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

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-Krasovskii functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov-Krasovskii 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.

Automated summary

This paper considers a class of systems of linear and non-linear delay differential equations of first order with time-varying delay. 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 are obtained. Two appropriate Lyapunov-Krasovskii functionals are constructed as main tools in the proofs, eliminating the use of Gronwall's inequality. The established conditions are more general, less restrictive, and optimal for applications compared to recent results in the literature.