General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions

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Springer, Jun 2, 2014 - Science - 146 pages
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.
 

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Contents

1 Introduction
1
2 Preliminaries
11
3 WellPosedness of the VectorValued BSEEs
23
4 WellPosedness Result for the OperatorValued BSEEs with Special Data
36
5 Sequential BanachAlaogluType Theorems in the Operator Version
49
6 WellPosedness of the OperatorValued BSEEs in the General Case
60
7 Some Properties of the Relaxed Transposition Solutions to the OperatorValued BSEEs
91
8 Necessary Condition for Optimal Controls the Case of Convex Control Domains
108
9 Necessary Condition for Optimal Controls the Case of Nonconvex Control Domains
115
References
145
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