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 function T is continuous we show that hypo(T ) satisfies a weak external sphere condition. From here, our results are divided into two parts. In the first part: if the normal cone to the hypograph of T , N hypo(T) is pointed then 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. This work is based on Clarke generalized gradient of T . An example is provided, showing that if N hypo(T) is not pointed then the result may fail. The second part is motivated by the first part. In order to remove the pointedness assumption of the hypograph of T , we prove the following general result. If the hypograph of a continuous function satisfies a weak external sphere condition then it ”essentially” satisfies a strong external sphere condition.