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In this paper, we deal with a minimum time problem in presence of a timedelay $\tau.$ The value function of the considered optimal control problem isno longer defined in a subset of $\mathbb{R}^{n}$, as it happens in theundelayed case, but its domain is a subset of the Banach space$C([-\tau,0];\mathbb{R}^{n})$. For the undelayed minimum time problem, it isknown that the value function associated with it is semiconcave in a subset ofthe reachable set and is a viscosity solution of a suitableHamilton-Jacobi-Belmann equation. The Hamilton-Jacobi theory for optimalcontrol problems involving time delays has been developed by several authors.Here, we are rather interested in investigating the regularity properties ofthe minimum time functional. Extending classical arguments, we are able toprove that the minimum time functional is semiconcave in a suitable subset ofthe reachable set.

Semiconcavity for the value function of a minimum time problem with time delay