the attainability relation Holding and transient sets Equivalence classes Linear Systems Autonomous linear Systems Examples of linear Systems Systems in canonical form Controllability Linear Systems with unbounded controls Necessary and sufficient condition The controllability matrix 33 3 Optimal Control and Related Results Optimal control Bolza, Lagrange and Mayer problems Pontryagin's maximum principle Maximum principle for linear autonomous Systems Examples The maximum principle in the general case The existenceof optimal controls A counterexample The compactness of the attainable set Cesari's "property Q" Results for linear Systems Invariant sets Invariance in control Systems Rest points Stabilityof invariant sets Stability for control Systems Asymptotic stability Finite stability Attractors and repellers Attractors Repellers 55
4 CONTENTS vii 3.6 Viability Viability kemel 55 4 Typical Behavior of Control Systems Restpoints Rest points of control Systems Example Attracting, repelling and saddle holding sets Attracting holding sets Repelling holding sets Saddle holding sets Examples Linear Systems Non-linear Systems Periodic orbits and tubes of periodic orbits Periodic orbits Tubes of periodic orbits The index for control Systems The index in the plane Index of a circuit with respect to a vector field Index of a point with respect to a vector field Index with respect to a control system 77 5 Conversion of Dynamical Systems into Control Systems Additive control Adding control to a dynamical system "Spreading" control Systems Random noise Multiplicative control. Bilinear Systems Bilinear control Systems Solution of (5.9) for bang-bang controls The multiplicative integral The problem of the attainable set Parameter control Direct parameter control Examples Indirect parameter control Systems coupled by control Coupling by control Examples : 92
5 vüi CONTENTS 5.5 Expansion and fusion of holding sets Natural ordering of control Systems Examples The controlled Lienard equation The Lienard equation The Cartwright-Littlewood equation The controlled van der Pol equation Control of chaotic Systems Chaos in dynamical Systems Control of discrete Systems Effect of control on chaos Control of Systems on manifolds Dynamical Systems on manifolds A control System on the sphere A control System on the cylinder Control Systems on the torus Control Systems on the cone Applications The regulator problem A thermostat problem Chattering controls The general problem ofregulation Stabilization Stable linear Systems Stabilization by linear feedback Region of stabilization Population problems Controllable exponential growth The logistic equation The problem of coexisting species The "chemostat" The mathematical model Some approximations to the model Mainresults Selective growth and survival Fishery management Management of natural resources The logistic equation in fisheries Fishing effort and the Schaefer model Economic considerations Several species 137
6 CONTENTS ix 6.6 Predator-prey Systems Predator-prey interaction. Lotka-Volterra model Additive control for the Lotka-Volterra model Parameter control of the Lotkta-Volterra model Systems of related type A problem of Cancer chemotherapy Control Systems in medical treatment Cancer chemotherapy. The Gompertzian model The dynamics of the Gompertzian model Other optimality considerations 147 Bibliographie References 151 Solutions to Problems and Exercises 157 Index 175
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