Convergence with rates for a Riccati-based discretization of SLQ problems with SPDEs

Abstract

Abstract We consider a new discretization in space (parameter $h>0$) and time (parameter $\tau>0$) of a stochastic optimal control problem, where a quadratic functional is minimized subject to a linear stochastic heat equation with linear noise. Its construction is based on the perturbation of a generalized difference Riccati equation to approximate the related feedback law. We prove a convergence rate of almost ${\mathcal O}(h^{2}+\tau )$ for its solution, and conclude from it a rate of almost ${\mathcal O}(h^{2}+\tau )$ resp. ${\mathcal O}(h^{2}+\tau ^{1/2})$ for computable approximations of the optimal state and control with additive resp. multiplicative noise.