its resolvent operator. For the remaining cases satisfying (35) the au-thors suggest in the conclusion solving numerically an equation that is related to the eigenvalues of the system. Although Theorem III.2 ex-cludes the casek21 = h1v2=v1h2andk22 = h2v1=v2h1, it has the advantage of treating all the remaining cases satisfying (35) using only a matrix inequality.
V. CONCLUSION
We provided tools that facilitate checking the exponentially stability property of a class of BCS. We showed that by using results of [14] and [16] it is easy to select the input and outputs of a BCS. Therefore, we use those results on boundary port Hamiltonian systems to define inputs and outputs for our class of BCS. Once this is done, checking for exponential stability follows easily. The main idea behind the proof consists in using a multiplier common to the whole class of BCS. This multiplier only depends on the norm of the co-energy variables at the boundary of the spatial domain. In this way one avoids searching for different multipliers every time the system or the boundary conditions are changed. This simplifies drastically the verification of the expo-nential stability property, as can be seen already from the examples in Section IV. Also the proof of the results of [22] and [23] can be sim-plified by using our results.
Even though the results are only valid for a class of one-dimensional systems, the authors believe that the approach has potential to be ex-tended to 2-D and 3-D systems. The key point being the definition and selection of the boundary port variables. Some ideas about this are pre-sented in [16, Ch. 8]. However, this still requires more research.
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Delay-Dependent Exponential Stability of Neutral Stochastic Delay Systems
Lirong Huang and Xuerong Mao
Abstract—This technical note studies stability of neutral stochastic delay systems by linear matrix inequality approach. Delay-dependent criterion for exponential stability is presented and numerical examples are conducted to verify the effectiveness of the proposed method.
Index Terms—Exponential stability, linear matrix inequalitys (LMIs), neutral systems, stochastic systems, time delay.
I. INTRODUCTION
Many dynamical systems are described with neutral functional differential equations that include neutral delay differential equations [19]. These systems are called neutral-type systems or neutral systems. Motivated by chemical engineering systems as well as theory of aero elasticity, studies on deterministic neutral systems have been of research interest over the past decades [3]–[11], [21]. As stochastic modelling has come to play an important role in many branches of science and industry, neutral stochastic delay systems have been intensively studied over recent year [10]–[17]. Mao [14]–[17] initiated the study of exponential stability of neutral stochastic functional equations, developed the Razumikhin-type theorems further for exponential stability of neutral stochastic functional equations and studied asymptotic properties of neutral stochastic delay differential equations [1]. More recently, Luoet al.[12] proposed new criteria on exponential stability of neutral stochastic delay differential equations while Chenet al.[2] studied delay-dependent stability of neutral sto-chastic delay systems. However, the stability result in [2] employed an
Manuscript received July 22, 2008; revised September 03, 2008 and September 23, 2008. Current version published January 14, 2009. This work was supported by UK ORSAS and the University of Strathclyde. Recom-mended by Associate Editor J.-F. Zhang.
The authors are with the Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH, U.K. (e-mail: [email protected]; [email protected]).
Digital Object Identifier 10.1109/TAC.2008.2007178
assumption on the difference operator matrix, which is also assumed in other results [4] and [18] but may be restrictive in many cases (see Examples 1 and 2). As is known, delay-independent results may be conservative when the size of time delay is small. This technical note studies problem of delay-dependent stability of neutral stochastic delay systems. An exponential stability criterion is established by linear matrix inequality (LMI) approach. Numerical examples are conducted to verify the effectiveness of our proposed method.
II. PROBLEMSTATEMENT
Throughout the technical note, unless otherwise specified, we will employ the following notation. Let(; F; fFtgt0; )be a complete probability space with a natural filtrationfFtgt0and [1]be the ex-pectation operator with respect to the probability measure. Letw(t) be a scalar Brownian motion defined on the probability space. IfAis a vector or matrix, its transpose is denoted byAT. IfP is a square matrix,P > 0(P < 0) means thatP is a symmetric positive (nega-tive) definite matrix of appropriate dimensions whileP 0(P 0) is a symmetric positive (negative) semidefinite matrix.Istands for the identity matrix of appropriate dimensions. Denote byM(1)andm(1)
the maximum and minimum eigenvalue of a matrix respectively. Letj1j denote the Euclidean norm of a vector and its induced norm of a matrix. Unless explicitly specified, matrices are assumed to have real entries and compatible dimensions. Leth 0andC([0h; 0]; Rn)denote the family of all continuousRn-valued functions'on[0h; 0]with the normk'k = supfj'()j : 0h 0g. LetCFb ([0h; 0]; Rn) be the family of allF0-measurable boundedC([0h; 0]; Rn)-valued random variables = f() : 0h 0g.
Let us consider ann-dimensional neutral stochastic delay system d[x(t) 0 Cx(t 0 h1)]
= [A0x(t) + A1x(t 0 h1) + A2x(t 0 h2)] dt
+ [H0x(t) + H1x(t 0 h1)+H2x(t 0 h2)]dw(t) (1) on t 0with initial datax0 = fx() : 0h 0g = 2 Cb
F ([0h; 0]; Rn), wherex(t) 2 Rnis the state vector; positive scalar
constantsh1,h2are time delays of the system andh = maxfh1; h2g; C,AiandHi,i = 0, 1, 2, are known matrices.
Denote
f(t) = A0x(t) + A1x(t 0 h1) + A2x(t 0 h2)
g(t) = H0x(t) + H1x(t 0 h1) + H2x(t 0 h2) (2) for allt 0. One can observe that
jf(t)j2 K
fkxtk2; jg(t)j2 Kgkxtk2 (3) for all t 0, wherext = fx(t + ) : 0h 0g, Kf = 3 2
i=0jAij2andKg= 3 2i=0jHij2. This implies that bothf('; t)
andg('; t)satisfy the local Lipschitz condition and the linear growth condition. It is easy to verify, by the way of induction proposed in the proof of Theorem 3.1, p208, [16], that there exists a unique continuous solution denoted byx(t; )to neutral stochastic delay differential (1). The objective of this technical note is to establish sufficient con-ditions for robust exponential stability of system (1). It should be pointed out that, for simplicity only, we do not consider uncertainties in our models. The proposed method can be easily extended to those cases with norm-bounded uncertainties in parametersAiandHi. The method can also be applied to systems with multiple and distributed delays.
At the end of this section, let us introduce the following definitions and lemmas that are useful for the development of our results.
Definition 1: The neutral stochastic delay system (1) is said to be exponentially stable in mean square if there is a positive constant such that [16]
lim sup
t!1
1
tlog jx(t; )j2 0 : (4) Definition 2: The neutral stochastic delay system (1) is said to be almost surely exponentially stable if there is a positive constantsuch that [16]
lim sup
t!1
1
tlog jx(t; )j 0 : (5) Lemma 1: ([20]) For any constant matrixM 2 Rq2l, inequality
2uTMv ruTMGMTu + 1
rvTG01v ; u 2 Rq; v 2 Rl holds for any pair of symmetric positive definite matrixG 2 Rl2land positive numberr > 0.
Lemma 2: ([6]) For any pair of symmetric positive definite constant matrixG 2 Rl2land scalarr > 0, if there exists a vector function v : [0; r] ! Rl such that integrals r
0 vT(s)Gv(s)dsand 0rv(s)ds
are well defined, then the following inequality holds:
r r
0 v
T(s)Gv(s)ds r 0 v(s)ds
T
G r
0 v(s)ds :
III. DELAY-DEPENDENTEXPONENTIALSTABILITY
Delay-dependent stability of neutral deterministic delay systems has been intensively studied over recent years [3]–[5], [8], [11], [18]. How-ever, relatively little is known about delay-dependent stability of neu-tral stochastic delay systems. DenoteA0 = A0,A1 = A0C + A1,
A2 = A2,H0 = H0,H1 = H0C + H1,H2 = H2,A = 2i=0Ai
andH = 2i=0Hi. Sufficient conditions for delay-dependent expo-nential stability of system (1) are proposed as follows.
Theorem 1: The neutral stochastic delay system (1) is mean-square exponentially stable and is also almost surely exponentially stable pro-vided that there exist matricesP11 > 0,Qk > 0,Rk > 0,S > 0,
Tk> 0,P21,P22,P23,P31,P32,P33andk = 1, 2 such that LMI is (6), shown at the bottom of the next page, where
011= P21TA + ATP21+ P31TH + HTP31+ S + T1+ T2
012= ATP22+ HTP32+ P110 P21T
013= ATP23+ HTP330 P31T ; 018= (S + T1+ T2)C
022= 0 P22T 0 P22+ h1Q1+ h2Q2; 023= 0P230 P32T
033= 0 P33T 0 P33+ P11+ h1R1+ h2R2
088= 0 S + CT(S + T1+ T2)C
L11= P21TA1+ P31TH1; L12= P21TA2+ P31TH2
L21= P22TA1+ P32TH1; L22= P22TA2+ P32TH2
L31= P23TA1+ P33TH1; L32= P23TA2+ P33TH2
and entries denoted by3can be readily inferred from symmetry of the matrix.
Proof: To simplify the expression, we define
for allt 0. With notations (2) and (7), we can rewrite the unforced system (1) as
d(t) = f(t)dt + g(t)dw(t) (8) ont 0with initial data.
So we have
(t2) 0 (t1) = t
t f(s)ds + g(s)dw(s)
(9)
for allt2 t1 0.
By (2) and (9), we can observe that
f(t) = 2
i=0
Ai(t) 0
2
i=1
Ai (t) 0 (t 0 hi)
+
2
i=1
AiCx(t 0 h10 hi)
= A (t) 0 2
i=1
Ai
t
t0h f(s)ds + g(s)dw(s)
+ 2
i=1
AiCx(t 0 h10 hi); (10)
g(t) = H(t) 0
2
i=1
Hi
t
t0h f(s)ds + g(s)dw(s)
+ 2
i=1
HiCx(t 0 h10 hi) (11)
for allt h. Choose a Lyapunov–Krasovskii functional candidate for system (8) as follows:
V (t) = 5
j=1
Vj(t) ; t h (12)
where
V1(t) = T(t)P11(t)
V2(t) = 2
i=1 t
t0h (s 0 t + hi)f T(s)Q
if(s)ds
V3(t) = 2
i=1 t
t0h (s 0 t + hi)g T(s)R
ig(s)ds
V4(t) = t
t0h x
T(s)Sx(s)ds
V5(t) = 2
i=1 t
t0h 0h x T(s)T
ix(s)ds :
By Itô’s lemma, we have
dV (t) = LV (t)dt + (t)dw(t) (13) where
LV (t) = 5
j=1
LVj(t) = 2T(t)P11f(t) + gT(t)P11g(t)
+ 5
j=2
_Vj(t)
(t) = 2T(t)P
11g(t) : (14)
Denote
y(t) = (t) f(t) g(t)
and P =
P11 0 0
P21 P22 P23
P31 P32 P33
: (15)
By equalities (10) and (11), we have 2T(t)P
11f(t)
= yT(t)(PTA+ATP )y(t)02yT(t) 2 i=1
PT[ 0 AT i HiT]T
1 t
t0h f(s)ds + g(s)dw(s) + Cx(t 0 h10 hi)
(16)
where
A =
0 I 0
A 0I 0
H 0 0I
PTA + ATP =
PA1 PA2 PA3
3 0PT
220 P22 0P32T 0 P23
3 3 0PT
330 P33
with PA1 = P21TA + ATP21 + P31TH + HTP31, PA2 =
ATP
22+ HTP32+ P110 P21T,PA3= ATP23+ HTP330 P31T and
PT[ 0 AT
i HiT]T = [ LT1i L2iT LT3i]T fori = 1, 2.
0 =
011 012 013 h1L11 h2L12 L11 L12 018 L11C L12C
3 022 023 h1L21 h2L22 L21 L22 0 L21C L22C
3 3 033 h1L31 h2L32 L31 L32 0 L31C L32C
3 3 3 0h1Q1 0 0 0 0 0 0
3 3 3 0 0h2Q2 0 0 0 0 0
3 3 3 0 0 0R1 0 0 0 0
3 3 3 0 0 0 0R2 0 0 0
3 0 0 0 0 0 0 088 0 0
3 3 3 0 0 0 0 0 0T1 0
3 3 3 0 0 0 0 0 0 0T2
Direct computations with Lemma 2 and (7) give
_V2(t) 2
i=1
fT(t)h
iQif(t) 0 t
t0h
1 hif
T(s)ds 1 (h iQi)
1 t
t0h
1
hif(s)ds ; (17)
_V3(t) = 2
i=1
gT(t)h
iRig(t) 0 t
t0h g T(s)R
ig(s)ds ; (18)
_V4(t) = x(t0h(t) 1)
T S SC
CTS 0S+CTSC x(t0h(t) 1)
(19)
_V5(t) = 2
i=1
(t) x(t 0 h1)
x(t 0 h10 hi)
T T
i TiC 0
CTT
i CTTiC 0
0 0 0Ti
1
(t) x(t 0 h1)
x(t 0 h10 hi)
: (20)
By isometry property, fori = 1, 2, we have
t
t0h gT(s)Rig(s)ds
= t
t0h g
T(s)R
ig(s) ds
= t
t0h g
T(s)dw(s) R i
t
t0h g(s)dw(s) :
Therefore, substituting inequalities (16)–(20) into (14) and taking ex-pectation on the both sides of (14) yield
LV (t) zT(t)0z(t) (21)
where
zT(t) = T(t) fT(t) gT(t) 1
h1 t
t0h f
T(s)ds 1
h2
1 t
t0h f
T(s)ds t t0h g
T(s)dw(s) t t0h g
T(s)dw(s)
1 xT(t 0 h
1) xT(t 0 2h1) xT(t 0 h10 h2) T
:
By LMI (6), we have
LV (t) 00 jz(t)j2 00 j(t)j2+ jx(t 0 h1)j2 (22)
with0 = m(00)and
CTSC 0 S < 0 : (23)
For any 2 (0; 1), (7), inequalities (22)–(23) and Lemma 1 give LV (t)
0 (10)0 j(t)j200 j(t)j2+ 1 jx(t0h1)j2
0 (10)0 j(t)j20 0
M(S) x(t)0Cx(t 0 h1) T
S
1 x(t) 0 Cx(t 0 h1) + 1xT(t 0 h1)Sx(t 0 h1)
0 (1 0 )0 j(t)j20 0
M(S) x
T(t)Sx(t)
0 2xT(t)SCx(t0h
1)+ 1+ xT(t0h1)CTSCx(t0h1)
0 (1 0 )0 j(t)j20 (1 + )0m(S)
M(S) jx(t)j 2:
0 0 j(t)j2+ jx(t)j2
where0= min (1 0 )0; 0m(S)[(1 + )M(S)]01 > 0. It is obvious from the definition ofV (t)that
0j(t)j2 V (t) 1j(t)j2+ 2 t
t02hjx(s)j
2ds (24)
where0 = m(P11),1 = M(P11)
2= 2
i=1
hi[M(Qi)Kf+M(Ri)Kg]+M(S) + 2
i=1
M(Ti):
Choose" > 0such that
maxf"1; 2h"2e2h"g 0 and e2h"CTSC 0 S < 0: (25) By Itô’s lemma, we have
d[e"sV (s)] = e"s["V (s) + LV (s)]ds
+e"s(s)dw(s); 8s 0: (26)
Lett0= h, then integrating fromt0totand taking expectation on (28) give
e"t V (t)0e"t V (t 0)
= t
t e
"s "V (s)+LV (s) ds
t
t e "s "
1j(s)j2+"2 s
s02hjx(v)j 2dv
00(j(s)j2+jx(s)j2) ds
t
t e "s "
2 s
s02hjx(v)j
2dv0
0jx(s)j2 ds: (27)
Since
t
t e "sds s
s02hjx(v)j 2dv
t
t 02hjx(v)j
2dv v+2h
v e
"sds
2he2h" t
2he2h" t t jx(s)j
2e"sds + 2he2h" t
t 02hjx(s)j 2ds
it follows:
0e"t j(t)j2 e"t V (t) 0Ch or j(t)j2 Che0"t (28)
where Ch = hsup0hh jx()j2 with h = 010 e"h[1 +
2h2(1 + 2h"e2h")] 1. Since neutral stochastic delay differential (1) has a unique continuous solution,Chis a nonnegative finite number for any0 h < 1.
Sincee2"hCTSC < S, there exists a number 2 (0; 1)such that e2"hCTSC < S < S : (29) Note that
T(t)S(t) = xT(t)Sx(t)
02xT(t)SCx(t 0 h
1) + xT(t 0 h1)CTSCx(t 0 h1)
for allt 0. By Lemma 1, we have e"txT(t)Sx(t) e"t
1 0 T(t)S(t) + e"t
xT(t 0 h1)CTSCx(t 0 h1): (30) Letbe any nonnegative real number. For all0 t , we have
e"t xT(t)Sx(t) 11 0 sup
0t e
"tT(t)S(t)
+ 1 sup
0t e
"txT(t 0 h
1)CTSCx(t 0 h1)
11 0 M(S) sup
0t [e
"tj(t)j2]
+ e"h sup
0h t e
"txT(t)CTSCx(t)
11 0 M(S)Ch+ e0"h sup
0ht e
"t xT(t)Sx(t) :
But this holds for all0h t . So sup
0ht e
"t xT(t)Sx(t) M(S)Ch
(1 0 e0"h)(1 0 ): (31)
Sinceis an arbitary nonnegative number, we have jx(t)j2 (1 0 e0"hM(S)C)(1 0 )he0"t
m(S); 8 t 0h : (32)
The mean-square exponential stability has been proven.
Now let us proceed to discuss the almost sure exponential stability. Let 2 (0; ") be arbitrary. We claim that there is a finite positive numberthsuch that for allt th
j(t)j2 e0("0)t a:s: (33)
Therefore, for allt th, inequality (30) implies e("0)txT(t)Sx(t)
M(S)e("0)t
1 0 + e
("0)t
xT(t 0 h1)CTSCx(t 0 h1) a:s:
Using the similar reasoning as above and letting ! 0, we have jx(t)j2
M(S)e0"t[(10 e0"h)(10 )m(S)]01 a:s:for allt th0 h. This implies immediately
lim sup
t!1
1
tlog jx(t)j 0 "2 a:s:
We complete the proof by showing that inequality (33) is true. Note that
jf(t)j2 Kf sup
0h0 jx(t 0 )j 2
and jg(t)j2 K
g sup
0h0 jx(t 0 )j 2
for allt 0. For any integerk 1, by Hölder’s inequality and Burkholder–Davis–Gundy inequality, one can derive that
sup0hj(kh + )j2
3 j(kh)j2+ h (k+1)h kh jf(s)j
2ds
+ sup
0h kh+
kh g(s)dw(s)
he0kh" (34)
whereh = 3Ch(1 + Kfh2eh"+ 4Kgheh"). But, by Chebyshev’s inequality, this implies
! : sup
0hj(kh + )j
2> e0("0)kh
he0kh:
By Borel–Cantelli lemma, there is a finite integerk0such that sup
0hj(kh + )j
2 e0("0)kh a:s:
for allk k0. Therefore, inequality (33) holds withth k0h. Remark 1: From the proof of Theorem 1, it is observed that, letting
zT(t) = T(t) fT(t) gT(t) 1
h1 t
t0h f
T(s)ds 1
h2
1 t
t0h f
T(s)ds t t0h g
T(s)dw(s) t t0h g
T(s)dw(s)
1 xT(t 0 h
1) xT(t02h1)CTxT(t0h10h2)CT T
we can have a corollary derived from Theorem 1 with 0Ti CTWi
WiC 0Wi 0; i = 1; 2 (35)
whereWi > 0. This corollary can be easily applied to problems of stabilization by the approach of LMIs.
IV. EXAMPLES
Example 1: Let us look at the following neutral stochastic delay system:
d[x(t) 0 Cx(t 0 h)] = [A0x(t) + A1x(t 0 h)]dt
TABLE I
h BYDIFFERENTMETHODS
with
C = 00:21 0:20 ; A0= 0:50 0:30 ; A1= 0101 010
H0= 0:20 0:20 ; H1= 0:30 0:30 :
It is easy to verify that the existing results [2], [10], [12]–[17] do not work. But, by Theorem 1, the upper bounds of time delay for exponen-tial stability of system (36) ishmax= 0:35.
Example 2: Deterministic systems may be regarded a special class of stochastic systems, e.g., the following deterministic neutral system is exactly system (1) withA0 = A,A1 = BandA2 = H0 = H1=
H2 = 0, i.e.:
_x(t) 0 C _x(t 0 h) = Ax(t) + Bx(t 0 h) (37)
for allt 0, where
A= 00:90:1 00:90:2 ; B =0 1:1 0:20:1 1:1 ; C = 00:20:2 00:1
andis a constant real number.
The case of = 0has been studied by many works [4], [8] and [11]. However, results of [2], [4], [9], [11], and [18] are not (conveniently) applicable whenjj 1. For 2, the criterion in [5] does not work, but the upper boundshmaxfor exponential stability of (37) by other methods are listed in Table I, which shows that the results obtained by the methods proposed in this technical note are less conservative in these cases.
V. CONCLUSION
In this technical note, delay-dependent criterion for stability of neu-tral stochastic delay systems has been presented by approach of LMIs. Numerical examples have been given to verify the effectiveness of the method proposed in this technical note. Particularly, Example 2 demon-strates that our result developed for stochastic systems is competitive even when it is specialized to the deterministic cases.
ACKNOWLEDGMENT
The authors wish to thank the associate editor and reviewers for their comments and suggestions.
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