Mathematical Control Theory for Stochastic Partial Differential Equations

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.

Autorentext

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.

Inhalt
1 Introduction.- 2 Some Preliminaries in Stochastic Calculus.- 3 Stochastic Evolution Equations.- 4 Backward Stochastic Evolution Equations.- 5 Control Problems in Stochastic Distributed Parameter Systems.- 6 Controllability for Stochastic Differential Equations in Finite Dimensions.- 7 Controllability for Stochastic Linear Evolution Equations.- 8 Exact Controllability for Stochastic Transport Equations.- *9 Controllability and Observability of Stochastic Parabolic Systems.- 10 Exact Controllability for a Refined Stochastic Wave Equation.- *11 Exact Controllability for Stochastic Schrödinger Equations.- 12 Pontryagin-Type Stochastic Maximum Principle.- 13 Linear Quadratic Optimal Control Problems.- References.- Index.