B. G. Grebenshchikov


The paper studies differential equations with linear delay of neutral type. Equations with linear delay occur in problems of mechanics, biology, and economics. A feature of such equations is that the delay is unbounded, which significantly reduces the applicability of traditional methods for studying stability problems for similar systems. One of the approaches to studying asymptotic properties is to replace the argument. The system is reduced to a system with a constant delay, but in this case an exponential factor appears on the right side of the system obtained, and the right side of the resulting system becomes unbounded as $t>\infty$. The asymptotic properties of systems without neutral terms were studied by the authors earlier. Taking into account the asymptotic properties of these systems (without neutral terms on the right side), an analysis of the asymptotic properties (boundedness, stability and asymptotic stability) of some systems of a neutral type is carried out. Since the property of stability is a more subtle property than the property of asymptotic stability, we study a system of neutral type with perturbations, which is simply stable when unperturbed.


stability; asymptotic stability; Lyapunov-Krasovsky functionals


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