Computation of Lyapunov functions for nonlinear differential equations via a Massera-type construction

TL;DR

This paper introduces a novel Massera-type method for computing Lyapunov functions for nonlinear differential equations, relaxing exponential stability assumptions and enabling finite-time integration to estimate system stability and domain of attraction.

Contribution

The paper develops a new Massera-type construction for Lyapunov functions that relaxes exponential stability assumptions and allows finite-time analysis for nonlinear systems.

Findings

Efficient computation of Lyapunov functions for nonlinear systems.

Improved estimates of the domain of attraction.

Application to biological systems demonstrating practical utility.

Abstract

An approach for computing Lyapunov functions for nonlinear continuous-time differential equations is developed via a new, Massera-type construction. This construction is enabled by imposing a finite-time criterion on the integrated function. By means of this approach, we relax the assumptions of exponential stability on the system dynamics, while still allowing integration over a finite time interval. The resulting Lyapunov function can be computed based on any K infinity function of the norm of the solution of the system. In addition, we show how the developed converse theorem can be used to construct an estimate of the domain of attraction. Finally, a range of examples from literature and biological applications such as the genetic toggle switch, the repressilator and the HPA axis are worked out to demonstrate the efficiency and improvement in computations of the proposed approach.