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Journal of Function Spaces
Volume 2016, Article ID 8901563, 12 pages
http://dx.doi.org/10.1155/2016/8901563
Review Article

Input-to-State Stability of Linear Stochastic Functional Differential Equations

1Dagestan Research Center, The Russian Academy of Sciences and Department of Mathematics, Dagestan State University, Makhachkala 367005, Russia
2Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P.O. Box 5003, 1432 Ås, Norway

Received 11 February 2016; Accepted 16 March 2016

Academic Editor: Kishin Sadarangani

Copyright © 2016 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 show how asymptotic properties, first of all stochastic Lyapunov stability, of linear stochastic functional differential equations can be studied via the property of solvability of the equation in certain pairs of spaces of stochastic processes, the property which we call input-to-state stability with respect to these spaces. Input-to-state stability and hence the desired asymptotic properties can be effectively verified by means of a special regularization, also known as “the -method” in the literature. How this framework provides verifiable conditions of different kinds of stochastic stability is shown.