Abstract
This paper deals with non-perturbed and perturbed systems of nonlinear differential systems of first order with multiple time-varying delays. Here, for the considered systems, easily verifiable and applicable uniformly asymptotic stability, integrability, and boundedness criteria are obtained via defining an appropriate Lyapunov–Krasovskiĭ functional (LKF) and using the Lyapunov–Krasovskiĭ method (LKM). Comparisons with a former result that can be found in the literature illustrate the novelty of the stability theorem and show new contributions to the qualitative theory of solutions. A discussion of two illustrative examples and the obtained results are presented.
Highlights
Criteria for Delay Differential Systems.The research of systems of delay differential equations (DDEs) with multiple constant and time-varying delays is always a challenging field of study
This is due to the fact that the system of DDEs can be frequently found in many fields such as mechanics, artificial neural networks power systems, medicine, physics, biology, population ecology, engineering, and so forth
We study the uniformly asymptotic stability of zero solution and the integrability of the norm of solutions of the following unperturbed nonlinear system of DDEs via Theorem 3 and Theorem 4, respectively: ẋ (t) = A(t) x (t) + BF ( x (t − h1 (t))) + CG ( x (t − h2 (t)))
Summary
In Tunç [19,20,21] and Tunç and Tunç [22,23,24,25], stability, boundedness, and some other properties of solutions of various non-linear differential systems of second order without or with delay are investigated by the second Lyapunov method and integration techniques. On the basis of the obtained results, the author proves an analogue of the Krein’s theorem on stability of solutions to the linear system of differential equations with distributed delay. In Zhang and Jiang [28], by constructing a suitable Lyapunov functional and using some analytical techniques, the authors obtain sufficient conditions for the global exponential stability of zero solution to a class of differential systems of first order with delay.
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