Inner approximations of the maximal positively invariant set for polynomial dynamical systems

TL;DR

This paper introduces a method using the Lasserre hierarchy to compute inner approximations of the maximal positively invariant set for polynomial dynamical systems, with proven convergence in volume under certain conditions.

Contribution

It extends previous work by focusing on inner approximations in infinite time, providing volume convergence guarantees unlike prior methods.

Findings

Proves convergence in volume of the hierarchy under growth conditions.

Develops a method for inner approximations in infinite time.

Addresses limitations of previous outer and finite-time inner approximation methods.

Abstract

The Lasserre or moment-sum-of-square hierarchy of linear matrix inequality relaxations is used to compute inner approximations of the maximal positively invariant set for continuous-time dynamical systems with polynomial vector fields. Convergence in volume of the hierarchy is proved under a technical growth condition on the average exit time of trajectories. Our contribution is to deal with inner approximations in infinite time, while former work with volume convergence guarantees proposed either outer approximations of the maximal positively invariant set or inner approximations of the region of attraction in finite time.