Ensemble approximations for constrained dynamical systems using Liouville equation

For a class of state-constrained dynamical systems described by evolution variational inequalities, we study the time probability measure which describes distribution state over set. In contrast to smooth ordinary differential equations, where this is Liouville flow map associated with nonsmooth inclusion not necessarily invertible and one cannot directly derive continuity equation describe states. Instead, consider Lipschitz approximation our original system construct sequence measures obtained from equations corresponding these approximations. This converges in weak-star topology describing states for system. allows us approximate numerically moments (up some finite order) system, using solver that uses order moment approximations equation. Our approach illustrated help two academic examples.