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 di ff erential inclusion (x) over dot is an element of- N-C(t) (x) + G(t, x) starting from x(0) is an element of C(t(0)) in the minimum time T(t(0), x(0)). The dynamics is called a perturbed sweeping (or Moreau) process. We give su ffi cient conditions for T to be fi nite 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) is an element of C(t) and is not Lipschitz continuous with respect to x.