The asymptotic stability of systems with delay

Attraction Domains of Delay Systems: Construction by the Lyapunov Function Method

“…10 2005 Since such a set is the maximal set satisfying Definition 1, it is called the complete attraction domain. The invariance condition is not contained in the definition of the attraction domain given in [3,8]. As a rule, whether this condition is included or not depends on the properties of the applied methods.…”

“…The Lyapunov function of the corresponding system without delay is usually used as a Lyapunov-Razumikhin function [6,8,10,19]. In [19], the Lyapunov-Razumikhin function is constructed by the method, in which this function is determined as a quadratic form with a matrix satisfying the Lyapunov inequality for some system without delay.…”

“…In this paper, we use a method that is close to the method of [7][8][9][10] in idea. Since the attraction domain in the papers cited above is defined differently, we give a general definition of the attraction domain and its modification that is convenient for the application of the methods we shall use.…”

“…The attraction domain of a delay system describing a nuclear reactor with two linked active zones is estimated in [7] using the square of the Euclidean norm as a Lyapunov-Razumikhin function. In [8], the attraction domain of a general delay system is estimated from below by the attraction domain of the corresponding system without delay. The maximal delay guaranteeing the existence of such an estimate is determined.…”

“…In [7,8], attraction domains of delay systems are constructed with a function satisfying the conditions of the Razumikhin asymptotic stability theorem (Theorem 5 in [6]). We refer to this function as the Lyapunov-Razumikhin function.…”