Exploiting Term Sparsity in Moment-SOS hierarchy for Dynamical Systems

TL;DR

This paper extends the term sparsity sum-of-squares (TSSOS) algorithm to dynamical systems, enabling more efficient computation for stability and attractor analysis while maintaining theoretical guarantees.

Contribution

It introduces a dynamical system version of TSSOS that exploits algebraic structure for scalable SOS-based analysis of dynamical systems.

Findings

Significant computational savings demonstrated.

Preserves convergence guarantees.

Effective in analyzing regions of attraction and invariant sets.

Abstract

In this paper, we develop a dynamical system counterpart to the term sparsity sum-of-squares (TSSOS) algorithm proposed for static polynomial optimization. This allows for computational savings and improved scalability while preserving convergence guarantees when sum-of-squares methods are applied to problems from dynamical systems, including the problems of approximating region of attraction, the maximum positively invariant set, and the global attractor. At its core, the method exploits the algebraic structure of the data, thereby complementing existing methods that exploit causality relations among the states of the dynamical system. The procedure encompasses sign symmetries of the dynamical system as was already revealed for polynomial optimization. Numerical examples demonstrate the efficiency of the approach in the presence of this type of sparsity.