The classical Pontryagin maximum principle (addressed to deterministic finite dimensional control systems) is one of the three milestones in modern control theory. The corresponding theory is by now well-developed in the deterministic infinite dimensional setting and for the stochastic differential equations. However, very little is known about the same problem but for controlled stochastic (infinite dimensional) evolution equations when the diffusion term contains the control variables and the control domains are allowed to be non-convex. Indeed, it is one of the longstanding unsolved problems in stochastic control theory to establish the Pontryagin type maximum principle for this kind of general control systems: this book aims to give a solution to this problem. This book will be useful for both beginners and experts who are interested in optimal control theory for stochastic evolution equations.

First monograph on Pontryagin-type maximum principle for stochastic evolution equations Provides useful approach to the topic, useful for both beginners and experts Provides detailed proof for most of results faced in the book Includes supplementary material: sn.pub/extras

Autorentext

Xiaoyu Fu is Professor of Mathematics in the School of Mathematics, Sichuan University, Chengdu, China. Her main research interest is control theory of partial differential equations. Qi Lü is a Professor in the School of Mathematics, Sichuan University, Chengdu, China. His main research interest is Mathematical Control Theory, including in particular control theory of deterministic and stochastic partial differential equations. Xu Zhang is Cheung Kong Scholar Distinguished Professor in the School of Mathematics, Sichuan University, Chengdu, China. His main research interests include mathematical control theory and related partial differential equations and stochastic analysis.

Inhalt
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.