  In this paper, we deal with a minimum time problem in presence of a time
delay $\tau.$ The value function of the considered 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-Belmann 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.
