Abstract
AbstractWe consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above $1/2$ in space and $1/4$ in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions the convergence order is further decreased. Our results only assume that the related (deterministic) differential Riccati equation can be approximated with a certain convergence order, which is easy to achieve in practice. We confirm these theoretical results through a numerical experiment in a two-dimensional domain.
Highlights
This paper is devoted to a numerical scheme for a linear quadratic control problem constrained by the stochastic heat equation with non-homogeneous Neumann boundary conditions
We prove optimal convergence orders for a full discretization which combines a linear implicit Euler method in time and a finite element discretization in space
To keep a desired heat profile, it is necessary to regulate the system. This task can be formulated as a linear quadratic control problem constrained by the stochastic heat equation, where controls and additive noise terms are located inside the domain as well as on the boundary
Summary
This paper is devoted to a numerical scheme for a linear quadratic control problem constrained by the stochastic heat equation with non-homogeneous Neumann boundary conditions. The theory of fractional powers to closed operators can still be applied, and we use this to prove optimal convergence orders under the assumption that the associated Riccati equation can be well approximated. We make such an assumption mainly because there is a lack of temporal error analyses applicable to the current situation, and providing such a proof is out of the scope of this paper.
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