HJB equations and stochastic control on half-spaces of Hilbert spaces

TL;DR

This paper extends the theory of mild solutions for Hamilton-Jacobi-Bellman equations in Hilbert spaces to half-space domains, establishing existence, uniqueness, and regularity results with applications to optimal control problems.

Contribution

It introduces a novel framework for solving HJB equations on half-spaces in Hilbert spaces, including existence, uniqueness, and regularity of solutions.

Findings

Proved existence and uniqueness of solutions to HJB equations on half-spaces.

Demonstrated the regularity of the value function in an optimal control setting.

Provided an illustrative example demonstrating the theoretical results.

Abstract

In this paper we study a first extension of the theory of mild solutions for HJB equations in Hilbert spaces to the case when the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear Hamilton-Jacobi-Bellman (HJB) equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, that are continuously differentiable in the space variable. We also provide an application of our results to an exit time optimal control problem and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.