Exponential stability of linear delayed differential systems

Permanence for Nonautonomous Differential Systems with Delays in the Linear and Nonlinear Terms

“…It is worthwhile mentioning that, in the last few years, the stability of nonautonomous linear DDEs has received a great deal of attention, and several methods have been used to obtain explicit sufficient conditions for the asymptotic and exponential asymptotic stability of a general linear system (4), see e.g., [21,31] and references therein. Actually, both delay independent and delay-dependent criteria for the stability of linear DDEs were given in e.g., [21,31], the latter also with possible infinite delays. Since the exponential stability of ( 4) is a key ingredient to show the permanence of (8), this leads us to two natural lines of future research, explained below.…”

Exponential Stability for a Class of Linear Delay Differential Systems Under Logic Impulsive Control

“…The diagonal terms condition a k ii (t) ≥ 0, i = 1, 2, • • • , n is always assumed, which means the dominance of diagonal elements (see [4], [5]). This situation was improved in [6] and [7].…”

Stability for nonautonomous linear differential systems with infinite delay

“…For the last few years, there has been a renewed interest in the analysis of stability of nonautonomous linear DDEs, and several methods and tools have been proposed, see e.g. [2,4,5,14,15,17,18,21,28,29] and references therein. The main goal of this paper is to obtain new explicit sufficient conditions for the asymptotic and exponential asymptotic stability of a general linear system (1.1), which improve and generalize some criteria in recent literature.…”

“…Two types of criteria will be obtained, depending on whether system (1.1) possesses diagonal terms without delay which dominate the effect of the delayed terms, or not. The latter case is not often treated in the literature, although there have been some recent interesting developments in this area [2,4,5,14,33], following different approaches: 3/2-stability conditions, Lyapounov functionals, theory of monotone systems, asymptotic equivalence to linear ODEs, etc.…”

“…We shall show that some of the results in [17,18,21,28] are a simple consequence of the stability criteria established here for ndimensional linear DDEs. On the other hand, there are several recent works where explicit conditions for the exponential asymptotic stability of linear DDEs depending on the size of delays were found, see Berezansky and Braverman [2,3] for the scalar case and Berezansky et al [4,5] for n-dimensional systems, as well as references therein. However, in [2,4] only the situation of time-varying bounded and discrete delays was considered -two constraints removed in this work.…”