Stochastic Linear-Quadratic Optimal Control Theory: Open-Loop and Closed-Loop Solutions

✍ Scribed by Jingrui Sun, Jiongmin Yong

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 129
• Series: SpringerBriefs in Mathematics
• Category: Library

✦ Synopsis

This book gathers the most essential results, including recent ones, on linear-quadratic optimal control problems, which represent an important aspect of stochastic control. It presents the results in the context of finite and infinite horizon problems, and discusses a number of new and interesting issues. Further, it precisely identifies, for the first time, the interconnections between three well-known, relevant issues – the existence of optimal controls, solvability of the optimality system, and solvability of the associated Riccati equation. Although the content is largely self-contained, readers should have a basic grasp of linear algebra, functional analysis and stochastic ordinary differential equations. The book is mainly intended for senior undergraduate and graduate students majoring in applied mathematics who are interested in stochastic control theory. However, it will also appeal to researchers in other related areas, such as engineering, management, finance/economics and the social sciences.

✦ Table of Contents

Preface......Page 7
Contents......Page 9
About the Authors......Page 11
I. Notation for Euclidean Spaces and Matrices......Page 12
II. Sets and Spaces of Functions and Processes......Page 13
1.1 Why Linear-Quadratic Problems?......Page 14
1.2 Standard Results for Deterministic LQ Problems......Page 17
1.3 Quadratic Functionals in a Hilbert Space......Page 19
2 Linear-Quadratic Optimal Controls in Finite Horizons......Page 24
2.1 Formulation of the Problem......Page 26
2.2 Representation of the Cost Functional......Page 31
2.3 Open-Loop Solvability and FBSDEs......Page 39
2.4 Closed-Loop Solvability and Riccati Equation......Page 41
2.5 Uniform Convexity of the Cost Functional......Page 49
2.6 Finiteness and Solvability Under Other Conditions......Page 62
2.7 An Example......Page 70
3 Linear-Quadratic Optimal Controls in Infinite Horizons......Page 73
3.1 Formulation of the Problem......Page 74
3.2 Stability......Page 75
3.3 Stabilizability......Page 79
3.3.1 Definition and Characterization......Page 80
3.3.2 The Case of One-Dimensional State......Page 85
3.4 Solvability and the Algebraic Riccati Equation......Page 87
3.5 A Study of Problem (SLQ)0infty......Page 90
3.5.1 A Finite Horizon Approach......Page 92
3.5.2 Open-Loop and Closed-Loop Solvability......Page 96
3.6 Nonhomogeneous Problems......Page 101
3.7 The One-Dimensional Case......Page 110
A.1 The Moore-Penrose Pseudoinverse......Page 116
A.2 Linear BSDEs in Infinite Horizons......Page 119
BookmarkTitle:......Page 125
Index......Page 128

✦ Subjects

Stochastic, Control Theory