Abstract

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, Nhypo(T ), be pointed, we show that hypo(T) is ϕ-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 Nhypo(T ) is not pointed, then the result may fail.