Optimal Control of Differential and Functional Equations

Optimal Control of Differential and Functional Equations presents a mathematical theory of deterministic optimal control, with emphasis on problems involving functional-integral equations and functional restrictions. The book reviews analytical foundations, and discusses deterministic optimal control problems requiring original, approximate, or relaxed solutions. Original solutions involve mathematicians, and approximate solutions concern engineers. Relaxed solutions yield a complete theory that encompasses both existence theorems and necessary conditions. The text also presents general optimal control problems, optimal control of ordinary differential equations, and different types of functional-integral equations. The book discusses control problems defined by equations in Banach spaces, the convex cost functionals, and the weak necessary conditions for an original minimum. The text illustrates a class of ordinary differential problems with examples, and explains some conflicting control problems with relaxed adverse controls, as well as conflicting control problems with hyper-relaxed adverse controls. The book is intended for mature mathematicians, graduate students in analysis, and practitioners of optimal control whose primary interests and training are in science or engineering.

Preface

Part One Foundations

Chapter I. Analytical Foundations

1.1 Sets, Functions, Sequences

1.2 Topology

1.3 Topological Vector Spaces

1.4 Measures, Measurable Functions, arid Integrals

1.5 The Banach Spaces C(S, ℋ) and Lp(S, Σ, μ, ℋ)

1.5.A The Metric Space C(S, X) and the Banach Space C(S, ℋ)

1.5.B The Space Lp(S, Σ, μ, ℋ)

1.5.C Special Spaces

1.6 Convex Sets

1.7 Measurable Set-Valued Mappings

Notes

Chapter II. Functional Equations

II.1 Definitions and Background

II.2 Brouwer's, Schauder's, and TychonofTs Fixed Point Theorems

II.3 Derivatives and the Implicit Function Theorem

11.4 Ordinary Differential Equations

11.5 Functional-Integral Equations in C(T, Rn)

11.6 Functional-Integral Equations in Lp(T, Rn)

Notes

Part Two Optimal Control

Chapter III. Basic Problems and Concepts, and Heuristic Considerations

111.1 The Subject of the Optimal Control Theory

111.2 Original, Approximate, and Relaxed Solutions

111.3 Measure-Valued Control Functions

111.4 Necessary Conditions for a Minimum

111.5 Minimizing Original Solutions

Chapter IV. Original and Relaxed Control Functions

IV.0 Summary

IV.1 The Spaces C(R) and L1(T, C(R)) and Their Conjugate Spaces

IV.2 The Sets ℛ and ℓ

IV.3 The Sets ℛ# and ℓ# and Abundant Sets

Notes

Chapter V. Control Problems Defined by Equations in Banach Spaces

V.0 Formulation of the Optimal Control Problem

V.1 Existence of Minimizing Relaxed and Approximate Solutions

V.2 Necessary Conditions for a Relaxed Minimum

V.3 Necessary Conditions for an Original Minimum

V.4 Convex Cost Functionals

V.5 Weak Necessary Conditions for an Original Minimum

V.6 An Illustration—A Class of Ordinary Differential Problems and Examples

V.7 State-Dependent Controls

Notes

Chapter VI. Optimal Control of Ordinary Differential Equations

VI.0 Formulation of the "Standard" Problem

VI. 1 Existence of Minimizing Relaxed and Approximate Solutions

VI.2 Necessary Conditions for a Minimum

VI.3 Contingent Equations and Equivalent Control Functions

VI.4 Unbounded Contingent Sets and Compactified Parametric Problems

VI.5 Variable Initial Conditions, Free Time, Infinite Time, Staging, Advance-Delay Differential Problems

Notes

Chapter VII. Optimal Control of Functional-Integral Equations in C(T, Rn)

VII.0 Formulation of the Problem

VII.1 Existence of Minimizing Solutions

VII.2 Necessary Conditions for a Relaxed Minimum

VII.3 Necessary Conditions for a Relaxed Minimum in Unilateral and Related Problems

VII.4 Necessary Conditions for an Original Minimum

VII.5 Problems with Pseudodelays

Notes

Chapter VIII. Optimal Control of Functional-Integral Equations in Lp( T, Rn)

VIII.0 Formulation of the Problem

VIII.1 Existence of Minimizing Solutions

VIII.2 Necessary Conditions for a Relaxed Minimum

VIII.3 Necessary Conditions for an Original Minimum

VIII.4 Problems with Pseudodelays

Notes

Chapter IX. Conflicting Control Problems with Relaxed Adverse Controls

IX.0 Formulation of the Problem

IX.1 Existence and Necessary Conditions for Optimal Controls

IX.2 Conflicting Control Problems Defined by Functional Equations. Additively Coupled Conflicting Controls. A Counterexample

IX.3 An Evasion Problem

IX.4 Zero-Sum Games with Control Strategies

Notes

Chapter X. Conflicting Control Problems with Hyperrelaxed Adverse Controls

X.0 Formulation of the Problem

X.1 Existence of Minimizing Relaxed and Approximate Controls

X.2 Necessary Conditions for a Relaxed Minimum

X.3 Hyperrelaxed and Relaxed Adverse Controls in Ordinary Differential Equations

Notes

References

Notation Index

Subject Index