The minimum time function for the controlled Moreau's Sweeping Process

Let $C(t)$, $t\geq0$ be a Lipschitz set-valued map with closed and (mildly non-)convex values and $f(t, x,u)$ be a map, Lipschitz continuous w.r.t. $x$. We consider the problem of reaching a target $S$ within the graph of $C$ subject to the differential inclusion $$(\star)\qquad \dot{x} \in -N_{C(t)}(x) + G(t,x)$$ starting from $x_{0}\in C(t_{0})$ in the minimum time $T(t_{0},x_{0})$. The dynamics $(\star)$ is called a perturbed sweeping (or Moreau) process. We give sufficient conditions for $T$ to be finite and continuous and characterize $T$ through Hamilton-Jacobi inequalities. Crucial tools for our approach are characterizations of weak and strong flow invariance of a set $S$ subject to $(\star)$. Due to the presence of the normal cone $N_{C(t)}(x)$, the right hand side of $(\star)$ contains implicitly the state constraint $x(t)\in C(t)$ and is not Lipschitz continuous with respect to $x$.