Spatio-Temporal Decomposition of Sum-of-Squares Programs for the Region of Attraction and Reachability

TL;DR

This paper introduces a spatio-temporal decomposition technique for sum-of-squares programs to efficiently compute the region of attraction and reachability in controlled polynomial dynamical systems, reducing memory usage and increasing precision.

Contribution

It proposes a novel splitting method that decomposes large SDPs into smaller interconnected problems, improving scalability and accuracy over previous approaches.

Findings

Retains convergence and outer-approximation guarantees

Achieves higher precision in less time

Uses smaller memory footprint

Abstract

This paper presents a method for calculating Region of Attraction of a target set (not necessarily an equilibrium) for controlled polynomial dynamical systems, using a hierarchy of semidefinite programming problems (SDPs). Our approach builds on previous work and addresses its main issue, the fast-growing memory demands for solving large-scale SDPs. The main idea in this work is in dissecting the original resource-demanding problem into multiple smaller, interconnected, and easier to solve problems. This is achieved by spatio-temporal splitting akin to methods based on partial differential equations. We show that the splitting procedure retains the convergence and outer-approximation guarantees of the previous work, while achieving higher precision in less time and with smaller memory footprint.