We propose a time-implicit, finite-element based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+\sigma(t)]dW(t)$, and prove optimal convergence w.r.t. both, space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work [16], where the error analysis was restricted to additive noise.
翻译:我们建议对由 $[X(t) ⁇ sigma(t)]dW(t)$ 型线性噪音驱动的随机热方程式产生的随机线性线性赤道最佳控制问题,对必要和充分的最佳控制条件进行时间上隐含的、以空间和时间分解参数为基础的空间-时间分解,并证明最佳的趋同,特别是,我们使用随机线性里卡蒂方程式作为处理线性噪音的适当分析工具,从而扩大早先的工作[16]的可适用性,因为错误分析仅限于添加噪音。