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.
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21 June 2022
The original version of the book was inadvertently published with incorrect abstracts in the chapters. This has now been amended.
In addition to this, the affiliation of author Dr. Bertram Tschiderer has been changed to Faculty of Mathematics, University of Vienna in the online version of Chapter 10.
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Bensoussan, A., Cheung, H., Yam, S.C.P. (2022). Control in Hilbert Space and First-Order Mean Field Type Problem. In: Yin, G., Zariphopoulou, T. (eds) Stochastic Analysis, Filtering, and Stochastic Optimization. Springer, Cham. https://doi.org/10.1007/978-3-030-98519-6_1
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DOI: https://doi.org/10.1007/978-3-030-98519-6_1
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Publisher Name: Springer, Cham
Print ISBN: 978-3-030-98518-9
Online ISBN: 978-3-030-98519-6
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