基于时空交互作用算子的SPDE随机最优控制问题
A stochastic optimal control problem governed by SPDEs via a spatial-temporal interaction operator
摘要：
在本文中，我们首先引入一个新的时空相互作用算子来描述时空相关的不定常现象。然后，我们考虑一个新的系统的随机最优控制的随机偏微分方程与时空相互作用算子。为了解决这类随机最优控制问题，通过定义一个Hamilton泛函，导出了一个具有时空依赖性的伴随倒向随机偏微分方程，并给出了充分必要的（Pontryagin-Bismut-Bensoussan型）最大值原理.并证明了相应的伴随倒向随机偏微分方程解的存在唯一性。最后，将所得结果应用于研究具有时空不定常现象的种群增长问题。
英文摘要：
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.
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