## General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite DimensionsThe 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 | |

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 |

145 | |

### Other editions - View all

General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic ... Qi Lu,Xu Zhang No preview available - 2014 |

General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic ... Qi Lü,Xu Zhang No preview available - 2014 |

### Common terms and phrases

Assume Author(s Banach-Alaoglu Theorem bounded linear operator BSDE Cauchy sequence Chap Chapter choose completes the proof conclude consider the following deﬁned linear operators denote deterministic dominated convergence theorem easy Equations in Inﬁnite ﬁnd ﬁrst gnmozl Gronwall’s inequality H-valued Hence holds implies infinite dimensional Inﬁnite Dimensions Jacques-Louis Lions Lebesgue measurable Lemma LfFo lim lim martingale representation theorem normed vector space obtain operator Q operator-valued BSEEs optimal control optimal pair pointwisely deﬁned linear Pontryagin-Type Stochastic Maximum Principle and Backward proof of Theorem reﬂexive Banach space relaxed transposition solution resp right hand side Rnds satisﬁes separable Hilbert space sequential sequential compactness Similarly solution to 1.10 solves the Eq SpringerBriefs in Mathematics Step stochastic differential equations Stochastic Evolution Equations Stochastic Maximum Principle Theorem 9.1 tnOIl well-posedness result