Converging approximations of attractors via almost Lyapunov functions and semidefinite programming

TL;DR

This paper introduces a novel method combining two existing approaches to approximate attractors with positively invariant sets using convex optimization, providing guaranteed convergence and practical applicability.

Contribution

It merges infinite-dimensional linear programming and positive invariance techniques into a unified, computationally tractable framework using sum-of-squares methods.

Findings

Converging outer approximations of attractors are achieved.

The method guarantees positive invariance of the approximations.

Numerical examples demonstrate ease of use and effectiveness.

Abstract

In this paper we combine two existing approaches for approximating attractors. One of them approximates the attractors arbitrarily well by sublevel sets related to solutions of infinite dimensional linear programming problems. A downside there is that these sets are not necessarily positively invariant. On the contrary, the second method provides supersets of the attractor which are positively invariant. Their method on the other hand has the disadvantage that the underlying optimization problem is not computationally tractable without the use of heuristics - and incorporating them comes at the price of losing guaranteed convergence. In this paper we marry both approaches by combining their techniques and we get converging outer approximations of the attractor consisting of positively invariant sets based on convex optimization via sum-of-squares techniques. The method is easy to use and illustrated by numerical examples.