# Assessing the Finite-Time Stability of Nonlinear Systems by means of Physics-Informed Neural Networks

## TL;DR

This paper introduces a novel method using Physics-Informed Neural Networks to assess finite-time stability in nonlinear systems by approximating Lyapunov-like functions, enabling practical verification and control applications.

## Contribution

It proposes a neural network-based approach to verify finite-time stability conditions for nonlinear systems, extending existing methods with a flexible, data-driven technique.

## Key findings

- Successfully verified FTS in numerical examples
- Provided a closed-form Lyapunov-like function
- Enhanced verification flexibility for nonlinear systems

## Abstract

In this paper, the problem of assessing the Finite-Time Stability (FTS) property for general nonlinear systems is considered. First, some necessary and sufficient conditions that guarantee the FTS of general nonlinear systems are provided; such conditions are expressed in terms of the existence of a suitable Lyapunov-like function. Connections of the main theoretical result of given in this article with the typical conditions based on Linear Matrix Inequalities (LMI) that are used for Linear Time-Varying (LTV) systems are discussed. An extension to the case of discrete time systems is also provided. Then, we propose a method to verify the obtained conditions for a very broad class of nonlinear systems. The proposed technique leverages the capability of neural networks to serve as universal function approximators to obtain the Lyapunov-like function. The network training data are generated by enforcing the conditions defining such function in a (large) set of collocation points, as in the case of Physics-Informed Neural Networks. To illustrate the effectiveness of the proposed approach, some numerical examples are proposed and discussed. The technique proposed in this paper allows to obtain the required Lyapunov-like function in closed form. This has the twofold advantage of a) providing a practical way to verify the considered FTS property for a very general class of systems, with an unprecedented flexibility in the FTS context, and b) paving the way to control applications based on Lyapunov methods in the framework of Finite-Time Stability and Control.