HJB equations in infinite dimensions under weak regularizing properties

TL;DR

This paper develops methods to solve Hamilton-Jacobi-Bellman equations in infinite-dimensional spaces with weak regularity assumptions, enabling solutions to complex stochastic control problems like controlled heat equations.

Contribution

It introduces a novel approach to solving HJB equations in infinite dimensions under weak regularizing conditions, expanding applicability to models with limited regularity.

Findings

Successfully solves HJB equations in Hilbert and Banach spaces with Lipschitz conditions.

Applies results to stochastic heat equations with control and noise in subdomains.

Demonstrates the approach's effectiveness in complex stochastic control models.

Abstract

We solve in mild sense Hamilton Jacobi Bellman equations, both in an infinite dimensional Hilbert space and in a Banach space, with lipschitz Hamiltonian and lipschitz continuous final condition, and asking only a weak regularizing property on the transition semigroup of the corresponding state equation. The results are applied to solve stochastic optimal control problems; the models we can treat include a controlled stochastic heat equation in space dimension one and with control and noise on a subdomain.