Semiconcavity for the minimum time problem in presence of time delay effects

In this paper, we deal with a minimum time problem in presence of a time delay $ \tau. $ The value function of the optimal control problem is no longer defined in a subset of $ \mathbb{R}^{n} $, as it happens in the undelayed case, but its domain is a subset of the Banach space $ C([-\tau, 0];\mathbb{R}^{n}) $. For the undelayed minimum time problem, it is known that the value function associated with it is semiconcave in a subset of the reachable set and is a viscosity solution of a suitable Hamilton-Jacobi-Bellman equation. The Hamilton-Jacobi theory for optimal control problems involving time delays has been developed by several authors. Here, we are rather interested in investigating the regularity properties of the minimum time functional. Extending classical arguments, we are able to prove that the minimum time functional is semiconcave in a suitable subset of the reachable set.