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

Salvatore Federico (1) , Ben Goldys (2) , Fausto Gozzi (3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 School of Mathematics and Statistics
3 Dipartimento di Scienze Economiche e Aziendali

This paper, which is the natural continuation of a paper by the same authors, studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. In the first paper the problem is embedded in a suitable Hilbert space H and the regularity of the associated Hamilton-Jacobi-Bellman (HJB) equation is studied. Therein the main result is that the value function V is a viscosity solution to the associated HJB equation and has continuous classical derivative in the direction of the "present". The goal of the present paper is to exploit this regularity result 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-tobuild models. In particular, we show the existence and uniqueness of optimal controls and their characterization as feedbacks.

Mots clés: Delay Differential Equations; Hamilton-Jacobi-Bellman equation; Optimal Control; Verification Theorem

Domaines: Optimisation et contrôle [math.OC]

Salvatore Federico, Ben Goldys, Fausto Gozzi. HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, II: Verification and Optimal Feedbacks. 2011. ⟨hal-00583124⟩