Dissipativity and Exponential Stability of $\theta$-Methods for Singularly Perturbed Delay Differential Equations with a Bounded Lag

Abstract

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small $\epsilon>0$. We will study the numerical solution defined by the linear $\theta-$method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small $\epsilon>0$ if and only if $\theta=1$.