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A Class of Optimal Control Problems of Forward–Backward Systems with Input Constraint

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Abstract

In this paper, we consider a new class of optimal control problems with admissibility constraint, where the state is driven by a fully coupled forward backward stochastic differential equation (FBSDE) with mixed initial-terminal condition. Different from the classical control problems, both dynamic process control and static initial-terminal perturbations are considered. Moreover, all control/perturbation components are subject to input constraint in terms of closed convex sets and partial information in terms of some sub-filtration for randomness evolution. We first study the nonlinear case of aforementioned FBSDE optimal control by deriving stochastic maximum principle. Next, we consider the linear quadratic case with explicit representation of the optimal admissible controls. More specifically, a new Hamiltonian system involving three projection operators and conditional expectation is derived. Finally, we apply obtained maximum principle to study a general class of large-population system and provide a unified framework to analyze related mean-field game (MFG). Our result includes considerable existing MFG results as its special cases and provides some new features such as recursive functional or input delay average.

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Notes

  1. In fact, the assumption (H2) directly follows from (6.25) and (6.26) of [46].

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People’s Republic of China

    Jianhui Huang & Hanyu Zhao

  2. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, People’s Republic of China

    Wenqiang Li

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  1. Jianhui Huang
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  2. Wenqiang Li
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  3. Hanyu Zhao
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Correspondence to Wenqiang Li.

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Communicated by Mihai Sirbu.

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The first author acknowledges support from RGC PolyU P0035957, P0038384, P0037987, P0031044, and NSFC 12171407. The second author acknowledges support from the NSF of P.R. China (No. 12101537, 12271304) and Doctoral Scientific Research Fund of Yantai University (No. SX17B09).

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Huang, J., Li, W. & Zhao, H. A Class of Optimal Control Problems of Forward–Backward Systems with Input Constraint. J Optim Theory Appl 199, 1050–1084 (2023). https://doi.org/10.1007/s10957-023-02314-0

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