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 \(x\).