Lyapunov-Razumikhin techniques for state-dependent delay differential equations

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 Arzelà-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.

Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

“…This leads to numerically verifiable conditions for the global asymptotic stability of the steady state of (2) which are given in Theorem 3.7. This establishes global asymptotic stability of the steadystate solution in most of the region where we previously [15] showed local asymptotic stability 2…”

“…We also assume that µ + σ < 0, which ensures that the deviating argument α(t, u(t)) = t − a − cu(t) t, i.e. the delayed term does not become an advance [15].…”

“…In this work we restrict attention to the case where µ < 0, which in [15] was shown to ensure that all solutions of the SDDE remain bounded; this is a necessary property for the techniques that we will deploy. This result is stated as Theorem 2.1 below.…”

“…Many of the results and examples are tailored to constant delay problems, but amongst others [19,25,30] include results for time-dependent delays. In [15] we present Razumikhin-type Lyapunov and asymptotic stability results tailored to SDDEs.…”

“…Mallet-Paret and Nussbaum investigate the existence and form of the slowly oscillating periodic solutions of a singularly perturbed version of (2) in detail in [21,22,23,24]. This DDE is also known as the sawtooth equation and has also been studied in [13,14,15,18].…”