A Stochastic Optimal Control Problem for the Heat Equation on the Halfline with Dirichlet Boundary-noise and Boundary-control

TL;DR

This paper formulates and analyzes a stochastic optimal control problem for a heat equation on the halfline with boundary noise and control, using backward stochastic differential equations to find optimal strategies.

Contribution

It introduces a novel control framework for parabolic PDEs with boundary noise, linking boundary control problems to backward stochastic differential equations.

Findings

Established existence of optimal controls under boundary noise conditions

Derived backward stochastic differential equations characterizing optimal solutions

Extended control theory to boundary-noise parabolic PDEs

Abstract

We consider a controlled state equation of parabolic type on the halfline $(0,+\infty)$ with boundary conditions of Dirichlet type in which the unknown is equal to the sum of the control and of a white noise in time. We study finite horizon and infinite horizon optimal control problem related by menas of backward stochastic differential equations.