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Mathematical Control Theory for Stochastic Partial Differential Equations

  • Book
  • © 2021

Overview

Authors:
  1. Qi Lü

    1. School of Mathematics, Sichuan University, Chengdu, China

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    You can also search for this author in PubMed   Google Scholar

  2. Xu Zhang

    1. School of Mathematics, Sichuan University, Chengdu, China

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    You can also search for this author in PubMed   Google Scholar

  • Control theory for stochastic distributed parameters is a new branch
  • New tools presented in this book
  • Required reading for anyone who wants to study quantum control systems mathematically

Part of the book series: Probability Theory and Stochastic Modelling (PTSM, volume 101)

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Table of contents (13 chapters)

  1. Front Matter

    Pages I-XIII
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  2. Introduction

    • Qi Lü, Xu Zhang
    Pages 1-23

  3. Some Preliminaries in Stochastic Calculus

    • Qi Lü, Xu Zhang
    Pages 25-130

  4. Stochastic Evolution Equations

    • Qi Lü, Xu Zhang
    Pages 131-156

  5. Backward Stochastic Evolution Equations

    • Qi Lü, Xu Zhang
    Pages 157-188

  6. Control Problems for Stochastic Distributed Parameter Systems

    • Qi Lü, Xu Zhang
    Pages 189-208

  7. Controllability for Stochastic Differential Equations in Finite Dimensions

    • Qi Lü, Xu Zhang
    Pages 209-227

  8. Controllability for Stochastic Linear Evolution Equations

    • Qi Lü, Xu Zhang
    Pages 229-262

  9. Exact Controllability for Stochastic Transport Equations

    • Qi Lü, Xu Zhang
    Pages 263-272

  10. Controllability and Observability of Stochastic Parabolic Systems

    • Qi Lü, Xu Zhang
    Pages 273-330

  11. Exact Controllability for a Refined Stochastic Wave Equation

    • Qi Lü, Xu Zhang
    Pages 331-361

  12. Exact Controllability for Stochastic Schrödinger Equations

    • Qi Lü, Xu Zhang
    Pages 363-385

  13. Pontryagin-Type Stochastic Maximum Principle and Beyond

    • Qi Lü, Xu Zhang
    Pages 387-475

  14. Linear Quadratic Optimal Control Problems

    • Qi Lü, Xu Zhang
    Pages 477-565

  15. Back Matter

    Pages 567-592
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Keywords

About this book

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.



Reviews

“This book of about 600 pages presents in an appealing manner how control theory can be applied for stochastic partial differential equations. I recommend it to all students and researchers interested in these topics.” (‪Gheorghe Tigan, zbMATH 1497.93001, 2022)

Authors and Affiliations

  • School of Mathematics, Sichuan University, Chengdu, China

    Qi Lü, Xu Zhang

About the authors

Qi Lü is a professor at School of Mathematics, Sichuan University, Chengdu, China. He is currently an associate editor/editorial board member of several journals including Systems & Control Letters.  His research interests include control theory for deterministic and stochastic partial differential equations and stochastic analysis.  

Xu Zhang is a Cheung Kong Scholar Distinguished Professor at School of Mathematics, Sichuan University, Chengdu, China. He is a sectional speaker at International Congress of Mathematicians (Control Theory & Optimization Section, 2010). He is/was the editor in chief/corresponding editor/associate editor for several journals including Mathematical Control and Related Fields, ESAIM: Control, Optimisation and Calculus of Variations, and SIAM Journal on Control and Optimization. His research interests include control theory, partial differential equations and stochastic analysis.   

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