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Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates
Applied Mathematics and Computation 218 (2012) 5769–5781
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Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates Junkang Tian a,b,c,⇑, Yongming Li a, Jinzhou Zhao a, Shouming Zhong c a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China c School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China b
a r t i c l e
i n f o
Keywords: Stochastic stability Markovian jumping neural networks Time-varying delays
a b s t r a c t In this paper, the problem of stochastic stability criterion of Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates is considered. Some new delay-dependent stability criteria are derived by choosing a new class of Lyapunov functional. The obtained criteria are less conservative because freeweighting matrices method and a convex optimization approach are considered. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. 2011 Elsevier Inc. All rights reserved.
1. Introduction In recent decades, neural networks have been investigated extensively because of their successful applications in various areas such as pattern recognition, image processing, associative memory and combinatorial optimization. However, these successful applications are greatly dependent on the dynamic behaviors of neural networks. As is well known now, stability is one of the main properties of neural networks, which is a crucial feature in the design of neural networks. On the other hand, it has been recognized that the time delays often occur in various neural networks, and may cause undesirable dynamic network behaviors such as oscillation and instability. Therefore, the stability analysis for delayed neural networks has become a topic of great theoretic and practical importance in recent years [1–27]. Recently, systems with Marvokian jumps have been attracting increasing research attention. This class of systems are the hybrid systems with two components in the state. The first one refers to the mode, which is described by a continuous–time finite-state Markovian process, and the second one refers to the state which is represented by a system of differential equations. The Markovian jump systems have the advantage of modeling the dynamic systems subject to abrupt variation in their structures, such as component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, and operating in different points of a nonlinear plant [28]. Recently, there has been a growing interest in the study of neural networks with Markovian jumping parameters [29–38]. In [29], the problem of stochastic robust stability for uncertain delayed neural networks with Markovian jumping parameters is investigated. The state estimation problem for a class of Markovian neural networks with discrete and distributed time-delays is studied in [30]. Without assuming the boundedness, monotonicity and differentiability of the activation functions, some results for delay-dependent stochastic stability criteria for the Markovian jumping Hopfield neural networks with time-delay are developed in [31]. Some new delay-dependent
⇑ Corresponding author at: State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China. E-mail address: [email protected] (J. Tian). 0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.087
5770
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
stochastic stability criteria for BAM neural networks with Markovian jumping parameters are derived in [32] based on delay partitioning idea. To the best of our knowledge, the stochastic stability analysis for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates has never been tackled, and such a situation motivates our present study. In this paper, the problem of stochastic stability criterion for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates is considered. By choosing a new class of Lyapunov functional, some new delay-dependent stochastic stability criteria are derived to guarantee the stochastic stability of Markovian jumping neural networks. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization approach are considered. Finally, a numerical example is given to show the effectiveness of the derived method. 2. Problem formulation Consider the following delayed neural network:
_ xðtÞ ¼ AxðtÞ þ BgðxðtÞÞ þ Cgðxðt hðtÞÞÞ þ D
Z
t
gðxðsÞÞds þ l;
ð1Þ
tdðtÞ
0; xðtÞ ¼ UðtÞ; t 2 ½h;
ð2Þ T
T
where xðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ 2 Rn is the neuron state vector, gðxðÞÞ ¼ ½g 1 ðx1 ðÞÞ; g 2 ðx2 ðÞÞ; . . . ; g n ðxn ðÞÞ 2 Rn denotes the neuron activation function, and l ¼ ðl1 ; l2 ; . . . ; ln ÞT 2 Rn is a constant input vector. B; C; D 2 Rnn are the connection weight matrix and the delayed connection weight matrix,respectively. A = diag(a1, a2, . . . , an) with ai > 0, i = 1, 2, . . . , n. h(t), _ d(t) are time-varying continuous functions that satisfy 0 6 hðtÞ 6 h; 0 6 dðtÞ 6 d; hðtÞ 6 u, where h, d and u are constants. 0, where h ¼ maxfd; hg. In addition, it is assumed that each neuron The initial vector U(t) is continuously differential on ½h; activation function gi(), i = 1, 2, . . . , n is bounded and satisfies the following condition:
ci 6
g i ðxÞ g i ðyÞ 6 cþi ; xy
8x; y 2 R;
x – y;
i ¼ 1; 2; . . . ; n;
ð3Þ
þ where c i ; ci ; i ¼ 1; 2; . . . ; n are constants. Note that by using the Brouwers fixed-point theorem, T it can be easily proven that there exists at least one equilibrium point for system (1). Assuming that x ¼ x1 ; x2 ; . . . ; xn is the equilibrium point of (1) and using the transformation z() = x() x⁄, (1) can be converted to the following system:
z_ ðtÞ ¼ AzðtÞ þ Bf ðzðtÞÞ þ Cf ðzðt hðtÞÞÞ þ D
Z
t
f ðzðsÞÞds;
ð4Þ
tdðtÞ
where z(t) = [z1(t), z2(t), . . . , zn(t)]T, f(z()) = [f1(z1()), f2(z2()), . . . , fn(zn())]T and fi ðzi ðÞÞ ¼ g i zi ðÞ þ xi g i xi ; i ¼ 1; 2; . . . ; n. According to the inequality (3), one can obtain that:
ci 6
fi ðzi ðtÞÞ 6 cþi f i ð0Þ ¼ 0; zi ðtÞ
i ¼ 1; 2; . . . ; n:
ð5Þ
Given probability space (X, !,P) where X is sample space, ! is r algebra of subset of the sample space, and P is the probability measure defined on !. Let S = {1, 2, . . . , N} and the random form process {r(t), t 2 [0, +1)} be a homogeneous, finitestate Markovian process with right continuous trajectories with generator P = (pij)NN and transition probability from mode i at time t to mode j at time t + Dt, i, j 2 S:
Pfrðt þ DtÞ ¼ jjrðtÞ ¼ ig ¼
pij Dt þ oðDtÞ j – i; 1 þ pii Dt þ oðDtÞ j ¼ i;
ð6Þ
P with transition rates pij P 0 for i, j 2 S, j – i and pii ¼ Nj¼1;j–i pij , where Dt > 0 and limDt!0 oðDDttÞ ¼ 0. In addition, the transition rates of the Markovian chain are considered to be partially available, namely, some elements in matrix P are time-invariant but unknown. For instance, a system with three operation modes may have the transition rates matrix P as follows:
2
p11
P¼6 4 ?
p31
?
?
3
p22 ? 7 5; p32 p33
where ‘‘?’’ represents the inaccessible element. For notation clarity, "i 2 S, we denote S ¼ Sikn [ Siuk with Sikn , fj; if pij is knowng, Siuk , fj; if pij is unknowng; pikn P , j2Si pij throughout the paper. Furthermore, we assume the diagonal elements of P are known. Note that the set S comkn prises the various operational modes of the system under study. In this paper, we consider the Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates described by the following nonlinear differential equations:
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
_ xðtÞ ¼ AðrðtÞÞxðtÞ þ BðrðtÞÞgðxðtÞÞ þ CðrðtÞÞgðxðt hðrðtÞ; tÞÞÞ þ DðrðtÞÞ
Z
5771
t
gðxðsÞÞds þ l
ð7Þ
tdðrðtÞ;tÞ
Similar to the above analysis,we can obtain:
z_ ðtÞ ¼ AðrðtÞÞzðtÞ þ BðrðtÞÞf ðzðtÞÞ þ CðrðtÞÞf ðzðt hðrðtÞ; tÞÞÞ þ DðrðtÞÞ
Z
t
f ðzðsÞÞds;
ð8Þ
tdðrðtÞ;tÞ
when r(t) = i 2 S, the matrices A(r(t)), B(r(t)), C(r(t)), D(r(t)) are represented by Ai, Bi, Ci, Di. In the system (8) hi(t), di(t) denote ~ ¼ max fh g; d ~¼ the mode-dependent time-varying delays which satisfy 0 6 hi ðtÞ 6 hi ; 0 6 di ðtÞ 6 di ; h_ i ðtÞ 6 ui ; h j2S j ~ dg. ~ max fd g; s ¼ maxfh; j2S
j
The initial conditions of system (8) is of the following form:
zðtÞ ¼ uðtÞ;
t 2 ½s; 0;
rð0Þ ¼ r 0 :
ð9Þ
Definition 1. Markovian system (8) is said to be stochastically stable, if for any u(t) defined on [s, 0] and r(0) 2 S, the following condition is satisfied:
lim E
Z
t!1
t
zT ðsÞzðsÞdsju; r 0
< 1:
ð10Þ
0
Definition 2. Let U1 ; U2 ; . . . ; UN : Rm ! Rn be a given finite number of functions such that they have positive values in an open subset D of Rm . Then, a reciprocally convex combination of these functions over D is a function of the form:
1
a1
U1 þ
1
a2
1
U2 þ þ
aN
UN : D ! Rn ;
ð11Þ P
where the real numbers ai satisfy ai > 0 and i ai ¼ 1. The following Lemma 1 suggests a lower bound for a reciprocally convex combination of scalar positive functions Ui = fi. Lemma 1 [39]. Let f1 ; f2 ; . . . ; fN : Rm ! R have positive values in an open subset D of Rm . Then, the reciprocally convex combination of fi over D satisfies:
min P
fai jai >0;
X 1 a ¼1g i i
i
ai
fi ðtÞ ¼
X
fi ðtÞ þ max g i;j ðtÞ
i
(
X i–j
m
subject to :
g i;j : R ! R; g j;i ðtÞDg i;j ðtÞ;
g i;j ðtÞ; "
ð12Þ
fi ðtÞ
g i;j ðtÞ
g i;j ðtÞ
fj ðtÞ
#
) P0 :
ð13Þ
Lemma 2 [40]. For any constant matrix Z 2 Rnn ; Z ¼ Z T > 0, scalar h > 0, such that the following integration is well defined, then:
h
Z
t
xT ðsÞZxðsÞds 6
th
Z
t
th
xT ðsÞdsZ
Z
t
xðsÞds:
ð14Þ
th
3. Main results In this section, a new Lyapunov functional is constructed to derived a delay-dependent stochastic stability criterion for system (8) when the time-varying delays are mode-dependent and the transition rates are partially known. Theorem 1. For given scalars hi P 0, di > 0, ui, the system (8) with mode-dependent time-varying delays and partially known transition rates is stochastically stable if there exist symmetric positive definite matrices Pi, Q1i, Q2i, Q3i, Q4i, R1, R2, R3, R4, R5, R6, R7, positive diagonal matrices Ti1, Ti2 and any matrices S12, Li, Ni, Mi, with appropriate dimensions, for any i = 1, 2, . . . ,N, such that the following LMIs hold:
R7
S12
R7
P 0;
ð15Þ
5772
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
2 6 6 4
1 þ pikn ðR1 þ pii Q 1i Þ
P
pij Q 1j
j–i;j2Sikn
P
3 7 7 < 0;
pij Q 1j 5
ð16Þ
j–i;j2Sikn
R1 þ pii Q 1i
Q 1j
Q 1j
2 6 6 4
8j 2 Siuk ;
< 0;
1 þ pikn ðR2 þ pii Q 2i Þ
P
pij Q 2j
j–i;j2Sikn
P
ð17Þ 3 7 7 < 0;
pij Q 2j 5
ð18Þ
j–i;j2Sikn
R2 þ pii Q 2i
Q 2j
Q 2j
2 6 6 4
8j 2 Siuk ;
< 0;
1 þ pikn ðR3 þ pii Q 3i Þ
P
pij Q 3j
P
3
7 7 < 0; pij Q 3j 5
j–i;j2Sikn
ð19Þ
ð20Þ
j–i;j2Sikn
R3 þ pii Q 3i
Q 3j
Q 3j
2 6 6 4
8j 2 Siuk ;
< 0;
1 þ pikn ðR4 þ pii Q 4i Þ
P
pij Q 4j
j–i;j2Sikn
P
ð21Þ 3 7 7 < 0;
pij Q 4j 5
ð22Þ
j–i;j2Sikn
R4 þ pii Q 4i
Q 4j
Q 4j
2 6 6 4
< 0;
8j 2 Siuk ;
1 þ pikn ðR5 þ pii di Q 4i Þ
P
pij dj Q 4i
j–i;j2Sikn
P
ð23Þ 3 7 7 < 0;
pij dj Q 4i 5
ð24Þ
j–i;j2Sikn
R5 þ pii di Q 4i
dj Q 4i
dj Q 4i
2 6 6 4
1 þ pikn Ei
2
Ei 6 4
6 6 4
hi Li hi R6
1 þ pikn Ei
2
Ei 6 4
< 0;
hi 1 þ pikn Li hi 1 þ pikn R6
2
Euk 1i
7 0 5 < 0;
7 0 7 5 < 0; kn E2i
8j 2 Siuk ;
3 ð26Þ
ð27Þ
Euk 2i
hi R6
Ekn 1i
ð25Þ
3
hi 1 þ pikn Ni hi 1 þ pikn R6
hi N i
8j 2 Siuk ;
Euk 1i
3
7 0 7 5 < 0; kn E2i
ð28Þ
3
7 0 5 < 0;
Euk 2i
Ekn 1i
8j 2 Siuk ;
ð29Þ
5773
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
where
2
E11 6 6 6 6 6 Ei ¼ 6 6 6 6 4
E12 E22
2
Ekn 2i
E13 E23 E33
M i1 C i E24 0 E44
0
Xi1 6 0 6 ¼4 0 0
E15 0 0 C Ti M Ti2 E55 3 0 0 7 7; 0 5
0 0
Xi2 0 0
Xi3 0
3 3 2 Mi1 Di Xi1 0 0 0 0 7 6 0 7 0 Xi2 0 7 7 6 7 6 0 0 7 X 0 0 7 i3 7 6 7 kn 6 0 7; E1i ¼ 6 0 0 0 Xi4 7 7; 7 6 0 Mi2 Di 7 0 0 0 7 7 6 7 4 0 0 5 0 0 0 5 d1 Q 4i 0 0 0 0 3 2 i 0 0 0 Pj 6 0 hj Q 1i 2 0 0 7 7 6 0 Pj 7 60 Q 0 0 h j 2i 7 6 0 hj Q 1i 6 uk 7 6 6 ¼60 0 0 hj Q 3i 7; E2i ¼ 4 0 0 60 0 0 0 7 7 6 0 0 5 40 0 0 0 0 0 0 0
E16 0 0 0 Mi2 Bi E66
Euk 1i
Xi4
0 0 hj Q 2i 0
3 0 0 7 7 0 5 hj Q 3i
~ 1 þ R2 Þ R7 þ L þ LT M A AT M T 2C1 T C2 ; E12 ¼ R7 ST þ LT L þ N ; E11 ¼ pii Pi þ Q 1i þ Q 2i þ hðR i1 i1 i i1 i1 i1 12 i1 i1 i2 i E13 ¼ ST12 þ LTi3 Ni1 ; E15 ¼ Pi M i1 ATi MTi2 ; E16 ¼ M i1 Bi þ T i1 ðC1 þ C2 Þ; E22 ¼ 2R7 þ S12 þ ST12 ð1 ui ÞQ 1i Li2 LTi2 þ Ni2 þ NTi2 2C1 T i2 C2 ; E23 ¼ R7 ST12 LTi3 þ NTi3 Ni2 ; ~ 6 þh ~ 2 R7 M M T ; E24 ¼ T i2 ðC1 þ C2 Þ; E33 ¼ R7 Q 2i þ pii hi Q 2i Ni3 NTi3 ; E44 ¼ ð1 ui ÞQ 3i 2T i2 ; E55 ¼ hR i2 i2 2 ~ X X X d ~ 3 þ R4 þ dR ~ 5 2T i1 ; Xi1 ¼ pij Pj ; Xi2 ¼ pij hj Q 1i ; Xi3 ¼ pij hj Q 2i ; E66 ¼ Q 3i þ di Q 4i þ hR 2 j–i;j2Sikn j–i;j2Sikn j–i;j2Sikn X Xi4 ¼ pij hj Q 3i ; C1 ¼ diag c1 ; c2 ; .. .; cn ; C2 ¼ diag cþ1 ; cþ2 ; .. .; cþn ; j–i;j2Sikn
e1 ¼ ð I
T
0 0 0 0 0 0Þ ; 0 0 0 0 ÞT ;
e3 ¼ ð 0 0 I Ni ¼ N Ti1
NTi2
NTi3
e2 ¼ ð 0 I Li ¼ LTi1
0 0 0 0
T
;
0 0 0 0 0Þ LTi2
LTi3
M i ¼ M Ti1
T
0 0 0 0
T
0 0 0 M Ti2
0 0
T
Proof. Construct a new class of Lyapunov functional candidate as follow:
VðzðtÞ; iÞ ¼
X
9
V j ðzðtÞ; iÞ;
j¼1
where
V 1 ðzðtÞ; iÞ ¼ zT ðtÞPðrðtÞÞzðtÞ; Z t V 2 ðzðtÞ; iÞ ¼ zT ðsÞQ 1 ðrðtÞÞzðsÞds; thðrðtÞ;tÞ Z t V 3 ðzðtÞ; iÞ ¼ zT ðsÞQ 2 ðrðtÞÞzðsÞds; thðrðtÞÞ Z t V 4 ðzðtÞ; iÞ ¼ f T ðzðsÞÞQ 3 ðrðtÞÞf ðzðsÞÞds; thðrðtÞ;tÞ Z 0 Z t f T ðzðsÞÞQ 4 ðrðtÞÞf ðzðsÞÞdsdh; V 5 ðzðtÞ; iÞ ¼ dðrðtÞÞ tþh Z 0Z t Z 0Z t zT ðsÞðR1 þ R2 ÞzðsÞdsdh þ f T ðzðsÞÞR3 f ðzðsÞÞdsdh; V 6 ðzðtÞ; iÞ ¼ ~ ~ tþh h Z Z0h Ztþh Z 0 t 0 Z t f T ðzðsÞÞR4 f ðzðsÞÞdsdkdh þ f T ðzðsÞÞR5 f ðzðsÞÞdsdh; V 7 ðzðtÞ; iÞ ¼ ~ h ~ tþh d tþk d Z 0Z t z_ T ðsÞR6 z_ ðsÞdsdh; V 8 ðzðtÞ; iÞ ¼ ~ tþh h Z 0 Z t ~ z_ T ðsÞR7 z_ ðsÞdsdh; V 9 ðzðtÞ; iÞ ¼ h ~ h
tþh
ð30Þ
5774
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
where Pi, Q1i, Q2i, Q3i, Q4i, R1, R2, R3, R4, R5, R6, R7, i = 1, 2, . . . , N, are positive definite matrices and:
X
N
pij Q kj < Rk ; k ¼ 1; 2; 3; 4;
ð31Þ
pij dj Q 4i < R5 ;
ð32Þ
j¼1
X
N
j¼1
Remark 1 Our paper fully uses the information about mode-dependent time-varying delays and mode-dependent positive definite matrices, but [35] only use the information about mode-dependent positive definite matrices when constructing the Lyapunov functional V(z(t), i). So the Lyapunov functional V(z(t), i) in our paper is more general than that in [35], and the stability criteria in our paper may be more applicable. Let L be the infinitesimal generator of random process {zt, t P 0}, then for each r(t) = i, i 2 S, it can be shown that:
(Z
)
t
zT ðsÞQ 1 ðrðtÞÞzðsÞds
L
thðrðtÞ;tÞ
("Z # Z ) tþD t 1 E zT ðsÞQ 1 ðrðt þ DÞÞzðsÞdsjrðtÞ ¼ i zT ðsÞQ 1i zðsÞds D!0 D thi ðtÞ tþDhðrðtþDÞ;tþDÞ 8 " # >Z X 1 < tþD T N z ðsÞ Q 1i þ pij D þ oðDÞ zðsÞds ¼ limþ P D!0 D > : tþDhi ðtþDÞ N ðpij DþoðDÞÞhj ðtþDÞ j¼1 ¼ limþ
j¼1
Z
)
t
zT ðsÞQ 1i zðsÞds
thi ðtÞ
8 9 >Z > Z t = 1 < tþD zT ðsÞQ 1i zðsÞds zT ðsÞQ 1i zðsÞds ¼ limþ P > D!0 D > thi ðtÞ : tþDhi ðtþDÞ N ðpij DþoðDÞÞhj ðtþDÞ ; Z
1 þ limþ D!0 D 1 D
¼ limþ D!0
j¼1
tþD
tþDhi ðtþDÞ
Z
zT ðsÞ ðpij DþoðDÞÞhj ðtþDÞ
j¼1
tþD
zT ðsÞQ 1i zðsÞds þ limþ D!0
t
1 þ limþ D!0 D
PN
Z
1 D
X
N
ðpij D þ oðDÞÞQ 1j zðsÞds
j¼1
Z
thi ðtÞ
tþDhi ðtþDÞ
PN
ðpij DþoðDÞÞhj ðtþDÞ
zT ðsÞQ 1i zðsÞds
j¼1
tþD
tþDhi ðtþDÞ
PN
T
z ðsÞ ðpij DþoðDÞÞhj ðtþDÞ
j¼1
¼ zT ðtÞQ 1i zðtÞ ð1 h_ i ðtÞ
X
N
X
N
ðpij D þ oðDÞÞQ 1j zðsÞds
j¼1
pij hj ðtÞÞzT ðt hi ðtÞÞQ 1i zðt hi ðtÞÞ
j¼1
Z
þ
t
zT ðsÞ
thi ðtÞ
(Z L
Z
0
¼i
t
xT ðsÞQ ðrðtÞÞxðsÞdsdh
¼ limþ D!0
tþh
hðrðtÞÞ
Z
)
0
h½rðtÞ
Z
xT ðsÞQ ½rðtÞxðsÞds
1 E D
¼ limþ D!0
tþh
N
pij Q 1j zðsÞds
j¼1
)
t
!
X
(Z
h½rðtþDÞ
8 >Z 1<
D> :
Z
0
tþD
xT ðsÞQ ½rðt þ DÞxðsÞdsjrðtÞ
tþDþh
Z
0
PN j¼1
½pij DþoðDÞhj hi
tþD
xT ðsÞ tþDþh
X N j¼1
9 > Z 0 Z 0 Z t Z tþD = þ xT ðsÞQ i xðsÞdsdh xT ðsÞQ i xðsÞdsdh PN > hi hi tþh ; ½pij DþoðDÞhj tþDþh j¼1 P Z 0 Z t Z Z N X 1 hi j¼1 ½pij DþoðDÞhj t T N pij xT ðsÞQ j xðsÞdsdh limþ x ðsÞQ i xðsÞdsdh ¼ D!0 D hi hi tþh tþh j¼1 Z Z tþD Z t 1 0 þ limþ xT ðsÞQ i xðsÞds xT ðsÞQ i xðsÞds dh P N p DþoðDÞ h D!0 D hi tþh ½ ij j tþDþh j¼1
pij D þ oðDÞ Q j xðsÞds
5775
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
¼
X
N
pij
Z
¼
Z
N
¼
N
Z
pij hj limþ D!0
j¼1
pij
Z
T
Z
0
t
thi
PN
½pij DþoðDÞhj
xT ðsÞQ i xðsÞds
pij
Z
Z
0
t
xT ðsÞQ j xðsÞdsdh þ
X
N
Z
pij hj
t
xT ðsÞQ i xðsÞds þ
thi
j¼1
X
t
N
xT ðsÞQ j xðsÞdsdh 1
tþh
hi
j¼1
T
tþh
hi N
X
lim x ðt þ DÞQ i xðt þ DÞ x ðt þ h þ DÞQ i xðt þ h þ DÞ dh
j¼1
X
xT ðsÞQ j xðsÞdsdh þ
j¼1
0
þ hi D!0
X
t
tþh
hi
j¼1
þ
Z
0
pij hj
Z
0
xT ðtÞQ i xðtÞ xT ðt þ hÞQ i xðt þ hÞ dh
hi
!Z
t
xT ðsÞQ i xðsÞds þ hi xT ðtÞQ i xðtÞ thi
j¼1
Similar to the process above, we can obtain:
X LV 1 ðzðtÞ; iÞ ¼ 2zT ðtÞPi z_ ðtÞ þ zT ðtÞ N pij Pj zðtÞ;
ð33Þ
j¼1
LV 2 ðzðtÞ; iÞ ¼ zT ðtÞQ 1i zðtÞ ð1 h_ i ðtÞÞzT ðt hi ðtÞÞQ 1i zðt hi ðtÞÞ þ þ
Z
zT ðsÞ
N
thi ðtÞ
N
pij hj ðtÞzT ðt hi ðtÞÞQ 1i zðt hi ðtÞÞ
j¼1
!
X
t
X
pij Q 1j zðsÞds;
ð34Þ
j¼1
LV 3 ðzðtÞ; iÞ ¼ zT ðtÞQ 2i zðtÞ zT ðt hi ÞQ 2i zðt hi Þ þ
X
N
Z
pij hj zT ðt hi ÞQ 2i zðt hi Þ þ
t
zT ðsÞ
thi
j¼1
X
! N
pij Q 2j zðsÞds; ð35Þ
j¼1
LV 4 ðzðtÞ; iÞ ¼ f T ðzðtÞÞQ 3i f ðzðtÞÞ ð1 h_ i ðtÞÞf T ðzðt hi ðtÞÞÞQ 3i f ðzðt hi ðtÞÞÞ ! Z t X X N N T T pij hj ðtÞf ðzðt hi ðtÞÞÞQ 3i f ðzðt hi ðtÞÞÞ þ f ðzðsÞÞ pij Q 3j f ðzðsÞÞds: þ thi ðtÞ
j¼1
ð36Þ
j¼1
By Lemma 2, it is easy to obtain:
LV 5 ðzðtÞ; iÞ ¼
X
N
pij
Z
0
Z
di
j¼1
t
f T ðzðsÞÞQ 4j f ðzðsÞÞdsdh 1
tþh
X
N
pij dj
di
j¼1
1 di
Z
tþh
!T f ðzðsÞÞds
Q 4i
Z
tdi ðtÞ
t
f T ðzðsÞÞQ 4i f ðzðsÞÞds
!
t
f ðzðsÞÞds þ di f T ðzðtÞÞQ 4i f ðzðtÞÞ;
ð37Þ
tdi ðtÞ
~ T ðtÞðR þ R ÞzðtÞ þ hf ~ T ðzðtÞÞR f ðzðtÞÞ LV 6 ðzðtÞ; iÞ ¼ hz 1 2 3
LV 7 ðzðtÞ; iÞ ¼
f T ðzðsÞÞQ 4i f ðzðsÞÞds
tdi
j¼1
t
t tdi
þ di f T ðzðtÞÞQ 4i f ðzðtÞÞ Z 0 Z t Z X X N N pij f T ðzðsÞÞQ 4j f ðzðsÞÞdsdh þ pij dj 6 j¼1
!Z
Z
t ~ th
zT ðsÞðR1 þ R2 ÞzðsÞds
Z
t
~ th
f T ðzðsÞÞR3 f ðzðsÞÞds;
Z 0 Z t Z t ~2 d ~ T ðzðtÞÞR5 f ðzðtÞÞ f T ðzðsÞÞR4 f ðzðsÞÞdsdh f T ðzðsÞÞR5 f ðzðsÞÞds; f T ðzðtÞÞR4 f ðzðtÞÞ þ df ~ tþh ~ 2 d td
~z_ T ðtÞR6 z_ ðtÞ LV 8 ðzðtÞ; iÞ ¼ h
Z
t
~z_ T ðtÞR6 z_ ðtÞ z_ T ðsÞR6 z_ ðsÞds 6 h
~ th
z_ T ðsÞR6 z_ ðsÞds
Z
Z
ð39Þ
t
z_ T ðsÞR6 z_ ðsÞds;
ð40Þ
thi ðtÞ
t
~ th
~2 z_ T ðtÞR7 z_ ðtÞ h 6h i
thi ðtÞ
thi
Z
~ ~2 z_ T ðtÞR7 z_ ðtÞ h LV 9 ðzðtÞ; iÞ ¼ h
Z
ð38Þ
z_ T ðsÞR7 z_ ðsÞds
thi ðtÞ
z_ T ðsÞR7 z_ ðsÞds hi
thi
Z
t
z_ T ðsÞR7 z_ ðsÞds:
ð41Þ
thi ðtÞ
By the Newton–Leibniz formula and (8), for any appropriately dimensioned matrices Li, Ni, Mi, i = 1, 2, . . . , N, one can obtain: T
2f ðtÞLi zðtÞ zðt hi ðtÞÞ
Z
!
t
thi ðtÞ
z_ ðsÞds
¼0
ð42Þ
5776
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
2fT ðtÞNi zðt hi ðtÞÞ zðt hi Þ
Z
!
thi ðtÞ
z_ ðsÞds
¼0
ð43Þ
thi
T
Z
z_ ðtÞ Ai zðtÞ þ Bi f ðzðtÞÞ þ C i f ðzðt hi ðtÞÞÞ þ Di
2f ðtÞM i
!
t
f ðzðsÞÞds
¼0
ð44Þ
tdi ðtÞ
" T
T
T
T
T
_T
T
f ðtÞ ¼ z ðtÞ z ðt hi ðtÞÞ z ðt hi Þ f ðzðt hi ðtÞÞÞ z ðtÞ f ðzðtÞÞ
Z
#
t T
f ðzðsÞÞds tdi ðtÞ
It is easy to obtain that:
2fT ðtÞLi
Z
t
thi ðtÞ
2fT ðtÞNi
Z
thi ðtÞ
thi
Z
Z
T z_ ðsÞds 6 hi ðtÞfT ðtÞLi R1 6 Li fðtÞ þ
t
z_ T ðsÞR6 z_ ðsÞds
T z_ ðsÞds 6 ðhi hi ðtÞÞfT ðtÞNi R1 6 N i fðtÞ þ
t
z_ T ðsÞR7 z_ ðsÞds ¼ thi
Z
thi ðtÞ
z_ T ðsÞR7 z_ ðsÞds
thi
ð45Þ
thi ðtÞ
Z
Z
thi ðtÞ
z_ T ðsÞR6 z_ ðsÞds
ð46Þ
thi
t
z_ T ðsÞR7 z_ ðsÞds
ð47Þ
thi ðtÞ
The LV9(z(t), i) is upper-bounded by
hi hi T fT ðtÞðe2 e3 ÞR7 ðe2 e3 ÞT fðtÞ f ðtÞðe1 e2 ÞR7 ðe1 e2 ÞT fðtÞ hi hi ðtÞ hi ðtÞ " #T # " R7 S12 eT2 eT3 eT2 eT3 T 2 _T ~ _ fðtÞ 6 h z ðtÞR7 zðtÞ f ðtÞ T ST12 R7 e1 eT2 eT1 eT2
~2 z_ T ðtÞR7 z_ ðtÞ LV 9 ðzðtÞ; iÞ 6 h
ð48Þ ð49Þ
where the inequalities in (48) come from Lemma 2, and that of (49) from Lemma 1 as:
2 qffiffi
3T T ðe e Þ 2 3 a 6 7 R7 fT ðtÞ4 qffiffi 5 ST12 abðe1 e2 ÞT b
2 qffiffi 3 T b ðe e Þ 2 3 S12 6 a 7 4 qffiffi 5fðtÞ 6 0; R7 aðe1 e2 ÞT
ð50Þ
b
i ðtÞ where a ¼ hi h ; b ¼ hihðtÞ . Note that when hi(t) = hi or hi(t) = 0, one can obtain fT(t)(e2 e3) = 0 or fT(t)(e1 e2) = 0, respechi i tively. So the relation (49) also holds. Furthermore, there exist positive diagonal matrices Ti1, Ti2, such that the following inequalities hold based on (3):
2f T ðzðtÞÞT i1 f ðzðtÞÞ þ 2zT ðtÞT i1 ðC1 þ C2 Þf ðzðtÞÞ 2zT ðtÞC1 T i1 C2 zðtÞ P 0;
ð51Þ
2f T ðzðt hi ðtÞÞÞT i2 f ðzðt hi ðtÞÞÞ þ 2zT ðt hi ðtÞÞT i2 ðC1 þ C2 Þf ðzðt hi ðtÞÞÞ 2zT ðt hi ðtÞÞC1 T i2 C2 zðt hi ðtÞÞ P 0: ð52Þ From (30)–(52), one can obtain that:
LVðzðtÞ; iÞ 6 fT ðtÞRi fðtÞ;
ð53Þ
where T 1 T Ri ¼ Ei þ hi ðtÞLi R1 6 Li þ ðhi hi ðtÞÞN i R6 N i þ diagfW1i ; W2i ; W3i ; W4i ; 0; 0; 0g;
W1i ¼
X
pij Pj ; W2i ¼
j–i;j2S
X j–i;j2S
pij hj Q 1i ; W3i ¼
X
pij hj Q 2i ; W4i ¼
j–i;j2S
X
pij hj Q 3i :
j–i;j2S
T 1 T 1 T 1 T Note that 0 6 hi ðtÞ 6 hi ; hi ðtÞLi R1 6 Li þ ðhi hi ðtÞÞN i R6 N i can be seen as the convex combination of Li R6 Li and N i R6 N i on hi(t). Therefore, Ri < 0 holds if and only if:
T Ri1 ¼ Ei þ hi Li R1 6 Li þ diagfW1i ; W2i ; W3i ; W4i ; 0; 0; 0g < 0;
Ri2 ¼ Ei þ
T hi Ni R1 6 Ni
þ diagfW1i ; W2i ; W3i ; W4i ; 0; 0; 0g < 0:
Applying the Schur complement, (31),(32),(54) and (55 can be rewritten respectively as:
ð54Þ ð55Þ
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
2 6 6 4
1 þ pikn ðRk þ pii Q ki Þ
P
pij Q kj
j–i;j2Sikn
P
3
7 X Rk þ pii Q ki 7þ pij 5 pij Q kj i j2Suk
j–i;j2Sikn
2 6 6 4
1 þ pikn ðR5 þ pii di Q 4i Þ
P
pij dj Q 4i
P
2 6 6 4 2 6 6 4
1 þ pikn Ei
hi 1 þ pikn Li hi 1 þ pikn R6
1 þ pikn Ei
hi 1 þ pikn N i hi 1 þ pikn R6
3
2 E 7 X 6 i þ p 0 7 4 ij 5 j2Siuk kn E2i
Ekn 1i
3
2 E 7 X 6 i þ p 0 7 4 ij 5 j2Siuk kn E2i
Ekn 1i
< 0;
3
j2Suk
j–i;j2Sikn
Q kj
7 X R5 þ pii di Q 4i 7þ p ij 5 pij dj Q 4i i
j–i;j2Sikn
Q kj
hi Li hi R6
Euk 2i
hi N i
Euk 1i
hi R6
dj Q 4i dj Q 4i
3 Euk 1i 7 0 5 < 0;
5777
k ¼ 1; 2; 3; 4;
< 0;
ð56Þ
ð57Þ
ð58Þ
3
7 0 5 < 0:
ð59Þ
Euk 2i
If (16)–(29) hold, then (56)–(59) also hold. From (53)–(55), it is easy to obtain that:
Ri ¼
hi ðtÞ h hi ðtÞ Ri1 þ i Ri2 : hi hi
ð60Þ
Setting k1 = min{kmin(Ri1), kmin(Ri2), i 2 S}, then k1 > 0. For any t P 0, we have:
LVðzðtÞ; iÞ 6 k1 fT ðtÞfðtÞ 6 k1 zT ðtÞzðtÞ
ð61Þ
By Dynkin’s formula, one can obtain:
Z t EfVðzðtÞ; iÞg EfVðu; r 0 Þg 6 k1 E zT ðsÞzðsÞds ;
ð62Þ
0
and hence:
Z t 1 E zT ðsÞzðsÞds 6 EfVðu; r0 Þg; k1 0
tP0
ð63Þ
Based on Definition 1, the system (8) is stochastically stable when the time-varying delays are mode-dependent and the transition rates are partially known. The proof is completed. h
Remark 2. Theorem 1 develops a stochastic stability criterion of Markovian jumping neural networks with mode-dependent time varying delays and partially known transition rates. The result of Theorem 1 makes use of the information of the subsystems’ upper bounds of the time-varying delays, which may bring us less conservativeness. Moreover, by freeweighting matrices method, the upper bounds of ui are not restricted to be 1 in this paper. Therefore, our result is more natural and reasonable to Markovian jumping neural networks.
Ri < 0 is not simply guaranteed by Remark 3. From (53), it can be easily seen that T 1 T Ei þ hi Li R1 6 Li þ hi N i R6 N i þ diagfW1i ; W2i ; W3i ; W4i ; 0; 0; 0g < 0, but is evaluated by the LMIs in (54), (55), which can help reduce much conservatism than some existing results. Remark 4. Theorem 1 directly handles the inversely weighted convex combination of quadratic terms of integral quantities by utilizing the result of Lemma 1, which achieves performance behavior identical to the approaches based on Lemma 2. Now, the following corollary presents a sufficient condition for Markovian jumping neural networks with mode-dependent time varying delays and completely known transition rates. Corollary 1. For given scalars hi P 0, di > 0, ui, the system (8) with mode-dependent time varying delays and completely known transition rates is stochastically stable if there exist symmetric positive definite matrices Pi, Q1i, Q2i, Q3i, Q4i, R1, R2, R3, R4, R5, R6, R7, positive diagonal matrices Ti1, Ti2 and any matrices S12, Li, Ni, Mi, with appropriate dimensions, for any i = 1, 2, . . . ,N, such that LMIs (15), (31), (32) and the following LMIs hold:
5778
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
"
"
Ei
hi L i
hi R6
Ei
hi Ni hi R6
# < 0;
ð64Þ
< 0;
ð65Þ
#
where
2
E11 6 6 6 6 6 6 Ei ¼ 6 6 6 6 6 4
E12 E22
E13 E23
E33
0
0
0
E44
C Ti MTi2
0
E11 ¼ E11 þ
Mi1 C i E24
E15 0
E16 0
E55
M i2 Bi
E66
X
X
pij Pj ; E22 ¼ E22 þ
j–i;j2S
3 M i1 Di 0 7 7 7 0 7 7 0 7 7; 7 M i2 Di 7 7 7 0 5 1 di Q 4i
pij hj Q 1i ; E33 ¼ E33 þ
j–i;j2S
X
pij hj Q 2i ; E44 ¼ E44 þ
j–i;j2S
X
pij hj Q 3i :
j–i;j2S
Proof. By Theorem 1, the desired results can be obtained easily according to (54), (55). This completes the proof.
h
Remark 5. It is easy to obtain that the conditions (16)–(29) will reduce to 31, 32, 64, 65, respectively when the i th row of P are all available.
4. A Numerical example Consider the system (8) with the following parameters:
1 1 0:88 1 0:8 0:4 2:2 0 B1 ¼ ; C1 ¼ ; D1 ¼ ; A2 ¼ ; 0 2 1 1 1 1 0:5 0:6 0 1:5 1 0:6 1 0:1 1:2 0:7 2:3 0 0:3 0:2 B2 ¼ ; C2 ¼ ; D2 ¼ ; A3 ¼ ; B3 ¼ ; 0:1 0:3 0:1 0:2 0:6 0:4 0 2:5 0:4 0:1 0:5 0:7 0:5 O:3 0:2 0 0:4 0 ; D3 ¼ ; C1 ¼ ; C2 ¼ : C3 ¼ 0:7 0:4 0:2 1:2 0 0:1 0 0:8
A1 ¼
2 0
;
The three cases of the transition rates matrices are considered as:
2
0:3
0:8
6 Caseð1Þ : P ¼ 4 0:1
0:5
3
2
7 0:7 5; 0:7 0:4 1:1 2 3 0:8 ? ? 6 7 Caseð3Þ : P ¼ 4 ? 0:8 ? 5: 0:7 0:4 1:1
0:8
6 Caseð2Þ : P ¼ 4 0:1
0:8
0:7
? 0:8 0:4
?
3
7 0:7 5; 1:1
We assume condition 1: h1 = h2 = h3, u1 = u2 = u3 = 0.1, d1 = d2 = d3 = 0.5; condition 2: h1 = h2 = h3, u1 = u2 = u3 = 0.5, ~ which d1 = d2 = d3 = 0.8, and under the three cases above, respectively. Table 1 lists the corresponding upper bounds of h ~ decreases when can be computed by the method of Theorem 1 in this paper. Table 1 shows that the upper bounds of h the number of unknown elements increases. ~ ¼ 1:34, one can obtain: Solving LMIs (15)–(29) for the case (1), condition 1 and h
P1 ¼
0:0031 0:0014 0:0014 0:0084
Q 11 ¼ 1:0e004 Q 13 ¼ 1:0e004
;
0:1438
P2 ¼
0:0024 0:0012 0:0012 0:0072
0:1307
0:1307 0:1563 0:3876 0:3860 0:3860
0:5253
; ;
;
P3 ¼
Q 12 ¼ 1:0e004
0:0025 0:0013 0:0013 0:0074
;
0:6556
0:6800
0:6800
0:8977
;
5779
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
Q 21 ¼
0:0023 0:0019 0:0019 0:0072
Q 31 ¼ 1:0e003 Q 41 ¼
T 21 ¼
T 32 ¼
;
Q 42 ¼
; Q 32 ¼ 1:0e003
0:0108 0:0025 0:0025 0:0047
;
Q 23 ¼
0:0023 0:0019
;
0:0019 0:0072
0:4380 0:2729 ; Q 33 ¼ 1:0e003 ; 0:4896 0:3481 0:2729 0:2035 0:7800 0:4896
Q 43 ¼
0:0155 0:0036 0:0036 0:0092
;
0:4489 0:4449 0:9668 0:6602 ; R2 ¼ 1:0e004 ; R3 ¼ 1:0e003 ; 0:5361 0:7249 0:4449 0:5991 0:6602 0:4934
R5 ¼
;
0:0051
R7 ¼ 1:0e003
0:0019 0:0072
;
0:5223 0:5361
0:0062 0:0003 0:0003
0:0023 0:0019
0:1205 0:0703
0:0098 0:0150
Q 22 ¼
0:2258 0:1205
0:0201 0:0098
R1 ¼ 1:0e004
R4 ¼
;
0:1250 0:0362 0:0362 0:2403
0:0237
0
0
0:0090
0:0160
0
0
0:0072
;
T 22 ¼
0:0020
0:0014
0:0014
0:0010
;
T 11 ¼
;
R6 ¼
0:0546
0
0
0:0173
0:0092
0
0
0:0013
T 31 ¼
;
0:0008 0:0008
;
0:0008 0:0026
;
T 12 ¼
0:0441
0
0
0:0130
0:0170
0
0
0:0073
;
;
Therefore it follows from Corollary 1 that the system (6)with mode-dependent time-varying delays and completely known transition rates is stochastically stable. ~ ¼ 1:21, one can obtain: Solving LMIs (15)–(29) for the case (2), condition 1 and h
P1 ¼
0:0014 0:0007 0:0007 0:0041
Q 11 ¼ 1:0e004
Q 21 ¼
R4 ¼
0:0009
0:0009
;
0:0034 0:0064
Q 22 ¼
Q 42 ¼
; Q 12 ¼ 1:0e004
0:0008 0:0007
P3 ¼
;
0:0007 0:0027
0:0660 0:0411 ;
0:0006
0:0006 0:0031
0:1164 0:0660
0:0082 0:0034
R1 ¼ 1:0e004
0:0855 0:0877
0:0009 0:0034
P2 ¼
0:0877 0:1093
0:0010
Q 31 ¼ 1:0e003
Q 41 ¼
;
0:0011 0:0028
0:0010
0:0006
0:0006 0:0033
;
0:0963 0:1053 ; Q 13 ¼ 1:0e004 ; 0:1257 0:1785 0:1053 0:1517
0:0008 0:0008
Q 23 ¼
0:1958 0:1395
0:0008 0:0029
0:1395 0:1121
;
0:1157 0:1257
;
; Q 32 ¼ 1:0e003
0:0048 0:0011
Q 43 ¼
;
; Q 33 ¼ 1:0e003
0:0065 0:0014 0:0014 0:0044
0:1198 0:0797 0:0797 0:0630
;
0:0708 0:0544 0:2341 0:1832 ; R2 ¼ 1:0e004 ; R3 ¼ 1:0e003 ; 0:1155 0:1751 0:0544 0:2331 0:1832 0:1562 0:1026 0:1155
0:0020 0:0001 0:0001 0:0018
;
R5 ¼
0:0011 0:0001 0:0001 0:0008
;
R6 ¼
0:0004
0:0004
0:0004 0:0013
;
Table 1 ~ under the three cases. Allowable upper bound of h
Condition 1 Condition 2
Case 1
Case 2
Case 3
1.34 1.21
1.21 1.04
1.11 0.87
;
5780
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
R7 ¼ 1:0e003 T 21 ¼
T 32 ¼
0:0454 0:0192
0:0192 0:1158
0:0100
0
0
0:0045
0:0084
0
0
0:0035
;
T 11 ¼
;
T 22 ¼
0:0195
0
0
0:0068
0:0036
0
0
0:0007
;
T 12 ¼
T 31 ¼
;
0:0194
0
0
0:0063
;
0:0085
0
0
0:0036
;
~ ¼ 1:11, one can obtain: Solving LMIs (15)–(29) for the case (3), condition 1 and h
P1 ¼
0:0029 0:0014
0:0014 0:0087
Q 11 ¼ 1:0e004
Q 21 ¼
Q 42 ¼
;
;
P3 ¼
; Q 12 ¼ 1:0e003
0:0016 0:0016
0:8951 0:5838
; Q 13 ¼ 1:0e004
0:0049 0:0092
0:6257 0:6855 0:6855 0:9630
;
;
; Q 33 ¼ 1:0e003
0:0123 0:0049
0:0014 0:0051
0:5838 0:4315
Q 43 ¼
;
0:0014 0:0014
Q 23 ¼
;
0:0072 0:0063
0:0815 0:1134
0:0014 0:0076
;
; Q 32 ¼ 1:0e003
0:0153 0:0072
0:0024 0:0014
0:0737 0:0815
0:0016 0:0058
0:7917 0:5346 0:5346 0:4136
;
;
0:0003 0:0003 0:8490 0:6195 ; R2 ¼ 1:0e004 ; R3 ¼ 1:0e003 ; 0:5934 0:8668 0:0003 0:0012 0:6195 0:5010 0:5248 0:5934
0:0022
0:0022
0:0078
Q 22 ¼
0:5478 0:3271
0:0060
T 32 ¼
;
0:0110
R7 ¼ 1:0e003
0:0016 0:0084
0:3271 0:2202
0:0060
T 21 ¼
0:0146 0:0060
R1 ¼ 1:0e004
0:0026 0:0016
0:4740 0:6191
0:0016 0:0058
Q 41 ¼
0:4529 0:4740
0:0016 0:0016
Q 31 ¼ 1:0e003
R4 ¼
P2 ¼
;
;
R5 ¼
0:1266 0:0742 0:0742 0:2548
0:0390
0
0
0:0138
0:0090
0
0
0:0060
;
T 22 ¼
;
0:0027
0:0003
0:0003
0:0018
T 11 ¼
;
R6 ¼
0:0377
0
0
0:0135
0:0119
0
0
0:0015
;
T 31 ¼
;
0:0010
0:0009
0:0009 0:0031 T 12 ¼
;
0:0413
0
0
0:0136
0:0157
0
0
0:0094
;
;
Therefore it follows from Theorem 1 that the system (6) with mode-dependent time-varying delays and partially known transition rates is stochastically stable. 5. Conclusions In this paper, the problem of stochastic stability criterion of Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates has been proposed. By choosing a new class of Lyapunov functional, some new delay-dependent stochastic stability criteria are derived to guarantee the stochastic stability of Markovian jumping neural networks. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization approach are considered. Finally, a numerical example has been given to illustrate the effectiveness of the proposed method. Acknowledgments The authors thank the editors and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the demonstration project of oil and gas development for carbonate reservoirs in Tarim basin under Grant 2011ZX05049 and the National Basic Research Program of China under Grant 2010CB732501.
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Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Delay-dependent stochastic stability criteria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates Junkang Tian a,b,c,⇑, Yongming Li a, Jinzhou Zhao a, Shouming Zhong c a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China c School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China b
a r t i c l e
i n f o
Keywords: Stochastic stability Markovian jumping neural networks Time-varying delays
a b s t r a c t In this paper, the problem of stochastic stability criterion of Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates is considered. Some new delay-dependent stability criteria are derived by choosing a new class of Lyapunov functional. The obtained criteria are less conservative because freeweighting matrices method and a convex optimization approach are considered. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. 2011 Elsevier Inc. All rights reserved.
1. Introduction In recent decades, neural networks have been investigated extensively because of their successful applications in various areas such as pattern recognition, image processing, associative memory and combinatorial optimization. However, these successful applications are greatly dependent on the dynamic behaviors of neural networks. As is well known now, stability is one of the main properties of neural networks, which is a crucial feature in the design of neural networks. On the other hand, it has been recognized that the time delays often occur in various neural networks, and may cause undesirable dynamic network behaviors such as oscillation and instability. Therefore, the stability analysis for delayed neural networks has become a topic of great theoretic and practical importance in recent years [1–27]. Recently, systems with Marvokian jumps have been attracting increasing research attention. This class of systems are the hybrid systems with two components in the state. The first one refers to the mode, which is described by a continuous–time finite-state Markovian process, and the second one refers to the state which is represented by a system of differential equations. The Markovian jump systems have the advantage of modeling the dynamic systems subject to abrupt variation in their structures, such as component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, and operating in different points of a nonlinear plant [28]. Recently, there has been a growing interest in the study of neural networks with Markovian jumping parameters [29–38]. In [29], the problem of stochastic robust stability for uncertain delayed neural networks with Markovian jumping parameters is investigated. The state estimation problem for a class of Markovian neural networks with discrete and distributed time-delays is studied in [30]. Without assuming the boundedness, monotonicity and differentiability of the activation functions, some results for delay-dependent stochastic stability criteria for the Markovian jumping Hopfield neural networks with time-delay are developed in [31]. Some new delay-dependent
⇑ Corresponding author at: State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China. E-mail address: [email protected] (J. Tian). 0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.087
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J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
stochastic stability criteria for BAM neural networks with Markovian jumping parameters are derived in [32] based on delay partitioning idea. To the best of our knowledge, the stochastic stability analysis for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates has never been tackled, and such a situation motivates our present study. In this paper, the problem of stochastic stability criterion for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates is considered. By choosing a new class of Lyapunov functional, some new delay-dependent stochastic stability criteria are derived to guarantee the stochastic stability of Markovian jumping neural networks. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization approach are considered. Finally, a numerical example is given to show the effectiveness of the derived method. 2. Problem formulation Consider the following delayed neural network:
_ xðtÞ ¼ AxðtÞ þ BgðxðtÞÞ þ Cgðxðt hðtÞÞÞ þ D
Z
t
gðxðsÞÞds þ l;
ð1Þ
tdðtÞ
0; xðtÞ ¼ UðtÞ; t 2 ½h;
ð2Þ T
T
where xðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ 2 Rn is the neuron state vector, gðxðÞÞ ¼ ½g 1 ðx1 ðÞÞ; g 2 ðx2 ðÞÞ; . . . ; g n ðxn ðÞÞ 2 Rn denotes the neuron activation function, and l ¼ ðl1 ; l2 ; . . . ; ln ÞT 2 Rn is a constant input vector. B; C; D 2 Rnn are the connection weight matrix and the delayed connection weight matrix,respectively. A = diag(a1, a2, . . . , an) with ai > 0, i = 1, 2, . . . , n. h(t), _ d(t) are time-varying continuous functions that satisfy 0 6 hðtÞ 6 h; 0 6 dðtÞ 6 d; hðtÞ 6 u, where h, d and u are constants. 0, where h ¼ maxfd; hg. In addition, it is assumed that each neuron The initial vector U(t) is continuously differential on ½h; activation function gi(), i = 1, 2, . . . , n is bounded and satisfies the following condition:
ci 6
g i ðxÞ g i ðyÞ 6 cþi ; xy
8x; y 2 R;
x – y;
i ¼ 1; 2; . . . ; n;
ð3Þ
þ where c i ; ci ; i ¼ 1; 2; . . . ; n are constants. Note that by using the Brouwers fixed-point theorem, T it can be easily proven that there exists at least one equilibrium point for system (1). Assuming that x ¼ x1 ; x2 ; . . . ; xn is the equilibrium point of (1) and using the transformation z() = x() x⁄, (1) can be converted to the following system:
z_ ðtÞ ¼ AzðtÞ þ Bf ðzðtÞÞ þ Cf ðzðt hðtÞÞÞ þ D
Z
t
f ðzðsÞÞds;
ð4Þ
tdðtÞ
where z(t) = [z1(t), z2(t), . . . , zn(t)]T, f(z()) = [f1(z1()), f2(z2()), . . . , fn(zn())]T and fi ðzi ðÞÞ ¼ g i zi ðÞ þ xi g i xi ; i ¼ 1; 2; . . . ; n. According to the inequality (3), one can obtain that:
ci 6
fi ðzi ðtÞÞ 6 cþi f i ð0Þ ¼ 0; zi ðtÞ
i ¼ 1; 2; . . . ; n:
ð5Þ
Given probability space (X, !,P) where X is sample space, ! is r algebra of subset of the sample space, and P is the probability measure defined on !. Let S = {1, 2, . . . , N} and the random form process {r(t), t 2 [0, +1)} be a homogeneous, finitestate Markovian process with right continuous trajectories with generator P = (pij)NN and transition probability from mode i at time t to mode j at time t + Dt, i, j 2 S:
Pfrðt þ DtÞ ¼ jjrðtÞ ¼ ig ¼
pij Dt þ oðDtÞ j – i; 1 þ pii Dt þ oðDtÞ j ¼ i;
ð6Þ
P with transition rates pij P 0 for i, j 2 S, j – i and pii ¼ Nj¼1;j–i pij , where Dt > 0 and limDt!0 oðDDttÞ ¼ 0. In addition, the transition rates of the Markovian chain are considered to be partially available, namely, some elements in matrix P are time-invariant but unknown. For instance, a system with three operation modes may have the transition rates matrix P as follows:
2
p11
P¼6 4 ?
p31
?
?
3
p22 ? 7 5; p32 p33
where ‘‘?’’ represents the inaccessible element. For notation clarity, "i 2 S, we denote S ¼ Sikn [ Siuk with Sikn , fj; if pij is knowng, Siuk , fj; if pij is unknowng; pikn P , j2Si pij throughout the paper. Furthermore, we assume the diagonal elements of P are known. Note that the set S comkn prises the various operational modes of the system under study. In this paper, we consider the Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates described by the following nonlinear differential equations:
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
_ xðtÞ ¼ AðrðtÞÞxðtÞ þ BðrðtÞÞgðxðtÞÞ þ CðrðtÞÞgðxðt hðrðtÞ; tÞÞÞ þ DðrðtÞÞ
Z
5771
t
gðxðsÞÞds þ l
ð7Þ
tdðrðtÞ;tÞ
Similar to the above analysis,we can obtain:
z_ ðtÞ ¼ AðrðtÞÞzðtÞ þ BðrðtÞÞf ðzðtÞÞ þ CðrðtÞÞf ðzðt hðrðtÞ; tÞÞÞ þ DðrðtÞÞ
Z
t
f ðzðsÞÞds;
ð8Þ
tdðrðtÞ;tÞ
when r(t) = i 2 S, the matrices A(r(t)), B(r(t)), C(r(t)), D(r(t)) are represented by Ai, Bi, Ci, Di. In the system (8) hi(t), di(t) denote ~ ¼ max fh g; d ~¼ the mode-dependent time-varying delays which satisfy 0 6 hi ðtÞ 6 hi ; 0 6 di ðtÞ 6 di ; h_ i ðtÞ 6 ui ; h j2S j ~ dg. ~ max fd g; s ¼ maxfh; j2S
j
The initial conditions of system (8) is of the following form:
zðtÞ ¼ uðtÞ;
t 2 ½s; 0;
rð0Þ ¼ r 0 :
ð9Þ
Definition 1. Markovian system (8) is said to be stochastically stable, if for any u(t) defined on [s, 0] and r(0) 2 S, the following condition is satisfied:
lim E
Z
t!1
t
zT ðsÞzðsÞdsju; r 0
< 1:
ð10Þ
0
Definition 2. Let U1 ; U2 ; . . . ; UN : Rm ! Rn be a given finite number of functions such that they have positive values in an open subset D of Rm . Then, a reciprocally convex combination of these functions over D is a function of the form:
1
a1
U1 þ
1
a2
1
U2 þ þ
aN
UN : D ! Rn ;
ð11Þ P
where the real numbers ai satisfy ai > 0 and i ai ¼ 1. The following Lemma 1 suggests a lower bound for a reciprocally convex combination of scalar positive functions Ui = fi. Lemma 1 [39]. Let f1 ; f2 ; . . . ; fN : Rm ! R have positive values in an open subset D of Rm . Then, the reciprocally convex combination of fi over D satisfies:
min P
fai jai >0;
X 1 a ¼1g i i
i
ai
fi ðtÞ ¼
X
fi ðtÞ þ max g i;j ðtÞ
i
(
X i–j
m
subject to :
g i;j : R ! R; g j;i ðtÞDg i;j ðtÞ;
g i;j ðtÞ; "
ð12Þ
fi ðtÞ
g i;j ðtÞ
g i;j ðtÞ
fj ðtÞ
#
) P0 :
ð13Þ
Lemma 2 [40]. For any constant matrix Z 2 Rnn ; Z ¼ Z T > 0, scalar h > 0, such that the following integration is well defined, then:
h
Z
t
xT ðsÞZxðsÞds 6
th
Z
t
th
xT ðsÞdsZ
Z
t
xðsÞds:
ð14Þ
th
3. Main results In this section, a new Lyapunov functional is constructed to derived a delay-dependent stochastic stability criterion for system (8) when the time-varying delays are mode-dependent and the transition rates are partially known. Theorem 1. For given scalars hi P 0, di > 0, ui, the system (8) with mode-dependent time-varying delays and partially known transition rates is stochastically stable if there exist symmetric positive definite matrices Pi, Q1i, Q2i, Q3i, Q4i, R1, R2, R3, R4, R5, R6, R7, positive diagonal matrices Ti1, Ti2 and any matrices S12, Li, Ni, Mi, with appropriate dimensions, for any i = 1, 2, . . . ,N, such that the following LMIs hold:
R7
S12
R7
P 0;
ð15Þ
5772
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
2 6 6 4
1 þ pikn ðR1 þ pii Q 1i Þ
P
pij Q 1j
j–i;j2Sikn
P
3 7 7 < 0;
pij Q 1j 5
ð16Þ
j–i;j2Sikn
R1 þ pii Q 1i
Q 1j
Q 1j
2 6 6 4
8j 2 Siuk ;
< 0;
1 þ pikn ðR2 þ pii Q 2i Þ
P
pij Q 2j
j–i;j2Sikn
P
ð17Þ 3 7 7 < 0;
pij Q 2j 5
ð18Þ
j–i;j2Sikn
R2 þ pii Q 2i
Q 2j
Q 2j
2 6 6 4
8j 2 Siuk ;
< 0;
1 þ pikn ðR3 þ pii Q 3i Þ
P
pij Q 3j
P
3
7 7 < 0; pij Q 3j 5
j–i;j2Sikn
ð19Þ
ð20Þ
j–i;j2Sikn
R3 þ pii Q 3i
Q 3j
Q 3j
2 6 6 4
8j 2 Siuk ;
< 0;
1 þ pikn ðR4 þ pii Q 4i Þ
P
pij Q 4j
j–i;j2Sikn
P
ð21Þ 3 7 7 < 0;
pij Q 4j 5
ð22Þ
j–i;j2Sikn
R4 þ pii Q 4i
Q 4j
Q 4j
2 6 6 4
< 0;
8j 2 Siuk ;
1 þ pikn ðR5 þ pii di Q 4i Þ
P
pij dj Q 4i
j–i;j2Sikn
P
ð23Þ 3 7 7 < 0;
pij dj Q 4i 5
ð24Þ
j–i;j2Sikn
R5 þ pii di Q 4i
dj Q 4i
dj Q 4i
2 6 6 4
1 þ pikn Ei
2
Ei 6 4
6 6 4
hi Li hi R6
1 þ pikn Ei
2
Ei 6 4
< 0;
hi 1 þ pikn Li hi 1 þ pikn R6
2
Euk 1i
7 0 5 < 0;
7 0 7 5 < 0; kn E2i
8j 2 Siuk ;
3 ð26Þ
ð27Þ
Euk 2i
hi R6
Ekn 1i
ð25Þ
3
hi 1 þ pikn Ni hi 1 þ pikn R6
hi N i
8j 2 Siuk ;
Euk 1i
3
7 0 7 5 < 0; kn E2i
ð28Þ
3
7 0 5 < 0;
Euk 2i
Ekn 1i
8j 2 Siuk ;
ð29Þ
5773
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
where
2
E11 6 6 6 6 6 Ei ¼ 6 6 6 6 4
E12 E22
2
Ekn 2i
E13 E23 E33
M i1 C i E24 0 E44
0
Xi1 6 0 6 ¼4 0 0
E15 0 0 C Ti M Ti2 E55 3 0 0 7 7; 0 5
0 0
Xi2 0 0
Xi3 0
3 3 2 Mi1 Di Xi1 0 0 0 0 7 6 0 7 0 Xi2 0 7 7 6 7 6 0 0 7 X 0 0 7 i3 7 6 7 kn 6 0 7; E1i ¼ 6 0 0 0 Xi4 7 7; 7 6 0 Mi2 Di 7 0 0 0 7 7 6 7 4 0 0 5 0 0 0 5 d1 Q 4i 0 0 0 0 3 2 i 0 0 0 Pj 6 0 hj Q 1i 2 0 0 7 7 6 0 Pj 7 60 Q 0 0 h j 2i 7 6 0 hj Q 1i 6 uk 7 6 6 ¼60 0 0 hj Q 3i 7; E2i ¼ 4 0 0 60 0 0 0 7 7 6 0 0 5 40 0 0 0 0 0 0 0
E16 0 0 0 Mi2 Bi E66
Euk 1i
Xi4
0 0 hj Q 2i 0
3 0 0 7 7 0 5 hj Q 3i
~ 1 þ R2 Þ R7 þ L þ LT M A AT M T 2C1 T C2 ; E12 ¼ R7 ST þ LT L þ N ; E11 ¼ pii Pi þ Q 1i þ Q 2i þ hðR i1 i1 i i1 i1 i1 12 i1 i1 i2 i E13 ¼ ST12 þ LTi3 Ni1 ; E15 ¼ Pi M i1 ATi MTi2 ; E16 ¼ M i1 Bi þ T i1 ðC1 þ C2 Þ; E22 ¼ 2R7 þ S12 þ ST12 ð1 ui ÞQ 1i Li2 LTi2 þ Ni2 þ NTi2 2C1 T i2 C2 ; E23 ¼ R7 ST12 LTi3 þ NTi3 Ni2 ; ~ 6 þh ~ 2 R7 M M T ; E24 ¼ T i2 ðC1 þ C2 Þ; E33 ¼ R7 Q 2i þ pii hi Q 2i Ni3 NTi3 ; E44 ¼ ð1 ui ÞQ 3i 2T i2 ; E55 ¼ hR i2 i2 2 ~ X X X d ~ 3 þ R4 þ dR ~ 5 2T i1 ; Xi1 ¼ pij Pj ; Xi2 ¼ pij hj Q 1i ; Xi3 ¼ pij hj Q 2i ; E66 ¼ Q 3i þ di Q 4i þ hR 2 j–i;j2Sikn j–i;j2Sikn j–i;j2Sikn X Xi4 ¼ pij hj Q 3i ; C1 ¼ diag c1 ; c2 ; .. .; cn ; C2 ¼ diag cþ1 ; cþ2 ; .. .; cþn ; j–i;j2Sikn
e1 ¼ ð I
T
0 0 0 0 0 0Þ ; 0 0 0 0 ÞT ;
e3 ¼ ð 0 0 I Ni ¼ N Ti1
NTi2
NTi3
e2 ¼ ð 0 I Li ¼ LTi1
0 0 0 0
T
;
0 0 0 0 0Þ LTi2
LTi3
M i ¼ M Ti1
T
0 0 0 0
T
0 0 0 M Ti2
0 0
T
Proof. Construct a new class of Lyapunov functional candidate as follow:
VðzðtÞ; iÞ ¼
X
9
V j ðzðtÞ; iÞ;
j¼1
where
V 1 ðzðtÞ; iÞ ¼ zT ðtÞPðrðtÞÞzðtÞ; Z t V 2 ðzðtÞ; iÞ ¼ zT ðsÞQ 1 ðrðtÞÞzðsÞds; thðrðtÞ;tÞ Z t V 3 ðzðtÞ; iÞ ¼ zT ðsÞQ 2 ðrðtÞÞzðsÞds; thðrðtÞÞ Z t V 4 ðzðtÞ; iÞ ¼ f T ðzðsÞÞQ 3 ðrðtÞÞf ðzðsÞÞds; thðrðtÞ;tÞ Z 0 Z t f T ðzðsÞÞQ 4 ðrðtÞÞf ðzðsÞÞdsdh; V 5 ðzðtÞ; iÞ ¼ dðrðtÞÞ tþh Z 0Z t Z 0Z t zT ðsÞðR1 þ R2 ÞzðsÞdsdh þ f T ðzðsÞÞR3 f ðzðsÞÞdsdh; V 6 ðzðtÞ; iÞ ¼ ~ ~ tþh h Z Z0h Ztþh Z 0 t 0 Z t f T ðzðsÞÞR4 f ðzðsÞÞdsdkdh þ f T ðzðsÞÞR5 f ðzðsÞÞdsdh; V 7 ðzðtÞ; iÞ ¼ ~ h ~ tþh d tþk d Z 0Z t z_ T ðsÞR6 z_ ðsÞdsdh; V 8 ðzðtÞ; iÞ ¼ ~ tþh h Z 0 Z t ~ z_ T ðsÞR7 z_ ðsÞdsdh; V 9 ðzðtÞ; iÞ ¼ h ~ h
tþh
ð30Þ
5774
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
where Pi, Q1i, Q2i, Q3i, Q4i, R1, R2, R3, R4, R5, R6, R7, i = 1, 2, . . . , N, are positive definite matrices and:
X
N
pij Q kj < Rk ; k ¼ 1; 2; 3; 4;
ð31Þ
pij dj Q 4i < R5 ;
ð32Þ
j¼1
X
N
j¼1
Remark 1 Our paper fully uses the information about mode-dependent time-varying delays and mode-dependent positive definite matrices, but [35] only use the information about mode-dependent positive definite matrices when constructing the Lyapunov functional V(z(t), i). So the Lyapunov functional V(z(t), i) in our paper is more general than that in [35], and the stability criteria in our paper may be more applicable. Let L be the infinitesimal generator of random process {zt, t P 0}, then for each r(t) = i, i 2 S, it can be shown that:
(Z
)
t
zT ðsÞQ 1 ðrðtÞÞzðsÞds
L
thðrðtÞ;tÞ
("Z # Z ) tþD t 1 E zT ðsÞQ 1 ðrðt þ DÞÞzðsÞdsjrðtÞ ¼ i zT ðsÞQ 1i zðsÞds D!0 D thi ðtÞ tþDhðrðtþDÞ;tþDÞ 8 " # >Z X 1 < tþD T N z ðsÞ Q 1i þ pij D þ oðDÞ zðsÞds ¼ limþ P D!0 D > : tþDhi ðtþDÞ N ðpij DþoðDÞÞhj ðtþDÞ j¼1 ¼ limþ
j¼1
Z
)
t
zT ðsÞQ 1i zðsÞds
thi ðtÞ
8 9 >Z > Z t = 1 < tþD zT ðsÞQ 1i zðsÞds zT ðsÞQ 1i zðsÞds ¼ limþ P > D!0 D > thi ðtÞ : tþDhi ðtþDÞ N ðpij DþoðDÞÞhj ðtþDÞ ; Z
1 þ limþ D!0 D 1 D
¼ limþ D!0
j¼1
tþD
tþDhi ðtþDÞ
Z
zT ðsÞ ðpij DþoðDÞÞhj ðtþDÞ
j¼1
tþD
zT ðsÞQ 1i zðsÞds þ limþ D!0
t
1 þ limþ D!0 D
PN
Z
1 D
X
N
ðpij D þ oðDÞÞQ 1j zðsÞds
j¼1
Z
thi ðtÞ
tþDhi ðtþDÞ
PN
ðpij DþoðDÞÞhj ðtþDÞ
zT ðsÞQ 1i zðsÞds
j¼1
tþD
tþDhi ðtþDÞ
PN
T
z ðsÞ ðpij DþoðDÞÞhj ðtþDÞ
j¼1
¼ zT ðtÞQ 1i zðtÞ ð1 h_ i ðtÞ
X
N
X
N
ðpij D þ oðDÞÞQ 1j zðsÞds
j¼1
pij hj ðtÞÞzT ðt hi ðtÞÞQ 1i zðt hi ðtÞÞ
j¼1
Z
þ
t
zT ðsÞ
thi ðtÞ
(Z L
Z
0
¼i
t
xT ðsÞQ ðrðtÞÞxðsÞdsdh
¼ limþ D!0
tþh
hðrðtÞÞ
Z
)
0
h½rðtÞ
Z
xT ðsÞQ ½rðtÞxðsÞds
1 E D
¼ limþ D!0
tþh
N
pij Q 1j zðsÞds
j¼1
)
t
!
X
(Z
h½rðtþDÞ
8 >Z 1<
D> :
Z
0
tþD
xT ðsÞQ ½rðt þ DÞxðsÞdsjrðtÞ
tþDþh
Z
0
PN j¼1
½pij DþoðDÞhj hi
tþD
xT ðsÞ tþDþh
X N j¼1
9 > Z 0 Z 0 Z t Z tþD = þ xT ðsÞQ i xðsÞdsdh xT ðsÞQ i xðsÞdsdh PN > hi hi tþh ; ½pij DþoðDÞhj tþDþh j¼1 P Z 0 Z t Z Z N X 1 hi j¼1 ½pij DþoðDÞhj t T N pij xT ðsÞQ j xðsÞdsdh limþ x ðsÞQ i xðsÞdsdh ¼ D!0 D hi hi tþh tþh j¼1 Z Z tþD Z t 1 0 þ limþ xT ðsÞQ i xðsÞds xT ðsÞQ i xðsÞds dh P N p DþoðDÞ h D!0 D hi tþh ½ ij j tþDþh j¼1
pij D þ oðDÞ Q j xðsÞds
5775
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
¼
X
N
pij
Z
¼
Z
N
¼
N
Z
pij hj limþ D!0
j¼1
pij
Z
T
Z
0
t
thi
PN
½pij DþoðDÞhj
xT ðsÞQ i xðsÞds
pij
Z
Z
0
t
xT ðsÞQ j xðsÞdsdh þ
X
N
Z
pij hj
t
xT ðsÞQ i xðsÞds þ
thi
j¼1
X
t
N
xT ðsÞQ j xðsÞdsdh 1
tþh
hi
j¼1
T
tþh
hi N
X
lim x ðt þ DÞQ i xðt þ DÞ x ðt þ h þ DÞQ i xðt þ h þ DÞ dh
j¼1
X
xT ðsÞQ j xðsÞdsdh þ
j¼1
0
þ hi D!0
X
t
tþh
hi
j¼1
þ
Z
0
pij hj
Z
0
xT ðtÞQ i xðtÞ xT ðt þ hÞQ i xðt þ hÞ dh
hi
!Z
t
xT ðsÞQ i xðsÞds þ hi xT ðtÞQ i xðtÞ thi
j¼1
Similar to the process above, we can obtain:
X LV 1 ðzðtÞ; iÞ ¼ 2zT ðtÞPi z_ ðtÞ þ zT ðtÞ N pij Pj zðtÞ;
ð33Þ
j¼1
LV 2 ðzðtÞ; iÞ ¼ zT ðtÞQ 1i zðtÞ ð1 h_ i ðtÞÞzT ðt hi ðtÞÞQ 1i zðt hi ðtÞÞ þ þ
Z
zT ðsÞ
N
thi ðtÞ
N
pij hj ðtÞzT ðt hi ðtÞÞQ 1i zðt hi ðtÞÞ
j¼1
!
X
t
X
pij Q 1j zðsÞds;
ð34Þ
j¼1
LV 3 ðzðtÞ; iÞ ¼ zT ðtÞQ 2i zðtÞ zT ðt hi ÞQ 2i zðt hi Þ þ
X
N
Z
pij hj zT ðt hi ÞQ 2i zðt hi Þ þ
t
zT ðsÞ
thi
j¼1
X
! N
pij Q 2j zðsÞds; ð35Þ
j¼1
LV 4 ðzðtÞ; iÞ ¼ f T ðzðtÞÞQ 3i f ðzðtÞÞ ð1 h_ i ðtÞÞf T ðzðt hi ðtÞÞÞQ 3i f ðzðt hi ðtÞÞÞ ! Z t X X N N T T pij hj ðtÞf ðzðt hi ðtÞÞÞQ 3i f ðzðt hi ðtÞÞÞ þ f ðzðsÞÞ pij Q 3j f ðzðsÞÞds: þ thi ðtÞ
j¼1
ð36Þ
j¼1
By Lemma 2, it is easy to obtain:
LV 5 ðzðtÞ; iÞ ¼
X
N
pij
Z
0
Z
di
j¼1
t
f T ðzðsÞÞQ 4j f ðzðsÞÞdsdh 1
tþh
X
N
pij dj
di
j¼1
1 di
Z
tþh
!T f ðzðsÞÞds
Q 4i
Z
tdi ðtÞ
t
f T ðzðsÞÞQ 4i f ðzðsÞÞds
!
t
f ðzðsÞÞds þ di f T ðzðtÞÞQ 4i f ðzðtÞÞ;
ð37Þ
tdi ðtÞ
~ T ðtÞðR þ R ÞzðtÞ þ hf ~ T ðzðtÞÞR f ðzðtÞÞ LV 6 ðzðtÞ; iÞ ¼ hz 1 2 3
LV 7 ðzðtÞ; iÞ ¼
f T ðzðsÞÞQ 4i f ðzðsÞÞds
tdi
j¼1
t
t tdi
þ di f T ðzðtÞÞQ 4i f ðzðtÞÞ Z 0 Z t Z X X N N pij f T ðzðsÞÞQ 4j f ðzðsÞÞdsdh þ pij dj 6 j¼1
!Z
Z
t ~ th
zT ðsÞðR1 þ R2 ÞzðsÞds
Z
t
~ th
f T ðzðsÞÞR3 f ðzðsÞÞds;
Z 0 Z t Z t ~2 d ~ T ðzðtÞÞR5 f ðzðtÞÞ f T ðzðsÞÞR4 f ðzðsÞÞdsdh f T ðzðsÞÞR5 f ðzðsÞÞds; f T ðzðtÞÞR4 f ðzðtÞÞ þ df ~ tþh ~ 2 d td
~z_ T ðtÞR6 z_ ðtÞ LV 8 ðzðtÞ; iÞ ¼ h
Z
t
~z_ T ðtÞR6 z_ ðtÞ z_ T ðsÞR6 z_ ðsÞds 6 h
~ th
z_ T ðsÞR6 z_ ðsÞds
Z
Z
ð39Þ
t
z_ T ðsÞR6 z_ ðsÞds;
ð40Þ
thi ðtÞ
t
~ th
~2 z_ T ðtÞR7 z_ ðtÞ h 6h i
thi ðtÞ
thi
Z
~ ~2 z_ T ðtÞR7 z_ ðtÞ h LV 9 ðzðtÞ; iÞ ¼ h
Z
ð38Þ
z_ T ðsÞR7 z_ ðsÞds
thi ðtÞ
z_ T ðsÞR7 z_ ðsÞds hi
thi
Z
t
z_ T ðsÞR7 z_ ðsÞds:
ð41Þ
thi ðtÞ
By the Newton–Leibniz formula and (8), for any appropriately dimensioned matrices Li, Ni, Mi, i = 1, 2, . . . , N, one can obtain: T
2f ðtÞLi zðtÞ zðt hi ðtÞÞ
Z
!
t
thi ðtÞ
z_ ðsÞds
¼0
ð42Þ
5776
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
2fT ðtÞNi zðt hi ðtÞÞ zðt hi Þ
Z
!
thi ðtÞ
z_ ðsÞds
¼0
ð43Þ
thi
T
Z
z_ ðtÞ Ai zðtÞ þ Bi f ðzðtÞÞ þ C i f ðzðt hi ðtÞÞÞ þ Di
2f ðtÞM i
!
t
f ðzðsÞÞds
¼0
ð44Þ
tdi ðtÞ
" T
T
T
T
T
_T
T
f ðtÞ ¼ z ðtÞ z ðt hi ðtÞÞ z ðt hi Þ f ðzðt hi ðtÞÞÞ z ðtÞ f ðzðtÞÞ
Z
#
t T
f ðzðsÞÞds tdi ðtÞ
It is easy to obtain that:
2fT ðtÞLi
Z
t
thi ðtÞ
2fT ðtÞNi
Z
thi ðtÞ
thi
Z
Z
T z_ ðsÞds 6 hi ðtÞfT ðtÞLi R1 6 Li fðtÞ þ
t
z_ T ðsÞR6 z_ ðsÞds
T z_ ðsÞds 6 ðhi hi ðtÞÞfT ðtÞNi R1 6 N i fðtÞ þ
t
z_ T ðsÞR7 z_ ðsÞds ¼ thi
Z
thi ðtÞ
z_ T ðsÞR7 z_ ðsÞds
thi
ð45Þ
thi ðtÞ
Z
Z
thi ðtÞ
z_ T ðsÞR6 z_ ðsÞds
ð46Þ
thi
t
z_ T ðsÞR7 z_ ðsÞds
ð47Þ
thi ðtÞ
The LV9(z(t), i) is upper-bounded by
hi hi T fT ðtÞðe2 e3 ÞR7 ðe2 e3 ÞT fðtÞ f ðtÞðe1 e2 ÞR7 ðe1 e2 ÞT fðtÞ hi hi ðtÞ hi ðtÞ " #T # " R7 S12 eT2 eT3 eT2 eT3 T 2 _T ~ _ fðtÞ 6 h z ðtÞR7 zðtÞ f ðtÞ T ST12 R7 e1 eT2 eT1 eT2
~2 z_ T ðtÞR7 z_ ðtÞ LV 9 ðzðtÞ; iÞ 6 h
ð48Þ ð49Þ
where the inequalities in (48) come from Lemma 2, and that of (49) from Lemma 1 as:
2 qffiffi
3T T ðe e Þ 2 3 a 6 7 R7 fT ðtÞ4 qffiffi 5 ST12 abðe1 e2 ÞT b
2 qffiffi 3 T b ðe e Þ 2 3 S12 6 a 7 4 qffiffi 5fðtÞ 6 0; R7 aðe1 e2 ÞT
ð50Þ
b
i ðtÞ where a ¼ hi h ; b ¼ hihðtÞ . Note that when hi(t) = hi or hi(t) = 0, one can obtain fT(t)(e2 e3) = 0 or fT(t)(e1 e2) = 0, respechi i tively. So the relation (49) also holds. Furthermore, there exist positive diagonal matrices Ti1, Ti2, such that the following inequalities hold based on (3):
2f T ðzðtÞÞT i1 f ðzðtÞÞ þ 2zT ðtÞT i1 ðC1 þ C2 Þf ðzðtÞÞ 2zT ðtÞC1 T i1 C2 zðtÞ P 0;
ð51Þ
2f T ðzðt hi ðtÞÞÞT i2 f ðzðt hi ðtÞÞÞ þ 2zT ðt hi ðtÞÞT i2 ðC1 þ C2 Þf ðzðt hi ðtÞÞÞ 2zT ðt hi ðtÞÞC1 T i2 C2 zðt hi ðtÞÞ P 0: ð52Þ From (30)–(52), one can obtain that:
LVðzðtÞ; iÞ 6 fT ðtÞRi fðtÞ;
ð53Þ
where T 1 T Ri ¼ Ei þ hi ðtÞLi R1 6 Li þ ðhi hi ðtÞÞN i R6 N i þ diagfW1i ; W2i ; W3i ; W4i ; 0; 0; 0g;
W1i ¼
X
pij Pj ; W2i ¼
j–i;j2S
X j–i;j2S
pij hj Q 1i ; W3i ¼
X
pij hj Q 2i ; W4i ¼
j–i;j2S
X
pij hj Q 3i :
j–i;j2S
T 1 T 1 T 1 T Note that 0 6 hi ðtÞ 6 hi ; hi ðtÞLi R1 6 Li þ ðhi hi ðtÞÞN i R6 N i can be seen as the convex combination of Li R6 Li and N i R6 N i on hi(t). Therefore, Ri < 0 holds if and only if:
T Ri1 ¼ Ei þ hi Li R1 6 Li þ diagfW1i ; W2i ; W3i ; W4i ; 0; 0; 0g < 0;
Ri2 ¼ Ei þ
T hi Ni R1 6 Ni
þ diagfW1i ; W2i ; W3i ; W4i ; 0; 0; 0g < 0:
Applying the Schur complement, (31),(32),(54) and (55 can be rewritten respectively as:
ð54Þ ð55Þ
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
2 6 6 4
1 þ pikn ðRk þ pii Q ki Þ
P
pij Q kj
j–i;j2Sikn
P
3
7 X Rk þ pii Q ki 7þ pij 5 pij Q kj i j2Suk
j–i;j2Sikn
2 6 6 4
1 þ pikn ðR5 þ pii di Q 4i Þ
P
pij dj Q 4i
P
2 6 6 4 2 6 6 4
1 þ pikn Ei
hi 1 þ pikn Li hi 1 þ pikn R6
1 þ pikn Ei
hi 1 þ pikn N i hi 1 þ pikn R6
3
2 E 7 X 6 i þ p 0 7 4 ij 5 j2Siuk kn E2i
Ekn 1i
3
2 E 7 X 6 i þ p 0 7 4 ij 5 j2Siuk kn E2i
Ekn 1i
< 0;
3
j2Suk
j–i;j2Sikn
Q kj
7 X R5 þ pii di Q 4i 7þ p ij 5 pij dj Q 4i i
j–i;j2Sikn
Q kj
hi Li hi R6
Euk 2i
hi N i
Euk 1i
hi R6
dj Q 4i dj Q 4i
3 Euk 1i 7 0 5 < 0;
5777
k ¼ 1; 2; 3; 4;
< 0;
ð56Þ
ð57Þ
ð58Þ
3
7 0 5 < 0:
ð59Þ
Euk 2i
If (16)–(29) hold, then (56)–(59) also hold. From (53)–(55), it is easy to obtain that:
Ri ¼
hi ðtÞ h hi ðtÞ Ri1 þ i Ri2 : hi hi
ð60Þ
Setting k1 = min{kmin(Ri1), kmin(Ri2), i 2 S}, then k1 > 0. For any t P 0, we have:
LVðzðtÞ; iÞ 6 k1 fT ðtÞfðtÞ 6 k1 zT ðtÞzðtÞ
ð61Þ
By Dynkin’s formula, one can obtain:
Z t EfVðzðtÞ; iÞg EfVðu; r 0 Þg 6 k1 E zT ðsÞzðsÞds ;
ð62Þ
0
and hence:
Z t 1 E zT ðsÞzðsÞds 6 EfVðu; r0 Þg; k1 0
tP0
ð63Þ
Based on Definition 1, the system (8) is stochastically stable when the time-varying delays are mode-dependent and the transition rates are partially known. The proof is completed. h
Remark 2. Theorem 1 develops a stochastic stability criterion of Markovian jumping neural networks with mode-dependent time varying delays and partially known transition rates. The result of Theorem 1 makes use of the information of the subsystems’ upper bounds of the time-varying delays, which may bring us less conservativeness. Moreover, by freeweighting matrices method, the upper bounds of ui are not restricted to be 1 in this paper. Therefore, our result is more natural and reasonable to Markovian jumping neural networks.
Ri < 0 is not simply guaranteed by Remark 3. From (53), it can be easily seen that T 1 T Ei þ hi Li R1 6 Li þ hi N i R6 N i þ diagfW1i ; W2i ; W3i ; W4i ; 0; 0; 0g < 0, but is evaluated by the LMIs in (54), (55), which can help reduce much conservatism than some existing results. Remark 4. Theorem 1 directly handles the inversely weighted convex combination of quadratic terms of integral quantities by utilizing the result of Lemma 1, which achieves performance behavior identical to the approaches based on Lemma 2. Now, the following corollary presents a sufficient condition for Markovian jumping neural networks with mode-dependent time varying delays and completely known transition rates. Corollary 1. For given scalars hi P 0, di > 0, ui, the system (8) with mode-dependent time varying delays and completely known transition rates is stochastically stable if there exist symmetric positive definite matrices Pi, Q1i, Q2i, Q3i, Q4i, R1, R2, R3, R4, R5, R6, R7, positive diagonal matrices Ti1, Ti2 and any matrices S12, Li, Ni, Mi, with appropriate dimensions, for any i = 1, 2, . . . ,N, such that LMIs (15), (31), (32) and the following LMIs hold:
5778
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
"
"
Ei
hi L i
hi R6
Ei
hi Ni hi R6
# < 0;
ð64Þ
< 0;
ð65Þ
#
where
2
E11 6 6 6 6 6 6 Ei ¼ 6 6 6 6 6 4
E12 E22
E13 E23
E33
0
0
0
E44
C Ti MTi2
0
E11 ¼ E11 þ
Mi1 C i E24
E15 0
E16 0
E55
M i2 Bi
E66
X
X
pij Pj ; E22 ¼ E22 þ
j–i;j2S
3 M i1 Di 0 7 7 7 0 7 7 0 7 7; 7 M i2 Di 7 7 7 0 5 1 di Q 4i
pij hj Q 1i ; E33 ¼ E33 þ
j–i;j2S
X
pij hj Q 2i ; E44 ¼ E44 þ
j–i;j2S
X
pij hj Q 3i :
j–i;j2S
Proof. By Theorem 1, the desired results can be obtained easily according to (54), (55). This completes the proof.
h
Remark 5. It is easy to obtain that the conditions (16)–(29) will reduce to 31, 32, 64, 65, respectively when the i th row of P are all available.
4. A Numerical example Consider the system (8) with the following parameters:
1 1 0:88 1 0:8 0:4 2:2 0 B1 ¼ ; C1 ¼ ; D1 ¼ ; A2 ¼ ; 0 2 1 1 1 1 0:5 0:6 0 1:5 1 0:6 1 0:1 1:2 0:7 2:3 0 0:3 0:2 B2 ¼ ; C2 ¼ ; D2 ¼ ; A3 ¼ ; B3 ¼ ; 0:1 0:3 0:1 0:2 0:6 0:4 0 2:5 0:4 0:1 0:5 0:7 0:5 O:3 0:2 0 0:4 0 ; D3 ¼ ; C1 ¼ ; C2 ¼ : C3 ¼ 0:7 0:4 0:2 1:2 0 0:1 0 0:8
A1 ¼
2 0
;
The three cases of the transition rates matrices are considered as:
2
0:3
0:8
6 Caseð1Þ : P ¼ 4 0:1
0:5
3
2
7 0:7 5; 0:7 0:4 1:1 2 3 0:8 ? ? 6 7 Caseð3Þ : P ¼ 4 ? 0:8 ? 5: 0:7 0:4 1:1
0:8
6 Caseð2Þ : P ¼ 4 0:1
0:8
0:7
? 0:8 0:4
?
3
7 0:7 5; 1:1
We assume condition 1: h1 = h2 = h3, u1 = u2 = u3 = 0.1, d1 = d2 = d3 = 0.5; condition 2: h1 = h2 = h3, u1 = u2 = u3 = 0.5, ~ which d1 = d2 = d3 = 0.8, and under the three cases above, respectively. Table 1 lists the corresponding upper bounds of h ~ decreases when can be computed by the method of Theorem 1 in this paper. Table 1 shows that the upper bounds of h the number of unknown elements increases. ~ ¼ 1:34, one can obtain: Solving LMIs (15)–(29) for the case (1), condition 1 and h
P1 ¼
0:0031 0:0014 0:0014 0:0084
Q 11 ¼ 1:0e004 Q 13 ¼ 1:0e004
;
0:1438
P2 ¼
0:0024 0:0012 0:0012 0:0072
0:1307
0:1307 0:1563 0:3876 0:3860 0:3860
0:5253
; ;
;
P3 ¼
Q 12 ¼ 1:0e004
0:0025 0:0013 0:0013 0:0074
;
0:6556
0:6800
0:6800
0:8977
;
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J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
Q 21 ¼
0:0023 0:0019 0:0019 0:0072
Q 31 ¼ 1:0e003 Q 41 ¼
T 21 ¼
T 32 ¼
;
Q 42 ¼
; Q 32 ¼ 1:0e003
0:0108 0:0025 0:0025 0:0047
;
Q 23 ¼
0:0023 0:0019
;
0:0019 0:0072
0:4380 0:2729 ; Q 33 ¼ 1:0e003 ; 0:4896 0:3481 0:2729 0:2035 0:7800 0:4896
Q 43 ¼
0:0155 0:0036 0:0036 0:0092
;
0:4489 0:4449 0:9668 0:6602 ; R2 ¼ 1:0e004 ; R3 ¼ 1:0e003 ; 0:5361 0:7249 0:4449 0:5991 0:6602 0:4934
R5 ¼
;
0:0051
R7 ¼ 1:0e003
0:0019 0:0072
;
0:5223 0:5361
0:0062 0:0003 0:0003
0:0023 0:0019
0:1205 0:0703
0:0098 0:0150
Q 22 ¼
0:2258 0:1205
0:0201 0:0098
R1 ¼ 1:0e004
R4 ¼
;
0:1250 0:0362 0:0362 0:2403
0:0237
0
0
0:0090
0:0160
0
0
0:0072
;
T 22 ¼
0:0020
0:0014
0:0014
0:0010
;
T 11 ¼
;
R6 ¼
0:0546
0
0
0:0173
0:0092
0
0
0:0013
T 31 ¼
;
0:0008 0:0008
;
0:0008 0:0026
;
T 12 ¼
0:0441
0
0
0:0130
0:0170
0
0
0:0073
;
;
Therefore it follows from Corollary 1 that the system (6)with mode-dependent time-varying delays and completely known transition rates is stochastically stable. ~ ¼ 1:21, one can obtain: Solving LMIs (15)–(29) for the case (2), condition 1 and h
P1 ¼
0:0014 0:0007 0:0007 0:0041
Q 11 ¼ 1:0e004
Q 21 ¼
R4 ¼
0:0009
0:0009
;
0:0034 0:0064
Q 22 ¼
Q 42 ¼
; Q 12 ¼ 1:0e004
0:0008 0:0007
P3 ¼
;
0:0007 0:0027
0:0660 0:0411 ;
0:0006
0:0006 0:0031
0:1164 0:0660
0:0082 0:0034
R1 ¼ 1:0e004
0:0855 0:0877
0:0009 0:0034
P2 ¼
0:0877 0:1093
0:0010
Q 31 ¼ 1:0e003
Q 41 ¼
;
0:0011 0:0028
0:0010
0:0006
0:0006 0:0033
;
0:0963 0:1053 ; Q 13 ¼ 1:0e004 ; 0:1257 0:1785 0:1053 0:1517
0:0008 0:0008
Q 23 ¼
0:1958 0:1395
0:0008 0:0029
0:1395 0:1121
;
0:1157 0:1257
;
; Q 32 ¼ 1:0e003
0:0048 0:0011
Q 43 ¼
;
; Q 33 ¼ 1:0e003
0:0065 0:0014 0:0014 0:0044
0:1198 0:0797 0:0797 0:0630
;
0:0708 0:0544 0:2341 0:1832 ; R2 ¼ 1:0e004 ; R3 ¼ 1:0e003 ; 0:1155 0:1751 0:0544 0:2331 0:1832 0:1562 0:1026 0:1155
0:0020 0:0001 0:0001 0:0018
;
R5 ¼
0:0011 0:0001 0:0001 0:0008
;
R6 ¼
0:0004
0:0004
0:0004 0:0013
;
Table 1 ~ under the three cases. Allowable upper bound of h
Condition 1 Condition 2
Case 1
Case 2
Case 3
1.34 1.21
1.21 1.04
1.11 0.87
;
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J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
R7 ¼ 1:0e003 T 21 ¼
T 32 ¼
0:0454 0:0192
0:0192 0:1158
0:0100
0
0
0:0045
0:0084
0
0
0:0035
;
T 11 ¼
;
T 22 ¼
0:0195
0
0
0:0068
0:0036
0
0
0:0007
;
T 12 ¼
T 31 ¼
;
0:0194
0
0
0:0063
;
0:0085
0
0
0:0036
;
~ ¼ 1:11, one can obtain: Solving LMIs (15)–(29) for the case (3), condition 1 and h
P1 ¼
0:0029 0:0014
0:0014 0:0087
Q 11 ¼ 1:0e004
Q 21 ¼
Q 42 ¼
;
;
P3 ¼
; Q 12 ¼ 1:0e003
0:0016 0:0016
0:8951 0:5838
; Q 13 ¼ 1:0e004
0:0049 0:0092
0:6257 0:6855 0:6855 0:9630
;
;
; Q 33 ¼ 1:0e003
0:0123 0:0049
0:0014 0:0051
0:5838 0:4315
Q 43 ¼
;
0:0014 0:0014
Q 23 ¼
;
0:0072 0:0063
0:0815 0:1134
0:0014 0:0076
;
; Q 32 ¼ 1:0e003
0:0153 0:0072
0:0024 0:0014
0:0737 0:0815
0:0016 0:0058
0:7917 0:5346 0:5346 0:4136
;
;
0:0003 0:0003 0:8490 0:6195 ; R2 ¼ 1:0e004 ; R3 ¼ 1:0e003 ; 0:5934 0:8668 0:0003 0:0012 0:6195 0:5010 0:5248 0:5934
0:0022
0:0022
0:0078
Q 22 ¼
0:5478 0:3271
0:0060
T 32 ¼
;
0:0110
R7 ¼ 1:0e003
0:0016 0:0084
0:3271 0:2202
0:0060
T 21 ¼
0:0146 0:0060
R1 ¼ 1:0e004
0:0026 0:0016
0:4740 0:6191
0:0016 0:0058
Q 41 ¼
0:4529 0:4740
0:0016 0:0016
Q 31 ¼ 1:0e003
R4 ¼
P2 ¼
;
;
R5 ¼
0:1266 0:0742 0:0742 0:2548
0:0390
0
0
0:0138
0:0090
0
0
0:0060
;
T 22 ¼
;
0:0027
0:0003
0:0003
0:0018
T 11 ¼
;
R6 ¼
0:0377
0
0
0:0135
0:0119
0
0
0:0015
;
T 31 ¼
;
0:0010
0:0009
0:0009 0:0031 T 12 ¼
;
0:0413
0
0
0:0136
0:0157
0
0
0:0094
;
;
Therefore it follows from Theorem 1 that the system (6) with mode-dependent time-varying delays and partially known transition rates is stochastically stable. 5. Conclusions In this paper, the problem of stochastic stability criterion of Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates has been proposed. By choosing a new class of Lyapunov functional, some new delay-dependent stochastic stability criteria are derived to guarantee the stochastic stability of Markovian jumping neural networks. The obtained criteria are less conservative because free-weighting matrices method and a convex optimization approach are considered. Finally, a numerical example has been given to illustrate the effectiveness of the proposed method. Acknowledgments The authors thank the editors and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the demonstration project of oil and gas development for carbonate reservoirs in Tarim basin under Grant 2011ZX05049 and the National Basic Research Program of China under Grant 2010CB732501.
J. Tian et al. / Applied Mathematics and Computation 218 (2012) 5769–5781
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