Asymptotic stability of solutions of a class of systems of nonlinear differential equations with delay (Q1759272)

The authors study systems of differential equations with delay whose right-hand sides are represented as sums of potential and gyroscopic components of vector fields. They assume that, in the absence of delay, zero solutions of the considered systems are asymptotically stable. By Lyapunov's direct method, using the Razumikhin approach, they prove that, in the case of essentially nonlinear equations, the asymptotic stability of the zero solution is preserved for any value of the delay. The duration of the transient processes for asymptotically stable systems with delay is estimated. The stability of the perturbed systems is studied.