Optimal Control of Differential and Functional Equations

PrefacePart 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) NotesPart 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 NotesReferencesNotation IndexSubject Index