# Converse Theorems for Safety and Barrier Certificates

## TL;DR

This paper establishes that every robustly safe dynamical system possesses a barrier certificate, providing a constructive method based on finite-time reachable states to prove safety.

## Contribution

It proves the existence of barrier certificates for all robustly safe ODEs and offers a constructive approach based on finite-time reachability.

## Key findings

- Every robustly safe ODE has a barrier certificate.
- Constructive method for barrier certificates based on finite-time reachability.
- Bridges safety verification with reachability analysis.

## Abstract

An important tool for proving safety of dynamical systems is the notion of a barrier certificate. In this paper we prove that every robustly safe ordinary differential equation has a barrier certificate. Moreover, we show a construction of such a barrier certificate based on a set of states that is reachable in finite time.