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

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.

Stability tests and solution estimates for non-linear differential equations

“…The obtained inequality demonstrates that the second condition, i.e., (A2), of ( [32, Theorem 4.2.9], Tunç et al [23,Theorem 1]) is satisfied.…”

“…Thus, the locally Lipschitz condition in x t is satisfied by the LKF ∆ 1 (t, x t ). Thus, condition (A1) of ( [32, Theorem 4.2.9], Tunç et al [23,Theorem 1]) is held. For the next step, by virtue of the definition of Z 1 (t, x t ) and condition (H2), we have…”

“…(10) Thus, it is obvious that d dt ∆ 1 (t, x t ) is negative definite. From the inequality (10), it follows that assumption (A3) of ( [32, Theorem 4.2.9], Tunç et al [23,Theorem 1]) is satisfied. Thus, all the assumptions of ( [32, Theorem 4.2.9], Tunç et al [23,Theorem 1]) are held.…”

“…In [13], the method of proof is depending upon the definition of a very interesting suitable LKF. Later, Tunç et al [23] dealt with the nonlinear system of DDEs: dx dt =A(t)x(t) + BF (x(t − h(t)))…”

“…In [23], three theorems, which have sufficient conditions, were proved on the UAS and integrability of solutions, when E(.) ≡ 0 in (2), and the boundedness of the solutions of (2), when E(.)…”