ABSTRACT

The aim of this paper is to derive a maximum principle for a control problem governed by a stochastic partial differential equation (SPDE) with locally monotone coefficients. To reach our goal we adapt the method which uses the relation between backward stochastic partial differential equation (BSPDE) and the maximum principle. In particular, necessary conditions for optimality for this stochastic optimal control problem are obtained. In spite of the fact that the method used here was used by several authors before, our adaptation is not immediate. It applies a trick which is used to get estimates for solutions of SPDE with Locally Monotone Coefficients as in the proof of the Lemmas 5.1 and 5.3. This adaptation permits us to apply our results to get a maximum principle for the optimal control to the cases when the system is governed by the 2D stochastic Navier-Stokes equation and by a stochastic Burgers' equation.

摘要

本文的目的是推导出由具有局部单调系数的随机偏微分方程 (SPDE) 控制的控制问题的最大值原理。为了达到我们的目标，我们采用了使用反向随机偏微分方程（BSPDE）和极大值原理之间关系的方法。特别地，获得了该随机最优控制问题的最优性的必要条件。尽管这里使用的方法以前被几位作者使用过，但我们的适应并不是立即的。它应用了一个技巧，该技巧用于获得具有局部单调系数的 SPDE 解的估计，如引理 5.1 和 5.3 的证明。