Projected Evolutionary Lifting and Well-Posedness of Stationary Hamilton-Jacobi-Bellman Equations in Infinite Dimensions

TL;DR

This paper proves the existence and uniqueness of solutions to stationary Hamilton-Jacobi-Bellman equations in infinite-dimensional spaces, extending previous work by removing discount factor restrictions using monotone operator theory.

Contribution

It introduces the Projected Evolutionary Lifting technique to handle challenges like lack of smoothing and unbounded controls, broadening the class of solvable infinite-dimensional HJB equations.

Findings

Established existence and uniqueness of solutions for all positive discount rates.

Extended previous results to settings with singular dynamics and state-dependent costs.

Removed the restriction on large discount factors in infinite-dimensional HJB equations.

Abstract

This paper establishes the existence and uniqueness of mild solutions to stationary Hamilton-Jacobi-Bellman (HJB) equations associated with infinite-horizon stochastic optimal control problems in separable Hilbert spaces. Our framework includes settings with a lack of global smoothing properties of the transition semigroup, singular dynamics involving unbounded control operators, and state-dependent running costs. We overcome these challenges by lifting the state space using the Projected Evolutionary Lifting technique. This work is an extension of G. Bolli and F. Gozzi, Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions, 2025, in which existence and uniqueness is proved via a contraction mapping argument and is consequently restricted to sufficiently large discount factors. We remove this restriction, proving existence and uniqueness for any discount rate $\lambda > 0$ using tools from the theory of maximally monotone operators.