In this paper a semiconcavity result is obtained for the value function of
an optimal exit time problem. The related state equation is of general form
(y) over dot (t) = f(y(t), u(t)), y(t) is an element of R-n, u(t) is an ele
ment of U subset of R-m.
However, suitable assumptions are needed relating f with the running and ex
it costs.
The semiconcavity property is then applied to obtain necessary optimality c
onditions, through the formulation of a suitable version of the Maximum Pri
nciple, and to study the singular set of the value function.