ON THE STRUCTURE OF THE MINIMUM TIME FUNCTION

A minimum time problem with a nonlinear smooth dynamics and a target satisfying an internal sphere condition is considered. Under the assumption that the minimum time $T$ be continuous and the normal cone to the hypograph of $T$, $N_{hypo(T)}$, be pointed, we show that $hypo(T)$ is $\varphi$-convex, i.e., satisfies a strong external sphere condition. Consequently, $T$ is a.e.~twice differentiable and satisfies some further regularity properties. Our results are based on a representation of Clarke generalized gradient of $T$. An example is provided, showing that if $N_{hypo(T)}$ is not pointed, then the result may fail