  In this paper, we study an optimal exit time problem with general running and
terminal costs and a target $\mathcal{S}\subset\mathbb{R}^d$ having an inner
ball property for a nonlinear control system that satisfies mild
controllability assumptions. In particular, Petrov's condition at the boundary
of $\mathcal{S}$ is not required and the value function $V$ may fail to be
locally Lipschitz. In such a weakened set-up, we first establish a
representation formula for proximal (horizontal) supergradients of $V$ by using
transported proximal normal vectors. This allows us to obtain an external
sphere condition for the hypograph of $V$ which yields several regularity
properties. In particular, $V$ is almost everywhere twice differentiable and
the Hausdorff dimension of its singularities is not greater than $d-1/2$.
Furthermore, besides optimality conditions for trajectories of the optimal
control problem, we extend the analysis to propagation of singularities and
differentiability properties of the value function. An upper bound for the
Hausdorff measure of the singular set is also studied, which implies that $V$
is a function of special bounded variation.
