An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise

TL;DR

This paper addresses a linear quadratic control problem for the heat equation on a halfline with boundary control and noise, reformulating it as a stochastic evolution equation and solving it via Riccati equations.

Contribution

It introduces a novel reformulation of the LQ problem as a stochastic evolution equation in a weighted space, enabling regularity analysis and explicit feedback control solutions.

Findings

Reformulation as stochastic evolution equation in weighted L2 space

Regularity results for boundary terms in the infinite-dimensional setting

Explicit feedback control and value function derived from Riccati equation

Abstract

We study a linear quadratic problem for a system governed by the heat equation on a halfline with Dirichlet boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted L2 space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated non-stochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem.