Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

Cemil Tunç,Jen-Chih Yao,Yuanheng Wang,Osman Tunç

doi:10.3390/math9111196

Abstract

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

Highlights

It can be observed that numerous processes, both natural and human-made, in biology, interaction of species, population dynamics, microbiology, distributed networks, learning models, mechanics, medicine, nuclear reactors, chemistry, distributed networks, epidemiology, physics, engineering, economics, physiology, viscoelasticity, as well as many others, involve time delays
It is worth mentioning that especially delay differential equations (DDEs) of first and second order with constant and time-varying delays can be encountered intensively during investigations and applications
To the best of available information, it should be noted that from the theory of DDEs, we know that analytically solving DDEs with time-varying delays is a very difficult mathematical task

Summary

Introduction

It can be observed that numerous processes, both natural and human-made, in biology, interaction of species, population dynamics, microbiology, distributed networks, learning models, mechanics, medicine, nuclear reactors, chemistry, distributed networks, epidemiology, physics, engineering, economics, physiology, viscoelasticity, as well as many others, involve time delays. The Lyapunov’s second method, Lyapunov–Krasovskiımethod, Razumikhin method and fixed point method can be effectively used to investigate the stability and some other properties of solutions of ODEs, DDEs, neutral and advanced functional differential equations. In this paper, motivated by the system of DDEs (1), the result of Tian and Ren ([15], Theorem 1) and those in the bibliography of this paper, as an alternative to the linear system of DDEs (1), we consider a nonlinear system of DDEs with time-varying delay as follows:. We summarize the aim of this paper by the following items, respectively: We investigate the uniformly asymptotically stability of zero solution of the system of DDEs (1), see Theorem 3 To investigate this problem, we define a very different LKF from that in Tian and Ren [15]. Two new examples and graphs of their solutions are provided to show applications of Theorems 3–6

Basic Result

Asymptotic Stability

Uniformly Asymptotic Stability and Integrability

Boundedness

Discussion and Contribution

Conclusions