Optimal Control of Differential and Functional Equations

Optimal Control of Differential and Functional Equations

1st Edition - January 28, 1972

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  • Author: J. Warga
  • eBook ISBN: 9781483259192

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Description

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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