Discretization of a distributed optimal control problem with a stochastic parabolic equation driven by multiplicative noise

Abstract

A discretization of an optimal control problem of a stochastic parabolic equation driven by multiplicative noise is analyzed. The state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. The convergence rate O(τ 1/2 +h 2 ) is rigorously derived.

Analysis of a discretization of a distributed control problem with a stochastic evolution equation

“…Thirdly, let us derive a first order optimality condition of problem (12). Following the proof of [8,Lemma 4.19], we can easily obtain that, for any…”

“…In this section, we will use the ideas in [8] to prove Theorem 3.1. For convenience, we use the following conventions: J > 2 and τ < 2/5; δW j := W (t j+1 ) − W (t j ) for all 0 j < J; the norm • L 2 (Ω;L 2 (0,T ;L 2 (O))) is abbreviated to • ; Q h be the L 2 (O)-orthogonal projection operator onto V h ; I means the identity mapping; a b means a Cb, where C is a generic positive constant, independent of h and τ , and its value may differ in different places.…”

“…Assume that g ∈ L 2 F (Ω; L 2 (0, T ; Ḣ2 (O))). Note that (∆ h p h , ∆ h z h ) is the solution of equation ( 27) with g being replaced by ∆ h Q h g and that (∆ h P, ∆ h Z) is the solution of (25) with g being replaced by ∆ h Q h g. Hence, by [8,Lemma 4.14] we have…”

“…This is the second part of our series works on the numerical analysis of stochastic parabolic optimal control problems. In [8] we analyzed a distributed control problem with a stochastic parabolic equation driven by multiplicative noise, where the diffusion term does not contain the control variable. To our best knowledge, no numerical result is available for the case that the diffusion term contains the control variable.…”

“…In this paper, we will use the techniques in [8] to establish the convergence of a discrete stochastic optimal control problem, where the state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. We obtain the first-order spatial accuracy, and, under the condition that y d ∈ L 2 F (Ω; L 2 (0, T ; Ḣβ (O))) with 0 < β 1, we prove that the temporal accuracy is dominated by…”