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 , Nhypo(T ), be pointed, we show that hypo(T ) is -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. Keywords and phrases: normal vectors, -convex (prox-regular, positive reach) sets, internal sphere condition, small time controllability, adjoint flow.

• Type: Journal
• Year: 2010
• Where: SIAMCO
• Authors: Giovanni Colombo, Khai T. Nguyen
• Keywords: Internal Sphere Condition, Minimum Time, SIAMCO 2010, Sphere Condition