  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 $x$.
