HJB EQUATIONS FOR THE OPTIMAL CONTROL OF DIFFERENTIAL EQUATIONS WITH DELAYS AND STATE CONSTRAINTS, II: VERIFICATION AND OPTIMAL FEEDBACKS

This paper, which is the natural continuation of part I [S. Federico, B. Goldys, and F. Gozzi, SIAM J. Control Optim., 48 (2010), pp. 4910-4937], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. In part I the problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman equation is studied. The goal of the present paper is to exploit the regularity result of part I to prove a verification theorem and find optimal feedback controls for the problem. While it is easy to define a feedback control formally following the classical case, 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 study 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.