Convex computation of extremal invariant measures of nonlinear dynamical systems and Markov processes

TL;DR

This paper introduces a convex optimization framework using semidefinite programming to compute extremal invariant measures of polynomial dynamical systems and Markov processes, enabling approximation of measures and their supports.

Contribution

It develops a novel convex-optimization-based method to compute and approximate extremal invariant measures, including physical, ergodic, and atomic measures, for nonlinear dynamical systems.

Findings

Framework characterizes invariant measures as solutions to an infinite-dimensional LP.

Finite-dimensional SDP hierarchy approximates the measures and their supports.

Method can be adapted to compute eigenmeasures of the Perron-Frobenius operator.

Abstract

We propose a convex-optimization-based framework for computation of invariant measures of polynomial dynamical systems and Markov processes, in discrete and continuous time. The set of all invariant measures is characterized as the feasible set of an infinite-dimensional linear program (LP). The objective functional of this LP is then used to single-out a specific measure (or a class of measures) extremal with respect to the selected functional such as physical measures, ergodic measures, atomic measures (corresponding to, e.g., periodic orbits) or measures absolutely continuous w.r.t. to a given measure. The infinite-dimensional LP is then approximated using a standard hierarchy of finite-dimensional semidefinite programming problems (SDPs), the solutions of which are truncated moment sequences, which are then used to reconstruct the measure. In particular, we show how to approximate the support of the measure as well as how to construct a sequence of weakly converging absolutely continuous approximations. The presented framework, where a convex functional is minimized or maximized among all invariant measures, can be seen as a generalization of and a computational method to carry out the so called ergodic optimization, where linear functionals are optimized over the set of invariant measures. Finally, we also describe how the presented framework can be adapted to compute eigenmeasures of the Perron-Frobenius operator.