Robust stochastic stability of discrete-time fuzzy Markovian jump neural networks.

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Research Article

Robust stochastic stability of discrete-time fuzzy Markovian jump neural networks A. Arunkumar a, R. Sakthivel b,c,n, K. Mathiyalagan d, Ju H. Park d a

Department of Mathematics, Anna University-Regional Centre, Coimbatore 641047, India Department of Mathematics, Sri Ramakrishna Institute of Technology, Coimbatore 641010, India c Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea d Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 712-749, Republic of Korea b

art ic l e i nf o

a b s t r a c t

Article history: Received 25 March 2013 Received in revised form 9 March 2014 Accepted 6 May 2014 This paper was recommended for publication by Prof. Y. Chen

This paper focuses the issue of robust stochastic stability for a class of uncertain fuzzy Markovian jumping discrete-time neural networks (FMJDNNs) with various activation functions and mixed time delay. By employing the Lyapunov technique and linear matrix inequality (LMI) approach, a new set of delay-dependent sufficient conditions are established for the robust stochastic stability of uncertain FMJDNNs. More precisely, the parameter uncertainties are assumed to be time varying, unknown and norm bounded. The obtained stability conditions are established in terms of LMIs, which can be easily checked by using the efficient MATLAB-LMI toolbox. Finally, numerical examples with simulation result are provided to illustrate the effectiveness and less conservativeness of the obtained results. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Discrete-time neural networks Stochastic stability Linear matrix inequality Markovian jump Various activation functions

1. Introduction Neural networks have been proposed for optimization in a variety of application areas such as design and layout of very large scale integrated circuits. In the past two decades, several types of neural networks have been extensively studied due to their widespread applications in many areas such as image processing, pattern recognition, engineering optimization, signal processing, optimization solvers, and associative memory. In these applications, neural networks require the knowledge of dynamical behaviors of the designed neural network such as stability of an equilibrium point. Therefore, there has been increasing research interest in analyzing the stability problems of neural networks [2,3,7]. Moreover, in practice, the discrete time neural network becomes more important than the continuous time counterparts when implementing the neural networks in a digital way [8]. In order to investigate the dynamical characteristics with respect to digital signal transmission, it is essential to formulate the discretetime analog of neural networks, and a large number of studies have been available for the stability analysis of discrete-time neural networks [17,27].

n Corresponding author at: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea. E-mail address: [email protected] (R. Sakthivel).

Moreover, in both artificial and biological neural systems, time delays are ubiquitous due to delay transmission line, partial element equivalent circuit, integration and communication. Time delays often become a source of oscillation, divergence, and instability and hence delayed neural networks have become a focus of research. Further, there may exist either a distribution of conduction velocities along these pathways or a distribution of propagation delays over a period of time in some cases, which may cause another type of time delays, namely, distributed time delays in discrete neural networks [31]. Therefore, it is of both theoretical and practical importance to investigate the problem of stability analysis of discrete-time neural networks with distributed time delays. Hence, the stability analysis problem for discrete-time neural networks with various types of time delay such as constant, time-varying and distributed delays become increasingly significant, and some results related to this problem have been reported in [10,13]. Further, it is well known that in practical situations, uncertainties have an effect on the performance of the neural networks. In neural networks, the connection weights of the neurons depend on certain resistance and capacitance values that include modeling errors or uncertainties [6]. In particular, when modeling neural networks, the parameter uncertainties should be taken into consideration. The problem of exponential stability has been studied for a class of discrete-time stochastic neural networks with time-varying delay and norm-bounded uncertainties based on Lyapunov stability theory and stochastic approaches [16].

http://dx.doi.org/10.1016/j.isatra.2014.05.002 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Arunkumar A, et al. Robust stochastic stability of discrete-time fuzzy Markovian jump neural networks. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.05.002i

2

A. Arunkumar et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Moreover, fuzzy technique has been widely and successfully used for the approximation of complex nonlinear systems. Among the various fuzzy modeling techniques, the Takagi–Sugeno (TS) fuzzy model [28] is a popular and convenient tool to analyze and synthesize complex nonlinear systems. It should be noted that the TS model can give an effective way to represent nonlinear systems, in which a nonlinear plant is first approximated by a TS fuzzy linear model, and then the analysis will be based on the T–S linear model. Also, one of the key issues for dynamic fuzzy systems is the analysis of their stability. Recently, a great number of results for the stability analysis and control of TS fuzzy neural networks have been reported in [5,18,22]. Based on the Lyapunov stability theory and the linear matrix inequality technique, the global exponential stability problem for a class of fuzzy cellular neural networks with time varying delays has been studied in [14]. Tang et al. [26] derived a delay-dependent synchronization criterion for a fuzzy stochastic discrete-time complex network described by TS fuzzy model with discrete and distributed time-varying delays by employing Lyapunov theory and stochastic analysis technique. In [20], the problem of robust stability is addressed for uncertain fuzzy Markovian jumping Cohen–Grossberg BAM neural networks with time varying delays, and a new delay-dependent stability condition is derived based on the linear matrix inequality for obtaining the stability result. Markovian jump system is a special class of hybrid systems which has the advantage of modeling the dynamical systems subject to abrupt variation in their structures, such as repairs, component failures, sudden environmental disturbances, and changing subsystem interconnections [21]. It should be mentioned that the switching between different modes can be governed by a Markov chain, and neural network sometimes has finite modes that switch from one to another at different times [9]. With the use of the linear matrix inequality approach, a number of important and interesting results have been published on various types of neural networks with Markovian jumping parameters [30,33]. The stability of Markovian jumping neural networks with modedependent time-varying delays and partially known transition rates can be found in [29], where some LMI-based conditions have been proposed for obtaining required result. In discrete time setting, Wu et al. [31] have established delay-dependent stability criteria for discrete-time Markovian jump neural networks with discrete and infinity distributed time delays by using LMI approach. The problem of robust stochastic stability analysis of uncertain discrete-time recurrent neural networks with Markovian jumping and time-varying delays has been investigated in [25]. Very recently, in [4], a unified LMI approach has been developed to solve stability analysis problem for a class of fuzzy bidirectional associative memory neural networks with timevarying interval delays and Markovian jumping parameters. In dealing delay-dependent stability analysis, the well-known index for testing the conservatism of the results is to find a maximum delay bound for which the concerned network ensures the required result. Further, the main issue in studying the delay-dependent stability criteria is how to reduce the possible conservatism and obtain the less conservative result. The less conservativeness in Lyapunov stability theory heavily depends on the introduction of the Lyapunov–Krasovskii functional when dealing with time delays, which leaves much room for researchers to propose new analysis techniques and approaches. Thus, many results on the development of some new approaches have been placed by many researchers to improve the feasibility region. Moreover, in studying the stability analysis, the conditions are determined by the characteristics of activation functions as well as network parameters. Thus, generalization of activation functions will play an important role for neural network designs and applications. Also, to smooth the progress of the design of neural

networks, it is necessary and important to consider them with various activation functions [19]. Despite some results on the stability analysis of delayed neural networks with various activation functions [24], the stability issue for discrete-time fuzzy neural networks has not been fully explored in the existing literature. To the best of author's knowledge, up to now, no result has been reported on stochastic stability for the uncertain fuzzy Markovian jump neural networks with mixed time delay in the discrete case. Motivated by this consideration, the main objective of this paper is to study the stochastic stability for a class of TS fuzzy Markovian jump discrete-time neural networks with various activation functions, discrete and distributed time varying delays. A new set of delay dependent sufficient conditions are derived for stability results by implementing the proper Lyapunov functional, free weighting matrix technique together with the linear matrix inequality approach. The parameter uncertainties are assumed to be time varying and norm bounded. Also, the derived conditions are established in terms of LMIs which can be easily calculated by MATLAB-LMI toolbox. Finally, a numerical example with simulation result is provided to illustrate the applicability and effectiveness of the developed result.

2. Problem formulation and preliminaries In this section, we start by introducing notations, definitions and basic results that will be used in this paper. The superscripts T and ( 1) stand for matrix transposition and matrix inverse respectively; Rnn denotes the n n-dimensional Euclidean space; P 4 0 means that P is real, symmetric and positive definite; I and 0 denote the identity matrix and zero matrix with compatible dimensions; diagfg denotes the block-diagonal matrix; we use an asterisk (n) to represent a term that is induced by symmetry. S denotes the set including zero and positive integers. Let ðΩ; F ; fF t gt Z 0 ; PÞ be a complete probability space with a filtration fF t gt Z 0 satisfying the usual conditions (i.e., the filtration contains all Pnull sets and is right continuous). Also, E½ stands for the mathematical expectation with respect to the given probability measure P. Let S ¼ f1; 2; …; Ng and fr t ; t A Zg be a homogeneous, finite-state Markovian process with right continuous trajectories on the probability space ðΩ; F ; fF t gt Z 0 ; PÞ with generator Γ ¼ ðpij Þnn , pij ¼ Prðr t þ 1 ¼ jjr t ¼ iÞ with pij Z 0 for i; j A S and ∑N j ¼ 1 pij ¼ 1. In this paper, we consider the discrete-time neural network with Markovian jumping parameters and mixed time delay in the following form: xðt þ 1Þ ¼ Aðr t ÞxðtÞ þ Bðr t Þf ðxðtÞÞ þ Cðr t Þgðxðt τðtÞÞÞ π

þ Dðr t Þ ∑ μm hðxðt mÞÞ; m¼1

xðtÞ ¼ ϕðtÞ

for every t A ½ τ; 0; τ ¼ max½τM ; π ;

ð1Þ

where xðtÞ A Rn is the state vector of the neural networks; Aðr t Þ ¼ diagfa1 ðr t Þ; a2 ðr t Þ; …; an ðr t Þg is the positive diagonal matrix that represents the self-feedback term with jaðr t Þj o 1; Bðr t Þ ¼ ½bðr t Þnn , Cðr t Þ ¼ ½cðr t Þnn and Dðr t Þ ¼ ½dðr t Þnn are known real constant matrices with appropriate dimensions. The functions f ðxðtÞÞ ¼ ½f 1 ðx1 ðtÞÞ; f 2 ðx2 ðtÞÞ; …; f n ðxn ðtÞÞT ; gðxðtÞÞ ¼ ½g 1 ðx1 ðtÞÞ; g 2 ðx2 ðtÞÞ; …; g n ðxn ðtÞÞT and hðxðtÞÞ ¼ ½h1 ðx1 ðtÞÞ; h2 ðx2 ðtÞÞ; …; hn ðxn ðtÞÞT denote the neuron activation functions. The positive integer τðtÞ denotes that the time-varying discrete delay satisfies τm r τðtÞ r τM , where τm and τM are positive integers. The scalar μm Z 0, where m is the positive upper bound of distributed delays. ϕðtÞ is the initial function, which is continuous and defined on ½ τ; 0.



In the presence of parameter uncertainties, the neural network model (1) can be written as xðt þ 1Þ ¼ ðAðr t Þ þ ΔAðt; r t ÞÞxðtÞ þ ðBðr t Þ þ ΔBðt; r t ÞÞf ðxðtÞÞ þ ðCðr t Þ þ ΔCðt; r t ÞÞ π

gðxðt τðtÞÞÞ þ ðDðr t Þ þ ΔDðt; r t ÞÞ ∑ μm hðxðt mÞÞ:

Assumption (II). The constant μm Z0 satisfies the following convergent condition: π

ð2Þ

3

∑ μm o 1

and

m¼1

π

∑ mμm o 1:

m¼1

m¼1

Next, we will consider uncertain FMJDNNs with mixed delay, which is represented by a TS fuzzy model composed of a set of fuzzy implications and each implication is expressed as a linear system model [26]. Consider the uncertain Takagi–Sugeno fuzzy model with time-delay which is described by the following IF–THEN rules: k k k Plant Rule Rlrt : IF fu1 ðtÞ is θrt 1 g, fu2 ðtÞ is θrt 2 g; …; fup ðtÞ is θrt p g THEN xðt þ 1Þ ¼ ðAk ðr t Þ þ ΔAk ðt; r t ÞÞxðtÞ þ ðBk ðr t Þ þ ΔBk ðt; r t ÞÞf ðxðtÞÞ

Definition 2.1. For every initial condition ðϕ; r 0 Þ, then the FMJDNNs (5) are said to be stochastically stable if satisfying the following condition: 1 E ∑ ‖xt ðϕ; r 0 Þ‖2 o 1; t¼0

where xt ðϕ; r 0 Þ denotes the solution of (5) at time t under the initial conditions ϕ and r0.

þ ðC k ðr t Þ þ ΔC k ðt; r t ÞÞgðxðt τðtÞÞÞþ ðDk ðr t Þ π

þ ΔDk ðt; r t ÞÞ ∑ μm hðxðt mÞÞ;

ð3Þ

m¼1

where θrt j ðk ¼ 1; 2; …; q; j ¼ 1; 2; …; pÞ are the fuzzy sets, ðu1 ðtÞ; u2 ðtÞ; …; up ðtÞÞ is the premise variable vector and r is the number of IF–THEN rules. Further r t ¼ i; i A S; Ak ðr t Þ ¼ Aik ; Bk ðr t Þ ¼ Bik ; C k ðr t Þ ¼ C ik and Dk ðr t Þ ¼ Dik , and we assume that the parameter uncertainties ΔAk ðt; rt Þ ¼ ΔAik , ΔBk ðt; r t Þ ¼ ΔBik , ΔC k ðt; r t Þ ¼ ΔC ik and ΔDk ðt; rt Þ ¼ ΔDik are time varying and described by k

½ΔAik ΔBik ΔC ik ΔDik ¼ Ek F k ðtÞ½N1ik N 2ik N 3ik N 4ik ;

ð4Þ

where N 1ik ; N2ik ; N 3ik ; N 4ik and Ek are known constant matrices of appropriate dimensions which characterize the structures of uncertainties and Fik(t) is a known time varying matrix with Lebesgue measurable elements bounded by F Tik ðtÞF ik ðtÞ r I, where I is the identity matrix with appropriate dimension. Let λik ðξðtÞÞ be the normalized membership function of the inferred fuzzy set βik ðξðtÞÞ, then the defuzzified form of FMJDNNs becomes q

The following definition and lemmas will be used in the proof of main results.

Lemma 2.2 (Li et al. [8]). For any symmetric positive-definite matrix Q A Rnn , integers τM and τm, ðτM Z τm Þ, xðkÞ : fτm ; τm þ 1 ; …; τM g -Rn , such that the sums in the following are well defined, then !T ! τM

∑ xðtÞ

t ¼ τm

Q

τM

τM

t ¼ τm

t ¼ τm

∑ xðtÞ r τ ∑ xT ðtÞQxðtÞ

holds, where τ ¼ τM τm þ 1. Lemma 2.3 (Boyd et al. [1]). Given constant matrices Ω1 ; Ω2 and Ω3, where Ω1 ¼ ΩT1 4 0 and Ω2 ¼ ΩT2 4 0, then Ω1 þ ΩT3 Ω2 1 Ω3 o0 ΩT3 1 if and only if ½Ω Ω3 Ω2 o 0.

Lemma 2.4 (Luo et al. [16]). For any vector x; y A Rn , matrices A; P; D; E and F are real matrices of appropriate dimensions with P 4 0; F T F r I, and scaler ϵ 4 0, the following inequalities hold: (i) 2xT DFEy r ϵ 1 xT DDT x þ ϵyT ET Ey. (ii) If P ϵDDT 4 0, then ðA þ DFEÞT P 1 ðA þ DFEÞ rAT ðP ϵDDT Þ 1 A þ ϵ 1 ET E.

"

xðt þ 1Þ ¼ ∑ λik ðξðtÞÞ ðAik þ ΔAik ÞxðtÞ þ ðBik þ ΔBik Þf ðxðtÞÞ:

3. Stability results

k¼1

#

π

þ ðC ik þ ΔC ik Þgðxðt τðtÞÞÞ þ ðDik þ ΔDik Þ ∑ μm hðxðt mÞÞ ; m¼1

ð5Þ where λik ðξðtÞÞ ¼ β ik ðξðtÞÞ=∑qk ¼ 1 β ik ðξðtÞÞ, βik ðξðtÞÞ ¼ ∏pj¼ 1 θij ðξj ðtÞÞ k

here θij ðξj ðtÞÞ is the grade of the membership function of ξj ðtÞ in θkij. We assume that βik ðξðtÞÞ Z0; k ¼ 1; 2; …; q, ∑qk ¼ 1 βi ðξðtÞÞ Z 0 and λik ðξðtÞÞ satisfies λik ðξðtÞÞ Z0; k ¼ 1; 2; …; q, and ∑qk ¼ 1 λi ðξðtÞÞ ¼ 1 for any ξðtÞ. k

Assumption (I). For s A f1; 2; …; ng, the neuron activation functions f s ðÞ; g s ðÞ and hs ðÞ in (5) are continuous and bounded, satisfying

F s r

f s ðx1 Þ f s ðx2 Þ r F sþ ; x1 x2

H s r

hs ðx1 Þ hs ðx2 Þ r H sþ ; x1 x2

Gs r

g s ðx1 Þ g s ðx2 Þ r Gsþ ; x1 x2

8 x1 ; x2 A R x1 a x2 ;

where F s , F sþ , Gs , Gsþ , H s and H sþ are constants.

In this section, we establish the robust stochastic stability of uncertain FMJDNNs (5). First, we study the stochastic stability of FMJDNNs (5) when the uncertainties ΔAik ¼ 0, ΔBik ¼ 0, ΔC ik ¼ 0 and ΔDik ¼ 0. Specifically, a set of sufficient condition is first derived for stochastic stability based on the Lyapunov–Krasovskii functional theory and LMI optimization approach. Further, the result is extended to the uncertain case. For presentation convenience, we denote F 1 ¼ diag F 1 F 1þ ; F 2 F 2þ ; …; F n F nþ ; F þ F 1þ F 2 þ F 2þ F þ F nþ ; ; …; n ; F 2 ¼ diag 1 2 2 2 þ þ G1 ¼ diag G1 G1 ; G2 G2 ; …; Gn Gnþ ; G þG1þ G2 þ G2þ G þ Gnþ ; ; …; n ; G2 ¼ diag 1 2 2 2 þ þ þ H 1 ¼ diag H 1 H 1 ; H 2 H 2 ; …; H n H n ; H 1 þ H 1þ H 2 þ H 2þ H þ H nþ ; ; …; n : H 2 ¼ diag 2 2 2 Theorem 3.1. Under Assumptions (I) and (II), the FMJDNNs (5) without uncertainties are stochastically stable, if there exist matrices P i 4 0, Q s ¼ ½Qns1 QQ s2 4 0; s ¼ 1; 2; 3; R1 4 0; R2 4 0; S 4 0; U s 4 0; s3 s ¼ 1; 2; 3, and any matrices M s ; s ¼ 1; 2; 3; 4, with appropriate



4

dimensions, such that the following LMIs hold for k ¼ 1; 2; …; q; i A S: " #

Ψ 12;12 Ψ 1 o 0; n Ψ2

Ψ¼

∑

s ¼ t τm

ð6Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Ψ 1 ¼ ½τm M 1 ; τM τ m M 2 ; τM τm M 3 , ðR1 þ R2 Þ; R1 g

Ψ

t 1

V 2 ðtÞ ¼

τ

Ψ 2 ¼ diagf R2 ;

τm

V 3 ðtÞ ¼

τ

þ 2M T42 ðATik IÞT ;

Ψ 1;4 ¼ M T14 2M21 ; Ψ 1;5 ¼ Q 12 þQ 22 þ Q 32 þ ðτM τm ÞQ 22 þ 2MT15 2M41 þ 2ðAik IÞT MT43 ; Ψ Ψ Ψ Ψ Ψ 1;10 ¼ ATik P i C ik þ MT110 þ 2M 41 C ik ; Ψ 1;11 ¼ H 2 U 3 þ MT111 ; Ψ 1;12 ¼ ATik P i Dik þ MT112 þ 2M 41 Dik ; T T T 1;6 ¼ M 16 ; 1;7 ¼ M 17 ; 1;8 ¼ M 18 ; T T 1;9 ¼ Aik P i Bik þ F 2 U 1 þ M 19 þ2M 41 Bik ;

N

P i ¼ ∑ pij P j ; j¼1

1

S;

τm 1

t1

∑ ηT ðsÞR1 ηðsÞ þ

∑

j ¼ τM s ¼ t þ j

π

t1

m¼1

s ¼ tm

1

t 1

∑ ηT ðsÞR2 ηðsÞ

∑

j ¼ τM s ¼ t þ j

T

∑

h ðxðsÞÞShðxðsÞÞ;

E½V 1 ðX t þ 1 ; jÞ V 1 ðxt ; iÞ ¼ E½xT ðt þ 1ÞP i xðt þ 1Þ xT ðtÞP i xðtÞ;

t

¼E

∑

s ¼ t þ 1 τm

þ þ

λT ðsÞQ 1 λðsÞ

t

∑

s ¼ t þ 1 τ ðt þ 1Þ t

∑

s ¼ t þ 1 τM

t1

∑

s ¼ t τm

∑

s ¼ t τðtÞ

t1


∑

s ¼ t τM

"

λT ðsÞQ 2 λðsÞ #


r E λ ðtÞðQ 1 þ Q 2 þ Q 3 ÞλðtÞ λ ðt τm ÞQ 1 λðt τm Þ T

T

λ ðt τðtÞÞQ 2 λðt τðtÞÞ λ ðt τM ÞQ 3 λðt τM Þ # T

þ

T

t τm

∑

s ¼ t þ 1 τM

λT ðsÞQ 2 λðsÞ ;

ð9Þ

E½V 3 ðX t þ 1 ; jÞ V 3 ðxt ; iÞ " ( τm

¼E

j ¼ τM þ 1

þ

þ

τm 1

(

∑

j ¼ τM 1

∑

s ¼ t þ1þj t

∑

(

∑

j ¼ τM

"

λT ðsÞQ 2 λðsÞ ∑ λT ðsÞQ 2 λðsÞ s ¼ t þj

t1

s ¼ tþj

∑

s ¼ t þ1þj

s ¼ tþj

t τm

T

∑

j ¼ t τM þ 1

t τm 1

∑

j ¼ t τM

t1

λT ðjÞQ 2 λðjÞ

ηT ðjÞR1 ηðjÞ #

η ðjÞR2 ηðjÞ ; T

∑

j ¼ t τM

E½V 4 ðX t þ 1 ; jÞ V 4 ðxt ; iÞ " ( π

∑ μm

m¼1

m¼1

)#

ηT ðsÞR2 ηðsÞ ∑ ηT ðsÞR2 ηðsÞ

¼ E ðτM τm Þλ ðtÞQ 2 λðtÞ

¼E

T

t1

t

þ τM η ðtÞR2 ηðtÞ

μ m ¼ ∑ μm

)

η ðsÞR1 ηðsÞ ∑ η ðsÞR1 ηðsÞ T

s ¼ t þ1þj

)

t 1

t

∑

T

π

ð8Þ


t 1


þðτM τm ÞηT ðtÞR1 ηðtÞ

P i ¼ Pðr t Þ;


here λðtÞ ¼ ½xT ðtÞ ηT ðtÞT and ηðtÞ ¼ xðt þ 1Þ xðtÞ. Let us define the forward difference of Vi(t) as ΔV i ðtÞ ¼ V i ðt þ 1Þ V i ðtÞ then we have

Ψ 5;9 ¼ M 43 Bik Ψ 5;10 ¼ M43 C ik ; Ψ 5;12 ¼ M43 Dik ; Ψ 6;6 ¼ Q 13 ; Ψ 7;7 ¼ Q 23 ; Ψ 8;8 ¼ Q 33 ; Ψ 9;9 ¼ DTik P i Dik U 1 ; Ψ 9;10 ¼ DTik P i C ik ; Ψ 9;12 ¼ DTik P i C ik ; Ψ 10;10 ¼ C Tik P i C ik U 2 ; Ψ 10;12 ¼ C Tik P i Dik ; μm

∑

s ¼ t τM

T

E½V 2 ðX t þ 1 ; jÞ V 2 ðxt ; iÞ "

Ψ 2;2 ¼ Q 11 þ 2M32 ; Ψ 2;3 ¼ 2M12 þ 2M 22 þ 2MT33 2M32 ; Ψ 2;4 ¼ 2M22 þ2MT34 ; Ψ 2;5 ¼ 2MT35 ; Ψ 2;6 ¼ Q 12 þ 2MT36 ; Ψ 2;7 ¼ 2MT37 ; Ψ 2;8 ¼ 2MT38 ; Ψ 2;9 ¼ 2MT39 ; Ψ 2;10 ¼ 2MT310 ; Ψ 2;11 ¼ 2MT311 ; Ψ 2;12 ¼ 2MT312 ; Ψ 3;3 ¼ Q 21 G1 U 2 2M13 þ 2M23 2M33 ; Ψ 3;4 ¼ 2MT14 þ2MT24 þ 2M23 2MT34 ; Ψ 3;5 ¼ 2MT15 þ2MT25 þ 2M42 2MT35 ; Ψ 3;6 ¼ 2MT16 þ2MT26 2MT36 ; Ψ 3;7 ¼ Q 22 2MT17 þ 2MT27 2MT37 ; Ψ 3;8 ¼ 2MT18 þ2MT28 2MT38 ; Ψ 3;9 ¼ 2MT19 þ2MT29 2MT39 þ2M42 Bik ; Ψ 3;10 ¼ G2 U 2 2MT110 þ 2MT210 2MT310 þ2M42 C ik ; Ψ 3;11 ¼ 2M T111 þ 2M T211 2MT311 ; Ψ 3;12 ¼ 2M T112 þ 2M T212 2MT312 þ 2M42 Dik ; Ψ 4;4 ¼ Q 31 2MT24 ; Ψ 4;5 ¼ 2MT25 ; Ψ 4;6 ¼ 2MT26 ; Ψ 4;7 ¼ 2MT27 ; Ψ 4;8 ¼ Q 32 2MT28 ; Ψ 4;9 ¼ 2MT29 ; Ψ 4;10 ¼ 2MT210 ; Ψ 4;11 ¼ 2MT211 ; Ψ 4;12 ¼ 2M T212 ; Ψ 5;5 ¼ Q 13 þQ 23 þ Q 33 þ ðτM τm ÞQ 23 þ ðτM τm ÞR1 þ τM R2 2M43 ;

Ψ 11;11 ¼ μ m S U 3 ; Ψ 12;12 ¼ DTik P i Dik

s ¼ t τðtÞ

t 1

∑ λ ðsÞQ 2 λðsÞ

V 4 ðtÞ ¼ ∑ μm

Ψ 13 ¼ 2MT13 2M11 þ 2M21 2M31

λT ðsÞQ 2 λðsÞ þ

∑

j ¼ τM þ 1 s ¼ t þ j

þ

F 1 U 1 H 1 U 3 þ2M 11 þ 2M 41 ðAik IÞ;

t1

∑

T 1;1 ¼ Aik P i Aik P i þQ 11 þ Q 21 þ Q 31 þ ð M m ÞQ 21

Ψ 1;2 ¼ 2MT12 þ M31 ;

t1

λT ðsÞQ 1 λðsÞ þ

t

∑

T

h ðxðsÞÞShðxðsÞÞ

s ¼ t þ1m

ð10Þ

t 1

∑

s ¼ tm

)# T

h ðxðsÞÞShðxðsÞÞ

π T T ¼ E μ m h ðxðtÞÞShðxðsÞÞ ∑ μm h ðxðt mÞÞShðxðt mÞÞ

and the other parameters are zero.

4

VðtÞ ¼ ∑ V n ðtÞ;

r E μ m h ðxðtÞÞShðxðsÞÞ T

V 1 ðtÞ ¼ xT ðtÞP i xðtÞ;

1

μm

π ∑ μm hðxðt mÞÞ :

ð7Þ

n¼1

where

m¼1

"

Proof. In order to prove the FMJDNNs (5) are stochastically stable, we choose the following Lyapunov–Krasovskii functional candidate

π T ∑ μm hðxðt mÞÞ S m¼1

ð11Þ

m¼1

ζ ðtÞ ¼ ½xT ðtÞ xT ðt τm Þ xT ðt τðtÞÞ xT ðt τM Þ ηT ðtÞ ηT ðt τm Þ π η ðt τðtÞÞ ηT ðt τM Þ f T ðxðtÞÞ gT ðxðt τðtÞÞÞ hT ðxðtÞÞ ∑ μm hT ðxðt mÞÞT .

Define T

m¼1



i þ R1 ηðiÞgT R1 1 fM T3 ζ ðtÞ þ R1 ηðiÞg " n q T r E ∑ λik ðξðtÞÞζ ðtÞ Ψ 12;12 þ τM M 1 R2 1 M T1

On the other hand, for any appropriately dimensioned matrices M s ; s ¼ 1; 2; 3, the following inequalities hold: " 2ζ ðtÞM 1 xðtÞ xðt τðtÞÞ

#

t1

T

ηðiÞ ¼ 0;

∑

i ¼ t τ ðtÞ

"

t τ ðtÞ 1

2ζ ðtÞM 2 xðt τðtÞÞ xðt τM Þ T

∑

i ¼ t τM

" 2ζ ðtÞM 3 xðt τm Þ xðt τðtÞÞ

∑

i ¼ t τ ðtÞ

ηðiÞ ¼ 0;

ð13Þ

ηðiÞ ¼ 0:

ð14Þ

π

ηðtÞ ¼ ∑ λik ðξðtÞÞ ðAik IÞxðtÞ þ Bik f ðxðtÞÞþ C ik gðxðt τðtÞÞÞ þ Dik ∑ μm hðxðt mÞÞ ; m¼1

k¼1

then for any matrices M 4 ¼ ½M 41 0 M 42 0 M 43 0n;7n , we have ζ ðtÞM4 ∑ λik ðξðtÞÞ ðAik IÞxðtÞ þ Bik f ðxðtÞÞ þ C ik gðxðt τðtÞÞÞ k¼1 π þ Dik ∑ μm hðxðt mÞÞÞ ηðkÞ ¼ 0:

ð15Þ

ðf s ðxs ðtÞÞ F s xs ðtÞÞðf s ðxs ðtÞÞ F sþ xs ðtÞÞ r 0

xðtÞ

H1 U 3 H2 U 3

3"

F s þ F sþ es eTs 2

5

es eTs

xðtÞ f ðxðtÞÞ

#

F 2U1

#"

∑

t τ ðtÞ 1

∑

j ¼ t τM

E½VðX 2 ; r 2 =x1 ; r 1 r Vðx1 ; r 1 Þ λ0 xT ð1Þxð1Þ:

ð23Þ

T

E½VðX T þ 1 ; r T þ 1 =x0 ; r 0 rVðx0 ; r 0 Þ λ0 ∑ E½xT ðlÞxðlÞ=x0 ; r 0

#

xðtÞ

r 0:

f ðxðtÞÞ

U1

H2 U 3 U3

#"

xðtÞ hðxðtÞÞ

r 0:

T

fM T1 ζ ðtÞ þR2 ηðiÞg R2 1 fM T1 ζ ðtÞ þ R2 ηðiÞg T

fM T2 ζ ðtÞ þ ðR1 þ R2 ÞηðiÞg ðR1 þ R2 Þ 1 fM T2 ζ ðtÞ

þðR1 þ R2 ÞηðiÞg

t τm 1

∑

j ¼ t τ ðtÞ

ð16Þ

ð17Þ

#

k¼1

t1

ð22Þ

l¼0

n q T ∑ λik ðξðtÞÞζ ðtÞ Ψ 12;12 þ τM M 1 R2 1 M T1

j ¼ t τðtÞ

Setting t¼ 0 and t¼1 in (21) yields

Then, one can continue the iterative procedure (21) to obtain r 0;

þðτM τm ÞM 2 ðR1 þ R2 Þ 1 M T2 : o þðτM τm ÞM 3 R1 1 M T3 ζ ðtÞ

ð21Þ

1

Combining (7)–(18), we get " E½ΔV i ðtÞ r E

E½VðX t þ 1 ; r t þ 1 ¼ jÞ=xt ; r t ¼ i r Vðxt ; r t ¼ iÞ λ0 xT ðtÞxðtÞ:

l¼0

Similar to the above, one can get #" " #T " # xðt τðtÞÞ xðt τðtÞÞ G1 U 2 G2 U 2 r0: gðxðt τðtÞÞÞ gðxðt τðtÞÞÞ G2 U 2 U2

hðxðtÞÞ

Let λ0 ¼ minfΨ ; i A Lg, then λ0 4 0 due to Ψ . Finally from (20) we obtain that for any t Z 0

E½VðX 2 ; r 2 =x0 ; r 0 r Vðx0 ; r 0 Þ λ0 ∑ E½xT ðlÞxðlÞ=x0 ; r 0 :

where es denotes the unit column vector having element 1 on its sth row and zeros elsewhere. Let U 1 ¼ diagfu11 ; u12 ; …; u1n g; U 2 ¼ diagfu21 ; u22 ; …; u2n g and U 3 ¼ diagfu31 ; u32 ; …; u3n g. Then 3" # " #T 2 þ F þF þ n F s F s es eTs s 2 s es eTs xðtÞ xðtÞ 4 5 ∑ rs r 0: þ f ðxðtÞÞ f ðxðtÞÞ F s þ2 F s es eTs es eTs s¼1

#T "

ð20Þ

Taking expectation E½=X 0 ; r 0 on both sides of (23) with the aid of (22), we obtain

From Assumption (I), for any s ¼ 1; 2; …; n, we have

"

Ψ 12;12 þ τM M1 R2 1 MT1 þ ðτM τm ÞM2 ðR1 þR2 Þ 1 M T2 þ ðτM τm ÞM 3 R1 1 M T3 o 0:

and

m¼1

That is, " #T " xðtÞ F 1 U1 f ðxðtÞÞ F 2U1

From the above inequality and Lemma 2.3, it follows that

E½VðX 1 ; r 1 Þ=x0 ; r 0 r Vðx0 ; r 0 Þ λ0 xT ð0Þxð0Þ

q

which is equivalent to " #T 2 þ F F es eT xðtÞ 4 s s þ s F þF f ðxðtÞÞ s 2 s es eTs

o i þ ðτM τm ÞM 2 ðR1 þ R2 Þ 1 M T2 : þðτM τm ÞM 3 R1 1 M T3 ζ ðtÞ : ð19Þ

#

In addition we have, ηðtÞ ¼ xðt þ 1Þ xðtÞ, q

k¼1

ð12Þ

#

t τm 1

T

5

fM T3 ζ ðtÞ

ð18Þ

which implies that T

1

l¼0

λ0

∑ E½xT ðlÞxðlÞ=x0 ; r 0 r

Vðx0 ; r 0 Þ o 1:

Hence, from the Lyapunov stability theory that the dynamics of the FMJDNNs (5) without uncertainties is stochastically stable in the sense of Definition 2.1, which completes the proof. □ In the previous theorem, we developed criteria for stochastic stability of FMJDNNs (5) with various activation functions, discrete and distributed delays based on LMI. Next, we obtain the robust stochastic stability of uncertain FMJDNNs by using the results of the previous theorem. Now, using Lemma 2.4, we extend Theorem 3.1 to the uncertain case and give the following robust stochastic stability criterion. Theorem 3.2. 'Under Assumptions (I) and (II), the uncertain FMJDNNs (5) are robustly stochastically stable, if there exist matrices P i 4 0, Q s ¼ ½Qns1 QQ s2 4 0; s ¼ 1; 2; 3; R1 4 0; R2 4 0; S 4 0; U s 4 0; s3 s ¼ 1; 2; 3, any matrices M s ; s ¼ 1; 2; 3; 4, with appropriate dimensions and positive scalars ϵ1ik ; ϵ2ik such that the following LMIs hold for k ¼ 1; 2; …; q; i A S: 2 3 Ψ ΘT1 ϵ1ik ΘT2 Θ3 P i Eik ϵ2ik ΘT2 6 7 6 n ϵ1ik I 0 0 0 7 6 7 6 7 ð24Þ Θ¼6 n n ϵ1ik I 0 0 7o0 6 7 T 6n 7 n n ðϵ2ik I Eik P i Eik Þ 0 5 4 n n n n ϵ2ik I

Θ1 ¼ ½ETk MT41 0 ETk MT42 0 ETk MT43 0n;7n ; Θ2 ¼ ½N1ik 0n;7n N 2ik N3ik 0 N 4ik 0n;3n ; Θ3 ¼ ½Aik 0n;7n Bik C ik 0 Dik 0n;3n and other parameters are defined as in Theorem 3.1.



6

Remark 3.3. In the absence of distributed delay and Markovian jump, the discrete time neural network (1) can be written as

dimensions and positive scalars ϵ1 ; ϵ2 , such that the following LMI holds: 2 3 T T T Ξ Ξ^ 1 ϵ1 Ξ^ 2 Ξ^ 3 P 1 E ϵ2 Ξ^ 2 7 6 6 n ϵ I 0 0 0 7 1 6 7 6 7 ^ Ξ ¼6 n ð27Þ 7 o0; n ϵ I 0 0 1 6 7 T 6n n n ðϵ2 I E P 1 EÞ 0 7 4 5 n n n n ϵ2 I

xðt þ 1Þ ¼ ðA þ ΔAÞxðtÞ þ ðB þ ΔBÞf ðxðtÞÞ þðC þ ΔCÞgðxðt τðtÞÞÞ

Ξ^ 1 ¼ ½ET M T41 0 ET M T42 0 ET MT43 0n;5n , Ξ^ 2 ¼ ½N 1 0n;7n N 2 N 3 , Ξ^ 3 ¼

Proof. The proof immediately follows by replacing Aik ; Bik ; C ik and Dik with Aik þ Eik F ik ðtÞN1ik ; Bik þ Eik F ik ðtÞN 2ik ; C ik þ Eik F ik ðtÞN 3ik and Dik þ Eik F ik ðtÞN 4ik in Theorem 3.1. By applying Lemma 2.4, we can obtain (24). It can be concluded from Lyapunov stability theory that the dynamics of FMJDNNs (5) is robustly stochastically stable. The proof is completed. □

ð25Þ

½A 0n;7n B C and other parameters are defined as in Corollary 3.4.

In order to show the less conservatism of the proposed theory, we will provide the following corollary for the asymptotic stability of neural network (25) based on Theorem 3.1. Corollary 3.4. Under Assumption (I), the discrete time neural networks (25) without uncertainties are asymptotically stable, if there exist matrices P 1 4 0, Q s ¼ ½Qns1 QQ s2 4 0; s ¼ 1; 2; 3; R1 4 0; R2 4 0; U s 4 s3 0; s ¼ 1; 2, and any matrices M s ; s ¼ 1; 2; 3; 4, with appropriate dimensions, such that the following LMI holds: " #

Ξ¼

Ξ 10;10 Ξ 1 o0; n Ξ2

where Ξ 1 ¼ ½τ m M 1 ; ðR1 þ R2 Þ; R1 g

ð26Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τM τm M2 ; τM τm M3 ; Ξ 2 ¼ diagf R2 ;

Ξ 1;1 ¼ AT P A P 1 þ Q 11 þ Q 21 þ Q 31 þ ðτM τm ÞQ 21

4. Numerical simulation In this section, we present a numerical example with a simulation result to demonstrate the effectiveness of the proposed results in Theorems 3.1 and 3.2. Example 4.1. Consider the nominal form of 2-mode FMJDNNs model together with the R-th rule given as follows: 1 Plant Rule R11 : IF ξ1 ðtÞ ¼ x2 ðtÞ is θ11 ðx2 ðtÞÞ THEN π

xðt þ 1Þ ¼ A11 xðtÞ þB11 f ðxðtÞÞ þ C 11 gðxðt τ ðtÞÞÞ þ D11 ∑ μm hðxðt mÞÞ; m¼1

Plant Rule R21 : IF ξ1 ðtÞ ¼ x2 ðtÞ is θ11 ðx2 ðtÞÞ THEN 2

π


F 1 U 1 þ 2M 11 þ 2M 41 ðA IÞ;

Ξ 1;2 ¼ MT12 þM 31 ;

Ξ 13 ¼ MT13 M11 þ M21 M31 þ MT42 ðAT IÞT ;

Ξ Ξ 1;5 ¼ Q 12 þ Q 22 þ Q 32 þ ðτM τm ÞQ 22 þ MT15 M 41 þ 2ðA IÞT MT43 ; T 1;4 ¼ M 14 M 21 ;

Ξ 1;6 ¼ MT16 ; Ξ 1;7 ¼ M T17 ; Ξ 1;8 ¼ MT18 ; Ξ 1;9 ¼ AT P 1 B þ F 2 U 1 þM T19 þ 2M 41 B; Ξ 1;10 ¼ AT P 1 C þ MT110 þM 41 C; Ξ Ξ Ξ Ξ Ξ Ξ

Ξ Ξ

T 2;2 ¼ Q 11 þ 2M 32 ; 2;3 ¼ 2M 12 þ2M 22 þ 2M 33 2M 32 ; T T Q 12 þ 2M T36 ; 2;4 ¼ 2M 22 þ 2M 34 ; 2;5 ¼ 2M 35 ; 2;6 ¼ T T T 2;7 ¼ 2M 37 ; 2;8 ¼ 2M 38 ; 2;9 ¼ 2M 39 ; T ¼ 2M ; ¼ Q G 2;10 3;3 1 U 2 2M 13 þ2M 23 2M 33 ; 21 310 T T T 3;4 ¼ M 14 þ M 24 þ M 23 M 34 ; T T T T T T 3;5 ¼ 2M 15 þ 2M 25 þ 2M 42 2M 35 ; 3;6 ¼ M 16 þM 26 M 36 ;

Ξ

Ξ

Ξ

Ξ

Ξ

Ξ 3;7 ¼ Q 22 MT17 þ M T27 MT37 ; Ξ 3;8 ¼ M T18 þ MT28 MT38 ; Ξ 3;9 ¼ MT19 þ MT29 M T39 þ M42 B; Ξ 3;10 ¼ G2 U 2 MT110 þ MT210 M T310 þ M42 C; Ξ 4;4 ¼ Q 31 2M T24 ; Ξ 4;5 ¼ M T25 ; Ξ 4;6 ¼ MT26 ;

Ξ 4;7 ¼ M T27 ;

Ξ 4;8 ¼ Q 32 MT28 ;

Ξ 4;9 ¼ Ξ 4;10 ¼ Ξ 5;5 ¼ Q 13 þ Q 23 þ Q 33 þ ðτM τm ÞQ 23 þ ðτM τm ÞR1 þ τM R2 2M43 ; 2M T29 ;

M T210 ;

Ξ 5;9 ¼ M43 B Ξ 5;10 ¼ M43 C; Ξ 5;12 ¼ M43 D; Ξ 6;6 ¼ Q 13 ; Ξ 7;7 ¼ Q 23 ; Ξ 8;8 ¼ Q 33 ; Ξ 9;9 ¼ DT P 1 D U 1 ; Ξ 9;10 ¼ DT P 1 C; Ξ 10;10 ¼ C T P 1 C U 2 ;


π



π


with the parameters as follows: 2 3 2 0:1 0 0 0:03 6 7 6 A11 ¼ 4 0 0:2 0 5; B11 ¼ 4 0:02 0 0 0:3 0 2 3 0:04 0:02 0:01 6 7 0:02 0:03 5; C 11 ¼ 4 0 0:01 0 0:02 2

0:02

6 D11 ¼ 4 0:02 0 2 0:04 6 B21 ¼ 4 0:04 0 2 6 C 21 ¼ 4

and the other parameters are zero.

0:01

0

3

0:03

0:02 7 5;

0:2 0:05

0:2 0:052

0:02 0:03

0:05

0:01

0

0:056

0:04

0

2

0:01

0:02

0:02

0

0:01

0:04

0:2

6 A21 ¼ 4 0 0 3

0

0

3 7 5;

3

0 7 5; 0:4

0:3 0

7 0 5; 0:04

0:04

3

7 0:02 5; 0:01

2

0:04

6 D21 ¼ 4 0:04 0

0:02

3

0

0:03

7 0:01 5:

0:5

0:1

Next the results for the robust asymptotical stability of uncertain discrete time neural network (25) is presented in the following corollary. Corollary 3.5. Under Assumption (I), the uncertain discrete time neural networks (25) are robustly asymptotically stable, if there exist matrices P 1 4 0, Q s ¼ ½Qns1 QQ s2 4 0; s ¼ 1; 2; 3, R1 4 0; R2 4 0; U s 4 s3 0; s ¼ 1; 2, and any matrices M s ; s ¼ 1; 2; 3; 4, with appropriate

Table 1 Time delay upper bound τM for various values of τm. τm τM

2 34

4 36

5 37

7 39

9 41

10 42

11 43

13 45

15 47



2

3

0

2

2 1.5

3

2

0:03

0

0:06

0:06

0:07

0

6 D12 ¼ 4 0:03 0 2 0:05 6 B22 ¼ 4 0:08 0

1

0.5

2

0

0

0:01

6 7 6 A12 ¼ 4 0 0:3 0 5; B12 ¼ 4 0:06 0 0 0:3 0 2 3 0:05 0:01 0:03 6 7 0:03 0:05 5; C 12 ¼ 4 0

2.5

Mode

0:1

7

0

10

20

30

40

50

60

70

80

90

100

Time ’t’

0:06 6 C 22 ¼ 4 0 0:03

7 0:04 5;

0:1 0:02

0:8 0:056

0:01

0

0:07

0:03

0:01 0:01 0

0:2

0

0:03

0:01

0:1

6 A22 ¼ 4 0 0 3

0

0

0:1 0

3 7 5;

3

7 0 5; 0:2

7 5;

3 0:07 0:04 7 5;

2

0:04 6 D22 ¼ 4 0:05 0

0:01 1

1

0:03 0:07

3 0 0:03 7 5:

2

1

0:1

0:5

2

1

0.8

x1(t)

0.6

x2(t)

State Responses

-0.2

0.2 0 -0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8 40

60

80

x3(t)

0.4

0

20

x2(t)

0.6

0.2

0

x1(t)

0.8

x3(t)

0.4 State Responses

0:03

2

0:02

The fuzzy sets of θ11 ; θ11 ; θ21 ; θ21 are represented by the

Fig. 1. Jumping between modes during simulation.

-1

3

0:03

-1

100

0

20

40

Time ’t’

60

80

100

Time ’t’

Fig. 2. State responses of the FMJDNNs without uncertainties for k ¼1 and k ¼ 2.

Table 2 Time delay upper bound τM for various values of τm. τm τM

2 24

4 26

5 27

7 29

9 31

11 33

13 35

0.8

x1(t)

0.8

x1(t)

0.6

x2(t)

0.6

x2(t)

0.4

x3(t)

0.4

x3(t)

0.2 0 -0.2

0.2 0 -0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

15 37

1

State Responses

State Responses

1

10 32

0

20

40

60 Time ’t’

80

100

-1

0

20

40

60

80

100

Time ’t’

Fig. 3. State responses of the FMJDNNs with uncertainties for k ¼1 and k ¼ 2.



8

Table 3 Calculated time delay upper bound τM for various values of τm. Method

τm ¼ 1

τm ¼ 4

τm ¼ 8

τ m ¼ 15

τm ¼ 25

[11,12] Corollary 3.4

3 10

6 13

10 17

17 24

27 34

Table 4 Calculated time delay upper bound τM for various values of τm. Method

τm ¼ 2

τm ¼ 4

τm ¼ 6

τm ¼ 8

τm ¼ 10

[12,23,32,15] [11] Corollary 3.5

Infeasible 18 22

Infeasible 20 24

Infeasible 22 26

Infeasible 24 28

Infeasible 26 30

following membership functions respectively:

θ111 ðx2 ðtÞ1 Þ ¼ θ121 ðx2 ðtÞ1 Þ ¼ 1 θ211 ðx2 ðtÞ1 Þ ¼ θ221 ðx2 ðtÞ1 Þ ¼

1 ; 1 þ e 2x2 ðtÞ

1 : 1 þ e 2x2 ðtÞ

Further, the activation functions are described by f ðxðtÞÞ ¼ ½tanhð0:8x1 ðtÞÞ tanhð0:6x2 ðtÞÞ tanhð 0:6x3 ðtÞÞT ; gðxðtÞÞ ¼ ½tanhð0:8x1 ðtÞÞ tanhð0:8x2 ðtÞÞ tanhð0:4x3 ðtÞÞT

and hðxðtÞÞ ¼ ½tanhð0:6x1 ðtÞÞ tanhð0:6x2 ðtÞÞ tanhð 0:4x3 ðtÞÞT . Moreover, we assume that ∑πm ¼ 1 μm ¼ ∑2m ¼ 1 ð0:9Þm , that is, the upper bound of the distributed delay is π ¼2. Given these parameters, it can 0:7 be verified that μ m ¼ 1:71 and Γ ¼ ½0:3 0:5 0:5. It can be easily seen from Assumption (I) that F 1 ¼ diagf0; 0; 0g; G1 ¼ diagf0; 0; 0g; H 1 ¼ diag f0; 0; 0g; F 2 ¼ diagf0:4; 0:3; 0:3g; G2 ¼ diagf0:4; 0:4; 0:2g; H 2 ¼ diag f0:3; 0:3; 0:2g. By solving the LMIs in Theorem 3.1 using Matlab LMI Toolbox, the feasible solutions can be obtained. For instance, if we set i, k¼1,2, by solving the LMIs in Theorem 3.1, the obtained time delay upper bound τM for different values of lower bound τm is given in Table 1. Fig. 1 depicts the possible jumping signal during the simulation result. However, when τm ¼ 2 is given, the upper bound of distributed delay obtained by our Theorem 3.1 is τM ¼ 34, for these values the state responses are depicted in Fig. 2 for the initial condition ½ 0:2 1:1 0:7T , from which we can see that all the state components converge to zero. Now, we proceed to obtain robust stochastic stability of the considered neural network with parameter uncertainties. In order to further verify the above claim, we have presented simulation result for the uncertain system with nonzero initial states. The parameters in Eq. (4) are supposed to be N 1ik ¼ diagf0:04; 0:02; 0:01g ; N 2ik ¼ diagf0:03; 0:04; 0:05g; N 3ik ¼ diagf0:02; 0:03; 0:03g ; N 4ik ¼ diagf0:03; 0:04; 0:05g, Eik ¼ 0:4I; F ik ðtÞ ¼ diagf sin ðtÞ; cos ðtÞ; sin ð tÞg. Now, if we take i, k¼ 1,2, solving the LMIs in Theorem 3.2 by using Matlab LMI Toolbox, the obtained time delay upper bounds τM for various values of lower bound τm are presented in Table 2. Suppose that the upper bound of the interval for the delay is τM ¼ 24, the lower bound of the interval for the delay is τm ¼ 2, and for these values, the state responses for the initial value xð0Þ ¼ ½ 0:2 1:1 0:7T is given in Fig. 3. The simulation result reveals that the state vector x(t) converging to the equilibrium point zero. From Fig. 3, we can observe that the proposed condition guarantees robust stochastic stability of uncertain FMJDNNs for any admissible parameter uncertainty.

Example 4.2. Consider the discrete time neural network (25) with the following parameters: 0:8 0 0:001 0 0:1 0:01 A¼ ; B¼ ; C¼ ; 0 0:7 0 0:005 0:2 0:1

The activation functions are assumed to be f i ðsÞ ¼ g i ðsÞ ¼ 0:5ðjs þ1j js 1jÞ. Obviously, F 1 ¼ G1 ¼ 1; F 1þ ¼ G1þ ¼ 1. By solving the LMI in Corollary 3.4, the calculated maximum upper bound delay τM for different values of τm is presented in Table 3 which is compared with the results obtained in [11,12]. It is concluded from Table 3 that our results are less conservative than the results obtained in [11,12]. Example 4.3. For the convenience of comparison, consider a delayed uncertain discrete-time recurrent neural network in (25) with parameters given by [15] 2 3 2 3 0:4 0 0 0:3 0:1 0:2 6 0 0:5 0 7 6 0 0:3 0:2 7 A¼4 5; B ¼ 4 5; 0 0 0:4 0:1 0:1 0:2 3 2 0:2 0:1 0:1 6 0:2 0:3 0:1 7 C¼4 5; 0:1 0:2 0:3 2

0:1 6 E ¼ N1 ¼ N2 ¼ N3 ¼ 4 0 0

0 0:1 0

3 0 0 7 5: 0:1

The activation functions are described by f 1 ðsÞ ¼ tanhð0:2sÞ; f 2 ðsÞ ¼ tanhð0:4sÞ; f 3 ðsÞ ¼ tanhð0:2sÞ, g 1 ðsÞ ¼ tanhð0:12sÞ; g 2 ðsÞ ¼ tanhð0:2sÞ; g 3 ðsÞ ¼ tanhð0:4sÞ. Therefore, we can obtain 2 3 2 3 0:1 0 0 0:06 0 0 6 0 0:2 0 7 6 0 0:1 0 7 F 1 ¼ G1 ¼ 0; F 2 ¼ 4 5; G2 ¼ 4 5: 0 0 0:1 0 0 0:2 It can be verified that the LMI (27) is feasible. For different values of τm, Table 4 gives out the allowable upper bound τM of the timevarying delay. It is clear that, the delay-dependent stability result proposed in Corollary 3.5 provides less conservative results than those obtained in [11,12,15,23,32].

5. Conclusion In this paper, the problem of robust stochastic stability of uncertain FMJDNNs with various activation functions and mixed time delays has been studied. By employing a novel Lyapunov–Krasovskii functional together with the LMI technique, a new set of delay dependent sufficient conditions are established which ensures the stochastic stability of uncertain FMJDNNs. More precisely, the stability conditions are expressed in terms of the solutions to a set of LMIs, which can be solved very effectively by using standard LMI tool box. Finally, a numerical example has been presented to demonstrate the effectiveness and less conservativeness of the proposed theoretical result. However, in the stability criteria, one of the main issues is how to reduce the possible conservatism induced by the introduction of the Lyapunov functional when dealing with time varying delays. Further, a large number of decision variables in linear matrix inequality make the stability criteria more complex, since the computational burden is large and obtaining the solution by solving concerned LMI is much time consuming. In this connection, future work will focus on improving the existing techniques or finding new methods to reduce conservatism and computational complexity. Further, the problem of



control issues such as H 1 and reliable control for discrete-time stochastic Takagi–Sugeno (TS) fuzzy neural network with Markovian jumping is an untreated topic. These issues will also be investigated in future. Acknowledgments The research of J.H. Park was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10005201). References [1] Boyd S, Ghoui LE, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM; 1994. [2] Kwon OM, Lee SM, Park JH, Cha EJ. New approaches on stability criteria for neural networks with interval time-varying delays. Appl Math Comput 2012;218:9953–64. [3] Park MJ, Kwon OM, Park JH, Lee SM, Cha EJ. Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay. J Frankl Inst 2012;349:1699–720. [4] Balasubramaniam P, Rakkiyappan R, Sathy R. Delay dependent stability results for fuzzy BAM neural networks with Markovian jumping parameters. Expert Syst Appl 2011;38:121–30. [5] Balasubramaniam P, Vembarasan V, Rakkiyappan R. Delay-dependent robust asymptotic state estimation of Takagi–Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays. Expert Syst Appl 2012;39:472–81. [6] Lakshmanan S, Park JH, Jung HY, Balasubramaniam P. Design of state estimator for neural networks with leakage, discrete and distributed delays. Appl Math Comput 2012;218:11297–310. [7] Lee SM, Kwon OM, Park JuH. A new approach to stability analysis of neural networks with time-varying delay via novel Lyapunov–Krasovskii function. Chin Phys B 2010;19:050507. [8] Li H, Wang C, Shi P, Gao H. New passivity results for uncertain discrete-time stochastic neural networks with mixed time delays. Neurocomputing 2010;73: 3291–9. [9] Liu H, Zhao L, Zhang Z, Ou Y. Stochastic stability of Markovian jumping Hopfield neural networks with constant and distributed delays. Neurocomputing 2009;72:3669–74. [10] Liu Y, Wang Z, Liu X. State estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delays. Phys Lett A 2008;372: 7147–55. [11] Liu Y, Wang Z, Liu X. Robust stability of discrete-time stochastic neural networks with time-varying delays. Neurocomputing 2008;71:823–33. [12] Liu Y, Wang Z, Serrano A, Liu X. Discrete-time recurrent neural networks with time-varying delays: exponential stability analysis. Phys Lett A 2007;362:480–8. [13] Liu Y, Wang Z, Liu X. Asymptotic stability for neural networks with mixed time-delays: the discrete-time case. Neural Netw 2009;22:67–74.

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