The Exponential Formula for the Reachable Set of a Lipschitz Differential Inclusion

Abstract

The main goal of this paper is to prove a formula for the reachable set of a Lipschitz differential inclusion with convex values. The formula involves a Kuratowski limit of sets that resembles a standard approach of defining the exponential of a matrix—this explains the title. The proof of the main theorem partially relies on a $C^1 $ approximation result due to Filippov, for which a new proof is given. A new approach of characterizing the value function associated with a Mayer optimal control problem is given as an application.