Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 380304, 12 pages
http://dx.doi.org/10.1155/2012/380304
Research Article

Practical Stability in the 𝑝 th Mean for Itô Stochastic Differential Equations

Enguang Miao,1 Huisheng Shu,1 and Yan Che2,3

1Department of Applied Mathematics, Donghua University, Shanghai 201620, China
2Department of Electronics and Information Engineering, Putian University, Fujian, Putian 351100, China
3College of Information Sciences and Technology, Donghua University, Shanghai 201620, China

Received 29 June 2011; Accepted 6 September 2011

Academic Editor: Zidong Wang

Copyright © 2012 Enguang Miao et al. 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 𝑝 th mean practical stability problem is studied for a general class of Itô-type stochastic differential equations over both finite and infinite time horizons. Instead of the comparison principle, a function 𝜂 ( 𝑡 ) which is nonnegative, nondecreasing, and differentiable is cooperated with the Lyapunov-like functions to analyze the practical stability. By using this technique, the difficulty in finding an auxiliary deterministic stable system is avoided. Then, some sufficient conditions are established that guarantee the 𝑝 th moment practical stability of the considered equations. Moreover, the practical stability is compared with traditional Lyapunov stability; some differences between them are given. Finally, the results derived in this paper are demonstrated by an illustrative example.