  We study the time optimal control problem for differential inclusions with a
general closed target. We first give the representation of the proximal
horizontal subgradients of the minimum time function $\mathcal{T}$ and then,
together with the representation of the proximal subgradients, we obtain some
relationships between the normal cones to the sublevel of $\mathcal{T}$ and the
normal cones to its epigraph. The 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.

Optimization and Control