Dynamic Programming for Optimal Control Problems with Delays in the Control Variable

We study a class of optimal control problems with state constraint, where the state equation is a differential equation with delays in the control variable. This class of problems arises in some economic applications, in particular in optimal advertising problems. The optimal control problem is embedded in a suitable Hilbert space, and the associated Hamilton--Jacobi--Bellman (HJB) equation is considered in this space. It is proved that the value function is continuous with respect to a weak norm and that it solves in the viscosity sense the associated HJB equation. The main results are the proof of a directional $C^1$-regularity for the value function and the feedback characterization of optimal controls