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

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. These features, combined together, prevent the use of the classical mild solution theory of HJB equation (see e.g. [Fabbri et al. (2017), Ch.4]). The problem has been studied in the evolutionary case in Gozzi and Masiero (2025) 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.