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

对于由演化变分不等式描述的一类状态约束动力系统，我们研究了描述状态在集合上的分布的概率测度的时间演化。与光滑常微分方程相比，这种概率测度的演化由 Liouville 方程描述，与非光滑微分包含相关的流图不一定是可逆的，并且不能直接导出连续性方程来描述分布的演化状态。相反，我们考虑原始非光滑系统的 Lipschitz 近似，并构建从与这些近似对应的 Liouville 方程获得的一系列测量。这一系列测度在弱星拓扑中收敛于描述原始非光滑系统状态分布演化的测度。这使我们能够使用使用 Liouville 方程的有限阶矩近似的求解器，以数值方式近似原始非光滑系统的矩（达到某个有限阶）的演化。我们的方法在两个学术例子的帮助下得到了说明。