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Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays Cheng-De Zheng a,n, Xiaoyu Zhang a, Zhanshan Wang b a b
School of Science, Dalian Jiaotong University, Dalian 116028, PR China School of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 January 2014 Received in revised form 2 September 2014 Accepted 15 November 2014 This paper was recommended for publication by Q.-G. Wang.
This paper investigates the stochastic stability of fuzzy Markovian jumping neural networks with mixed delays in mean square. The mixed delays include time-varying delay and continuously distributed delay. By using the Lyapunov functional method, Jensen integral inequality, the generalized Jensen integral inequality, linear convex combination technique and the free-weight matrix method, several novel sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point of the considered networks in mean square. The proposed results, which do not require the differentiability of the activation functions, can be easily checked via Matlab software. Finally, two numerical examples are given to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fuzzy neural networks Jensen integral inequality Markovian jumping Linear convex combination technique
1. Introduction It is well known that many neural networks models have been extensively investigated and successfully applied to various areas such as signal processing, pattern recognition, associative memory and optimization problems [18,16,19,23,21,20,9]. In such applications, it is of prime importance to ensure that the designed neural networks are stable. In hardware implementation, time delays are likely to be present due to the finite switching speed of amplifiers and communication time. It has also been shown that the processing of moving images requires the introduction of delay in the signal transmitted through the networks. The time delays are usually variable with time, which will affect the stability of designed neural networks and may lead to some complex dynamic behaviors such as oscillation, bifurcation, or chaos. Therefore, the study of neural dynamics with consideration of time delays becomes extremely important to manufacture high quality neural networks. As well known, in mathematical modeling of real world problems, we will encounter some other inconveniences, for instance, the complexity and the uncertainty or vagueness. Fuzzy theory is considered as a more suitable method for the sake of taking vagueness into consideration. Based on traditional neural
Corresponding author.
E-mail addresses: [email protected] (C.-D. Zheng), [email protected] (Z. Wang).
networks, Yang et al. [14] introduced fuzzy cellular neural network in 1996, which combines fuzzy logic with the structure of traditional neural networks and maintains local connectedness among cells. Unlike previous neural network structures, fuzzy neural network has fuzzy logic between its template and input and/or output deciding the sum of product operation, which allows us to combine the low of fuzzy systems. Fuzzy neural network is a useful paradigm for image processing problems and Euclidean distance transformation. In addition, fuzzy neural network has inherent connection to mathematical morphology, which is a cornerstone in image processing and pattern recognition. In recent years, various interesting results on the stability and other behaviors of fuzzy neural network have been reported [2,5,10]. Markovian jump systems introduced in [6] 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. And many researchers have made a lot of progress in Markovian jump control theory [2,7,3,22,24]. In [7], Li et al. established robust stability conditions of nonlinear delayed Hopfield neural networks with Markovian jumping parameters by the Takagi–Sugeno fuzzy model. However, to the best of our knowledge, up to today there is only one paper (see [5]) reported on the stochastic stability for fuzzy neural networks with Markovian jumping parameters. In [5], Han et al. proposed several LMI-based global exponential stability criteria in the mean square for a class of fuzzy cellular neural networks with
http://dx.doi.org/10.1016/j.isatra.2014.11.004 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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2
time-varying delays and Markovian jumping parameters. But the stability of fuzzy Markovian jumping neural networks with continuously distributed delay has not been addressed in the previous literatures. Motivated by the above discussion, in this paper our purpose is to present some new stochastic stability criteria for a class of fuzzy Markovian jumping neural networks with mixed delays in mean square. By using Jensen integral inequality, the generalized Jensen integral inequality [8], linear convex combination, linear matrix inequality (LMI) technique and the improved approximation method [13], several novel sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point of the considered networks in mean square. The proposed results, which do not require the differentiability of the activation functions, can be easily checked via Matlab LMI Toolbox. Finally, two numerical examples are given to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature. Notation: Throughout this paper, let Z þ denote the set of positive integers, W T and W 1 denote the transpose and the inverse of a square matrix W, respectively. W 4 0ð o 0Þ denotes a positive (negative) definite symmetric matrix, I denotes the identity matrix with compatible dimension, 0mn denotes the m n zero matrix, the symbol “n” denotes a block that is readily inferred by symmetry. The shorthand colfM 1 ; M 2 ; …; M k g denotes a column matrix with the matrices M 1 ; M 2 ; …; M k . diagfg stands for a diagonal or blockdiagonal matrix, N ¼ f1; 2; …; ng. For τ 4 0; Cð½ τ; 0; Rn Þ denotes the family of continuous functions ϕfrom ½ τ; 0 to Rn with the norm J ϕ J ¼ sup τ r s r 0 jϕðsÞj. Moreover, let ðΩ; F; PÞ be a complete probability space with a filtration fFt gt Z 0 satisfying the usual conditions and Efg representing the mathematical expectation. Denote by CpF0 ð½ τ; 0; Rn Þ the family of all bounded, F0 measurable, Cð½τ; 0; Rn Þvalued random variables ξ ¼ fξðsÞ : τ r s r 0g such that sup τ r s r 0 EjξðsÞjp o1. J J stands for the Euclidean norm; matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. Problem description and preliminaries
probability
(
Pðηðt þ δÞ ¼ jjηðtÞ ¼ iÞ ¼
π ij δ þ oðδÞ;
1 þ π ii δ þ oðδÞ;
i a j; i ¼ j;
where δ 4 0; limδ-0 þ oðδÞ=δ ¼ 0 and πij is the transition rate from mode i to mode j satisfying π ij Z0 for ia j with
π ii ¼
N
∑
j ¼ 1;j a i
π ij ;
i; j A N :
f i ðÞ is the activation function, τðt; ηðtÞÞ is the transmission delay. kj ðsÞ Z 0 is the feedback kernel and satisfies Z 1 kj ðsÞ ds ¼ 1; jA N: ð2Þ 0
Function φi ðsÞði A NÞ is continuous on ð 1; 0, the norm is defined by _ ðsÞ J : J φ J 1 ¼ max sup J φðsÞ J ; sup J φ 1osr0
1osr0
In this paper, we make the following assumptions. (H1) The transmission delay τðt; ηðtÞÞ is time-varying and satisfies 0 r τðt; ηðtÞÞ r τðηðtÞÞ r τ, τ_ ðt; ηðtÞÞÞ r τ0 ðηðtÞÞ o1, where τðηðtÞÞ; τ; τ0 ðηðtÞÞ are known constants. (H2) The activation function f ðxðtÞÞ ¼ ðf 1 ðx1 ðtÞÞ; f 2 ðx2 ðtÞÞ; …; f n ðxn ðtÞÞÞT A Rn is bounded and satisfies the following condition: 0r
f j ðs1 Þ f j ðs2 Þ r λj ; s1 s2
8 s1 ; s2 A R; s1 a s2 ;
where λj ðj ¼ 1; 2; …; nÞ are known real constants. For simplicity, we denote Λ ¼ diagfλ1 ; λ2 ; ⋯; λn g. For convenience, each possible value of ηðtÞ is denoted by mðm A N Þ in the sequel. Then we have dim ¼ di ðηðtÞÞ;
aijm ¼ aij ðηðtÞÞ;
bijm ¼ bij ðηðtÞÞ:
As well known, the Itô's formula plays important role in the stability analysis of Markovian systems and we cite some related results here [1]. Consider a general Markovian delay system z_ ðtÞ ¼ hðt; zðtÞ; zðt κ Þ; ηðtÞÞ;
Fuzzy recurrent neural network model with Markovian jump can be described by the following model: 8 n > > x_ i ðtÞ ¼ di ðηðtÞÞxi ðtÞ þ ∑ aij ðηðtÞÞf j ðxj ðtÞÞ > > > j¼1 > > > > n > > > þ ∑ bij ðηðtÞÞf j ðxj ðt τðt; ηðtÞÞÞÞ > > > > j¼1 > < Z t n n kj ðt sÞf j ðxj ðsÞÞ ds þ ∑ cij ϱj þ χ i þ ⋀ αij > > > 1 j¼1 j¼1 > > > Z t > > n n n > > > β kj ðt sÞf j ðxj ðsÞÞ ds þ ⋀ σ ij ϱj þ ⋁ δij ϱj ; þ ⋁ > ij > > 1 j¼1 j¼1 j¼1 > > > : x ðsÞ ¼ φ ðsÞ; s A ð 1; 0; i A N; i i ð1Þ where αij ; βij ; σ ij and δij are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively. di ðηðtÞÞ is a positive scalar representing the firing rate, aij ðηðtÞÞ and bij ðηðtÞÞ are elements of feedback template and cij are elements of feed-forward template. ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively. xi ðtÞ; ϱj and χi denote state, input and bias of the ith neurons, respectively. fηðtÞ; t Z0g is a homogeneous, finite-state Markovian process with right continuous trajectories and taking values in finite set N ¼ f1; 2; …; Ng based on given probability space ðΩ; F; PÞ and the initial model η0 . Let Π ¼ ½π ij NN denote the transition rate matrix with transition
ð3Þ
on t Z t 0 with initial value zðt 0 Þ ¼ z0 A R , where κ 4 0 is time delay, h : R þ Rn Rn N -Rn . Let C2;1 ðR þ Rn Rn N ; R þ Þ denote the family of all nonnegative functions V ðt; z; v; ηðtÞÞ on R þ Rn Rn N which are differentiable in t and continuously differentiable twice in z; v. Let L be the weak infinitesimal generator of the random process fzðtÞ; ηðtÞgt Z t 0 along the system (3) (see [11,15]), i.e. n
1 LVðt; zt ; vt ; mÞ≔ limþ ½E Vðt þ δ; zt þ δ ; vt þ δ ; ηðt þ δÞÞzt ; vt ; ηðtÞ ¼ m δ-0
δ
V ðt; zt ; vt ; ηðtÞ ¼ mÞ; then, by the Dynkin's formula [24,17], one can get Z t EVðt; zðtÞ; vðtÞ; mÞ ¼ EVðt 0 ; zðt 0 Þ; vðt 0 Þ; mÞ þ E LVðs; zðsÞ; vðsÞ; mÞ ds: t0
In addition, we use the following lemmas: Lemma 1 (See [12]). Let X; Y and P be real matrices of appropriate dimensions with P 4 0. Then for any positive scalar ε the following matrix inequality holds: X T Y þ Y T X r ε 1 X T P 1 X þ εY T PY: Lemma 2 (Jensen integral inequality, see [4]). For any constant matrix M 4 0, any scalars a and b with a o b, and a vector function χ ðtÞ : ½a; b-R such that the integrals concerned are well defined,
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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3. Main result
then the following inequalities hold: !T ! Z Z b
a
Z
b
χ ðsÞ ds
b
2
Z
b
θ
a
M a
χ ðsÞ ds r ðb aÞ
!T
χ ðsÞ ds dθ Z
r ðb aÞ2
b
Z
b
θ
a
Z
Z
b
M
b
θ
a
Z
b a
As usual, a vector xn ¼ ½xn1 ; xn2 ; …; xnn T is said to be an equilibrium point of system (1) if it satisfies
χ ðsÞT Mχ ðsÞ ds !
n
0 ¼ dim xni þ ∑ aijm f j ðxnj Þ
χ ðsÞ ds dθ
j¼1
n
n
þ ∑ bijm f j ðxnj Þ þ ∑ cij ϱj þ χ i j¼1
χ ðsÞT Mχ ðsÞ ds dθ:
n
þ ⋀ αij j¼1
Lemma 3 (See [14]). Let x and y be two states of system (1), then we have n n n α f ðx Þ ⋀ α f ðy Þ r ∑ j α jjf ðx Þ f ðy Þ ⋀ ij j j ij j j ij j j j j ; j ¼ 1 j¼1 j¼1 n n n ⋁ β ij f j ðxj Þ ⋁ β ij f j ðyj Þ r ∑ jβij jjf j ðxj Þ f j ðyj Þ: j ¼ 1 j¼1 j¼1 Lemma 4 (See [10]). For any x A Rn , any constant matrix A ¼ ½aij nn with aij Z 0, the following matrix inequality holds: xT AT Ax rnxT ATs As x; where As ¼ diagf∑ni¼ 1 ai1 ; ∑ni¼ 1 ai2 ; …; ∑ni¼ 1 ain g. Similar to Lemma 2 of Ref. [18], we have the following linear convex combination result. Lemma 5. For any symmetric matrices E; E 1 ; E 2 with appropriate dimensions, scalars ϵ1 r ϵ2 , and a real-valued function ϵðtÞ : R þ ½ϵ1 ; ϵ2 , then ðϵðtÞ ϵ1 ÞE 1 þ ðϵ2 ϵðtÞÞE 2 þ E o0 holds for any t 4 0 if and only if the following two matrix inequalities hold simultaneously: ðϵ2 ϵ1 ÞE 1 þ E o0;
ðϵ2 ϵ1 ÞE 2 þ E o 0:
b
a
b
θ
b
χ ðsÞ ds dθ Z
r ðb aÞ3
b
a
b
M
θ
a
χ ðsÞ ds dθ
χ ðsÞT Mχ ðsÞ ds:
Similar to Lemma 2, it is easy to see the following Jensen-type integral inequality holds. Lemma 7. For any constant matrix M 4 0, any scalars a and b with a o b, and a vector function χ ðtÞ : ½a; b-R such that the integrals concerned are well defined, then the following inequality holds bZ
Z 6 a
θ
bZ
!T
b
α
r ðb aÞ3
χ ðsÞ ds dα dθ
Z a
bZ
bZ
θ
b
α
Z
bZ
M a
θ
Z
n
þ ⋁ β ij
j¼1
Z
t
t 1
j¼1
kj ðt sÞf j ðxnj Þ ds
1
kj ðt sÞf j ðxnj Þ ds
n
n
j¼1
j¼1
þ ⋀ σ ij ϱj þ ⋁ δij ϱj : In this paper, we always assume that some conditions are satisfied so that system (1) has a unique equilibrium point. For the purpose of simplicity, we shift the intended equilibrium xn to the origin by letting yi ðtÞ ¼ xi ðtÞ xni , and then the system (1) can be transformed into 8 n n > > _ > > y i ðtÞ ¼ dim yi ðtÞ þ ∑ aijm g j ðyj ðtÞÞ þ ∑ bijm g j ðyj ðt τm ðtÞÞÞ > j¼1 j¼1 > > > Z t > > n n > > < k ðt sÞf ðy ðsÞ þxn Þ ds ⋀ α f ðxn Þ þ ⋀α j¼1
> > > > > > > > > > > > : y ðsÞ
ij
n
þ ⋁ β ij
Z
j¼1
t
j
j
j
j¼1
ij j
j
n
1
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds ⋁ βij f j ðxnj Þ; j¼1
ψ i ðsÞ≔φi ðsÞ xni ;
¼
i
j
1
s A ð 1; 0; ð4Þ
n
n
where g j ðyj ðÞÞ ¼ f j ðyj ðÞ þ xj Þ f j ðxj Þ. For convenience, we denote Dm ¼ diagfd1m ; d2m ; …; dnm g; Bm ¼ ðbijm Þnn ; C ¼ ðcij Þnn ;
Am ¼ ðaijm Þnn ;
Δ ¼ ðδij Þnn ; Σ ¼ ðσ ij Þnn ; ϱ ¼ ½ϱ1 ; ϱ2 ; …; ϱn T ; yðtÞ ¼ ½y1 ðtÞ; …; yn ðtÞT :
Lemma 6 (Generalized Jensen integral inequality, see [8]). For any constant matrix M 4 0, any scalars a and b with a o b, and a vector function χ ðtÞ : ½a; b-R such that the integrals concerned are well defined, then the following inequality holds: !T ! Z Z Z Z 2
3
bZ
b
α
χ ðsÞT Mχ ðsÞ ds dα dθ:
!
χ ðsÞ ds dα dθ
Now for system (3), we give our main result about the stability of the equilibrium point. Theorem 1 (See Appendix I for a proof). Assume that conditions (H1)–(H2) hold, then the unique equilibrium point of model (3) is globally asymptotically stable in mean square if there exist positive definite matrices P 4m ; P 6m ; Q i ; Q im , Rim ði ¼ 1; 4; 6Þ, Sm ; S; U m ; U; T m ; T, Ej ðj ¼ 1; 2; 3Þ, positive diagonal matrices Γ ; H; P 1m , Lm ; X m ; Y m ; W m ; Z j , any real matrices F; P lm ; Q l ; Q lm , Rlm ðl ¼ 2; 3; 5Þ of appropriate dimensions such that 2 3 P 1m P 2m P 3m 6 P 4m P 5m 7 ð5Þ Pm 4 n 5 40; n n P 6m 2
Q2
Q1
6 Q4 n
n
Q 1m
6 Qm 4 n n
2
R1m 6 Rm 4 n n
3
Q5 7 5 Z 0;
Q4
n
2
Q3
ð6Þ
Q6 Q 2m Q 4m n
R2m R4m n
Q 3m
3
Q 5m 7 5 Z0;
ð7Þ
Q 6m R3m
3
R5m 7 5 Z 0;
ð8Þ
R6m
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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4 N
∑ π 0mj Qj r Q;
ð9Þ
j¼1 N
∑ π 0mj ðRj þ τm τj diagfτ2m E1 þ Sm ; U m ; 0gÞ r Q;
ð10Þ
j¼1
τm ∑ π 0mj ðSj þ τm τj E2 Þ r S;
ð11Þ
j¼1
j¼1
ϕ44m ¼ ATm Γ þ Γ Am þQ 6m þ 2τQ 6 þ R6m þ H 2W m ;
ϕ45m ¼ Γ Bm ;
N
ϕ55m ¼ ð1 τ0m ÞQ 6m 2X m þ ∑ π 0mj τj Q 6m ; j¼1
N
∑ π 0mj ½τm U j þ τj ðE3 þ τ2m T m Þ r U;
ð12Þ
j¼1
N
τ2m ∑ π 0mj T j r τ3 T;
ð13Þ
j¼1
6 6 6 6 6 6 4
ϕ3;11m ¼ P 4m R2m þ ∑ π mj τj R2m ;
ϕ49m ¼ Q T5m þ 2τQ T5 þRT5m þ ATm Lm ; N
2
N
ϕ39m ¼ P T2m ;
N
ϕ59m ¼ BTm Lm ;
ϕ5;10m ¼ ð1 τ0m ÞQ T5m þ ∑ π 0mj τj Q T5m ; j¼1
N
ϕ66m ¼ R6m 2Y m þ ∑ π mj τj R6m ; j¼1
Φm Φm k
1
Ψm
2
Ψm
3
Ψm
4
Ψm
n
nZ 1 0
0 nZ 2
0 0
0 0
n
0
0
nZ 3
0
n
0
0
0
2E3
n
3
ϕ6;11m ¼
7 7 7 7 o 0; 7 7 5
k ¼ 1; 2;
ð14Þ
RT5m þ
N
∑ π mj τj RT5m ;
j¼1
N
ϕ77m ¼ Sm 2T m þ ∑ π mj P 6j ; j¼1
where
ϕ7;11m ¼ P T5m ;
Φm ¼ ½ϕijm 12n12n ; 1 Φm ¼ ðϖ 1 ϖ 2 ÞT U m ðϖ 1 ϖ 2 Þ; 2 Φm ¼ ðϖ 2 ϖ 3 ÞT U m ðϖ 2 ϖ 3 Þ;
τ2 τ2 τ4 τ6 ϕ99m ¼ Q 4m þ 2τQ 4 þ R4m þ τ2m U m þ U þ m E3 þ m T m þ T 2Lm ;
1
Ψ m ¼ colfnP 1m Υ 011nn g;
3
Ψ m ¼ colf08nn nLm Υ 03nn g;
2
Ψ m ¼ colf03nn nΓΥ 08nn g
ϕ7;12m ¼ F;
ϕ79m ¼ P T3m ;
ϕ88m ¼ H þ Z 1 þ Z 2 þ Z 3 ; 2
2
2
3
N
ϕ10;10m ¼ ð1 τ0m ÞQ 4m þ ∑ π 0mj τj Q 4m ; j¼1
4
Ψ m ¼ colf011nn τm F T g;
N
ϕ11;11m ¼ R4m þ ∑ π mj τj R4m ;
with
j¼1
ϕ11m ¼ 2P 1m Dm þ P 3m þ P T3m þ Q 1m þ 2τQ 1 þ R1m τ2 τ4 þ τ2m ðτ2m E1 þ Sm Þ þ S þ m E2 2τ2m T m
and π 0mj ¼ maxf0; π mj g, other parameters ϕijm ð1 r i o jr 12Þ are all equal to zeros. ϖ i ði ¼ 1; 2; …; 12Þ is an n 12n row vector with block matrix entries, i.e., the ith block is an identity matrix and the others are zero blocks, such that yðtÞ ¼ ϖ 1 ζ ðtÞ; yðt τm ðtÞÞ ¼ ϖ 2 ζ ðtÞ, and so on, and ( )
2 2 N 3 4 τ T U m þ ∑ π mj P 1j ; 2 j¼1
ϕ12m ¼ U m ;
N
α ¼ ðαij Þnn ;
ϕ13m ¼ P 3m þ P T5m þ ∑ π mj P 2j ;
jαjs ¼ diag
j¼1
ϕ14m ¼ P 1m Am Dm Γ þ Q 3m þ 2τQ 3 þ R3m þW m Λ;
ϕ12;12m ¼ 2E1 2E2 6T;
ϕ15m ¼ P 1m Bm ;
N
(
β ¼ ðβij Þnn ;
jβ js ¼ diag
n
n
n
i¼1
i¼1
i¼1
n
n
n
i¼1
i¼1
i¼1
∑ jαi1 j; ∑ jαi2 j; …; ∑ jαin j ; )
∑ jβi1 j; ∑ jβ i2 j; …; ∑ jβ in j ;
ϕ17m ¼ P 6m þ 2τm T m þ ∑ π mj P 3j ;
Υ ¼ jαjs þ jβjs ;
ϕ19m ¼ Q 2m þ2τQ 2 þR2m Dm Lm ;
ζ ðtÞ ¼ col yðtÞ; yðt τm ðtÞÞ; yðt τm Þ; gðyðtÞÞ; gðyðt τm ðtÞÞ; gðyðt τm Þ;
j¼1
ϕ1;11m ¼ P 2m ;
Z
ϕ1;12m ¼ 3τ2 T τm F;
t τm
N
ϕ22m ¼ ð1 τ0m ÞQ 1m 2U m þ ∑ π 0mj τj Q 1m ; N
ϕ25m ¼ ð1 τ0m ÞQ 3m þ X m Λ þ ∑ π 0mj τj Q 3m ; j¼1
N
ϕ2;10m ¼ ð1 τ0m ÞQ 2m þ ∑ π 0mj τj Q 2m ; j¼1
N
ϕ33m ¼ P 5m P T5m R1m U m þ ∑ π mj ðτj R1m þ P 4j Þ; j¼1
N
ϕ36m ¼ R3m þ Y m Λ þ ∑ π mj τj R3m ; j¼1
N
ϕ37m ¼ P 6m þ ∑ π mj P 5j ; j¼1
t
yðsÞ ds;
_ τm Þ; yðt
j¼1
ϕ23m ¼ U m ;
Z
t
Z
t
1 Z t
t τm
θ
_ _ τm ðtÞÞ; kðt sÞgðyðsÞÞ ds; yðtÞ; yðt yðsÞ ds dθ :
Remark 1. When τ_ m ðtÞ is unknown or τm ðtÞ is non-differentiable, we can verify the stability of model (4) by setting Qm ¼ 0 in Theorem 1. Remark 2. Theorem 1 provides an LMI-based sufficient condition for the stability of the neural network (4). One advantage of the LMI approach is that the LMI condition can be checked numerically very efficiently by using the interior-point algorithms, which have been developed in solving LMIs by employing the Matlab standard software. While the other conditions, which are based on the theory of M-matrix, matrix norm or algebraic inequality, ignore the signs of the weights, the neuron's excitatory and inhibitory effects, hence they result in conservativeness than LMI-based criteria.
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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If there is no Markovian jumping in system (3), similar to Theorem 1, we have the following result. Theorem 2 (See Appendix II for a proof). Assume that conditions (H1)–(H2) hold, then the unique equilibrium point of model (3) with N ¼1 is globally asymptotically stable if there exist positive definite matrices P 4 ; P 6 ; Q i , Ri ði ¼ 1; 4; 6Þ, S; U; T; Ej ðj ¼ 1; 2; 3Þ; J, positive diagonal matrices Γ ; H; P 1 ; L, X; Y; W; Z j , any real matrices F; P l ; Q l , Rl ðl ¼ 2; 3; 5Þ of appropriate dimensions such that (6) and the following LMIs hold: 2 3 P1 P2 P3 6 7 P 4 n P 4 P 5 5 40; ð15Þ n n P6 2
R1
6 R4 n
R2
R3
3
R5 7 5 Z 0;
n
R6
2 ~ k Φ Φ 6 n 6 6 6 n 6 6 n 4
1
n
Ψ
2
ð16Þ
Ψ
3
Ψ
4
Ψ
nZ 1
0
0
0
nZ 2
0
0
0
0
nZ 3
0
0
0
0
2E3
0
3 7 7 7 7 o 0; 7 7 5
k ¼ 1; 2;
where
Φ ¼ ðϖ 1 ϖ 2 ÞT Uðϖ 1 ϖ 2 Þ;
1
Ψ ¼ colfnP 1 Υ 011nn g;
2
2
Φ ¼ ðϖ 2 ϖ 3 ÞT Uðϖ 2 ϖ 3 Þ;
Ψ ¼ colf03nn nΓΥ 08nn g
Ψ ¼ colf09nn nLΥ 02nn g;
4
Ψ ¼ colf011nn τF T g;
with ~ ~ ¼ ½ϕ Φ ij 12n12n ;
ϕ~ 11 ¼ 2P 1 D þ P 3 þP T3 þQ 1 þ R1 þ τ2 ðτ2 E1 þ SÞ τ4 3 þ E2 2τ 2 T τ4 J U; 2
2
ϕ~ 12 ¼ U; ϕ~ 13 ¼ P 3 þP T5 ; ϕ~ 14 ¼ P 1 A DΓ þ Q 3 þ R3 þ W Λ;
ϕ~
15
¼ P 1 B;
ϕ~ 17 ¼ P 6 þ 2τT; ϕ~ 19 ¼ Q 2 þ R2 DL; ϕ~ 1;11 ¼ P 2 ; ϕ~ 1;12 ¼ 3τ2 J τF; ϕ~ 22 ¼ ð1 τ0 ÞQ 1 2U; ϕ~ 23 ¼ U; ϕ~ 25 ¼ ð1 τ0 ÞQ 3 þ X Λ; ϕ~ 2;10 ¼ ð1 τ0 ÞQ 2 ; ϕ~
33
¼
ϕ~ 37 ¼
P 5 P T5 R1 U; P 6 ; ~ 39 ¼ P T2 ;
ϕ~
36
¼ R3 þ Y Λ;
ϕ
ϕ~ 3;11 ¼ P 4 R2 ; ϕ~ 45 ¼ Γ B;
ϕ~ 12;12 ¼ 2E1 2E2 6J;
and other parameters are all defined in Theorem 1. Remark 3. In Theorem 2, the number of the decision variables involved is 17:5nðn þ1Þ. While in [10], there are 3n2 þ7n þ2 decision variables involved. With the increased decision variables, our proposed stability conditions will be less conservative and more effective than those in [10]. This will be addressed by two illustrating examples. Remark 4. When τ_ ðtÞ is unknown or τðtÞ is non-differentiable, we can verify the stability of model (3) without Markovian jump by setting Q ¼ 0 in Theorem 2. _ ¼ DyðtÞ þ AgðyðtÞÞ þ Bgðyðt τðtÞÞÞ; yðtÞ
ð17Þ
Corollary 1. Assume that conditions (H1)–(H2) hold, then the unique equilibrium point of model (18) is globally asymptotically stable if there exist positive definite matrices P i ; Q i ; Ri ði ¼ 1; 4; 6Þ, S; U; T; Ej ðj ¼ 1; 2; 3Þ; J, positive diagonal matrices Γ ; X; Y; W, any real matrices F; L; P l ; Q l , Rl ðl ¼ 2; 3; 5Þ of appropriate dimensions such that (6), (15), (16) and the following LMIs hold: " # b kΦ Φ Ψ o 0; k ¼ 1; 2; ð19Þ n 2E3 where
Ψ ¼ colf010nn τF T g; with ^ b ¼ ½ϕ Φ ij 11n11n ;
ϕ^ 4;4 ¼ AT Γ Γ A þ Q 6 þ R6 W;
ϕ^ ij ¼ ϕ~ ij ði; j r 7Þ;
ϕ^ ij ¼ ϕ~ i;j þ 1 ði r 7; j Z 8Þ;
4. Illustrative examples In this section, we provide two numerical examples to demonstrate the effectiveness of our delay-dependent stability criteria. Example 1. Consider system (3) with n ¼ N ¼ 2 and 3:8 0 4:1 0 D1 ¼ ; D2 ¼ ; 0 3:4 0 3:5 1 0 C¼Δ¼Σ ¼ ; 0 2 A1 ¼ B1 ¼ B2 ¼
ϕ~ 49 ¼ Q T5 þ RT5 þ AT L; ϕ~ 55 ¼ ð1 τ0 ÞQ 6 2X; ϕ~ 59 ¼ BT L; ϕ~ 5;10 ¼ ð1 τ0 ÞQ T5 ; ϕ~ 66 ¼ R6 2Y; ϕ~ 6;11 ¼ RT5 ; ϕ~ 77 ¼ S 2T; ϕ~ 79 ¼ P T3 ; ϕ~ 7;11 ¼ P T5 ; ϕ~ 7;12 ¼ F; ϕ~ 88 ¼ H þZ 1 þ Z 2 þ Z 3 ; τ2 τ4 τ6 ϕ~ 99 ¼ Q 4 þR4 þ τ2 U þ E3 þ T þ J 2L; 2
6
ϕ^ ij ¼ ϕ~ i þ 1;j þ 1 ði; j Z 8Þ;
and other parameters are all defined in Theorem 2.
ϕ~ 44 ¼ AT Γ þ Γ A þQ 6 þ R6 þ H 2W;
2
ð18Þ
by setting H ¼0 in Theorem 2, we have the following result.
1
3
ϕ~ 11;11 ¼ R4 ;
Especially, for the following time-varying neural networks:
R4
n
ϕ~ 10;10 ¼ ð1 τ0 ÞQ 4 ;
5
1:4 1:8 1:5 1:5 1:9
0:8
;
1 1:25 ; 1:25 1:8
1:2 1:65 1 1 1 ; β¼ 32 1 1
ϱ ¼ ½2; 1T ;
;
A2 ¼
α¼
1:1
0:7
0:9
1:2
1 1 32 1
1 1
;
τ1 ðtÞ ¼ 0:26 þ 0:13 sin ð2tÞ;
f i ðxÞ ¼ 12 ðx þ1 x 1Þ;
;
τ2 ðtÞ ¼ 0:15 þ0:15 cos ð2tÞ;
ki ðsÞ ¼ expð sÞ; i ¼ 1; 2:
It is easy to see that assumptions (H1) and (H2) are satisfied with
τ ¼ τ1 ¼ 0:39; τ2 ¼ 0:3, τ01 ¼ 0:26; τ02 ¼ 0:3; L ¼ I. Set π 11 ¼ 0:6;
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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6
π 12 ¼ 0:6, π 21 ¼ 0:3; π 22 ¼ 0:3, by using the Matlab LMI Toolbox, the LMIs (5)–(14) are feasible. Thus the unique equilibrium point of model (3) is globally asymptotically stable. The global optimum of the convex optimization problem is tmin¼ 2.8213 10 5 and one solution as follows:
0:6341 0 0:0192 0:0047 ; P 21 ¼ ; P 11 ¼ 0 0:4980 0:0524 0:0210
0:0126 0:0520 P 31 ¼ ; 0:0248 0:1186 P 41 ¼ P 61 ¼
0:0213
0:0202
0:0202 0:0515
0:0294 0:0702 ; 0:0702 0:1999
P 51 ¼
0:0447
P 42 ¼
Q 31 ¼
2:5990
0:5170 1:2437 0:5465
0:0017
0:0119
0:0199 0:0337
P 52 ¼
0:0685
0:0360
0:5170
;
1:2715
0:5328 ; 0:6541
Q 21 ¼
0:0170
0:0581
0:0287
0:0953
0:0029 0:0029 0:0220 ; Q 51 ¼ 0:0029 0:0216 0:0917
1:5732 0:0686 Q 61 ¼ ; 0:0686 1:2039 0:1271
0:0076
;
Q 22 ¼
0:0076 0:2899
0:0608 0:0534 Q 32 ¼ ; 0:0434 0:1998
Q 42 ¼ Q 62 ¼ Q1 ¼ Q3 ¼ Q4 ¼ Q6 ¼
R31 ¼
R41 ¼
0:3435 0:2238 0:2196
;
0:2238
;
0:0178
0:0099
0:5057
0:0930 0:0230 0:0208
0:2671
0:3546
0:2671 0:1092
0:4482
0:0015
0:0088
0:0381
0:0535
0:0764
R22 ¼
;
0:0535 0:1260
0:1804
0:1804 0:3413
0:7599 0:4004 ; 0:4004 0:2576
R52 ¼
;
S2 ¼
;
T1 ¼
0:0494
0:0719 ; 0:2582
0:9118
0:0578
0:0377
0:0497
0:0543
;
;
;
0:0248
0:1540 ; 0:5995
0:1348 ; 0:5691
0:0922 ; 0:3766
0:0208 ; 0:0445
X 2 ¼ diagf0:6014; 0:7517g; Y 2 ¼ diagf0:0164; 0:0022g;
0:0032 ; 0:0061
0:0163 0:0021
0:0087
0:0074
0:0074
0:0243
0:1082
0:1503
0:1503
0:3400
;
;
0:0079
S¼ T¼
0:0107
0:0020
0:0190
0:0051
;
0:0037
Q2 ¼
0:0033
;
0:0517
0:0275
0:0420
0:0527
F¼
0:0038
0:0038 0:0148
1:1990 0:5986 ; 0:5986 0:4426
E1 ¼
0:0175
Y 1 ¼ diagf0:0269; 0:0133g;
W 1 ¼ diagf2:1450; 1:6579g; Γ ¼ diagf0:0577; 0:1738g;
E3 ¼
0:0823
0:1348 0:4896
T2 ¼
Q 52 ¼
0:4652
0:1087 ; 0:3528
0:1540
0:0005
;
0:0378 ; 0:0760
0:0042
0:0005 0:0032
0:1740 0:1932 ; 0:1932 0:6387
0:0099 0:2594
R11 ¼
0:0010
;
L1 ¼ diagf0:0484; 0:1355g; L2 ¼ diagf0:1307; 0:0439g; X 1 ¼ diagf1:7254; 0:5013 g;
;
0:4279
0:0719
0:0446
0:1233
0:0106 0:0063 ; 0:0063 0:0074
0:1039 0:0162 ; R62 ¼ 0:0162 0:0399
U1 ¼
0:0834
0:1233
;
0:7252
R42 ¼
0:0181 ; 0:0269
Q 41 ¼
Q 12 ¼
;
R32 ¼
U2 ¼
0:0395 0:0379
0:2536 0:1132 P 62 ¼ ; 0:1132 0:0602
Q 11 ¼
0:0418
0:0395
0:0169
R12 ¼
S1 ¼
0 ; P 22 ¼ 0 0:2156
0:0113 0:0095 ; P 32 ¼ 0:0526 0:0262
P 12 ¼
0:3542
;
R61 ¼
;
U¼
0:1781
0:0052
0:0052 0:0056
0:0280
0:0003
0:0003
0:0046
0:0464
0:0324
0:0049
0:0298
E2 ¼
;
0:2183
0:0380
0:0380
0:0907
0:0308
0:0005
0:0005
0:0610
0:0319
0:0207
0:0059
0:0377
;
;
;
H ¼ diagf0:1220; 0:1472g; Z 2 ¼ diagf0:0108; 0:0318g;
; 0.6
y1 y2
0.4
Q5 ¼
;
Z 1 ¼ diagf0:0750; 0:0867g; Z 3 ¼ diagf0:0353; 0:0270g:
W 2 ¼ diagf0:8464; 0:9914g;
0.2
;
0 −0.2
R21 ¼
0:0522
0:0621
0:0456
0:1309
;
−0.4 −0.6 −0.8
R51 ¼
0:0120
0:0181
0:0470
0:0533
;
0
2
4
6
8
10
t Fig. 1. The state trajectory of the mode 1 with initial value ð0:4; 0:4ÞT in Example 1.
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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0.8
differentiability of the activation functions, can be easily checked via Matlab LMI Toolbox. Finally, two numerical examples are given to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature.
y1 y2
0.6
7
0.4 0.2
Acknowledgments
0 −0.2
This work was supported by the National Natural Science Foundation of China 61034005, 61074073, and 61273022, Program for New Century Excellent Talents in University of China (NCET-100306), and the Fundamental Research Funds for the Central Universities under Grants N110504001 and N100104102.
−0.4 −0.6 −0.8
0
1
2
3
4
5
6
7
t
Appendix A. Proof of Theorem 1
Fig. 2. The state trajectory of the mode 2 with initial value ð 0:4; 0:4ÞT in Example 1.
Construct a Lyapunov–Krasovskii functional in the following form: For numerical simulation, we consider the initial state ð0:4; 0:4ÞT in mode 1 and ð 0:4; 0:4ÞT in mode 2. Figs. 1 and 2 depict the time responses of state variables y1 ðtÞ and y2 ðtÞ in mode 1 and mode 2 with step 0.01 respectively. It confirms that the proposed condition in Theorem 1 leads to globally asymptotic stable equilibrium point for the model (3). In addition, as there exists Markovian jump in the considered neural networks, it is easy to see that the condition of Ref. [5] cannot be applied to verify the stability. Thus, we can say that for this system of Example 1, the results in this paper are much effective and less conservative than those in [5].
5
V m ðt; yðtÞÞ ¼ ∑ V im ðt; yðtÞÞ; i¼1
where n
V 1m ðt; yðtÞÞ ¼ ρðtÞT P m ρðtÞ þ 2 ∑ γ i i¼1
Z V 2m ðt; yðtÞÞ ¼
1 1 α¼ 16 1
1 ; 1
1 1 β¼ 16 1
1 f i ðxÞ ¼ ðjx þ 1j jx 1jÞ; 2
ki ðsÞ ¼
1 ; 1 2
1
π 1 þ s2
τðtÞ ¼ 0:39 þ 0:15 sin ð2tÞ;
Z
t τ t
Z
t τm
V 3m ðt; yðtÞÞ ¼ τm
Z
yi ðtÞ 0
g i ðsÞ ds;
ηðsÞT Qm ηðsÞ ds
t τ m ðtÞ Z t
þ
t
θ
ηðsÞT Q ηðsÞ ds dθ
ηðsÞT Rm ηðsÞ ds; Z
t t τm
t
θ
fyðsÞT ðτ2m E1 þ Sm ÞyðsÞ
_ _ T U m yðsÞg ds dθ þ yðsÞ Z t Z tZ t _ T U yðsÞg _ þ fyðsÞT SyðsÞ þ yðsÞ ds dα dθ;
; i ¼ 1; 2:
tτ
Z V 4m ðt; yðtÞÞ ¼
It is easy to see that assumptions (H1) and (H2) are satisfied with τ ¼ 0:54; τ0 ¼ 0:30; L ¼ I. By utilizing the Matlab LMI Toolbox, the LMIs (6) and (15)–(17) are feasible. Thus from Theorem 2 we conclude that the unique equilibrium point of model (3) is globally asymptotically stable. However, as there exists continuously distributed delay in the considered neural networks, it is easy to see that none of the conditions in [5,10] can be applied to verify the stability. Therefore, we can say that for this system of Example 2, the results in this paper are much effective and less conservative than those in [5,10].
t
þ2
Example 2. Consider system (3) with n ¼ 2; N ¼ 1 and 2:8 0 1:7 0:9 1:3 0:6 D¼ ; A¼ ; B¼ ; 0 2:9 1:3 0:8 2:5 0:8
Z
t
t τm
θ
Z
t
α
Z
θ
t
α
fτ2m yðsÞT E2 yðsÞ
_ T ðE3 þ τ2m T m ÞyðsÞg _ þ yðsÞ ds dα dθ Z t Z tZ tZ t _ T T yðsÞ _ yðsÞ þ 2τ 3 ds dκ dα dθ; tτ
n
V 5m ðt; yðtÞÞ ¼ ∑ hj j¼1
Z
1 0
θ
α
kj ðθÞ
κ
Z
t tθ
g 2j ðyj ðsÞÞ ds dθ;
Rt _ with ρðtÞ ¼ ½yðtÞ; yðt τm Þ; t τm yðsÞ dsT ; ηðsÞ ¼ ½yðsÞ; yðsÞ; g T ðyðsÞÞ . The weak infinitesimal operator LV m ðt; yðtÞÞ along the system (3) is given by 5
5. Conclusion
LV m ðt; yðtÞÞ ¼ ∑ LV im ðt; yðtÞÞ;
ð20Þ
i¼1
This paper investigates the stochastic stability of fuzzy Markovian jumping neural networks with mixed delays in mean square. The mixed delays include time-varying delays and continuously distributed delays. By using Jensen integral inequality, the generalized Jensen integral inequality, linear convex combination, LMI technique and the improved approximation method, several novel sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point of the considered networks in mean square. The proposed results, which do not require the
where LV 1m ðt; yðtÞÞ N
_ ¼ 2ρðtÞT P m ρ_ ðtÞ þ ∑ π mj ρðtÞT P j ρðtÞ þ 2gðyðtÞÞT Γ yðtÞ; j¼1
¼ 2½yðtÞP 1m þ gðyðtÞÞΓ T f Dm yðtÞ þ Am gðyðtÞÞ þ Bm gðyðt τm ðtÞÞÞg n
þ 2 ∑ ½p1im yi ðtÞ þ γ i g i ðyi ðtÞÞ i¼1
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8
n
⋀ αij
Z
j¼1
t 1
kj ðt sÞf j ðyj ðsÞ þxnj Þ ds
n
n
j¼1
j¼1
⋀ αij f j ðxnj Þ þ ⋁ βij
Z
t 1
þ n
j¼1
N
_ τm Þ þ P 3m ½yðtÞ yðt τm Þg þ ∑ π mj ρðtÞT P j ρðtÞ þ 2yðtÞT fP 2m yðt j¼1
þ 2yðt τ Z t þ2
τ
τ
T _ _ m Þ þ P 5m ½yðtÞ yðt m Þg þ P 4m yðt m Þ fP 2m yðtÞ
T _ þ P T5m yðt _ m Þ þ P 6m ½yðtÞ yðt m Þg; yðsÞ ds fP T3m yðtÞ
τ
t τm
o _ T U yðtÞ _ yðtÞT SyðtÞþ yðtÞ
2
τ
ð21Þ
t τ m ðtÞ
j¼1 N
t τm
þτ
θ
N
∑ π mj
2 m
j¼1
N
þ ∑ π mj τj
Z
Z
j¼1
þ 2τηðtÞT QηðtÞ 2
Z
t
tτ
ηðsÞT QηðsÞ ds T
3 τ2 n m
Z
t t τm
þ
j¼1
r ηðtÞT Qm ηðtÞ ð1 τ_ m ðtÞÞηðt τm ðtÞÞT Qm ηðt τm ðtÞÞ Z t N þ ∑ π 0mj ηðsÞT Qj ηðsÞ ds N
∑ π 0mj
j¼1
j¼1
Z
t
tτ
tτ
ð22Þ
j¼1
t τm
j¼1 N
þ τm ∑ π mj τj
Z
t
θ
n
t t τm
j¼1
n
o _ _ T U j yðsÞ ds dθ yðsÞT Sj yðsÞ þ yðsÞ
o _ _ T U m yðsÞ ds yðsÞT ðτ2m E1 þ Sm ÞyðsÞ þ yðsÞ
o _ T U yðtÞ _ yðtÞ SyðtÞ þ yðtÞ o t Z tn _ T U yðsÞ _ yðsÞT SyðsÞ þ yðsÞ ds dθ nt τ θ o _ T U m yðtÞ _ r τ 2m yðtÞT ðτ2m E1 þ Sm ÞyðtÞ þ yðtÞ Z t n o _ T U m yðsÞ _ yðsÞT ðτ2m E1 þ Sm ÞyðsÞ þ yðsÞ ds τm þ
τ2 n
Z
t
2 Z
t τm
þ τm ∑ π j¼1 N
0 mj
Z
þ τm ∑ π 0mj τj j¼1
Z
t tτ
Z
t t τ
t
θ
n
n
Z
t
tτ
t
Z
θ
t
α
_ T T yðsÞ _ yðsÞ ds dα dθ
o
Z
Z
o
Z
t tτ
θ
t
t
Z
tτ
Z
t
n
θ
j¼1 n
o
_ T ðE3 þ τ2m T m ÞyðsÞ _ τ2m yðsÞT E2 yðsÞ þ yðsÞ ds dθ Z
t
tτ
Z
n
LV 5m ðt; yðtÞÞ ¼ ∑ hj
_ T T j yðsÞ _ yðsÞ ds dα dθ
α
Z
3
t
1 0
Z
t
θ
Z
t
α
_ T T yðsÞ _ yðsÞ ds dα dθ;
ð24Þ
kj ðθÞg 2j ðyj ðtÞÞ dθ
1
0
kj ðθÞg 2j ðyj ðt θÞÞ dθ;
Based on Eq. (2) and Lemma 3, we obtain the following inequalities: Z t n n n n kj ðt sÞf j ðyj ðsÞ þxj Þ ds ⋀ αij f j ðxj Þ ⋀ αij j¼1 1 j¼1 Z t n kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds f j ðxnj Þ r ∑ jαij j j¼1
1
n
t
j¼1
1
kj ðt sÞg j ðyj ðsÞÞ ds;
Z t n n n n kj ðt sÞf j ðyj ðsÞ þ xj Þ ds ⋁ βij f j ðxj Þ ⋁ β ij j¼1 1 j¼1 Z t n n n kj ðt sÞf j ðyj ðsÞ þ xj Þ ds f j ðxj Þ r ∑ jβ ij j 1 j¼1 Z t n k ðt sÞg j ðyj ðsÞÞ ds: ¼ ∑ jβij j 1 j j¼1
n
2 ∑ ½p1im yi ðtÞ þ γ i g i ðyi ðtÞÞ i¼1
n
⋀ αij
j¼1
o _ _ ds dθ yðsÞ Sj yðsÞ þ yðsÞ U j yðsÞ T
o
_ T ðE3 þ τ2m T m ÞyðsÞ _ τ2m yðsÞT E2 yðsÞ þ yðsÞ ds dθ
By applying Lemmas 1 and 4, the following inequalities hold for any positive diagonal matrices Z 1 , Z 2
T
N
_ T T j yðsÞ _ yðsÞ ds dα dθ
_ T ðE3 þ τ 2m T m ÞyðsÞ _ τ2m yðsÞT E2 yðsÞ þ yðsÞ ds dθ
Z ¼ ∑ jαij j
LV 3m ðt; yðtÞÞ n o _ T U m yðtÞ _ ¼ τ2m yðtÞT ðτ2m E1 þ Sm ÞyðtÞ þ yðtÞ Z t n o _ T U m yðsÞ _ yðsÞT ðτ2m E1 þ Sm ÞyðsÞ þ yðsÞ ds τm Z
t
α
with P 1m ¼ diagfp11m ; p12m ; …; p1nm g, Γ ¼ diagfγ 1 ; γ 2 ; …; γ n g.
ηðsÞT QηðsÞ ds
þ ∑ π mj τj ηðt τm ÞT Rm ηðt τm Þ;
þ τm ∑ π mj
Z
j¼1
N
t τm
n
θ
∑ hj
þ ηðtÞT Rm ηðtÞ ηðt τm ÞT Rm ηðt τm Þ Z t N þ ∑ π 0mj ηðsÞT Rj ηðsÞ ds j¼1
t
τ _ T _ yðtÞ T yðtÞ 2τ3
T
þ 2τηðtÞT QηðtÞ 2
t
6
þ ∑ π τ ηðt τm ðtÞÞ Qm ηðt τm ðtÞÞ 0 mj j
n
θ
N
2 m
tτ
j¼1
θ
t τm
j¼1
N
N
t
N
þ ∑ π mj τj ηðt τm ÞT Rm ηðt τm Þ
o _ T U yðsÞ _ yðsÞT SyðsÞ þ yðsÞ ds dθ;
_ T ðE3 þ τ2m T m ÞyðtÞ _ τ2m yðtÞT E2 yðtÞ þ yðtÞ
þ ∑ π 0mj τj
t τm
j¼1
t τm
Z
Z
t
τ _ T _ yðtÞ T yðtÞ 2τ3
þτ
þ ηðtÞ Rm ηðtÞ ηðt τm Þ Rm ηðt τm Þ Z t N þ ∑ π mj ηðsÞT Rj ηðsÞ ds T
þ
Z
t
6
2 Z
þ ∑ π mj τj ðtÞηðt τm ðtÞÞ Qm ηðt τm ðtÞÞ
θ
n
LV 4m ðt; yðtÞÞ o τ2 n _ T ðE3 þ τ2m T m ÞyðtÞ _ ¼ m τ2m yðtÞT E2 yðtÞ þ yðtÞ 2 Z t Z tn o _ T ðE3 þ τ 2m T m ÞyðsÞ _ τ2m yðsÞT E2 yðsÞ þ yðsÞ ds dθ
r
T
t
ð23Þ
j¼1
LV 2m ðt; yðtÞÞ ¼ ηðtÞT Qm ηðtÞ ð1 τ_ m ðtÞÞηðt τm ðtÞÞT Qm ηðt τm ðtÞÞ Z t N ηðsÞT Qj ηðsÞ ds þ ∑ π mj
Z
t
t τ
!
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds ⋁ βij f j ðxnj Þ
T
Z
τ2 n
Z
t 1
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds
n
n
j¼1
j¼1
⋀ αij f j ðxnj Þ þ ⋁ β ij
T
o
_ _ T U m yðsÞ ds yðsÞT ðτ2m E1 þ Sm ÞyðsÞ þ yðsÞ
Z
t 1
n
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds ⋁ βij f j ðxnj Þ j¼1
n
r 2 ∑ ½p1im jyi ðtÞj þ γ i jg i ðyi ðtÞÞj i¼1
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
!
C.-D. Zheng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Z
Z t n ⋀ αij kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds j ¼ 1 1 n
n
⋀ αij f j ðxj Þj þ j ⋁ βij n
j¼1
Z
j¼1
! n n n kj ðt sÞf j ðyj ðsÞ þ xj Þ ds ⋁ βij f j ðxj Þ 1 j¼1 t
Z r 2ðjyðtÞjP 1m þ jgðyðtÞÞjΓ ÞT ðjαj þjβjÞ r jyðtÞj P 1m ðjαjþ jβ T
Z
kðt sÞgðyðsÞÞ ds
t 1
Z
αj þ jβjÞP 1m jyðtÞj
τ
2 m
Z
t τm
Z r 2
t
Z
t t τm
t
θ
_ yðsÞ ds dθ
Z Tm
Z
t
t τm
t
ð25Þ
θ
Z
τ3m τ
t
Z
t τm
Z 2 m
τ3
Z
t t τm
Z
tτ
Z
t
θ
Z r 6
t
α
Z
t
T
tτ
θ
Z
T 12 E2
ϖ 12 ζ ðtÞ;
t
Z
t
α
_ yðsÞ ds dα dθ
_ yðsÞ ds dα dθ
¼ τm
t τm ðtÞ
þ τm ¼ τm ðtÞ
Z
ð28Þ
ð29Þ
t τm t
Z
t τm ðtÞ
þ ðτm τm ðtÞÞ þ τm ðtÞ
Z
Z
t t τ m ðtÞ
t τm
t τ m ðtÞ
t τm
ð30Þ
_ T U m yðsÞ _ yðsÞ ds
_ T U m yðsÞ _ yðsÞ ds:
Using Lemma 2 and the Leibniz–Newton formulae again, we derive Z t _ T U m yðsÞ _ yðsÞ ds τm ðtÞ t τm ðtÞ
Z
t τm ðtÞ t τm
Z
ð32Þ
_ T U m yðsÞ _ yðsÞ ds
T Z _ yðsÞ ds U m
t τm ðtÞ t τm
_ yðsÞ ds ð33Þ
_ T U m yðsÞ _ yðsÞ ds
τm ðtÞ ðτm τm ðtÞÞ τm
θ
t
Z
t τm
¼ ζ ðtÞT
_ T U m yðsÞ _ yðsÞ ds
t τm ðtÞ
Z
t τ m ðtÞ t τm
T
_ T U m yðsÞ _ yðsÞ ds Z
ð34Þ
Based on Lemma 1, the following inequality holds for any real matrix F with appropriate dimension Z t Z t _ T E3 yðsÞ _ yðsÞ ds dθ r
_ T U m yðsÞ _ yðsÞ ds
Z
t τm
t τm
Z
_ T U m yðsÞ _ yðsÞ ds
þ ðτm τm ðtÞÞ
t τ m ðtÞ
t τ m ðtÞ
t τm
_ T U m yðsÞ _ yðsÞ ds
t τ m ðtÞ
Z
Z
On the other hand, we have Z t _ T U m yðsÞ _ yðsÞ ds τm t τm Z t
_ T U m yðsÞ _ yðsÞ ds
t τm ðtÞ
t τ m ðtÞ t τm ðtÞ τm ðtÞ _ _ yðsÞ ds U m yðsÞ ds τm t τm t τm τm ðtÞ T ¼ ζ ðtÞ ðϖ 2 ϖ 3 ÞT U m ðϖ 2 ϖ 3 Þζ ðtÞ: τm
T
3 ¼ ζ ðtÞT ðτ2 ϖ 1 2ϖ 12 ÞT Tðτ2 ϖ 1 2ϖ 12 Þζ ðtÞ: 2
t
Z
_ _ T T yðsÞ ds dα dθ yðsÞ
α t τ θ t Z tZ t
Z T
yðsÞ E2 yðsÞ ds dθ r 2ζ ðtÞ ϖ T
θ
Z
t
t
Z
¼ ζ ðtÞT ðϖ 2 ϖ 3 ÞT U m ðϖ 2 ϖ 3 Þζ ðtÞ;
ð27Þ
yðsÞT E1 yðsÞ ds r 2ζ ðtÞT ϖ T12 E1 ϖ 12 ζ ðtÞ;
ð31Þ
_ T U m yðsÞ _ yðsÞ ds
τm τm ðtÞ τm ðtÞ τm
Z
_ yðsÞ ds dθ ;
T
t t τm ðtÞ
ðτm τm ðtÞÞ
¼ 2ζ ðtÞ ðτm ϖ 1 ϖ 7 Þ T m ðτm ϖ 1 ϖ 7 Þζ ðtÞ; T
Z
Z
τm ðtÞ
T
t τ m ðtÞ
_ yðsÞ ds
T t t τm τm ðtÞ _ _ yðsÞ ds U m yðsÞ ds τm t τm ðtÞ t τ m ðtÞ τm τm ðtÞ T ¼ ζ ðtÞ ðϖ 1 ϖ 2 ÞT U m ðϖ 1 ϖ 2 Þζ ðtÞ; τm
_ T T m yðsÞ _ yðsÞ ds dθ
θ
t
Z
t τm t
t τ m ðtÞ
ðτm τm ðtÞÞ
By utilizing Lemmas 2 , 6, 7 and the Leibniz–Newton formulae, we obtain that Z t yðsÞT Sm yðsÞ ds r ζ ðtÞT ϖ T7 Sm ϖ 7 ζ ðtÞ; ð26Þ τm Z
T Z _ yðsÞ ds U m
t
¼ ζ ðtÞT ðϖ 1 ϖ 2 ÞT U m ðϖ 1 ϖ 2 Þζ ðtÞ;
jÞZ 1 1 ðj jÞZ 2 1 ðj
αjþ jβjÞΓ jgðyðtÞÞj þ jgðyðtÞÞjT Γ ðjαj þ jβ Z t Z t T kðt sÞgðyðsÞÞ ds ðZ 1 þ Z 2 Þ kðt sÞgðyðsÞÞ ds þ 1 1 n r ζ ðtÞT nϖ T1 P 1m Υ Z 1 1 Υ P 1m ϖ 1 þ nϖ T4 ΓΥ Z 2 1 Υ Γϖ 4 þ ϖ T8 ðZ 1 þ Z 2 Þϖ 8 ζ ðtÞ:
9
t
n
θ
o _ T F ϖ 12 ζ ðtÞ þ ζ ðtÞT ϖ T12 F T E3 1 F ϖ 12 ζ ðtÞ ds dθ 2yðsÞ
2ðτm ϖ 1 ϖ 7 ÞT F ϖ 12 þ
τ2m 2
ϖ T12 F T E3 1 F ϖ 12 ζ ðtÞ:
ð35Þ
From Eq. (2) and Cauchy–Schwarz inequality, we obtain that Z 1 Z 1 n kj ðθÞg 2j ðyj ðtÞÞ dθ ∑ hj kj ðθÞg 2j ðyj ðt θÞÞ dθ ∑ hj n
j¼1
n
j¼1 Z 1
j¼1
0
0
T
¼ gðyðtÞÞ HgðyðtÞÞ ∑ hj n
r gðyðtÞÞT HgðyðtÞÞ ∑ hj j¼1
¼ ζ ðtÞ ðϖ T
T 4H
ϖ4 ϖ
T 8H
Z
0
kj ðθÞ dθ
1 0
Z
1 0
kj ðθÞg 2j ðyj ðt θÞÞ dθ
kj ðθÞg j ðyj ðt θÞÞ dθ
2
ϖ 8 Þζ ðtÞ;
ð36Þ
where H ¼ diagfh1 ; h2 ; …; hn g. Moreover, based on (H2), the following matrix inequalities hold for any positive diagonal matrices W m ; X m ; and Y m 0 r 2yðtÞT W m ΛgðyðtÞÞ 2gðyðtÞÞT W m gðyðtÞÞ;
ð37Þ
0 r 2yðt τm ðtÞÞT X m Λgðyðt τm ðtÞÞÞ
2gðyðt τm ðtÞÞÞT X m gðyðt τm ðtÞÞÞ;
0 r 2yðt τm ÞT Y m Λgðyðt τm ÞÞ 2gðyðt τm ÞÞT Y m gðyðt τm ÞÞ:
ð38Þ ð39Þ
Again by utilizing Lemmas 1 and 4, we get the following inequality with any positive diagonal matrices Z 3 ; Lm ¼ diagfl1m ; l2m ; …; lnm g: " n
n
i¼1
j¼1
0 ¼ 2 ∑ y_ i ðtÞlim y_ i ðtÞ dim yi ðtÞ þ ∑ aijm g j ðyj ðtÞÞ
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
C.-D. Zheng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
10
Z
n
þ ∑ bijm g j ðyj ðt τðtÞÞÞ j¼1
Z
n
þ ⋀ αij j¼1
Z
n
⋀ αij j¼1
Z
n
þ ⋁ βij j¼1
Z
n
⋁ βij
t 1 t 1 t 1 t 1
j¼1
V 4 ðt; yðtÞÞ ¼
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds kj ðt sÞf j ðxnj Þ ds
n
j¼1
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds kj ðt sÞf j ðxnj Þ ds
LV m ðt; yðtÞÞ r ζ ðtÞT Φ m ζ ðtÞ;
ð40Þ
t 4 0;
ð41Þ
where
Φ m ¼ Φm þ nϖ T1 P 1m Υ Z 1 1 Υ P 1m ϖ 1 þ nϖ T4 ΓΥ Z 2 1 Υ Γϖ 4 τ2 þ nϖ T9 Lm Υ Z 3 1 Υ Lm ϖ 9 þ m ϖ T12 F T E3 1 F ϖ 12 2 τm τm ðtÞ ðϖ 1 ϖ 2 ÞT U m ðϖ 1 ϖ 2 Þ τm τm ðtÞ ðϖ 2 ϖ 3 ÞT U m ðϖ 2 ϖ 3 Þ: τm Based on Lemma 5, Φ m o 0 is true if and only if the following two inequalities hold simultaneously:
Φm 1 Φm þ nϖ T1 P 1m Υ Z 1 1 Υ P 1m ϖ 1 þ nϖ T4 ΓΥ Z 2 1 Υ Γϖ 4 τ2 þ nϖ T9 Lm Υ Z 3 1 Υ Lm ϖ 9 þ m ϖ T12 F T E3 1 F ϖ 12 o 0;
ð42Þ
Φm 2 Φm þ nϖ T1 P 1m Υ Z 1 1 Υ P 1m ϖ 1 þ nϖ T4 ΓΥ Z 2 1 Υ Γϖ 4 τ2 þ nϖ T9 Lm Υ Z 3 1 Υ Lm ϖ 9 þ m ϖ T12 F T E3 1 F ϖ 12 o 0:
ð43Þ
2
2
From the well-known Schur complement, we deduce that inequalities (42) and (43) are equivalent to inequalities (14) with k ¼1 and k ¼2 respectively. Therefore, if inequalities (14) hold, then from (41) we derive that LV m ðt; yðtÞÞ o 0;
8 t 4 0:
ð44Þ
Therefore, the system (4) is asymptotically stable in mean square, which implies that the equilibrium point of model (4) is globally asymptotically stable. This completes the proof of Theorem 1. Appendix B. Proof of Theorem 2 Construct a Lyapunov–Krasovskii functional in the following form: 5
Vðt; yðtÞÞ ¼ ∑ V i ðt; yðtÞÞ; i¼1
where n
V 1 ðt; yðtÞÞ ¼ ςðtÞT P ςðtÞ þ 2 ∑ γ i i¼1
t t τðtÞ t tτ
Z
yi ðtÞ 0
ηðsÞT QηðsÞ ds þ Z θ
t
Z
g i ðsÞ ds;
t tτ
Z
θ
t
α
_ T ðτ2 T þ E3 ÞyðsÞg _ fτ2 yðsÞT E2 yðsÞ þ yðsÞ ds dα dθ
Z
t tτ
Z 0
1
t
θ
Z
t
Z
α
k j ðθ Þ
t
κ
Z
_ T J yðsÞ _ yðsÞ ds dκ dα dθ;
t t θ
g 2j ðyj ðsÞÞ ds dθ;
References
Substituting (21)–(40) into (20) gives that
Z
Z
t
Rt with ςðtÞ ¼ ½yðtÞ; yðt τÞ; t τ yðsÞ dsT . Following the same line as in Theorem 1, we can prove that Theorem 2 is true.
#
1
V 3 ðt; yðtÞÞ ¼ τ
Z
V 5 ðt; yðtÞÞ ¼ ∑ hj
1
Z
tτ
þ τ3
_ T Lm f yðtÞ _ Dm yðtÞ þ Am gðyðtÞÞ þ Bm gðyðt τm ðtÞÞÞg r 2yðtÞ 1 T _ _ þ nyðtÞ Lm Υ Z 3 Υ Lm yðtÞ Z t
T Z t
kðt sÞgðyðsÞÞ ds Z 3 kðt sÞgðyðsÞÞ ds : þ
V 2 ðt; yðtÞÞ ¼
t
ηðsÞT RηðsÞ ds;
_ T U yðsÞg _ fyðsÞT ðτ2 E1 þ SÞyðsÞ þ yðsÞ ds dθ;
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Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays Cheng-De Zheng a,n, Xiaoyu Zhang a, Zhanshan Wang b a b
School of Science, Dalian Jiaotong University, Dalian 116028, PR China School of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 January 2014 Received in revised form 2 September 2014 Accepted 15 November 2014 This paper was recommended for publication by Q.-G. Wang.
This paper investigates the stochastic stability of fuzzy Markovian jumping neural networks with mixed delays in mean square. The mixed delays include time-varying delay and continuously distributed delay. By using the Lyapunov functional method, Jensen integral inequality, the generalized Jensen integral inequality, linear convex combination technique and the free-weight matrix method, several novel sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point of the considered networks in mean square. The proposed results, which do not require the differentiability of the activation functions, can be easily checked via Matlab software. Finally, two numerical examples are given to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fuzzy neural networks Jensen integral inequality Markovian jumping Linear convex combination technique
1. Introduction It is well known that many neural networks models have been extensively investigated and successfully applied to various areas such as signal processing, pattern recognition, associative memory and optimization problems [18,16,19,23,21,20,9]. In such applications, it is of prime importance to ensure that the designed neural networks are stable. In hardware implementation, time delays are likely to be present due to the finite switching speed of amplifiers and communication time. It has also been shown that the processing of moving images requires the introduction of delay in the signal transmitted through the networks. The time delays are usually variable with time, which will affect the stability of designed neural networks and may lead to some complex dynamic behaviors such as oscillation, bifurcation, or chaos. Therefore, the study of neural dynamics with consideration of time delays becomes extremely important to manufacture high quality neural networks. As well known, in mathematical modeling of real world problems, we will encounter some other inconveniences, for instance, the complexity and the uncertainty or vagueness. Fuzzy theory is considered as a more suitable method for the sake of taking vagueness into consideration. Based on traditional neural
Corresponding author.
E-mail addresses: [email protected] (C.-D. Zheng), [email protected] (Z. Wang).
networks, Yang et al. [14] introduced fuzzy cellular neural network in 1996, which combines fuzzy logic with the structure of traditional neural networks and maintains local connectedness among cells. Unlike previous neural network structures, fuzzy neural network has fuzzy logic between its template and input and/or output deciding the sum of product operation, which allows us to combine the low of fuzzy systems. Fuzzy neural network is a useful paradigm for image processing problems and Euclidean distance transformation. In addition, fuzzy neural network has inherent connection to mathematical morphology, which is a cornerstone in image processing and pattern recognition. In recent years, various interesting results on the stability and other behaviors of fuzzy neural network have been reported [2,5,10]. Markovian jump systems introduced in [6] 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. And many researchers have made a lot of progress in Markovian jump control theory [2,7,3,22,24]. In [7], Li et al. established robust stability conditions of nonlinear delayed Hopfield neural networks with Markovian jumping parameters by the Takagi–Sugeno fuzzy model. However, to the best of our knowledge, up to today there is only one paper (see [5]) reported on the stochastic stability for fuzzy neural networks with Markovian jumping parameters. In [5], Han et al. proposed several LMI-based global exponential stability criteria in the mean square for a class of fuzzy cellular neural networks with
http://dx.doi.org/10.1016/j.isatra.2014.11.004 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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2
time-varying delays and Markovian jumping parameters. But the stability of fuzzy Markovian jumping neural networks with continuously distributed delay has not been addressed in the previous literatures. Motivated by the above discussion, in this paper our purpose is to present some new stochastic stability criteria for a class of fuzzy Markovian jumping neural networks with mixed delays in mean square. By using Jensen integral inequality, the generalized Jensen integral inequality [8], linear convex combination, linear matrix inequality (LMI) technique and the improved approximation method [13], several novel sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point of the considered networks in mean square. The proposed results, which do not require the differentiability of the activation functions, can be easily checked via Matlab LMI Toolbox. Finally, two numerical examples are given to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature. Notation: Throughout this paper, let Z þ denote the set of positive integers, W T and W 1 denote the transpose and the inverse of a square matrix W, respectively. W 4 0ð o 0Þ denotes a positive (negative) definite symmetric matrix, I denotes the identity matrix with compatible dimension, 0mn denotes the m n zero matrix, the symbol “n” denotes a block that is readily inferred by symmetry. The shorthand colfM 1 ; M 2 ; …; M k g denotes a column matrix with the matrices M 1 ; M 2 ; …; M k . diagfg stands for a diagonal or blockdiagonal matrix, N ¼ f1; 2; …; ng. For τ 4 0; Cð½ τ; 0; Rn Þ denotes the family of continuous functions ϕfrom ½ τ; 0 to Rn with the norm J ϕ J ¼ sup τ r s r 0 jϕðsÞj. Moreover, let ðΩ; F; PÞ be a complete probability space with a filtration fFt gt Z 0 satisfying the usual conditions and Efg representing the mathematical expectation. Denote by CpF0 ð½ τ; 0; Rn Þ the family of all bounded, F0 measurable, Cð½τ; 0; Rn Þvalued random variables ξ ¼ fξðsÞ : τ r s r 0g such that sup τ r s r 0 EjξðsÞjp o1. J J stands for the Euclidean norm; matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. Problem description and preliminaries
probability
(
Pðηðt þ δÞ ¼ jjηðtÞ ¼ iÞ ¼
π ij δ þ oðδÞ;
1 þ π ii δ þ oðδÞ;
i a j; i ¼ j;
where δ 4 0; limδ-0 þ oðδÞ=δ ¼ 0 and πij is the transition rate from mode i to mode j satisfying π ij Z0 for ia j with
π ii ¼
N
∑
j ¼ 1;j a i
π ij ;
i; j A N :
f i ðÞ is the activation function, τðt; ηðtÞÞ is the transmission delay. kj ðsÞ Z 0 is the feedback kernel and satisfies Z 1 kj ðsÞ ds ¼ 1; jA N: ð2Þ 0
Function φi ðsÞði A NÞ is continuous on ð 1; 0, the norm is defined by _ ðsÞ J : J φ J 1 ¼ max sup J φðsÞ J ; sup J φ 1osr0
1osr0
In this paper, we make the following assumptions. (H1) The transmission delay τðt; ηðtÞÞ is time-varying and satisfies 0 r τðt; ηðtÞÞ r τðηðtÞÞ r τ, τ_ ðt; ηðtÞÞÞ r τ0 ðηðtÞÞ o1, where τðηðtÞÞ; τ; τ0 ðηðtÞÞ are known constants. (H2) The activation function f ðxðtÞÞ ¼ ðf 1 ðx1 ðtÞÞ; f 2 ðx2 ðtÞÞ; …; f n ðxn ðtÞÞÞT A Rn is bounded and satisfies the following condition: 0r
f j ðs1 Þ f j ðs2 Þ r λj ; s1 s2
8 s1 ; s2 A R; s1 a s2 ;
where λj ðj ¼ 1; 2; …; nÞ are known real constants. For simplicity, we denote Λ ¼ diagfλ1 ; λ2 ; ⋯; λn g. For convenience, each possible value of ηðtÞ is denoted by mðm A N Þ in the sequel. Then we have dim ¼ di ðηðtÞÞ;
aijm ¼ aij ðηðtÞÞ;
bijm ¼ bij ðηðtÞÞ:
As well known, the Itô's formula plays important role in the stability analysis of Markovian systems and we cite some related results here [1]. Consider a general Markovian delay system z_ ðtÞ ¼ hðt; zðtÞ; zðt κ Þ; ηðtÞÞ;
Fuzzy recurrent neural network model with Markovian jump can be described by the following model: 8 n > > x_ i ðtÞ ¼ di ðηðtÞÞxi ðtÞ þ ∑ aij ðηðtÞÞf j ðxj ðtÞÞ > > > j¼1 > > > > n > > > þ ∑ bij ðηðtÞÞf j ðxj ðt τðt; ηðtÞÞÞÞ > > > > j¼1 > < Z t n n kj ðt sÞf j ðxj ðsÞÞ ds þ ∑ cij ϱj þ χ i þ ⋀ αij > > > 1 j¼1 j¼1 > > > Z t > > n n n > > > β kj ðt sÞf j ðxj ðsÞÞ ds þ ⋀ σ ij ϱj þ ⋁ δij ϱj ; þ ⋁ > ij > > 1 j¼1 j¼1 j¼1 > > > : x ðsÞ ¼ φ ðsÞ; s A ð 1; 0; i A N; i i ð1Þ where αij ; βij ; σ ij and δij are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively. di ðηðtÞÞ is a positive scalar representing the firing rate, aij ðηðtÞÞ and bij ðηðtÞÞ are elements of feedback template and cij are elements of feed-forward template. ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively. xi ðtÞ; ϱj and χi denote state, input and bias of the ith neurons, respectively. fηðtÞ; t Z0g is a homogeneous, finite-state Markovian process with right continuous trajectories and taking values in finite set N ¼ f1; 2; …; Ng based on given probability space ðΩ; F; PÞ and the initial model η0 . Let Π ¼ ½π ij NN denote the transition rate matrix with transition
ð3Þ
on t Z t 0 with initial value zðt 0 Þ ¼ z0 A R , where κ 4 0 is time delay, h : R þ Rn Rn N -Rn . Let C2;1 ðR þ Rn Rn N ; R þ Þ denote the family of all nonnegative functions V ðt; z; v; ηðtÞÞ on R þ Rn Rn N which are differentiable in t and continuously differentiable twice in z; v. Let L be the weak infinitesimal generator of the random process fzðtÞ; ηðtÞgt Z t 0 along the system (3) (see [11,15]), i.e. n
1 LVðt; zt ; vt ; mÞ≔ limþ ½E Vðt þ δ; zt þ δ ; vt þ δ ; ηðt þ δÞÞzt ; vt ; ηðtÞ ¼ m δ-0
δ
V ðt; zt ; vt ; ηðtÞ ¼ mÞ; then, by the Dynkin's formula [24,17], one can get Z t EVðt; zðtÞ; vðtÞ; mÞ ¼ EVðt 0 ; zðt 0 Þ; vðt 0 Þ; mÞ þ E LVðs; zðsÞ; vðsÞ; mÞ ds: t0
In addition, we use the following lemmas: Lemma 1 (See [12]). Let X; Y and P be real matrices of appropriate dimensions with P 4 0. Then for any positive scalar ε the following matrix inequality holds: X T Y þ Y T X r ε 1 X T P 1 X þ εY T PY: Lemma 2 (Jensen integral inequality, see [4]). For any constant matrix M 4 0, any scalars a and b with a o b, and a vector function χ ðtÞ : ½a; b-R such that the integrals concerned are well defined,
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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3. Main result
then the following inequalities hold: !T ! Z Z b
a
Z
b
χ ðsÞ ds
b
2
Z
b
θ
a
M a
χ ðsÞ ds r ðb aÞ
!T
χ ðsÞ ds dθ Z
r ðb aÞ2
b
Z
b
θ
a
Z
Z
b
M
b
θ
a
Z
b a
As usual, a vector xn ¼ ½xn1 ; xn2 ; …; xnn T is said to be an equilibrium point of system (1) if it satisfies
χ ðsÞT Mχ ðsÞ ds !
n
0 ¼ dim xni þ ∑ aijm f j ðxnj Þ
χ ðsÞ ds dθ
j¼1
n
n
þ ∑ bijm f j ðxnj Þ þ ∑ cij ϱj þ χ i j¼1
χ ðsÞT Mχ ðsÞ ds dθ:
n
þ ⋀ αij j¼1
Lemma 3 (See [14]). Let x and y be two states of system (1), then we have n n n α f ðx Þ ⋀ α f ðy Þ r ∑ j α jjf ðx Þ f ðy Þ ⋀ ij j j ij j j ij j j j j ; j ¼ 1 j¼1 j¼1 n n n ⋁ β ij f j ðxj Þ ⋁ β ij f j ðyj Þ r ∑ jβij jjf j ðxj Þ f j ðyj Þ: j ¼ 1 j¼1 j¼1 Lemma 4 (See [10]). For any x A Rn , any constant matrix A ¼ ½aij nn with aij Z 0, the following matrix inequality holds: xT AT Ax rnxT ATs As x; where As ¼ diagf∑ni¼ 1 ai1 ; ∑ni¼ 1 ai2 ; …; ∑ni¼ 1 ain g. Similar to Lemma 2 of Ref. [18], we have the following linear convex combination result. Lemma 5. For any symmetric matrices E; E 1 ; E 2 with appropriate dimensions, scalars ϵ1 r ϵ2 , and a real-valued function ϵðtÞ : R þ ½ϵ1 ; ϵ2 , then ðϵðtÞ ϵ1 ÞE 1 þ ðϵ2 ϵðtÞÞE 2 þ E o0 holds for any t 4 0 if and only if the following two matrix inequalities hold simultaneously: ðϵ2 ϵ1 ÞE 1 þ E o0;
ðϵ2 ϵ1 ÞE 2 þ E o 0:
b
a
b
θ
b
χ ðsÞ ds dθ Z
r ðb aÞ3
b
a
b
M
θ
a
χ ðsÞ ds dθ
χ ðsÞT Mχ ðsÞ ds:
Similar to Lemma 2, it is easy to see the following Jensen-type integral inequality holds. Lemma 7. For any constant matrix M 4 0, any scalars a and b with a o b, and a vector function χ ðtÞ : ½a; b-R such that the integrals concerned are well defined, then the following inequality holds bZ
Z 6 a
θ
bZ
!T
b
α
r ðb aÞ3
χ ðsÞ ds dα dθ
Z a
bZ
bZ
θ
b
α
Z
bZ
M a
θ
Z
n
þ ⋁ β ij
j¼1
Z
t
t 1
j¼1
kj ðt sÞf j ðxnj Þ ds
1
kj ðt sÞf j ðxnj Þ ds
n
n
j¼1
j¼1
þ ⋀ σ ij ϱj þ ⋁ δij ϱj : In this paper, we always assume that some conditions are satisfied so that system (1) has a unique equilibrium point. For the purpose of simplicity, we shift the intended equilibrium xn to the origin by letting yi ðtÞ ¼ xi ðtÞ xni , and then the system (1) can be transformed into 8 n n > > _ > > y i ðtÞ ¼ dim yi ðtÞ þ ∑ aijm g j ðyj ðtÞÞ þ ∑ bijm g j ðyj ðt τm ðtÞÞÞ > j¼1 j¼1 > > > Z t > > n n > > < k ðt sÞf ðy ðsÞ þxn Þ ds ⋀ α f ðxn Þ þ ⋀α j¼1
> > > > > > > > > > > > : y ðsÞ
ij
n
þ ⋁ β ij
Z
j¼1
t
j
j
j
j¼1
ij j
j
n
1
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds ⋁ βij f j ðxnj Þ; j¼1
ψ i ðsÞ≔φi ðsÞ xni ;
¼
i
j
1
s A ð 1; 0; ð4Þ
n
n
where g j ðyj ðÞÞ ¼ f j ðyj ðÞ þ xj Þ f j ðxj Þ. For convenience, we denote Dm ¼ diagfd1m ; d2m ; …; dnm g; Bm ¼ ðbijm Þnn ; C ¼ ðcij Þnn ;
Am ¼ ðaijm Þnn ;
Δ ¼ ðδij Þnn ; Σ ¼ ðσ ij Þnn ; ϱ ¼ ½ϱ1 ; ϱ2 ; …; ϱn T ; yðtÞ ¼ ½y1 ðtÞ; …; yn ðtÞT :
Lemma 6 (Generalized Jensen integral inequality, see [8]). For any constant matrix M 4 0, any scalars a and b with a o b, and a vector function χ ðtÞ : ½a; b-R such that the integrals concerned are well defined, then the following inequality holds: !T ! Z Z Z Z 2
3
bZ
b
α
χ ðsÞT Mχ ðsÞ ds dα dθ:
!
χ ðsÞ ds dα dθ
Now for system (3), we give our main result about the stability of the equilibrium point. Theorem 1 (See Appendix I for a proof). Assume that conditions (H1)–(H2) hold, then the unique equilibrium point of model (3) is globally asymptotically stable in mean square if there exist positive definite matrices P 4m ; P 6m ; Q i ; Q im , Rim ði ¼ 1; 4; 6Þ, Sm ; S; U m ; U; T m ; T, Ej ðj ¼ 1; 2; 3Þ, positive diagonal matrices Γ ; H; P 1m , Lm ; X m ; Y m ; W m ; Z j , any real matrices F; P lm ; Q l ; Q lm , Rlm ðl ¼ 2; 3; 5Þ of appropriate dimensions such that 2 3 P 1m P 2m P 3m 6 P 4m P 5m 7 ð5Þ Pm 4 n 5 40; n n P 6m 2
Q2
Q1
6 Q4 n
n
Q 1m
6 Qm 4 n n
2
R1m 6 Rm 4 n n
3
Q5 7 5 Z 0;
Q4
n
2
Q3
ð6Þ
Q6 Q 2m Q 4m n
R2m R4m n
Q 3m
3
Q 5m 7 5 Z0;
ð7Þ
Q 6m R3m
3
R5m 7 5 Z 0;
ð8Þ
R6m
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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4 N
∑ π 0mj Qj r Q;
ð9Þ
j¼1 N
∑ π 0mj ðRj þ τm τj diagfτ2m E1 þ Sm ; U m ; 0gÞ r Q;
ð10Þ
j¼1
τm ∑ π 0mj ðSj þ τm τj E2 Þ r S;
ð11Þ
j¼1
j¼1
ϕ44m ¼ ATm Γ þ Γ Am þQ 6m þ 2τQ 6 þ R6m þ H 2W m ;
ϕ45m ¼ Γ Bm ;
N
ϕ55m ¼ ð1 τ0m ÞQ 6m 2X m þ ∑ π 0mj τj Q 6m ; j¼1
N
∑ π 0mj ½τm U j þ τj ðE3 þ τ2m T m Þ r U;
ð12Þ
j¼1
N
τ2m ∑ π 0mj T j r τ3 T;
ð13Þ
j¼1
6 6 6 6 6 6 4
ϕ3;11m ¼ P 4m R2m þ ∑ π mj τj R2m ;
ϕ49m ¼ Q T5m þ 2τQ T5 þRT5m þ ATm Lm ; N
2
N
ϕ39m ¼ P T2m ;
N
ϕ59m ¼ BTm Lm ;
ϕ5;10m ¼ ð1 τ0m ÞQ T5m þ ∑ π 0mj τj Q T5m ; j¼1
N
ϕ66m ¼ R6m 2Y m þ ∑ π mj τj R6m ; j¼1
Φm Φm k
1
Ψm
2
Ψm
3
Ψm
4
Ψm
n
nZ 1 0
0 nZ 2
0 0
0 0
n
0
0
nZ 3
0
n
0
0
0
2E3
n
3
ϕ6;11m ¼
7 7 7 7 o 0; 7 7 5
k ¼ 1; 2;
ð14Þ
RT5m þ
N
∑ π mj τj RT5m ;
j¼1
N
ϕ77m ¼ Sm 2T m þ ∑ π mj P 6j ; j¼1
where
ϕ7;11m ¼ P T5m ;
Φm ¼ ½ϕijm 12n12n ; 1 Φm ¼ ðϖ 1 ϖ 2 ÞT U m ðϖ 1 ϖ 2 Þ; 2 Φm ¼ ðϖ 2 ϖ 3 ÞT U m ðϖ 2 ϖ 3 Þ;
τ2 τ2 τ4 τ6 ϕ99m ¼ Q 4m þ 2τQ 4 þ R4m þ τ2m U m þ U þ m E3 þ m T m þ T 2Lm ;
1
Ψ m ¼ colfnP 1m Υ 011nn g;
3
Ψ m ¼ colf08nn nLm Υ 03nn g;
2
Ψ m ¼ colf03nn nΓΥ 08nn g
ϕ7;12m ¼ F;
ϕ79m ¼ P T3m ;
ϕ88m ¼ H þ Z 1 þ Z 2 þ Z 3 ; 2
2
2
3
N
ϕ10;10m ¼ ð1 τ0m ÞQ 4m þ ∑ π 0mj τj Q 4m ; j¼1
4
Ψ m ¼ colf011nn τm F T g;
N
ϕ11;11m ¼ R4m þ ∑ π mj τj R4m ;
with
j¼1
ϕ11m ¼ 2P 1m Dm þ P 3m þ P T3m þ Q 1m þ 2τQ 1 þ R1m τ2 τ4 þ τ2m ðτ2m E1 þ Sm Þ þ S þ m E2 2τ2m T m
and π 0mj ¼ maxf0; π mj g, other parameters ϕijm ð1 r i o jr 12Þ are all equal to zeros. ϖ i ði ¼ 1; 2; …; 12Þ is an n 12n row vector with block matrix entries, i.e., the ith block is an identity matrix and the others are zero blocks, such that yðtÞ ¼ ϖ 1 ζ ðtÞ; yðt τm ðtÞÞ ¼ ϖ 2 ζ ðtÞ, and so on, and ( )
2 2 N 3 4 τ T U m þ ∑ π mj P 1j ; 2 j¼1
ϕ12m ¼ U m ;
N
α ¼ ðαij Þnn ;
ϕ13m ¼ P 3m þ P T5m þ ∑ π mj P 2j ;
jαjs ¼ diag
j¼1
ϕ14m ¼ P 1m Am Dm Γ þ Q 3m þ 2τQ 3 þ R3m þW m Λ;
ϕ12;12m ¼ 2E1 2E2 6T;
ϕ15m ¼ P 1m Bm ;
N
(
β ¼ ðβij Þnn ;
jβ js ¼ diag
n
n
n
i¼1
i¼1
i¼1
n
n
n
i¼1
i¼1
i¼1
∑ jαi1 j; ∑ jαi2 j; …; ∑ jαin j ; )
∑ jβi1 j; ∑ jβ i2 j; …; ∑ jβ in j ;
ϕ17m ¼ P 6m þ 2τm T m þ ∑ π mj P 3j ;
Υ ¼ jαjs þ jβjs ;
ϕ19m ¼ Q 2m þ2τQ 2 þR2m Dm Lm ;
ζ ðtÞ ¼ col yðtÞ; yðt τm ðtÞÞ; yðt τm Þ; gðyðtÞÞ; gðyðt τm ðtÞÞ; gðyðt τm Þ;
j¼1
ϕ1;11m ¼ P 2m ;
Z
ϕ1;12m ¼ 3τ2 T τm F;
t τm
N
ϕ22m ¼ ð1 τ0m ÞQ 1m 2U m þ ∑ π 0mj τj Q 1m ; N
ϕ25m ¼ ð1 τ0m ÞQ 3m þ X m Λ þ ∑ π 0mj τj Q 3m ; j¼1
N
ϕ2;10m ¼ ð1 τ0m ÞQ 2m þ ∑ π 0mj τj Q 2m ; j¼1
N
ϕ33m ¼ P 5m P T5m R1m U m þ ∑ π mj ðτj R1m þ P 4j Þ; j¼1
N
ϕ36m ¼ R3m þ Y m Λ þ ∑ π mj τj R3m ; j¼1
N
ϕ37m ¼ P 6m þ ∑ π mj P 5j ; j¼1
t
yðsÞ ds;
_ τm Þ; yðt
j¼1
ϕ23m ¼ U m ;
Z
t
Z
t
1 Z t
t τm
θ
_ _ τm ðtÞÞ; kðt sÞgðyðsÞÞ ds; yðtÞ; yðt yðsÞ ds dθ :
Remark 1. When τ_ m ðtÞ is unknown or τm ðtÞ is non-differentiable, we can verify the stability of model (4) by setting Qm ¼ 0 in Theorem 1. Remark 2. Theorem 1 provides an LMI-based sufficient condition for the stability of the neural network (4). One advantage of the LMI approach is that the LMI condition can be checked numerically very efficiently by using the interior-point algorithms, which have been developed in solving LMIs by employing the Matlab standard software. While the other conditions, which are based on the theory of M-matrix, matrix norm or algebraic inequality, ignore the signs of the weights, the neuron's excitatory and inhibitory effects, hence they result in conservativeness than LMI-based criteria.
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
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If there is no Markovian jumping in system (3), similar to Theorem 1, we have the following result. Theorem 2 (See Appendix II for a proof). Assume that conditions (H1)–(H2) hold, then the unique equilibrium point of model (3) with N ¼1 is globally asymptotically stable if there exist positive definite matrices P 4 ; P 6 ; Q i , Ri ði ¼ 1; 4; 6Þ, S; U; T; Ej ðj ¼ 1; 2; 3Þ; J, positive diagonal matrices Γ ; H; P 1 ; L, X; Y; W; Z j , any real matrices F; P l ; Q l , Rl ðl ¼ 2; 3; 5Þ of appropriate dimensions such that (6) and the following LMIs hold: 2 3 P1 P2 P3 6 7 P 4 n P 4 P 5 5 40; ð15Þ n n P6 2
R1
6 R4 n
R2
R3
3
R5 7 5 Z 0;
n
R6
2 ~ k Φ Φ 6 n 6 6 6 n 6 6 n 4
1
n
Ψ
2
ð16Þ
Ψ
3
Ψ
4
Ψ
nZ 1
0
0
0
nZ 2
0
0
0
0
nZ 3
0
0
0
0
2E3
0
3 7 7 7 7 o 0; 7 7 5
k ¼ 1; 2;
where
Φ ¼ ðϖ 1 ϖ 2 ÞT Uðϖ 1 ϖ 2 Þ;
1
Ψ ¼ colfnP 1 Υ 011nn g;
2
2
Φ ¼ ðϖ 2 ϖ 3 ÞT Uðϖ 2 ϖ 3 Þ;
Ψ ¼ colf03nn nΓΥ 08nn g
Ψ ¼ colf09nn nLΥ 02nn g;
4
Ψ ¼ colf011nn τF T g;
with ~ ~ ¼ ½ϕ Φ ij 12n12n ;
ϕ~ 11 ¼ 2P 1 D þ P 3 þP T3 þQ 1 þ R1 þ τ2 ðτ2 E1 þ SÞ τ4 3 þ E2 2τ 2 T τ4 J U; 2
2
ϕ~ 12 ¼ U; ϕ~ 13 ¼ P 3 þP T5 ; ϕ~ 14 ¼ P 1 A DΓ þ Q 3 þ R3 þ W Λ;
ϕ~
15
¼ P 1 B;
ϕ~ 17 ¼ P 6 þ 2τT; ϕ~ 19 ¼ Q 2 þ R2 DL; ϕ~ 1;11 ¼ P 2 ; ϕ~ 1;12 ¼ 3τ2 J τF; ϕ~ 22 ¼ ð1 τ0 ÞQ 1 2U; ϕ~ 23 ¼ U; ϕ~ 25 ¼ ð1 τ0 ÞQ 3 þ X Λ; ϕ~ 2;10 ¼ ð1 τ0 ÞQ 2 ; ϕ~
33
¼
ϕ~ 37 ¼
P 5 P T5 R1 U; P 6 ; ~ 39 ¼ P T2 ;
ϕ~
36
¼ R3 þ Y Λ;
ϕ
ϕ~ 3;11 ¼ P 4 R2 ; ϕ~ 45 ¼ Γ B;
ϕ~ 12;12 ¼ 2E1 2E2 6J;
and other parameters are all defined in Theorem 1. Remark 3. In Theorem 2, the number of the decision variables involved is 17:5nðn þ1Þ. While in [10], there are 3n2 þ7n þ2 decision variables involved. With the increased decision variables, our proposed stability conditions will be less conservative and more effective than those in [10]. This will be addressed by two illustrating examples. Remark 4. When τ_ ðtÞ is unknown or τðtÞ is non-differentiable, we can verify the stability of model (3) without Markovian jump by setting Q ¼ 0 in Theorem 2. _ ¼ DyðtÞ þ AgðyðtÞÞ þ Bgðyðt τðtÞÞÞ; yðtÞ
ð17Þ
Corollary 1. Assume that conditions (H1)–(H2) hold, then the unique equilibrium point of model (18) is globally asymptotically stable if there exist positive definite matrices P i ; Q i ; Ri ði ¼ 1; 4; 6Þ, S; U; T; Ej ðj ¼ 1; 2; 3Þ; J, positive diagonal matrices Γ ; X; Y; W, any real matrices F; L; P l ; Q l , Rl ðl ¼ 2; 3; 5Þ of appropriate dimensions such that (6), (15), (16) and the following LMIs hold: " # b kΦ Φ Ψ o 0; k ¼ 1; 2; ð19Þ n 2E3 where
Ψ ¼ colf010nn τF T g; with ^ b ¼ ½ϕ Φ ij 11n11n ;
ϕ^ 4;4 ¼ AT Γ Γ A þ Q 6 þ R6 W;
ϕ^ ij ¼ ϕ~ ij ði; j r 7Þ;
ϕ^ ij ¼ ϕ~ i;j þ 1 ði r 7; j Z 8Þ;
4. Illustrative examples In this section, we provide two numerical examples to demonstrate the effectiveness of our delay-dependent stability criteria. Example 1. Consider system (3) with n ¼ N ¼ 2 and 3:8 0 4:1 0 D1 ¼ ; D2 ¼ ; 0 3:4 0 3:5 1 0 C¼Δ¼Σ ¼ ; 0 2 A1 ¼ B1 ¼ B2 ¼
ϕ~ 49 ¼ Q T5 þ RT5 þ AT L; ϕ~ 55 ¼ ð1 τ0 ÞQ 6 2X; ϕ~ 59 ¼ BT L; ϕ~ 5;10 ¼ ð1 τ0 ÞQ T5 ; ϕ~ 66 ¼ R6 2Y; ϕ~ 6;11 ¼ RT5 ; ϕ~ 77 ¼ S 2T; ϕ~ 79 ¼ P T3 ; ϕ~ 7;11 ¼ P T5 ; ϕ~ 7;12 ¼ F; ϕ~ 88 ¼ H þZ 1 þ Z 2 þ Z 3 ; τ2 τ4 τ6 ϕ~ 99 ¼ Q 4 þR4 þ τ2 U þ E3 þ T þ J 2L; 2
6
ϕ^ ij ¼ ϕ~ i þ 1;j þ 1 ði; j Z 8Þ;
and other parameters are all defined in Theorem 2.
ϕ~ 44 ¼ AT Γ þ Γ A þQ 6 þ R6 þ H 2W;
2
ð18Þ
by setting H ¼0 in Theorem 2, we have the following result.
1
3
ϕ~ 11;11 ¼ R4 ;
Especially, for the following time-varying neural networks:
R4
n
ϕ~ 10;10 ¼ ð1 τ0 ÞQ 4 ;
5
1:4 1:8 1:5 1:5 1:9
0:8
;
1 1:25 ; 1:25 1:8
1:2 1:65 1 1 1 ; β¼ 32 1 1
ϱ ¼ ½2; 1T ;
;
A2 ¼
α¼
1:1
0:7
0:9
1:2
1 1 32 1
1 1
;
τ1 ðtÞ ¼ 0:26 þ 0:13 sin ð2tÞ;
f i ðxÞ ¼ 12 ðx þ1 x 1Þ;
;
τ2 ðtÞ ¼ 0:15 þ0:15 cos ð2tÞ;
ki ðsÞ ¼ expð sÞ; i ¼ 1; 2:
It is easy to see that assumptions (H1) and (H2) are satisfied with
τ ¼ τ1 ¼ 0:39; τ2 ¼ 0:3, τ01 ¼ 0:26; τ02 ¼ 0:3; L ¼ I. Set π 11 ¼ 0:6;
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6
π 12 ¼ 0:6, π 21 ¼ 0:3; π 22 ¼ 0:3, by using the Matlab LMI Toolbox, the LMIs (5)–(14) are feasible. Thus the unique equilibrium point of model (3) is globally asymptotically stable. The global optimum of the convex optimization problem is tmin¼ 2.8213 10 5 and one solution as follows:
0:6341 0 0:0192 0:0047 ; P 21 ¼ ; P 11 ¼ 0 0:4980 0:0524 0:0210
0:0126 0:0520 P 31 ¼ ; 0:0248 0:1186 P 41 ¼ P 61 ¼
0:0213
0:0202
0:0202 0:0515
0:0294 0:0702 ; 0:0702 0:1999
P 51 ¼
0:0447
P 42 ¼
Q 31 ¼
2:5990
0:5170 1:2437 0:5465
0:0017
0:0119
0:0199 0:0337
P 52 ¼
0:0685
0:0360
0:5170
;
1:2715
0:5328 ; 0:6541
Q 21 ¼
0:0170
0:0581
0:0287
0:0953
0:0029 0:0029 0:0220 ; Q 51 ¼ 0:0029 0:0216 0:0917
1:5732 0:0686 Q 61 ¼ ; 0:0686 1:2039 0:1271
0:0076
;
Q 22 ¼
0:0076 0:2899
0:0608 0:0534 Q 32 ¼ ; 0:0434 0:1998
Q 42 ¼ Q 62 ¼ Q1 ¼ Q3 ¼ Q4 ¼ Q6 ¼
R31 ¼
R41 ¼
0:3435 0:2238 0:2196
;
0:2238
;
0:0178
0:0099
0:5057
0:0930 0:0230 0:0208
0:2671
0:3546
0:2671 0:1092
0:4482
0:0015
0:0088
0:0381
0:0535
0:0764
R22 ¼
;
0:0535 0:1260
0:1804
0:1804 0:3413
0:7599 0:4004 ; 0:4004 0:2576
R52 ¼
;
S2 ¼
;
T1 ¼
0:0494
0:0719 ; 0:2582
0:9118
0:0578
0:0377
0:0497
0:0543
;
;
;
0:0248
0:1540 ; 0:5995
0:1348 ; 0:5691
0:0922 ; 0:3766
0:0208 ; 0:0445
X 2 ¼ diagf0:6014; 0:7517g; Y 2 ¼ diagf0:0164; 0:0022g;
0:0032 ; 0:0061
0:0163 0:0021
0:0087
0:0074
0:0074
0:0243
0:1082
0:1503
0:1503
0:3400
;
;
0:0079
S¼ T¼
0:0107
0:0020
0:0190
0:0051
;
0:0037
Q2 ¼
0:0033
;
0:0517
0:0275
0:0420
0:0527
F¼
0:0038
0:0038 0:0148
1:1990 0:5986 ; 0:5986 0:4426
E1 ¼
0:0175
Y 1 ¼ diagf0:0269; 0:0133g;
W 1 ¼ diagf2:1450; 1:6579g; Γ ¼ diagf0:0577; 0:1738g;
E3 ¼
0:0823
0:1348 0:4896
T2 ¼
Q 52 ¼
0:4652
0:1087 ; 0:3528
0:1540
0:0005
;
0:0378 ; 0:0760
0:0042
0:0005 0:0032
0:1740 0:1932 ; 0:1932 0:6387
0:0099 0:2594
R11 ¼
0:0010
;
L1 ¼ diagf0:0484; 0:1355g; L2 ¼ diagf0:1307; 0:0439g; X 1 ¼ diagf1:7254; 0:5013 g;
;
0:4279
0:0719
0:0446
0:1233
0:0106 0:0063 ; 0:0063 0:0074
0:1039 0:0162 ; R62 ¼ 0:0162 0:0399
U1 ¼
0:0834
0:1233
;
0:7252
R42 ¼
0:0181 ; 0:0269
Q 41 ¼
Q 12 ¼
;
R32 ¼
U2 ¼
0:0395 0:0379
0:2536 0:1132 P 62 ¼ ; 0:1132 0:0602
Q 11 ¼
0:0418
0:0395
0:0169
R12 ¼
S1 ¼
0 ; P 22 ¼ 0 0:2156
0:0113 0:0095 ; P 32 ¼ 0:0526 0:0262
P 12 ¼
0:3542
;
R61 ¼
;
U¼
0:1781
0:0052
0:0052 0:0056
0:0280
0:0003
0:0003
0:0046
0:0464
0:0324
0:0049
0:0298
E2 ¼
;
0:2183
0:0380
0:0380
0:0907
0:0308
0:0005
0:0005
0:0610
0:0319
0:0207
0:0059
0:0377
;
;
;
H ¼ diagf0:1220; 0:1472g; Z 2 ¼ diagf0:0108; 0:0318g;
; 0.6
y1 y2
0.4
Q5 ¼
;
Z 1 ¼ diagf0:0750; 0:0867g; Z 3 ¼ diagf0:0353; 0:0270g:
W 2 ¼ diagf0:8464; 0:9914g;
0.2
;
0 −0.2
R21 ¼
0:0522
0:0621
0:0456
0:1309
;
−0.4 −0.6 −0.8
R51 ¼
0:0120
0:0181
0:0470
0:0533
;
0
2
4
6
8
10
t Fig. 1. The state trajectory of the mode 1 with initial value ð0:4; 0:4ÞT in Example 1.
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0.8
differentiability of the activation functions, can be easily checked via Matlab LMI Toolbox. Finally, two numerical examples are given to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature.
y1 y2
0.6
7
0.4 0.2
Acknowledgments
0 −0.2
This work was supported by the National Natural Science Foundation of China 61034005, 61074073, and 61273022, Program for New Century Excellent Talents in University of China (NCET-100306), and the Fundamental Research Funds for the Central Universities under Grants N110504001 and N100104102.
−0.4 −0.6 −0.8
0
1
2
3
4
5
6
7
t
Appendix A. Proof of Theorem 1
Fig. 2. The state trajectory of the mode 2 with initial value ð 0:4; 0:4ÞT in Example 1.
Construct a Lyapunov–Krasovskii functional in the following form: For numerical simulation, we consider the initial state ð0:4; 0:4ÞT in mode 1 and ð 0:4; 0:4ÞT in mode 2. Figs. 1 and 2 depict the time responses of state variables y1 ðtÞ and y2 ðtÞ in mode 1 and mode 2 with step 0.01 respectively. It confirms that the proposed condition in Theorem 1 leads to globally asymptotic stable equilibrium point for the model (3). In addition, as there exists Markovian jump in the considered neural networks, it is easy to see that the condition of Ref. [5] cannot be applied to verify the stability. Thus, we can say that for this system of Example 1, the results in this paper are much effective and less conservative than those in [5].
5
V m ðt; yðtÞÞ ¼ ∑ V im ðt; yðtÞÞ; i¼1
where n
V 1m ðt; yðtÞÞ ¼ ρðtÞT P m ρðtÞ þ 2 ∑ γ i i¼1
Z V 2m ðt; yðtÞÞ ¼
1 1 α¼ 16 1
1 ; 1
1 1 β¼ 16 1
1 f i ðxÞ ¼ ðjx þ 1j jx 1jÞ; 2
ki ðsÞ ¼
1 ; 1 2
1
π 1 þ s2
τðtÞ ¼ 0:39 þ 0:15 sin ð2tÞ;
Z
t τ t
Z
t τm
V 3m ðt; yðtÞÞ ¼ τm
Z
yi ðtÞ 0
g i ðsÞ ds;
ηðsÞT Qm ηðsÞ ds
t τ m ðtÞ Z t
þ
t
θ
ηðsÞT Q ηðsÞ ds dθ
ηðsÞT Rm ηðsÞ ds; Z
t t τm
t
θ
fyðsÞT ðτ2m E1 þ Sm ÞyðsÞ
_ _ T U m yðsÞg ds dθ þ yðsÞ Z t Z tZ t _ T U yðsÞg _ þ fyðsÞT SyðsÞ þ yðsÞ ds dα dθ;
; i ¼ 1; 2:
tτ
Z V 4m ðt; yðtÞÞ ¼
It is easy to see that assumptions (H1) and (H2) are satisfied with τ ¼ 0:54; τ0 ¼ 0:30; L ¼ I. By utilizing the Matlab LMI Toolbox, the LMIs (6) and (15)–(17) are feasible. Thus from Theorem 2 we conclude that the unique equilibrium point of model (3) is globally asymptotically stable. However, as there exists continuously distributed delay in the considered neural networks, it is easy to see that none of the conditions in [5,10] can be applied to verify the stability. Therefore, we can say that for this system of Example 2, the results in this paper are much effective and less conservative than those in [5,10].
t
þ2
Example 2. Consider system (3) with n ¼ 2; N ¼ 1 and 2:8 0 1:7 0:9 1:3 0:6 D¼ ; A¼ ; B¼ ; 0 2:9 1:3 0:8 2:5 0:8
Z
t
t τm
θ
Z
t
α
Z
θ
t
α
fτ2m yðsÞT E2 yðsÞ
_ T ðE3 þ τ2m T m ÞyðsÞg _ þ yðsÞ ds dα dθ Z t Z tZ tZ t _ T T yðsÞ _ yðsÞ þ 2τ 3 ds dκ dα dθ; tτ
n
V 5m ðt; yðtÞÞ ¼ ∑ hj j¼1
Z
1 0
θ
α
kj ðθÞ
κ
Z
t tθ
g 2j ðyj ðsÞÞ ds dθ;
Rt _ with ρðtÞ ¼ ½yðtÞ; yðt τm Þ; t τm yðsÞ dsT ; ηðsÞ ¼ ½yðsÞ; yðsÞ; g T ðyðsÞÞ . The weak infinitesimal operator LV m ðt; yðtÞÞ along the system (3) is given by 5
5. Conclusion
LV m ðt; yðtÞÞ ¼ ∑ LV im ðt; yðtÞÞ;
ð20Þ
i¼1
This paper investigates the stochastic stability of fuzzy Markovian jumping neural networks with mixed delays in mean square. The mixed delays include time-varying delays and continuously distributed delays. By using Jensen integral inequality, the generalized Jensen integral inequality, linear convex combination, LMI technique and the improved approximation method, several novel sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point of the considered networks in mean square. The proposed results, which do not require the
where LV 1m ðt; yðtÞÞ N
_ ¼ 2ρðtÞT P m ρ_ ðtÞ þ ∑ π mj ρðtÞT P j ρðtÞ þ 2gðyðtÞÞT Γ yðtÞ; j¼1
¼ 2½yðtÞP 1m þ gðyðtÞÞΓ T f Dm yðtÞ þ Am gðyðtÞÞ þ Bm gðyðt τm ðtÞÞÞg n
þ 2 ∑ ½p1im yi ðtÞ þ γ i g i ðyi ðtÞÞ i¼1
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8
n
⋀ αij
Z
j¼1
t 1
kj ðt sÞf j ðyj ðsÞ þxnj Þ ds
n
n
j¼1
j¼1
⋀ αij f j ðxnj Þ þ ⋁ βij
Z
t 1
þ n
j¼1
N
_ τm Þ þ P 3m ½yðtÞ yðt τm Þg þ ∑ π mj ρðtÞT P j ρðtÞ þ 2yðtÞT fP 2m yðt j¼1
þ 2yðt τ Z t þ2
τ
τ
T _ _ m Þ þ P 5m ½yðtÞ yðt m Þg þ P 4m yðt m Þ fP 2m yðtÞ
T _ þ P T5m yðt _ m Þ þ P 6m ½yðtÞ yðt m Þg; yðsÞ ds fP T3m yðtÞ
τ
t τm
o _ T U yðtÞ _ yðtÞT SyðtÞþ yðtÞ
2
τ
ð21Þ
t τ m ðtÞ
j¼1 N
t τm
þτ
θ
N
∑ π mj
2 m
j¼1
N
þ ∑ π mj τj
Z
Z
j¼1
þ 2τηðtÞT QηðtÞ 2
Z
t
tτ
ηðsÞT QηðsÞ ds T
3 τ2 n m
Z
t t τm
þ
j¼1
r ηðtÞT Qm ηðtÞ ð1 τ_ m ðtÞÞηðt τm ðtÞÞT Qm ηðt τm ðtÞÞ Z t N þ ∑ π 0mj ηðsÞT Qj ηðsÞ ds N
∑ π 0mj
j¼1
j¼1
Z
t
tτ
tτ
ð22Þ
j¼1
t τm
j¼1 N
þ τm ∑ π mj τj
Z
t
θ
n
t t τm
j¼1
n
o _ _ T U j yðsÞ ds dθ yðsÞT Sj yðsÞ þ yðsÞ
o _ _ T U m yðsÞ ds yðsÞT ðτ2m E1 þ Sm ÞyðsÞ þ yðsÞ
o _ T U yðtÞ _ yðtÞ SyðtÞ þ yðtÞ o t Z tn _ T U yðsÞ _ yðsÞT SyðsÞ þ yðsÞ ds dθ nt τ θ o _ T U m yðtÞ _ r τ 2m yðtÞT ðτ2m E1 þ Sm ÞyðtÞ þ yðtÞ Z t n o _ T U m yðsÞ _ yðsÞT ðτ2m E1 þ Sm ÞyðsÞ þ yðsÞ ds τm þ
τ2 n
Z
t
2 Z
t τm
þ τm ∑ π j¼1 N
0 mj
Z
þ τm ∑ π 0mj τj j¼1
Z
t tτ
Z
t t τ
t
θ
n
n
Z
t
tτ
t
Z
θ
t
α
_ T T yðsÞ _ yðsÞ ds dα dθ
o
Z
Z
o
Z
t tτ
θ
t
t
Z
tτ
Z
t
n
θ
j¼1 n
o
_ T ðE3 þ τ2m T m ÞyðsÞ _ τ2m yðsÞT E2 yðsÞ þ yðsÞ ds dθ Z
t
tτ
Z
n
LV 5m ðt; yðtÞÞ ¼ ∑ hj
_ T T j yðsÞ _ yðsÞ ds dα dθ
α
Z
3
t
1 0
Z
t
θ
Z
t
α
_ T T yðsÞ _ yðsÞ ds dα dθ;
ð24Þ
kj ðθÞg 2j ðyj ðtÞÞ dθ
1
0
kj ðθÞg 2j ðyj ðt θÞÞ dθ;
Based on Eq. (2) and Lemma 3, we obtain the following inequalities: Z t n n n n kj ðt sÞf j ðyj ðsÞ þxj Þ ds ⋀ αij f j ðxj Þ ⋀ αij j¼1 1 j¼1 Z t n kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds f j ðxnj Þ r ∑ jαij j j¼1
1
n
t
j¼1
1
kj ðt sÞg j ðyj ðsÞÞ ds;
Z t n n n n kj ðt sÞf j ðyj ðsÞ þ xj Þ ds ⋁ βij f j ðxj Þ ⋁ β ij j¼1 1 j¼1 Z t n n n kj ðt sÞf j ðyj ðsÞ þ xj Þ ds f j ðxj Þ r ∑ jβ ij j 1 j¼1 Z t n k ðt sÞg j ðyj ðsÞÞ ds: ¼ ∑ jβij j 1 j j¼1
n
2 ∑ ½p1im yi ðtÞ þ γ i g i ðyi ðtÞÞ i¼1
n
⋀ αij
j¼1
o _ _ ds dθ yðsÞ Sj yðsÞ þ yðsÞ U j yðsÞ T
o
_ T ðE3 þ τ2m T m ÞyðsÞ _ τ2m yðsÞT E2 yðsÞ þ yðsÞ ds dθ
By applying Lemmas 1 and 4, the following inequalities hold for any positive diagonal matrices Z 1 , Z 2
T
N
_ T T j yðsÞ _ yðsÞ ds dα dθ
_ T ðE3 þ τ 2m T m ÞyðsÞ _ τ2m yðsÞT E2 yðsÞ þ yðsÞ ds dθ
Z ¼ ∑ jαij j
LV 3m ðt; yðtÞÞ n o _ T U m yðtÞ _ ¼ τ2m yðtÞT ðτ2m E1 þ Sm ÞyðtÞ þ yðtÞ Z t n o _ T U m yðsÞ _ yðsÞT ðτ2m E1 þ Sm ÞyðsÞ þ yðsÞ ds τm Z
t
α
with P 1m ¼ diagfp11m ; p12m ; …; p1nm g, Γ ¼ diagfγ 1 ; γ 2 ; …; γ n g.
ηðsÞT QηðsÞ ds
þ ∑ π mj τj ηðt τm ÞT Rm ηðt τm Þ;
þ τm ∑ π mj
Z
j¼1
N
t τm
n
θ
∑ hj
þ ηðtÞT Rm ηðtÞ ηðt τm ÞT Rm ηðt τm Þ Z t N þ ∑ π 0mj ηðsÞT Rj ηðsÞ ds j¼1
t
τ _ T _ yðtÞ T yðtÞ 2τ3
T
þ 2τηðtÞT QηðtÞ 2
t
6
þ ∑ π τ ηðt τm ðtÞÞ Qm ηðt τm ðtÞÞ 0 mj j
n
θ
N
2 m
tτ
j¼1
θ
t τm
j¼1
N
N
t
N
þ ∑ π mj τj ηðt τm ÞT Rm ηðt τm Þ
o _ T U yðsÞ _ yðsÞT SyðsÞ þ yðsÞ ds dθ;
_ T ðE3 þ τ2m T m ÞyðtÞ _ τ2m yðtÞT E2 yðtÞ þ yðtÞ
þ ∑ π 0mj τj
t τm
j¼1
t τm
Z
Z
t
τ _ T _ yðtÞ T yðtÞ 2τ3
þτ
þ ηðtÞ Rm ηðtÞ ηðt τm Þ Rm ηðt τm Þ Z t N þ ∑ π mj ηðsÞT Rj ηðsÞ ds T
þ
Z
t
6
2 Z
þ ∑ π mj τj ðtÞηðt τm ðtÞÞ Qm ηðt τm ðtÞÞ
θ
n
LV 4m ðt; yðtÞÞ o τ2 n _ T ðE3 þ τ2m T m ÞyðtÞ _ ¼ m τ2m yðtÞT E2 yðtÞ þ yðtÞ 2 Z t Z tn o _ T ðE3 þ τ 2m T m ÞyðsÞ _ τ2m yðsÞT E2 yðsÞ þ yðsÞ ds dθ
r
T
t
ð23Þ
j¼1
LV 2m ðt; yðtÞÞ ¼ ηðtÞT Qm ηðtÞ ð1 τ_ m ðtÞÞηðt τm ðtÞÞT Qm ηðt τm ðtÞÞ Z t N ηðsÞT Qj ηðsÞ ds þ ∑ π mj
Z
t
t τ
!
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds ⋁ βij f j ðxnj Þ
T
Z
τ2 n
Z
t 1
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds
n
n
j¼1
j¼1
⋀ αij f j ðxnj Þ þ ⋁ β ij
T
o
_ _ T U m yðsÞ ds yðsÞT ðτ2m E1 þ Sm ÞyðsÞ þ yðsÞ
Z
t 1
n
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds ⋁ βij f j ðxnj Þ j¼1
n
r 2 ∑ ½p1im jyi ðtÞj þ γ i jg i ðyi ðtÞÞj i¼1
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
!
C.-D. Zheng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Z
Z t n ⋀ αij kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds j ¼ 1 1 n
n
⋀ αij f j ðxj Þj þ j ⋁ βij n
j¼1
Z
j¼1
! n n n kj ðt sÞf j ðyj ðsÞ þ xj Þ ds ⋁ βij f j ðxj Þ 1 j¼1 t
Z r 2ðjyðtÞjP 1m þ jgðyðtÞÞjΓ ÞT ðjαj þjβjÞ r jyðtÞj P 1m ðjαjþ jβ T
Z
kðt sÞgðyðsÞÞ ds
t 1
Z
αj þ jβjÞP 1m jyðtÞj
τ
2 m
Z
t τm
Z r 2
t
Z
t t τm
t
θ
_ yðsÞ ds dθ
Z Tm
Z
t
t τm
t
ð25Þ
θ
Z
τ3m τ
t
Z
t τm
Z 2 m
τ3
Z
t t τm
Z
tτ
Z
t
θ
Z r 6
t
α
Z
t
T
tτ
θ
Z
T 12 E2
ϖ 12 ζ ðtÞ;
t
Z
t
α
_ yðsÞ ds dα dθ
_ yðsÞ ds dα dθ
¼ τm
t τm ðtÞ
þ τm ¼ τm ðtÞ
Z
ð28Þ
ð29Þ
t τm t
Z
t τm ðtÞ
þ ðτm τm ðtÞÞ þ τm ðtÞ
Z
Z
t t τ m ðtÞ
t τm
t τ m ðtÞ
t τm
ð30Þ
_ T U m yðsÞ _ yðsÞ ds
_ T U m yðsÞ _ yðsÞ ds:
Using Lemma 2 and the Leibniz–Newton formulae again, we derive Z t _ T U m yðsÞ _ yðsÞ ds τm ðtÞ t τm ðtÞ
Z
t τm ðtÞ t τm
Z
ð32Þ
_ T U m yðsÞ _ yðsÞ ds
T Z _ yðsÞ ds U m
t τm ðtÞ t τm
_ yðsÞ ds ð33Þ
_ T U m yðsÞ _ yðsÞ ds
τm ðtÞ ðτm τm ðtÞÞ τm
θ
t
Z
t τm
¼ ζ ðtÞT
_ T U m yðsÞ _ yðsÞ ds
t τm ðtÞ
Z
t τ m ðtÞ t τm
T
_ T U m yðsÞ _ yðsÞ ds Z
ð34Þ
Based on Lemma 1, the following inequality holds for any real matrix F with appropriate dimension Z t Z t _ T E3 yðsÞ _ yðsÞ ds dθ r
_ T U m yðsÞ _ yðsÞ ds
Z
t τm
t τm
Z
_ T U m yðsÞ _ yðsÞ ds
þ ðτm τm ðtÞÞ
t τ m ðtÞ
t τ m ðtÞ
t τm
_ T U m yðsÞ _ yðsÞ ds
t τ m ðtÞ
Z
Z
On the other hand, we have Z t _ T U m yðsÞ _ yðsÞ ds τm t τm Z t
_ T U m yðsÞ _ yðsÞ ds
t τm ðtÞ
t τ m ðtÞ t τm ðtÞ τm ðtÞ _ _ yðsÞ ds U m yðsÞ ds τm t τm t τm τm ðtÞ T ¼ ζ ðtÞ ðϖ 2 ϖ 3 ÞT U m ðϖ 2 ϖ 3 Þζ ðtÞ: τm
T
3 ¼ ζ ðtÞT ðτ2 ϖ 1 2ϖ 12 ÞT Tðτ2 ϖ 1 2ϖ 12 Þζ ðtÞ: 2
t
Z
_ _ T T yðsÞ ds dα dθ yðsÞ
α t τ θ t Z tZ t
Z T
yðsÞ E2 yðsÞ ds dθ r 2ζ ðtÞ ϖ T
θ
Z
t
t
Z
¼ ζ ðtÞT ðϖ 2 ϖ 3 ÞT U m ðϖ 2 ϖ 3 Þζ ðtÞ;
ð27Þ
yðsÞT E1 yðsÞ ds r 2ζ ðtÞT ϖ T12 E1 ϖ 12 ζ ðtÞ;
ð31Þ
_ T U m yðsÞ _ yðsÞ ds
τm τm ðtÞ τm ðtÞ τm
Z
_ yðsÞ ds dθ ;
T
t t τm ðtÞ
ðτm τm ðtÞÞ
¼ 2ζ ðtÞ ðτm ϖ 1 ϖ 7 Þ T m ðτm ϖ 1 ϖ 7 Þζ ðtÞ; T
Z
Z
τm ðtÞ
T
t τ m ðtÞ
_ yðsÞ ds
T t t τm τm ðtÞ _ _ yðsÞ ds U m yðsÞ ds τm t τm ðtÞ t τ m ðtÞ τm τm ðtÞ T ¼ ζ ðtÞ ðϖ 1 ϖ 2 ÞT U m ðϖ 1 ϖ 2 Þζ ðtÞ; τm
_ T T m yðsÞ _ yðsÞ ds dθ
θ
t
Z
t τm t
t τ m ðtÞ
ðτm τm ðtÞÞ
By utilizing Lemmas 2 , 6, 7 and the Leibniz–Newton formulae, we obtain that Z t yðsÞT Sm yðsÞ ds r ζ ðtÞT ϖ T7 Sm ϖ 7 ζ ðtÞ; ð26Þ τm Z
T Z _ yðsÞ ds U m
t
¼ ζ ðtÞT ðϖ 1 ϖ 2 ÞT U m ðϖ 1 ϖ 2 Þζ ðtÞ;
jÞZ 1 1 ðj jÞZ 2 1 ðj
αjþ jβjÞΓ jgðyðtÞÞj þ jgðyðtÞÞjT Γ ðjαj þ jβ Z t Z t T kðt sÞgðyðsÞÞ ds ðZ 1 þ Z 2 Þ kðt sÞgðyðsÞÞ ds þ 1 1 n r ζ ðtÞT nϖ T1 P 1m Υ Z 1 1 Υ P 1m ϖ 1 þ nϖ T4 ΓΥ Z 2 1 Υ Γϖ 4 þ ϖ T8 ðZ 1 þ Z 2 Þϖ 8 ζ ðtÞ:
9
t
n
θ
o _ T F ϖ 12 ζ ðtÞ þ ζ ðtÞT ϖ T12 F T E3 1 F ϖ 12 ζ ðtÞ ds dθ 2yðsÞ
2ðτm ϖ 1 ϖ 7 ÞT F ϖ 12 þ
τ2m 2
ϖ T12 F T E3 1 F ϖ 12 ζ ðtÞ:
ð35Þ
From Eq. (2) and Cauchy–Schwarz inequality, we obtain that Z 1 Z 1 n kj ðθÞg 2j ðyj ðtÞÞ dθ ∑ hj kj ðθÞg 2j ðyj ðt θÞÞ dθ ∑ hj n
j¼1
n
j¼1 Z 1
j¼1
0
0
T
¼ gðyðtÞÞ HgðyðtÞÞ ∑ hj n
r gðyðtÞÞT HgðyðtÞÞ ∑ hj j¼1
¼ ζ ðtÞ ðϖ T
T 4H
ϖ4 ϖ
T 8H
Z
0
kj ðθÞ dθ
1 0
Z
1 0
kj ðθÞg 2j ðyj ðt θÞÞ dθ
kj ðθÞg j ðyj ðt θÞÞ dθ
2
ϖ 8 Þζ ðtÞ;
ð36Þ
where H ¼ diagfh1 ; h2 ; …; hn g. Moreover, based on (H2), the following matrix inequalities hold for any positive diagonal matrices W m ; X m ; and Y m 0 r 2yðtÞT W m ΛgðyðtÞÞ 2gðyðtÞÞT W m gðyðtÞÞ;
ð37Þ
0 r 2yðt τm ðtÞÞT X m Λgðyðt τm ðtÞÞÞ
2gðyðt τm ðtÞÞÞT X m gðyðt τm ðtÞÞÞ;
0 r 2yðt τm ÞT Y m Λgðyðt τm ÞÞ 2gðyðt τm ÞÞT Y m gðyðt τm ÞÞ:
ð38Þ ð39Þ
Again by utilizing Lemmas 1 and 4, we get the following inequality with any positive diagonal matrices Z 3 ; Lm ¼ diagfl1m ; l2m ; …; lnm g: " n
n
i¼1
j¼1
0 ¼ 2 ∑ y_ i ðtÞlim y_ i ðtÞ dim yi ðtÞ þ ∑ aijm g j ðyj ðtÞÞ
Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
C.-D. Zheng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
10
Z
n
þ ∑ bijm g j ðyj ðt τðtÞÞÞ j¼1
Z
n
þ ⋀ αij j¼1
Z
n
⋀ αij j¼1
Z
n
þ ⋁ βij j¼1
Z
n
⋁ βij
t 1 t 1 t 1 t 1
j¼1
V 4 ðt; yðtÞÞ ¼
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds kj ðt sÞf j ðxnj Þ ds
n
j¼1
kj ðt sÞf j ðyj ðsÞ þ xnj Þ ds kj ðt sÞf j ðxnj Þ ds
LV m ðt; yðtÞÞ r ζ ðtÞT Φ m ζ ðtÞ;
ð40Þ
t 4 0;
ð41Þ
where
Φ m ¼ Φm þ nϖ T1 P 1m Υ Z 1 1 Υ P 1m ϖ 1 þ nϖ T4 ΓΥ Z 2 1 Υ Γϖ 4 τ2 þ nϖ T9 Lm Υ Z 3 1 Υ Lm ϖ 9 þ m ϖ T12 F T E3 1 F ϖ 12 2 τm τm ðtÞ ðϖ 1 ϖ 2 ÞT U m ðϖ 1 ϖ 2 Þ τm τm ðtÞ ðϖ 2 ϖ 3 ÞT U m ðϖ 2 ϖ 3 Þ: τm Based on Lemma 5, Φ m o 0 is true if and only if the following two inequalities hold simultaneously:
Φm 1 Φm þ nϖ T1 P 1m Υ Z 1 1 Υ P 1m ϖ 1 þ nϖ T4 ΓΥ Z 2 1 Υ Γϖ 4 τ2 þ nϖ T9 Lm Υ Z 3 1 Υ Lm ϖ 9 þ m ϖ T12 F T E3 1 F ϖ 12 o 0;
ð42Þ
Φm 2 Φm þ nϖ T1 P 1m Υ Z 1 1 Υ P 1m ϖ 1 þ nϖ T4 ΓΥ Z 2 1 Υ Γϖ 4 τ2 þ nϖ T9 Lm Υ Z 3 1 Υ Lm ϖ 9 þ m ϖ T12 F T E3 1 F ϖ 12 o 0:
ð43Þ
2
2
From the well-known Schur complement, we deduce that inequalities (42) and (43) are equivalent to inequalities (14) with k ¼1 and k ¼2 respectively. Therefore, if inequalities (14) hold, then from (41) we derive that LV m ðt; yðtÞÞ o 0;
8 t 4 0:
ð44Þ
Therefore, the system (4) is asymptotically stable in mean square, which implies that the equilibrium point of model (4) is globally asymptotically stable. This completes the proof of Theorem 1. Appendix B. Proof of Theorem 2 Construct a Lyapunov–Krasovskii functional in the following form: 5
Vðt; yðtÞÞ ¼ ∑ V i ðt; yðtÞÞ; i¼1
where n
V 1 ðt; yðtÞÞ ¼ ςðtÞT P ςðtÞ þ 2 ∑ γ i i¼1
t t τðtÞ t tτ
Z
yi ðtÞ 0
ηðsÞT QηðsÞ ds þ Z θ
t
Z
g i ðsÞ ds;
t tτ
Z
θ
t
α
_ T ðτ2 T þ E3 ÞyðsÞg _ fτ2 yðsÞT E2 yðsÞ þ yðsÞ ds dα dθ
Z
t tτ
Z 0
1
t
θ
Z
t
Z
α
k j ðθ Þ
t
κ
Z
_ T J yðsÞ _ yðsÞ ds dκ dα dθ;
t t θ
g 2j ðyj ðsÞÞ ds dθ;
References
Substituting (21)–(40) into (20) gives that
Z
Z
t
Rt with ςðtÞ ¼ ½yðtÞ; yðt τÞ; t τ yðsÞ dsT . Following the same line as in Theorem 1, we can prove that Theorem 2 is true.
#
1
V 3 ðt; yðtÞÞ ¼ τ
Z
V 5 ðt; yðtÞÞ ¼ ∑ hj
1
Z
tτ
þ τ3
_ T Lm f yðtÞ _ Dm yðtÞ þ Am gðyðtÞÞ þ Bm gðyðt τm ðtÞÞÞg r 2yðtÞ 1 T _ _ þ nyðtÞ Lm Υ Z 3 Υ Lm yðtÞ Z t
T Z t
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t
ηðsÞT RηðsÞ ds;
_ T U yðsÞg _ fyðsÞT ðτ2 E1 þ SÞyðsÞ þ yðsÞ ds dθ;
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Please cite this article as: Zheng C-D, et al. Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.004i
