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 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. (c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

In this study, we investigate the time evolution of a probability measure for a class of state-constrained dynamical systems described by evolution variational inequalities. Unlike smooth ordinary differential equations, the evolution of this probability measure is not necessarily invertible due to the nonsmooth nature of the differential inclusion. Instead, we approximate the original nonsmooth system using Lipschitz approximation and construct a sequence of measures obtained from Liouville equations. This sequence converges to the measure describing the evolution of the distribution of states for the original nonsmooth system, allowing us to numerically approximate the evolution of moments.