HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, II: Verification and Optimal Feedbacks

This paper studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. The problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman (HJB) equation is studied. The goal of the present paper is to exploit the regularity result of a previous companion paper to prove a Verification Theorem and find optimal feedback controls for the problem. While it is easy to define formally a feedback control in classical sense, the proof of its existence and optimality is hard due to lack of full regularity of V and to the infinite dimensionality of the problem. The theory developed is applied to approach economic problems of optimal growth for nonlinear time-to-build models. In particular, we show the existence and uniqueness of optimal controls and their characterization as feedbacks.