Semidefinite Approximations of Invariant Measures for Polynomial Systems

TL;DR

This paper introduces semidefinite programming hierarchies to numerically approximate moments and supports of invariant measures for polynomial systems, covering both absolutely continuous and singular cases.

Contribution

It develops two Lasserre hierarchy-based methods for approximating invariant measures' densities and supports, applicable to continuous and discrete polynomial systems.

Findings

Hierarchies converge to the true moments and supports under certain conditions.

Numerical examples demonstrate effectiveness on various dynamical systems.

Methods handle both absolutely continuous and singular invariant measures.

Abstract

We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the approximation of the support of an invariant measure which is singular with respect to the Lebesgue measure. Each problem is handled through an appropriate reformulation into a linear optimization problem over measures, solved in practice with two hierarchies of finite-dimensional semidefinite moment-sum-of-square relaxations, also called Lasserre hierarchies. Under specific assumptions, the first Lasserre hierarchy allows to approximate the moments of an absolutely continuous invariant measure as close as desired and to extract a sequence of polynomials converging weakly to the density of this measure. The second Lasserre hierarchy allows to approximate as close as desired in the Hausdorff metric the support of a singular invariant measure with the level sets of the Christoffel polynomials associated to the moment matrices of this measure. We also present some application examples together with numerical results for several dynamical systems admitting either absolutely continuous or singular invariant measures.