Applied Functional Analysis

Here is a revised and enlarged version of this useful reference text on functional analysis and its application to problems of system optimi- zation: filtering and control. Written for the needs of engineers and scientists, the emphasis is on providing the most useful material as opposed to the most general. It covers such topics as Volterra and Hilbert-Schmidt operators; convex programming in a Hilbert space; semigroup theoretic models for systems governed by P. D. E., con- trollability, and observability of infinite dimensional systems; and stochastic (Kalman) filtering and control via rigorous white noise the- ory, including (asymptotic) steady state theory for infinite dimen- sional systems. It also includes, for instance, the kind of background theory needed for rigorous treatment of active stabilization of flex- ible flight structures. From a review of the first edition "wealth of examples, ,A tour de force" -Bulletin of the American Mathematical Society Contents: Chapter 1 Basic Properties of Hilbert Spaces 1.0 Introduction 1.1 Basic Definitions 1.2 Examples of Hilbert Spaces 1.3 Hilbert Spaces from Hilbert Spaces 1.4 Convex Sets and Projections 1.5 Orthogonality and Orthonormal Bases 1.6 Continuous Linear Functionals 1.7 Riesz Representation Theorem 1.8 Weak Convergence 1.9 . Nonlinear Functionals and Generalized Curves 1.10 The Hahn-Banach Theorem Chapter 2 Convex Sets and Convex Programming 2.0 Introduction 2.1 Elementary Notions 2.2 Support Functional of a Convex Set 2.3 Minkowski Functional 2.4 The Support Mapping 2.5 Separation Theorem 2.6 Application to Convex Programming 2.7 Generalization to Infinite Dimensional Inequalities 2,8 A Fundamental Result of Game Theory: Minimax Theorem 2.9 Application: Theorem of Farkas Chapter 3 Functions, Transformations, Operators 3.0 Introduction 3.1 Linear Operators and their Adjoints 3.2 Speetral Theory of Operators 3.3 Spectral Theory of Compaet Operators 3.4 Operators on Separable Hilbert Spaces 3.5 Lz Spaces over Hilbert Spaces 3.6 Multilinear Forms 3.7 Nonlinear Volterra Operators Chapter 4 Semigroups of Linear Operators 4.0 Introduction 4.1 Definitions and General Properties of Semigroups 4.2 Generation of Semi groups 4,3 Semi groups over Hilbert Spaces: Dissipative Semi groups 4.4 Compact Semi groups 4.5 Analytic (Holomorphic) Semigroups 4.6 Elementary Examples of Semi groups 4.7 Extensions 4.8 Differential Equations: Cauchy Problem 4.9 Controllability 4.10 State Reduction: Observability 4.11 Stability and Stabilizability 4.12 Boundary Input: An Example 4.13 Evolution Equations Chapter 5 Optimal Control Theory 5.0 Introduction 5.1 Preliminaries 5.2 Linear Quadratic Regulator Problem 5.3 Linear Quadratic Regulator Problem: Infinite Time Interval 5.4 Hard Constraints 5,5 Final Value Control 5.6 Time Optimal Control Problem Chapter 6 Stochastic Optimization Theory 6.0 Introduction 6.1 Preliminaries 6.2 Measures on Cylinder Sets 6.3 Characteristie Functions and Countable Additivity 6.4 Weak Random Variables 6.5 Random Variables 6.6 White Noise 6.7 Differential Systems 6.8 The Filtering Problem 6.9 Stochastic Control 6.10 Physical Random Variables 6.11 Radon-Nikodym Derivatives 6.12 Nonlinear Stochastic Equations Bibliography Index