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Optimal control problems governed by 1-D Kobayashi–Warren–Carter type systems
June  2021, 11(2): 291-312. doi: 10.3934/mcrf.2020037

## A stochastic optimal control problem governed by SPDEs via a spatial-temporal interaction operator

 1 Department of Mathematics, Sichuan University, Chengdu 610064, China 2 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China, and, Department of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author

Received  March 2020 Revised  June 2020 Published  August 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (11471230, 11671282)

In this paper, we first introduce a new spatial-temporal interaction operator to describe the space-time dependent phenomena. Then we consider the stochastic optimal control of a new system governed by a stochastic partial differential equation with the spatial-temporal interaction operator. To solve such a stochastic optimal control problem, we derive an adjoint backward stochastic partial differential equation with spatial-temporal dependence by defining a Hamiltonian functional, and give both the sufficient and necessary (Pontryagin-Bismut-Bensoussan type) maximum principles. Moreover, the existence and uniqueness of solutions are proved for the corresponding adjoint backward stochastic partial differential equations. Finally, our results are applied to study the population growth problems with the space-time dependent phenomena.

Citation: Zhun Gou, Nan-jing Huang, Ming-hui Wang, Yao-jia Zhang. A stochastic optimal control problem governed by SPDEs via a spatial-temporal interaction operator. Mathematical Control & Related Fields, 2021, 11 (2) : 291-312. doi: 10.3934/mcrf.2020037
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