
Optimal Control of Differential and Functional Equations
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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.
Table of Contents
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
Product details
- No. of pages: 546
- Language: English
- Copyright: © Academic Press 1972
- Published: January 28, 1972
- Imprint: Academic Press
- eBook ISBN: 9781483259192
About the Author
J. Warga
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