# Synthesizing Robust Domains of Attraction for State-Constrained Perturbed Polynomial Systems

## TL;DR

This paper introduces a semi-definite programming approach to compute robust domains of attraction for perturbed polynomial systems with state constraints, ensuring trajectories remain within safe bounds despite uncertainties.

## Contribution

It presents a novel SDP-based method that relaxes Zubov's equation to approximate the maximal robust domain of attraction for constrained systems.

## Key findings

- The method guarantees the existence of solutions under certain conditions.
- Solutions' sub-level sets inner-approximate the true domain of attraction.
- Illustrative examples demonstrate the effectiveness of the approach.

## Abstract

In this paper we propose a novel semi-definite programming based method to compute robust domains of attraction for state-constrained perturbed polynomial systems. A robust domain of attraction is a set of states such that every trajectory starting from it will approach an equilibrium while never violating a specified state constraint, regardless of the actual perturbation. The semi-definite program is constructed by relaxing a generalized Zubov's equation. The existence of solutions to the constructed semi-definite program is guaranteed and there exists a sequence of solutions such that their strict one sub-level sets inner-approximate the interior of the maximal robust domain of attraction in measure under appropriate assumptions. Some illustrative examples demonstrate the performance of our method.