Lyapunov-Razumikhin techniques for state-dependent delay differential equations

TL;DR

This paper develops Lyapunov-Razumikhin techniques to analyze stability of state-dependent delay differential equations, introducing new theorems and applying them to models including Hayes equation.

Contribution

It introduces novel Lyapunov-Razumikhin theorems for asymptotic stability of state-dependent DDEs, simplifying the analysis process.

Findings

New asymptotic stability theorem based on contradiction and Arzela-Ascoli.

Application to Hayes equation demonstrating stability regions.

Lower bounds on basin of attraction for the model.

Abstract

We present Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin methods. Our results apply to DDEs with multiple discrete state-dependent delays, which may be nonautonomous for the Lyapunov stability result, but autonomous (or periodically forced) for the asymptotic stability result. Our main technique is to replace the DDE by a nonautonomous ordinary differential equation (ODE) where the delayed terms become source terms in the ODE. The asymptotic stability result and its proof are entirely new, and based on a contradiction argument together with the Arzela-Ascoli theorem. This approach alleviates the need to construct auxiliary functions to ensure the asymptotic contraction, which is a feature of all other Lyapunov-Razumikhin asymptotic stability results of which we are aware. We apply our results to a state-dependent model equation which includes Hayes equation as a special case, to directly establish asymptotic stability in parts of the stability domain along with lower bounds on the size of the basin of attraction.