The minimum time function for the controlled Moreau's sweeping process

Let C(t), t ≥ 0 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 x ∈ −N_{C(t)} (x) + G(t, x) starting ̇from x_0 ∈ C(t_0 ) in the minimum time T (t_0 , x_0 ). The dynamics 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 the inclusion. Due to the presence of the normal cone N_{C(t)} (x), the right-hand side of the inclusion contains implicitly the state constraint x(t) ∈ C(t) and is not Lipschitz continuous with respect to x.