On the asymptotic stability of solutions of nonlinear systems with delay

Abstract

Under study are systems of homogeneous differential equations with delay. We assume that in the absence of delay the trivial solutions to the systems under consideration are asymptotically stable. Using the direct Lyapunov method and Razumikhin's approach, we show that if the order of homogeneity of the right-hand sides is greater than 1 then asymptotic stability persists for all values of delay. We estimate the time of transitions, study the influence of perturbations on the stability of the trivial solution, and prove a theorem on the asymptotic stability of a complex system describing the interaction of two nonlinear subsystems.

On estimation of rates of convergence in Lyapunov–Razumikhin approach

“…The previous evaluations by the Lyapunov-Razumikhin approach of the convergence rate for an asymptotically stable system have been presented in [3,12,14].…”

“…The latter method has been proven to be equivalent to the asymptotic stability property for some particular classes of the time-delay systems [4,15,16], and it can also be used to establish finite-time stability [13]. The former approach is only sufficient for the asymptotic stability [8,10], and it is less intuitive while obtaining the rate of solution convergence [3,6,14]. An advantage of Lyapunov-Razumikhin approach with respect to Lyapunov-Krasovskii one is that in many nonlinear cases it is more simple to find a Lyapunov-Razumikhin function than a Lyapunov-Krasovskii functional [5,7] (e.g., a Lyapunov function for the delay-free case can be tested).…”

Weighted Homogeneity for Time-Delay Systems: Finite-Time and Independent of Delay Stability

“…In those works an extension of the homogeneity theory for time-delay systems has been proposed. Applications of the conventional homogeneity framework for analysis of time-delay systems (considering delay as a kind of perturbation, for instance) have been carried out even earlier in [17], [18], [19], [20], [21].…”

Stability Analysis of Ships’ Movement Along Optimal Routes

“…The equalities eqns (7)-(9) are also true at points ∈ ⋅ ℎ, 1,2, … of the break of function . We are going to construct a complete type functional, positively defined in some neighborhood of zero, and which solve problem of the asymptotic stability of the zero solution of eqn (6).…”

“…In this article, we use the methodology of Lyapunov-Krasovskii functional to analyze the stability of a class of homogeneous differential-difference systems of neutral type. It is proved that if the corresponding homogeneous system with zero delays is asymptotically stable, then the trivial solution of a homogeneous retarded type system is also asymptotically stable for any limited delays [6], [7]. Basing on these results and results [8]- [11], we constructed the complete type Lyapunov-Krasovskii functional, which are suitable for analysis of every system within the class under consideration.…”