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Mathematics > Optimization and Control

arXiv:2108.11237 (math)
[Submitted on 25 Aug 2021]

Title:Control in Hilbert Space and First Order Mean Field Type Problem

Authors:Alain Bensoussan, Henry Hang Cheung, Sheung Chi Phillip Yam
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Abstract:We extend the work \cite{bensoussan2019control} by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic and investigated a novel form of the `lifting' technique proposed by P. L. Lions. In \cite{bensoussan2019control}, we only showed the local existence and uniqueness of solutions to the FBODEs in the Hilbert space which were associated to the control problems with drift function consisting of the control only. In this article, we establish the global existence and uniqueness of the solutions to the FBODEs in Hilbert space corresponding to control problems with separable drift function which is nonlinear in state and linear in control. We shall also prove the sufficiency of the Pontryagin Maximum Principle and derive the corresponding Bellman equation. Besides, we shall show an analogue in the stationary case. Finally, by using the `lifting' idea as in \cite{stochasticv2,stochasticv1}, we shall apply the result to solve the linear quadratic mean field type control problems, and to show the global existence of the corresponding Bellman equations.
Comments: Invited book chapter in Stochastic Analysis, Filtering, and Stochastic Optimization: A Commemorative Volume to Honor Mark H. A. Davies' Contributions
Subjects: Optimization and Control (math.OC)
Cite as: arXiv:2108.11237 [math.OC]
  (or arXiv:2108.11237v1 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.2108.11237

Submission history

From: Phillip Yam [view email]
[v1] Wed, 25 Aug 2021 13:49:26 UTC (24 KB)
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