Mathematical Control Theory for Stochastic Partial Differential Equations

✍ Scribed by Qi Lü, Xu Zhang

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 598
• Series: Probability Theory and Stochastic Modelling 101
• Edition: 1
• Category: Library

✦ Synopsis

This is the first book to systematically present control theory for stochastic distributed parameter systems, a comparatively new branch of mathematical control theory. The new phenomena and difficulties arising in the study of controllability and optimal control problems for this type of system are explained in detail. Interestingly enough, one has to develop new mathematical tools to solve some problems in this field, such as the global Carleman estimate for stochastic partial differential equations and the stochastic transposition method for backward stochastic evolution equations. In a certain sense, the stochastic distributed parameter control system is the most general control system in the context of classical physics. Accordingly, studying this field may also yield valuable insights into quantum control systems.

A basic grasp of functional analysis, partial differential equations, and control theory for deterministic systems is the only prerequisite for reading this book.

✦ Table of Contents

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