Variational Analysis and Regularity of the Minimum Time Function for Differential Inclusions

Abstract

We study the time optimal control problem for differential inclusions with a general closed target. We first give a representation of the proximal horizontal subgradients of the minimum time function $\mathcal{T}$ and then, together with a known representation of the proximal subgradients, we obtain some relationships between the normal cones to the sublevel set and to the epigraph of $\mathcal{T}$. These relationships allow us to get the propagation of the proximal subdifferential as well as of the proximal horizontal subdifferential of $\mathcal{T}$ along optimal trajectories. Finally, we show, under suitable assumptions, that the epigraph of $\mathcal{T}$ is $\varphi$-convex near the target. This is the first nonlinear $\varphi$-convexity result valid in any dimension.