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Control in Hilbert Space and First-Order Mean Field Type Problem

In: Stochastic Analysis, Filtering, and Stochastic Optimization

Author

Listed:
  • Alain Bensoussan

    (University of Texas at Dallas, International Center for Decision and Risk Analysis, Jindal School of Management
    City University of Hong Kong, School of Data Science)

  • Hang Cheung

    (City University of Hong Kong, School of Data Science)

  • Sheung Chi Phillip Yam

    (Chinese University of Hong Kong, Department of Statistics)

Abstract

We extend the work [9] 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 [9], 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. Finally, by using the ‘lifting’ idea as in [6, 7], 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.

Suggested Citation

  • Alain Bensoussan & Hang Cheung & Sheung Chi Phillip Yam, 2022. "Control in Hilbert Space and First-Order Mean Field Type Problem," Springer Books, in: George Yin & Thaleia Zariphopoulou (ed.), Stochastic Analysis, Filtering, and Stochastic Optimization, pages 1-32, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-98519-6_1
    DOI: 10.1007/978-3-030-98519-6_1
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