Research Article
The Stability and Stabilization of Stochastic Delay-Time Systems
Gang Li
1,2
and Ming Chen
1
1
School of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 265590, China
2
College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 265590, China
Correspondence should be addressed to Gang Li; ligangccm@163.com
Received 17 December 2013; Revised 17 April 2014; Accepted 1 May 2014; Published 3 June 2014
Academic Editor: Wuquan Li
Copyright © 2014 G. Li and M. Chen. is 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.
e aim of this paper is to investigate the stability and the stabilizability of stochastic time-delay deference system. To do this, we
use mainly two methods to give a list of the necessary and sucient conditions for the stability and stabilizability of the stochastic
time-delay deference system. One way is in term of the operator spectrum and -representation; the other is by Lyapunov equation
approach. In addition, we introduce the notion of unremovable spectrum of stochastic time-delay deference system, describe the
PBH criterion of the unremovable spectrum of time-delay system, and investigate the relation between the unremovable spectrum
and the stabilizability of stochastic time-delay deference system.
1. Introduction
e stochastic time-delay system is one of the fundamental
research branches in the theory of control systems, which is
usually applied in the elds of electronics, machinery, chemi-
cals, life sciences, economics, and so on. As is well known, the
stability is an essential concept in linear system theory, which
is relative to the system matrix root-clustering in subregions
of the complex plane, and also the spectral operator approach
iseectiveinthestudyoftheeigenvalueplacementofa
matrix (see [1]). Since two classic books [2, 3]appeared,
stochastic stability and stabilization of It
ˆ
o dierential systems
have been investigated by many researchers for several
decades; we refer the reader to [4–6] and the references
therein. More specically, for linear time-invariant stochastic
(LTIS) systems, most work is concentrated on the investi-
gation of mean square stabilization, which has important
applications in system analysis and design. Some necessary
and sucient conditions for the mean square stabilization
of LTIS systems have been obtained in terms of generalized
algebraic Riccati equation (GARE) or linear matrix inequality
(LMI) in [7–14] or spectra of some operators in [5, 9, 15].
For the stochastic delay-time systems, the present results
were mainly obtained by Lyapunov functional approach. We
concentrate our attention upon the stability and stabilization
of stochastic systems by the operator spectrum.
e structure of this paper is as follows. In Section 2,
with the aid of the operator spectrum, -representation, and
Lyapunov equation approach, some necessary and sucient
conditions are given for the stability and the stabilizability of
stochastic delay-time systems. In Section 3,theunremovable
spectrum of stochastic delay-time systems is introduced, and
PBH criterion of the stabilizability of stochastic delay-time
systems is presented.
For convenience, we adopt the following traditional
notations.
𝑛
: the set of all symmetric matrices, whose
components may be complex; ={0,1,2,...};
(Ker()):
the transpose (kernel space) of a matrix ; ≥0(>0)is a
positive semidenite (positive-denite) symmetric matrix ;
: identity matrix; (): spectral set of the operator or matrix
;
−
(
−0
): the open le (closed le) hand side complex
plane. (0,)={|<}; ⋅is the
2
-norm;
2
F
𝑡
(
+
,
𝑛
𝑥
): space of nonanticipative stochastic processes
() ∈
𝑛
𝑥
withrespecttoanincreasing-algebra {F
𝑡
}
𝑡≥0
satisfying ()
2
<∞. Finally, we make the assumption
throughout this paper that all systems have real coecients.
2. The Stability of Stochastic
Delay-Time Systems
In this section, we will investigate the stability and stabiliz-
ability of the stochastic time-delay deference system using
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 272745, 8 pages
http://dx.doi.org/10.1155/2014/272745
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