Asymptotic Stability of Solutions to a Class of Second-Order Delay Differential Equations

Gennadii V Demidenko,Inessa I Matveeva

doi:10.3390/math9161847

Abstract

We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.

Highlights

We consider the following second-order delay differential equation: y00 (ξ ) + αy0 (ξ ) + ( β + ωφ(ωξ ))ψ(y(ξ )) + γy(ξ − τ ) + σy0 (ξ − τ ) = 0, ξ > 0, (1)where α > 0, β < 0, ω > 0, γ, σ are parameters, φ(s) is a continuous T-periodic function such that the following equality holds: Citation: Demidenko, G.V.; ZTMatveeva, I.I
Using the results mentioned above, the authors investigated the asymptotic stability of the zero solution to the following nonlinear systems [6]: dx
On the basis of these studies, introducing special Lyapunov–Krasovskii functionals, the authors investigated the stability of solutions to some classes of delay differential equations with periodic coefficients in linear terms

Summary

Introduction

We consider the following second-order delay differential equation: y00 (ξ ) + αy0 (ξ ) + ( β + ωφ(ωξ ))ψ(y(ξ )) + γy(ξ − τ ) + σy0 (ξ − τ ) = 0, ξ > 0, (1) In [5], the following systems of linear differential equations were considered: creativecommons.org/licenses/by/ The following criterion for the asymptotic stability of the zero solution to (3) was established in terms of solvability of a special boundary value problem for the Lyapunov differential equation d

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Results

Conclusion