Lyapunov Stability of the Generalized Stochastic Pantograph Equation

Journal of Mathematics, Jun 2018

The purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different kinds of stochastic Lyapunov stability to specially chosen functional spaces. To prove stability, we check that the solutions of the equation belong to a suitable space of stochastic processes, instead of searching for an appropriate Lyapunov functional. This gives us possibilities to study moment stability, stability with probability 1, and many other stability properties in an efficient way. We show by examples how this approach works in practice, putting emphasis on delay-independent stability conditions for the generalized stochastic pantograph equation. The framework can be applied to any stochastic functional differential equation with finite dimensional initial conditions.

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Lyapunov Stability of the Generalized Stochastic Pantograph Equation

Lyapunov Stability of the Generalized Stochastic Pantograph Equation Ramazan Kadiev1 and Arcady Ponosov2 1Dagestan Research Center of the Russian Academy of Sciences and Department of Mathematics, Dagestan State University, Makhachkala 367005, Russia 2Norwegian University of Life Sciences, Faculty of Sciences and Technology, P.O. Box 5003, N-1432 Ås, Norway Correspondence should be addressed to Arcady Ponosov; on.ubmn@idakra Received 31 January 2018; Accepted 14 May 2018; Published 19 June 2018 Academic Editor: Qamar Din Copyright © 2018 Ramazan Kadiev and Arcady Ponosov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different kinds of stochastic Lyapunov stability to specially chosen functional spaces. To prove stability, we check that the solutions of the equation belong to a suitable space of stochastic processes, instead of searching for an appropriate Lyapunov functional. This gives us possibilities to study moment stability, stability with probability 1, and many other stability properties in an efficient way. We show by examples how this approach works in practice, putting emphasis on delay-independent stability conditions for the generalized stochastic pantograph equation. The framework can be applied to any stochastic functional differential equation with finite dimensional initial conditions. 1. Introduction In this paper we study Lyapunov stability of the stochastic pantograph equation (see, e.g., [1–3]):where , and its generalizations (see (21) in Section 4). A very good and comprehensive description of the role of the classical pantograph equation and its stochastic counterpart, including historical comments, can be found in the paper [2]. Let us only mention that generalizations of the pantograph equations have also attracted attention of many mathematicians; see, e.g., [4–11] and the references therein. Stability analysis of (1) and (21) has a special feature: the delay is unbounded, so that many methods, including those based on Lyapunov-Krasovskii functionals, are inapplicable. One uses therefore various special techniques, which can, e.g., be found in the papers [12, 13] (the stochastic case) and [5] (the deterministic case). These techniques help to produce verifiable stability criteria, mostly in the case of the classic pantograph equation (1). Our approach goes back to the framework developed in the monographs [14] (for linear differential equations in Banach spaces) and [15] (for linear deterministic functional differential equations), where Lyapunov stability is replaced by input-to-state stability, i.e., the property of the equation where its solutions belong to certain linear topological spaces and continuously depend (in the corresponding topology) on the initial data. In the stochastic case this approach is outlined in [16]. On the other hand, (1) and (21) possess a very specific property: their initial conditions are finite dimensional, i.e., identical to the ones for ordinary differential equations. This considerably simplifies the analysis of the input-to-state stability, as all linear finite dimensional operators are bounded, and we only need to prove that all solutions of the equation belong to a certain topological space. For brevity, we will call this property -stability keeping in mind that this is, in fact, a particular case of the input-to-state stability for linear equations with finite dimensional spaces of initial data. The idea of how to verify the property of input-to-state stability for linear deterministic functional differential equations goes back to the papers of N.V.Azbelev and his students (see [15] and the references therein) who call their technique the -method. It is somewhat similar to Lyapunov’s direct method. But instead of seeking a Lyapunov function(al) one aims to find a suitable reference equation which possesses the prescribed asymptotic property and which then is used to regularize the original equation. Like Lyapunov’s method, the -method also provides necessary and sufficient stability conditions. The -method proven to be rather efficient for many classes of delay equations, especially those where searching for Lyapunov functionals seems to be difficult. Equations with infinite delays can serve as a prominent example of such a class. In [17], the method was for the first time applied to linear stochastic functional differential equations and developed further by the authors in the series of publications (see the review article [16]). The first efficient stability conditions f (...truncated)


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Ramazan Kadiev, Arcady Ponosov. Lyapunov Stability of the Generalized Stochastic Pantograph Equation, Journal of Mathematics, 2018, 2018, DOI: 10.1155/2018/7490936