Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions

TL;DR

This paper extends a lifting and partial smoothing technique to solve stationary Hamilton-Jacobi-Bellman equations in infinite-dimensional Hilbert spaces, enabling the analysis of complex stochastic control problems with unbounded controls and state-dependent costs.

Contribution

It develops a new theoretical framework for existence, uniqueness, and verification of solutions to stationary HJB equations in infinite dimensions, broadening applicability to economic models.

Findings

Proves existence and uniqueness of regular mild solutions.

Provides a verification theorem for optimal controls.

Synthesizes optimal feedback control strategies.

Abstract

We study a family of stationary Hamilton-Jacobi-Bellman (HJB) equations in Hilbert spaces arising from stochastic optimal control problems. The main difficulties to treat such problems are: the lack of smoothing properties of the linear part of the HJB equation; the presence of unbounded control operators; the presence of state-dependent costs. This features, combined together, prevent the use of the classical mild solution theory of HJB equation (see e.g., Chapter 4 of G. Fabbri, F. Gozzi, A. Swiech, Stochastic Optimal Control in Infinite Dimensions: Dynamic Programming and HJB Equations, Springer, 2017). The problem has been studied in the evolutionary case in F. Gozzi, F. Masiero, Lifting Partial Smoothing to Solve HJB Equations and Stochastic Control Problems, SIAM Journal on Control and Optimization, 63(3), (2025), pp. 1515-1559 using a "lifting technique" (i.e. working in a suitable space of trajectories where a "partial smoothing" property of the linear part of the HJB equations holds. In this paper we extend such a theory to the case of infinite horizon optimal control problems, which are very common, in particular in economic applications. The main results are: the existence and uniqueness of a regular mild solution to the HJB equation; a verification theorem, and the synthesis of optimal feedback controls.