Viscosity solutions approach to economic models governed by DDEs

A family of economic and demographic models governed by linear delay differential equations is considered. They can be expressed as optimal control problems subject to delay differential equations (DDEs) characterized by some non-trivial mathematical difficulties: state/control constraints and delay in the control. The study is carried out rewriting the problem as an (equivalent) optimal control problem in infinite dimensions and then using the dynamic programming approach (DPA). Similar problems have been studied in the literature using classical and strong (approximating) solutions of the Hamilton-Jacobi-Bellman (HJB) equation. Here a more general formulation is treated thanks to the use of viscosity solutions approach. Indeed a general current objective function is considered and the concavity of the Hamiltonian is not required. It is shown that the value function is a viscosity solution of the HJB equation and a verification theorem in the framework of viscosity solutions is proved.