Note: Intro -- Preface -- Acknowledgments -- Contents -- 1 Introduction -- 2 Preliminaries -- 3 Well-Posedness of the Vector-Valued BSEEs -- 4 Well-Posedness Result for the Operator-Valued BSEEs with Special Data -- 5 Sequential Banach-Alaoglu-Type Theorems in the Operator Version -- 6 Well-Posedness of the Operator-Valued BSEEs in the General Case -- 7 Some Properties of the Relaxed Transposition Solutions to the Operator-Valued BSEEs -- 8 Necessary Condition for Optimal Controls, the Case of Convex Control Domains -- 9 Necessary Condition for Optimal Controls, the Case of Non-convex Control Domains -- References.

Note: Intro -- Preface -- Contents -- 1 Introduction -- 1.1 Why Stochastic Distributed Parameter Control Systems? -- 1.2 Two Fundamental Issues in Control Theory -- 1.3 Range Inclusion and the Duality Argument -- 1.4 Two Basic Methods in This Book -- 2 Some Preliminaries in Stochastic Calculus -- 2.1 Measures and Probability, Measurable Functions and Random Variables -- 2.2 Integrals and Expectation -- 2.3 Signed/Vector Measures, Conditional Expectation -- 2.3.1 Signed Measures -- 2.3.2 Distribution, Density and Characteristic Functions -- 2.3.3 Vector Measures -- 2.3.4 Conditional Expectation -- 2.4 A Riesz-Type Representation Theorem -- 2.4.1 Proof of the Necessity for a Special Case -- 2.4.2 Proof of the Necessity for the General Case -- 2.4.3 Proof of the Sufficiency -- 2.5 A Sequential Banach-Alaoglu-Type Theorem in the Operator Version -- 2.6 Stochastic Processes -- 2.7 Stopping Times -- 2.8 Martingales -- 2.8.1 Real Valued Martingales -- 2.8.2 Vector-Valued Martingales -- 2.9 Brownian Motions -- 2.9.1 Brownian Motions in Finite Dimensions -- 2.9.2 Construction of Brownian Motions in one Dimension -- 2.9.3 Vector-Valued Brownian Motions -- 2.10 Stochastic Integrals -- 2.10.1 Itô's Integrals w.r.t. Brownian Motions in Finite Dimensions -- 2.10.2 Itô's Integrals w.r.t. Vector-Valued Brownian Motions -- 2.11 Properties of Stochastic Integrals -- 2.11.1 Itô's Formula for Itô's Processes (in a Strong Form) -- 2.11.2 Burkholder-Davis-Gundy Inequality -- 2.11.3 Stochastic Fubini Theorem -- 2.11.4 Itô's Formula for Itô's processes in a Weak Form -- 2.11.5 Martingale Representation Theorem -- 2.12 Notes and Comments -- 3 Stochastic Evolution Equations -- 3.1 Stochastic Evolution Equations in Finite Dimensions -- 3.2 Well-Posedness of Stochastic Evolution Equations -- 3.2.1 Notions of Solutions -- 3.2.2 Well-Posedness in the Sense of Mild Solution. , 3.3 Regularity of Mild Solutions to Stochastic Evolution Equations -- 3.3.1 Burkholder-Davis-Gundy Type Inequality and Time Regularity -- 3.3.2 Space Regularity -- 3.4 Notes and Comments -- 4 Backward Stochastic Evolution Equations -- 4.1 The Case of Finite Dimensions and Natural filtration -- 4.2 The Case of Infinite Dimensions -- 4.2.1 Notions of Solutions -- 4.2.2 Well-Posedness in the Sense of Mild Solution for the Case of Natural Filtration -- 4.3 The Case of General Filtration -- 4.4 The Case of Natural Filtration Revisited -- 4.5 Notes and Comments -- 5 Control Problems for Stochastic Distributed Parameter Systems -- 5.1 An Example of Controlled Stochastic Differential Equations -- 5.2 Control Systems Governed by Stochastic Partial Differential Equations -- 5.3 Some Control Problems for Stochastic Distributed Parameter Systems -- 5.4 Notes and Comments -- 6 Controllability for Stochastic Differential Equations in Finite Dimensions -- 6.1 The Control Systems With Controls in Both Drift and Diffusion Terms -- 6.2 Control System With a Control in the Drift Term -- 6.3 Lack of Robustness for Null/Approximate Controllability -- 6.4 Notes and Comments -- 7 Controllability for Stochastic Linear Evolution Equations -- 7.1 Formulation of the Problems -- 7.2 Well-Posedness of Stochastic Systems With Unbounded Control Operators -- 7.3 Reduction to the Observability of Dual Problems -- 7.4 Explicit Forms of Controls for the Controllability Problems -- 7.5 Relationship Between the Forward and the Backward Controllability -- 7.5.1 The Case of Bounded Control Operators -- 7.5.2 The Case of Unbounded Control Operators -- 7.6 Notes and Comments -- 8 Exact Controllability for Stochastic Transport Equations -- 8.1 Formulation of the Problem and the Main Result -- 8.2 Hidden Regularity and a Weighted Identity. , 8.3 Observability Estimate for Backward Stochastic Transport Equations -- 8.4 Notes and Comments -- 9 Controllability and Observability of Stochastic Parabolic Systems -- 9.1 Formulation of the Problems -- 9.2 Controllability of a Class of Stochastic Parabolic Systems -- 9.2.1 Preliminaries -- 9.2.2 Proof of the Null Controllability -- 9.2.3 Proof of the Approximate Controllability -- 9.3 Controllability of a Class of Stochastic Parabolic Systems by one Control -- 9.3.1 Proof of the Null Controllability Result -- 9.3.2 Proof of the Negative Null Controllability Result -- 9.4 Carleman Estimate for a Stochastic Parabolic-Like Operator -- 9.5 Observability Estimate for Stochastic Parabolic Equations -- 9.5.1 Global Carleman Estimate for Stochastic Parabolic Equations, I -- 9.5.2 Global Carleman Estimate for Stochastic Parabolic Equations, II -- 9.5.3 Proof of the Observability Result -- 9.6 Null and Approximate Controllability of Stochastic Parabolic Equations -- 9.6.1 Global Carleman Estimate for Backward Stochastic Parabolic Equations -- 9.6.2 Proof of the Observability Estimate for Backward Stochastic Parabolic Equations -- 9.7 Notes and Comments -- 10 Exact Controllability for a Refined Stochastic Wave Equation -- 10.1 Formulation of the Problem -- 10.2 Well-Posedness of Stochastic Wave Equations With Boundary Controls -- 10.3 Main Controllability Results -- 10.4 A Reduction of the Exact Controllability Problem -- 10.5 A Fundamental Identity for Stochastic Hyperbolic-Like Operators -- 10.6 Observability Estimate for the Stochastic Wave Equation -- 10.7 Notes and Comments -- 11 Exact Controllability for Stochastic Schrödinger Equations -- 11.1 Formulation of the Problem and the Main Result -- 11.2 Well-Posedness of the Control System -- 11.3 A Fundamental Identity for Stochastic Schrödinger-Like Operators. , 11.4 Observability Estimate for Backward Stochastic Schrödinger Equations -- 11.5 Notes and Comments -- 12 Pontryagin-Type Stochastic Maximum Principle and Beyond -- 12.1 Formulation of the Optimal Control Problem -- 12.2 The Case of Finite Dimensions -- 12.3 Necessary Condition for Optimal Controls for Convex Control Regions -- 12.4 Operator-Valued Backward Stochastic Evolution Equations -- 12.4.1 Notions of Solutions -- 12.4.2 Preliminaries -- 12.4.3 Proof of the Uniqueness Results -- 12.4.4 Well-Posedness Result for a Special Case -- 12.4.5 Proof of the Existence and Stability for the General Case -- 12.4.6 A Regularity Result -- 12.5 Pontryagin-Type Maximum Principle -- 12.6 Sufficient Condition for Optimal Controls -- 12.6.1 Clarke's Generalized Gradient -- 12.6.2 A Sufficient Condition for Optimal Controls -- 12.7 Second Order Necessary Condition for Optimal Controls -- 12.8 Notes and Comments -- 13 Linear Quadratic Optimal Control Problems -- 13.1 Formulation of the Problem -- 13.2 Optimal Feedback for Deterministic LQ Problem in Finite Dimensions -- 13.3 Optimal Feedback for Stochastic LQ Problem in Finite Dimensions -- 13.3.1 Differences Between Deterministic and Stochastic LQ Problems in Finite Dimensions -- 13.3.2 Characterization of Optimal Feedbacks for Stochastic LQ Problems in Finite Dimensions -- 13.4 Finiteness and Solvability of Problem (SLQ) -- 13.5 Pontryagin-Type Maximum Principle for Problem (SLQ) -- 13.6 Transposition Solutions to Operator-Valued Backward Stochastic Riccati Equations -- 13.7 Existence of Optimal Feedback Operator for Problem (SLQ) -- 13.8 Global Solvability of Operator-Valued Backward Stochastic Riccati Equations -- 13.8.1 Some Preliminary Results -- 13.8.2 Proof of the Main Solvability Result -- 13.9 Some Examples -- 13.9.1 LQ Problems for Stochastic Wave Equations. , 13.9.2 LQ problems for Stochastic Schrödinger Equations -- 13.10 Notes and Comments -- References -- Index.