Robust Stochastic Stability of Discrete-Time Markovian Jump Neural Networks with Leakage Delay

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Robust Stochastic Stability of Discrete-Time

Markovian Jump Neural Networks with Leakage Delay

Mathiyalagan Kalidass^a, Hongye Su^a, and Sakthivel Rathinasamy^b,c

a National Laboratory of Industrial Control Technology, Institute of Cyber-System and Control, Zhejiang University, Yuquan Campus, Hangzhou 310027, China

b Department of Mathematics, Sungkyunkwan University, Suwon-440 746, South Korea

c Department of Mathematics, Sri Ramakrishna Institute of Technology, Coimbatore-641 010, India

Reprint requests to S. R; E-mail:krsakthivel@yahoo.com

Z. Naturforsch.69a,70 – 80 (2014) / DOI: 10.5560/ZNA.2013-0078

Received May 17, 2013 / revised September 18, 2013 / published online December 18, 2013 This paper presents a robust analysis approach to stochastic stability of the uncertain Markovian jumping discrete-time neural networks (MJDNNs) with time delay in the leakage term. By choos- ing an appropriate Lyapunov functional and using free weighting matrix technique, a set of delay dependent stability criteria are derived. The stability results are delay dependent, which depend on not only the upper bounds of time delays but also their lower bounds. The obtained stability criteria are established in terms of linear matrix inequalities (LMIs) which can be effectively solved by some standard numerical packages. Finally, some illustrative numerical examples with simulation results are provided to demonstrate applicability of the obtained results. It is shown that even if there is no leakage delay, the obtained results are less restrictive than in some recent works.

Key words:Discrete-Time Neural Networks; Stochastic Stability; Leakage Delays; Linear Matrix Inequality; Markovian Jump Systems.

1. Introduction

In the past two decades, neural networks have received considerable attention and have been success- fully applied in many areas such as image processing, pattern recognition, associative memories, signal processing, optimization solvers, and so on [1–3]. In particular, these applications heavily depend on the stability of the equilibrium of neural networks. More precisely, the stability analysis is a necessary step for the applications of neural networks. In practice, the discrete-time neural networks become more important than the continuous-time counterparts when im- plementing the neural networks in a digital way. Some important results related to the stability issue have been reported for discrete-time neural networks in [4–12].

On the other hand, both in biological and artificial neural networks, the interactions between neurons are gen- erally asynchronous which inevitably result in time delays. In electronic implementations of neural networks, the delays are usually time-varying due to the

finite switching speed of amplifiers. It is known that the existence of time-delay is often a source of in- stability, oscillation, and poor performance in neural networks [13]. Therefore, it is of both theoretical and practical importance to study the problem of stability analysis of discrete-time neural networks with time-varying delays. Moreover, neural networks with Markovian jumping parameters are of great signifi- cance in modelling a class of neural networks with finite network modes. Many interesting results about the stability of time-delayed neural networks with Marko- vian jumping parameters have been proposed via the linear matrix inequality (LMI) approach [14–19].

Moreover, in studying the applications of neural networks, the connection weights of the neurons depend on certain resistance and capacitance values which in- clude uncertainties. Thus, the stability problem of uncertain discrete-time neural networks has got much attention, and many interesting results have been proposed to guarantee the asymptotic, exponential and stochastic stability analysis of uncertain neural net-

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works with various types of time delays [20–22].

However, the conditions imposed on the neural network are determined by the characteristics of activation functions as well as network parameters. Thus, the generalization of activation functions will provide a wider scope for neural network design and applications [23].

Further, the time delay in the leakage term has also a great impact on the dynamical behaviour of neural networks. Thus, the stability analysis of neural networks with time delay in leakage also has primary sig- nificance in the research of neural networks. Although the stability analysis of neural networks has received much attention, the stability of stochastic neural networks with time delay in leakage has not been fully investigated. So far, few results have been reported for the stability analysis of neural networks with time delay in leakage (or “forgetting”) term [24–27]. In [28], the delay dependent sufficient conditions have been derived for the existence of a unique equilibrium as well as its asymptotic and exponential stability conditions for the bidirectional associative memory (BAM) neural networks with constant delay in the leakage term and pointed out that time delay in the stabilizing neg- ative feedback term could destabilize a system. Re- cently, Balasubramaniam et al. [29] studied the delay- dependent passivity analysis for Markovian jumping neural networks of neutral type with time delay in the leakage term and distributed delays. It is shown via simulation results that the neural network is stable when there is no leakage delay and it becomes unsta- ble when a small amount of leakage delay occurs. This indicates that the stability of neural networks may be affected due to the sensitiveness of the time delay in the leakage term. All the above mentioned results are derived for the continuous-time neural networks, to the best of authors knowledge, no result has been reported for the discrete-time case with time delay in the leakage term.

Motivated by the above discussion, in this paper we consider the stochastic stability analysis for Markovian jumping discrete-time neural networks (MJDNNs) with time delays in the leakage term. By employ- ing a Lyapunov–Krasovski functional (LKF) and LMI technique, some novel delay-dependent conditions are obtained to ensure the stochastic stability for a class of MJDNNs in the presence of time delays in the leakage term. Further, the result is extended to derive the stability conditions for uncertain discrete-time neu-

ral networks in which the parameter uncertainties are assumed to be time-varying but norm bounded. The derived conditions are established in terms of LMIs which can be easily calculated by MATLAB LMI tool- box. In particular, it is shown that in the absence of a leakage delay, the obtained results are less conserva- tive than some recent works. Finally, some numerical examples and their simulations are given to show the effectiveness of the obtained results.

Notations:Throughout this paper, unless otherwise specified, the superscripts ‘T’ and ‘−1’ stand for matrix transposition and matrix inverse respectively;R^n×n denotes then×n-dimensional Euclidean space; the no- tationP>0 means thatPis real, symmetric, and positive definite; diag{·} denotes the block-diagonal matrix; we use an asterisk (∗) to represent a term that is induced by symmetry. Matrices which are not ex- plicitly stated are assumed to be compatible for matrix multiplications. Zdenotes the set including zero and positive integers. Moreover, let(Ω,F,{F_t}t≥0,P)be a complete probability space with a filtration{F_t}t≥0

satisfying the usual conditions (i.e., the filtration con- tains allP-null sets and is right continuous).E[·]stands for the mathematical expectation with respect to the given probability measureP.

2. Problem Formulation and Preliminaries

Consider the following discrete-time neural network with leakage delay and Markovian jumping parameters;

x(k+1) = (A(r_k) +∆A(k,r_k))x(k−σ) + (B(r_k) +∆B(k,r_k))f(x(k)) + (C(r_k)

+∆C(k,r_k))f(x(k−τ(k))),

x(k) =φ(k) for everyk∈[−τ,0], τ=max[τ_M,σ], (1)

where x(k) ∈ Rⁿ is the state vector; A(r_k) = diag{a₁(r_k),a2(r_k), . . . ,an(r_k)} is the positive diagonal matrix that represents the self-feedback term with |a(r_k)| <1; B(r_k) = [b(r_k)]_n×n and C(r_k) = [c(r_k)]_n×n are known real constant matrices with appropriate dimensions; f(x(k)) = [f₁(x₁(k)),f₂(x₂(k)), . . . ,fn(x_n(k))]^Tdenotes the neuron activation function.

The positive integersσandτ(k)denote the leakage delay and time-varying discrete delay, respectively,τ(k) satisfiesτm≤τ(k)≤τMwhereτmandτMare positive integers.

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72 M. Kalidass et al.·Robust Stochastic Stability of Discrete-Time Markovian Jump Neural Networks Let S={1,2, . . . ,N} and {r_k,k∈Z} be a homo-

geneous, finite-state Markovian process with right continuous trajectories on the probability space (Ω,F,{F_k}_k≥0,P)with generatorΠ = [π_{i j}]n×n, i,j∈ S given by Pr(r_k+∆k= j/r_k=i) =π_{i j}, where π_{i j} is the transition rate from modei to mode j, satisfying πi j≥0 fori,j∈Sand∑^N_j=1πi j=1.

Further, r_k=i,i∈S,A_i=A(r_k),B_i=B(r_k),C_i= C(r_k),and we assume that the parameter uncertainties

∆A_i=∆A(k,r_k),∆B_i=∆B(k,r_k), and∆C_i=∆C(k,r_k) are time-varying and described by

[∆A_i ∆B_i ∆C_i] =E_iF_i(k) [N_1i N_2i N_3i], (2) whereN_1i,N_2i,N_3i, andE_iare known constant matrices of appropriate dimensions and Fi(k)is an known time-varying matrix with Lebegue measurable ele- ments bounded byF_i^T(k)F_i(k)≤I.

Assumption 1 ([30]). Fors∈ {1,2, . . . ,n}, the neu- ron activation functions f_s(·)in (1) are continuous and bounded, satisfying

F_s⁻≤ f_s(x₁)−f_s(x₂)

x₁−x₂ ≤F_s⁺ ∀x₁,x₂∈R, x₁6=x₂, whereF_s⁻andF_s⁺are constants.

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

Definition 1. For every initial condition (φ,r0), the MJDNNs (1) is said to be stochastically stable if satisfying the following condition:

E

"

∞

∑

k=0

kx_k(φ,r₀)k²

#

<∞,

where x_k(φ,r₀) denotes the solution of (1) at time k under the initial conditionsφandr₀.

Lemma 1([31]). For any symmetric constant matrix Z∈R^n×n, Z≥0,two scalarsτmandτMsatisfyingτm<

τM, and a vector valued function η(k) =x(k+1)− x(k),(k∈Z⁺), we have

−

k−τ_m−1 s=k−τ

∑

_M

η^T(s)Zη(s)≤−1 τ₂

k−τ_m−1 s=k−τ

∑

_M

η^T(s)Z

k−τ_m−1 s=k−τ

∑

_M

η(s), whereτ2=τM−τm.

Lemma 2([22]). For any vector x,y∈Rⁿ, matrices A,P,D,E, and F are real matrices of appropriate di- mensions with P>0,F^TF≤I, and scalerε>0, the following inequalities hold:

(i)2x^TDFEy≤ε⁻¹x^TDD^Tx+εy^TE^TEy ,

(ii) If P−εDD^T > 0, then (A+DFE)^TP⁻¹(A+ DFE)≤A^T(P−εDD^T)⁻¹A+ε⁻¹E^TE .

3. Stability Results

In this section, we derive the conditions for stochastic stability of MJDNNs in the presence of time delay in the leakage term. First, we study the stochastic stability of MJDNN (1) when the uncertainties∆A_i=0,

∆B_i=0, and∆C_i=0. For presentation convenience, we denote

F₁=diag

F₁⁻F₁⁺,F₂⁻F₂⁺, . . . ,F_n⁻F_n⁺ , F₂=diag

F₁⁻+F₁⁺

2 ,F₂⁻+F₂⁺

2 , . . . ,F_n⁻+F_n⁺ 2

.

Theorem 1. Under the Assumption1, the MJDNN (1) without uncertainties is stochastically stable if there exist matrices P_i > 0,Q_p >0,Q₄ >0,R_u>

0,u=1,2,3,4;Z_v>0,v=1,2,3 and diagonal ma- trices D_q>0, q=1,2,and M_l, l=1,2,3,4,5, with appropriate dimensions such that the following LMIs hold for all i,j∈S,i6= j:

Π=





 Π16,16

√ τMM1

√ τ2M2

√ τ2M3

∗ −Z₁ 0 0

∗ ∗ −Z₁−Z2 0

∗ ∗ ∗ −Z₂







<0, (3)

where M_j = [M^T_j1 M^T_j2 0 M^T_j3 0 M^T_j4 0_n,10n]^T, j = 1,2,3,4,5,

P_i=







P_11i P_12i P_13i P_14i

∗ P22i P23i P24i

∗ ∗ P_33i P_34i

∗ ∗ ∗ P_44i





 ,

Q_p=

Q_p1 Q_p2

∗ Qp3

, p=1,2,3,5,π¯=

N

∑

j=1

π_{i j},

Π1,1=Q₁₁+ (τ₂+1)Q₂₁+Q₃₁+Q₅₁−P_11i+τ_m²R₁ +τMR₂+τ2R₃+σR₄−F₁D₁+π¯P₁₁_j

−2M₅₁+2M₁₁−1 σZ₃,

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Π1,2=πP¯ _12j−P_12i+ 1

σZ₃+M₁₂^T +M₅₁A_i−M₅₂^T , Π1,3=πP¯ 13j−P_13i−M41+M31,

Π1,4=M₄₁−M₅₄^T −M₁₁+M₁₃^T +M₂₁−M₃₁, Π_1,5=πP¯ _14j−P_14i−M₂₁, Π_1,6=πP¯ ₁₁_j+Q₁₂

+ (τ₂+1)Q₂₂+Q₃₂−M₅₁−M₅₃^T +M₁₄^T, Π1,7=πP¯ _12j, Π1,8=πP¯ ₁₃_j, Π1,9=πP¯ ₁₄_j,

Π1,11=F₂D₁+M₅₁B_i, Π1,12=M₅₁C_i, Π1,13=M₄₁, Π1,14=−M₄₁, Π1,15=M₄₁,

Π2,2=πP¯ _22j−P_22i−Q₅₁−1

σZ₃+M₅₂A_i+A^T_iM₅₂^T , Π2,3=πP¯ 23j−P_23i−M42+M32,

Π_2,4=M₄₂−M₁₂−A^T_iM₅₄^T +M₂₂−M₃₂, Π2,5=πP¯ _24j−P_24i−M₂₂,

Π2,6=πP¯ _12j^T +A^T_iM₅₁^T −M₅₂,

Π2,7=πP¯ _22j, Π2,8=π¯P₂₃_j, Π2,9=π¯P₂₄_j, Π2,11=M₅₂B_i, Π2,12=M₅₂C_i, Π2,13=M₄₂, Π2,14=−M₄₂, Π2,15=M₄₂,

Π3,3=πP¯ 33j−P_33i−Q11, Π3,4=M₃₃^T−M₄₃^T , Π_3,5=πP¯ _34j−P_34i, Π_3,6=π¯P_13j^T −M₄₄^T +M₃₄^T , Π3,7=πP¯ _23j^T , Π3,8=π¯P₃₃_j−Q₁₂, Π3,9=πP¯ ₃₄_j, Π4,4=2M₄₃−2M₁₃+2M₂₃−2M₃₃−Q₂₁−F₁D₂, Π4,5=−M₂₃, Π4,6=M₄₄^T −M₅₄−M₁₄^T+M₂₄^T −M₃₄^T, Π4,9=−Q₂₂, Π4,11=M₅₄B_i, Π4,12=F₂D₂+M₅₄C_i, Π4,13=M₄₃, Π4,14=−M₄₃, Π4,15=M₄₃,

Π5,5=πP¯ 44j−P_44i−Q31, Π5,6=π¯P₁₄^T_j−M₂₄^T, Π5,7=πP¯ _24j^T , Π5,8=π¯P₃₄^T_j, Π5,9=π¯P44j, Π5,10=−Q₃₂, Π6,6=π¯P11j+Q13+ (τ₂+1)Q₂₃

+Q33+Q₅₃+τ2Z1+τMZ2+σZ3−2M₅₃, Π6,7=πP¯ 12j, Π6,8=π¯P13j, Π6,9=π¯P14j, Π6,11=M₅₃B_i, Π6,12=M₅₃C_i,Π6,13=M44, Π_6,14=−M₄₄, Π_6,15=M₄₄, Π7,7=πP¯ _22j−Q₅₃, Π7,8=πP¯ _23j, Π7,9=π¯P₂₄_j, Π8,8=π¯P₃₃_j−Q₁₃, Π_8,9=πP¯ _34j, Π_9,9=π¯P₄₄_j−Q₂₃, Π_10,10=−Q₃₃, Π_11,11= (τ₂+1)Q₄−D₁, Π_12,12=−Q₄−D₂, Π13,13=−R₁,Π14,14=− 1

τM

R₂,

Π_15,15=−1 τ2

R₃, Π_16,16=−1 σR₄, and the remaining terms are zero.

Proof. In order to prove the stochastic stability results, we consider the following LKF:

V(k) =

5 r=1

∑

Vr(k), (4)

where

V1(k) =ξ₁^T(k)P_iξ1(k), V₂(k) =

k−1 s=k−τ(k)

∑

λ^T(s)Q₁λ(s) +

k−1 s=k−τ

∑

_m

λ^T(s)Q₂λ(s)

+

k−1 s=k−τ

∑

_M

λ^T(s)Q₃λ(s) +

k−1 s=k−σ

∑

λ^T(s)Q₅λ(s)

+

k−1 s=k−τ(k)

∑

f^T(x(s))Q₄f(x(s)),

V₃(k) =

−τ_m j=−τ

∑

_M+1

k−1 s=k+j

∑

λ^T(s)Q₁λ(s)

+

−τ_m j=−τ

∑

_M+1

k−1 s=k+j

∑

f^T(x(s))Q₄f(x(s)),

V₄(k) =τm

−1 j=−τ

∑

_m

k−1 s=k+

∑

j

x^T(s)R₁x(s)

+

−1 j=−τ

∑

_M

k−1 s=k+j

∑

x^T(s)R₂x(s) +

−τ_m j=−τ

∑

_M+1

k−1 s=k+j

∑

x^T(s)R₃x(s)

+

−1 j=−σ

∑

k−1 s=k+

∑

j

x^T(s)R₄x(s),

V₅(k) =

−τ_m−1 j=−τ

∑

_M

k−1 s=k+j

∑

η^T(s)Z₁η(s)

+

−1 j=−τ

∑

_M

k−1 s=k+j

∑

η^T(s)Z₂η(s) +

−1 j=−σ

∑

k−1 s=k+j

∑

η^T(s)Z₃η(s), and λ(k) = [x^T(k) η^T(k)]^T, η(k) = x(k+1)− x(k), ξ1(k) = [x^T(k) x^T(k−σ) x^T(k−τm) x^T(k− τ_M)]^T.Calculating the difference ofV_r(k)by defining

∆Vr(k) =Vr(k+1)−Vr(k)along the trajectory of (1), we get

∆V₁(k) =ξ₁^T(k+1)

N j=1

∑

πi jP_jξ1(k+1)−ξ₁^T(k)P_iξ1(k)

=







x(k) +η(k)

x(k−σ) +η(k−σ) x(k−τm) +η(k−τ_m) x(k−τ_M) +η(k−τ_M)







T

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74 M. Kalidass et al.·Robust Stochastic Stability of Discrete-Time Markovian Jump Neural Networks

·πP¯ j







x(k) +η(k)

x(k−σ) +η(k−σ) x(k−τ_m) +η(k−τm) x(k−τ_M) +η(k−τ_M)







−





 x(k) x(k−σ) x(k−τm) x(k−τM)







T

P_i





 x(k) x(k−σ) x(k−τ_m) x(k−τ_M)







, (5)

∆V₂(k)≤λ^T(k)(Q₁+Q₂+Q₃+Q₅)λ(k)

−λ^T(k−τm)Q₁λ(k−τm)

−λ^T(k−τ(k))Q₂λ(k−τ(k))

−λ^T(k−τM)Q₃λ(k−τ_M)−λ^T(k−σ)Q₅λ(k−σ) +

k−τ_m s=k+1−τ

∑

_M

λ^T(s)Q₂λ(s) +f^T(x(k))Q₄f(x(k))

−f^T(x(k−τ(k)))Q₄f(x(k−τ(k))) +

k−τ_m s=k+1−τ

∑

_M

f^T(x(s))Q₄f(x(s))

≤x^T(k)(Q₁₁+Q₂₁+Q₃₁+Q₅₁)x(k) +2x^T(k)(Q₁₂+Q₂₂+Q₃₂+Q₅₂)η(k) +η^T(k)(Q₁₃+Q23+Q33+Q₅₃)η(k)

−x^T(k−τ_m)Q₁₁x(k−τ_m)−2x^T(k−τ_m)Q₁₂η(k−τ_m)

−η^T(k−τ_m)Q₁₃η(k−τm)

−x^T(k−τ(k))Q₂₁x(k−τ(k))

−2x^T(k−τ(k))Q₂₂η(k−τ(k))

−η^T(k−τ(k))Q₂₃η(k−τ(k))

−x^T(k−τM)Q₃₁x(k−τ_M)

−2x^T(k−τ_M)Q₃₂η(k−τ_M)

−η^T(k−τ_M)Q₃₃η(k−τ_M)−x^T(k−σ)Q₅₁x(k−σ)

−2x^T(k−σ)Q₅₂η(k−σ)−η^T(k−σ)Q₅₃η(k−σ) +

k−τ_m s=k+1−τ

∑

_M

λ^T(s)Q₂λ(s) +f^T(x(k))Q₄f(x(k))

−f^T(x(k−τ(k)))Q₄f(x(k−τ(k))) +

k−τ_m s=k+1−τ

∑

_M

f^T(x(s))Q₄f(x(s)), (6)

∆V₃(k) =τ2λ^T(k)Q₂λ(k) +τ2f^T(x(k))Q₄f(x(k))

−

k−τ_m s=k+1−τ

∑

_M

λ^T(s)Q₂λ(s)−

k−τ_m s=k+1−τ

∑

_M

f^T(x(s))Q₄f(x(s))

=τ₂(x^T(k)Q₂₁x(k) +2x^T(k)Q₂₂η(k)

+η^T(k)Q₂₃η(k) +f^T(x(k))Q₄f(x(k)))

−

k−τ_m s=k+1−τ

∑

_M

λ^T(s)Q₂λ(s)

−

k−τ_m s=k+1−τ

∑

_M

f^T(x(s))Q₄f(x(s)), (7)

∆V₄(k) =x^T(k) τ_m²R₁+τMR₂+τ2R₃+σR₄ x(k)

−τ_m

k−1 s=k−τ

∑

_m

x^T(s)R₁x(s)−

k−1 s=k−τ

∑

_M

x^T(s)R₂x(s)

−

k−τ_m s=k+1−τ

∑

_M

x^T(s)R₃x(s)−

k−1 s=k−σ

∑

x^T(s)R₄x(s).

By using Lemma1, we get

∆V4(k)≤x^T(k) τ_m²R1+τMR2+τ2R3+σR4 x(k)

− ( _k−1

s=k−τ

∑

_m

x^T(s)R₁

k−1 s=k−τ

∑

_m

x(s) )

− 1 τM

( _k−1

s=k−τ

∑

_M

x^T(s)R₂

k−1 s=k−τ

∑

_M

x(s) )

− 1 τ₂

( _k−τ

m

s=k+1−τ

∑

_M

x^T(s)R₃

k−τ_m s=k+1−τ

∑

_M

x(s) )

−1 σ

( _k−1

s=k−σ

∑

x^T(s)R₄

k−1 s=k−σ

∑

x(s) )

, (8)

∆V₅(k) =η^T(k) τ2Z₁+τMZ₂+σZ₃ η(k)

−

k−τ_m−1 s=k+1−τ

∑

_M

η^T(s)Z₁η(s)−

k−1 s=k−τ

∑

_M

η^T(s)Z₂η(s)

−

k−1 s=k−σ

∑

η^T(s)Z₃η(s),

−

k−1 s=k−σ

∑

η^T(s)Z₃η(s)≤ −1 σ

k−1 s=k−σ

∑

η^T(s)Z₃

k−1 s=k−σ

∑

η(s)

≤ −1 σ h

x(k)−x(k−σ)iT

Z3

h

x(k)−x(k−σ)i

. (9)

On the other hand, for any matricesMi, i=1,2,3,4, we have the following [32]:

2ξ^T(k)M₁

·

"

x(k)−x(k−τ(k))−

k−1

∑

s=k−τ(k)

x(s)

#

=0, (10)

(6)

2ξ^T(k)M₂

·

"

x(k−τ(k))−x(k−τM)−

k−τ(k)−1 s=k−τ

∑

_M

x(s)

#

=0, (11) 2ξ^T(k)M₃

·

"

x(k−τm)−x(k−τ(k))−

k−τ_m−1

∑

s=k−τ(k)

x(s)

#

=0, (12)

2ξ^T(k)M₄

" _k−τ

m

s=k+1−τ

∑

_M

x(s) +

k−1 s=k−τ

∑

_m

x(s)

−

k−1 s=k−τ

∑

_M

x(s)−x(k−τm) +x(k−τM)

#

=0,

(13)

τ_Mξ^T(k)M₁Z₁⁻¹M₁^Tξ(k)

−

k−1

∑

s=k−τ(k)

ξ^T(s)M₁Z⁻¹₁ M₁^Tξ(s) =0, (14) τ2ξ^T(k)M₂(Z₁+Z₂)⁻¹M₂^Tξ(k)

−

k−τ(k)−1 s=k−τ

∑

_M

ξ^T(s)M₂(Z₁+Z₂)⁻¹M₂^Tξ(s) =0, (15) τ2ξ^T(k)M₃Z₂⁻¹M₃^Tξ(k)

−

k−τ_m−1

∑

s=k−τ(k)

ξ^T(s)M₃Z⁻¹₂ M₃^Tξ(s) =0. (16) In addition, we have,η(k) =x(k+1)−x(k) =A_ix(k− σ) +B_if(x(k)) +C_if(x(k−τ(k)))−x(k).Then for any matrixM₅,we have

ζ^T(k)M₅

A_ix(k−σ)−x(k) +B_if(x(k)) +C_if(x(k−τ(k)))−η(k)

=0, (17)

where ζ(k) =

"

x^T(k)x^T(k−σ)x^T(k−τm)x^T(k−τ(k))

·x^T(k−τ_M)η^T(k)η^T(k−σ)η^T(k−τ_m)

·η^T(k−τ(k))η^T(k−τM)f^T(x(k)) f^T(x(k−τ(k))

·

k−1 s=k−τ

∑

_m

x^T(s)

k−1 s=k−τ

∑

_M

x^T(s)

k−τ_m s=k+1−τ

∑

_M

x^T(s)

k−1 s=k−σ

∑

x(s)iT

.

From Assumption1, for anys=1,2, . . . ,n,we have (f_s(x_s(k))−F_s⁻x_s(k))(f_s(x_s(k))−F_s⁺x_s(k))≤0, which is equivalent to

α(k)^T

"

F_s⁻F_s⁺e_se^T_s −^F^s⁻^+F₂ ^s⁺e_se^T_s

−^F^s⁻^+F₂ ^s⁺e_se^T_s e_se^T_s

#

α(k)≤0,

whereα^T(k) =

x^T(k) f^T(x(k))

, and es denotes the unit column vector having element 1 on its sth row and zeros elsewhere. LetD₁=diag{d₁₁,d₁₂, . . . ,d_1n}. Then

n s=1

∑

d1sα^T(k)

·

"

F_s⁻F_s⁺e_se^T_s −^F^s⁻^+F₂ ^s⁺e_se^T_s

−^F^s⁻^+F₂ ^s⁺e_se^T_s e_se^T_s

#

α(k)≤0, i.e., α^T(k)

F₁D₁ −F₂D₁

−F₂D1 D1

α(k)≤0.

(18)

Similarly forα₁^T(k) =

x^T(k−τ(k))f^T(x(k−τ(k))) , D₂=diag{d₂₁,d₂₂, . . . ,d_2n}, one can gets

α₁^T(k)

F1D2 −F₂D2

−F₂D₂ D₂

α1(k)≤0. (19) Combining (4) – (19) and taking mathematical expectation on both sides, we get

E[∆V(k)]≤E

"

ζ^T(k) (

Π+τMM₁Z₁⁻¹M₁^T +τ2M₂(Z₁+Z₂)⁻¹M₂^T+τ₂M₃Z₂⁻¹M₃^T

−

k−1

∑

s=k−τ(k)

M₁^Tξ(k) +Z₁η(s)T

Z₁⁻¹

M₁^Tξ(k) +Z₁η(s)

−

k−τ(k)−1 s=k−τ

∑

_M

M₂^Tξ(k) + (Z₁+Z₂)η(s)T

(Z₁+Z₂)⁻¹

·

M₂^Tξ(k) + (Z₁+Z₂)η(s)

−

k−τ_m−1

∑

s=k−τ(k)

M₃^Tξ(k) +Z2η(s)T

Z₂⁻¹

·

M₃^Tξ(k) +Z₂η(s) )

ζ(k)

#

. (20)

Letλ0=min{Ψ,i∈L}, thenλ0>0 due toΨ. Finally from (20), we obtain that for anyk≥0

E[V(X_k+1,r_k+1=j)/x_k,r_k=i]

≤V(x_k,r_k=i)−λ0x^T(k)x(k). (21) Settingk=0 andk=1 in (21) yields

E[V(X₁,r₁)/x₀,r₀]≤V(x₀,r₀)−λ0x^T(0)x(0) (22) and

E[V(X₂,r₂/x₁,r₁)]≤V(x₁,r₁)

−λ0x^T(1)x(1). (23)

(7)

76 M. Kalidass et al.·Robust Stochastic Stability of Discrete-Time Markovian Jump Neural Networks Taking expectation E[·/X₀,r0] on both sides of (23),

with the aid of (22), leads to E[V(X₂,r₂/x₀,r₀]≤V(x₀,r₀)

−λ₀

1 l=0

∑

E[x^T(l)x(l)/x₀,r₀].

Then one can continue the iterative procedure (21) to obtain

E[V(X_T+1,rT+1/x₀,r₀]≤V(x₀,r₀)

−λ₀

T l=0

∑

E[x^T(l)x(l)/x₀,r₀] which implies that

T l=0

∑

E[x^T(l)x(l)/x₀,r0]≤ 1

λ₀V(x₀,r0)<∞.

Hence the MJDNN (1) without uncertainties is stochastically stable in the sense of Definition1, and thus the proof is completed.

In the next theorem, we consider the robust stochastic stability of uncertain MJDNNs that exhibit paramet- ric uncertainty. In particular, we derive a sufficient condition in terms of LMIs for the robust stochastic stability of uncertain MJDNN (1) with time-varying delays by extending the results of the previous theorem.

Theorem 2. Under the Assumption1, the MJDNN (1) is robustly stochastically stable, if there exist matrices P_i>0, Q_p>0, p=1,2,3,5; Q₄>0, R_u>0, u= 1,2,3,4; Zv>0, v=1,2,3; and diagonal matrices D_q>0, q=1,2,and M_l, l=1,2,3,4,5,with appro- priate dimensions and positive scalarsε_isuch that the following LMIs hold for i∈S:





Π Θ₁^T εiΘ₂^T

∗ −ε_iI 0

∗ ∗ −ε_iI



<0 (24) Θ₁=

E_i^TM₅₁^T E_i^TM₅₂^T 0E_i^TM₅₃^T 0E_i^TM₅₄^T 0_n,10n , Θ2= [0N1i0_n,8nN2iN3i0_n,4n], and other parameters are defined as in Theorem1.

Proof. By replacing the matricesA_i, B_i, andC_i with A_i+E_iF_i(t)N_1i, B_i+E_iF_i(t)N_2i, and C_i+E_iF_i(t)N_3i in Theorem1 and applying Schur complement and Lemma2, we can obtain (24).

Next, we give a corollary to show that our main results cover some cases that have been studied in the literature. In the absence of leakage delay the uncertain discrete-time Markovian jumping neural network (1) becomes a standard uncertain MJDNN as follows:

x(k+1) = (A_i+∆A_i)x(k) + (B_i+∆B_i)f(x(k)) + (C_i+∆C_i)f(x(k−τ(k))). (25) In the following corollary, we will deduce the robust stability results to the model (25) from Theorem1and 2.

Corollary 1. Under the Assumption1, the MJDNN (25) is robustly stochastically stable if there exist ma- trices P_i>0;Q_p>0,Q₄>0,R_u>0,u=1,2,3;Z_v>

0, v=1,2,and diagonal matrices D_q>0, q=1,2, and M_l, l =1,2,3,4,5, with appropriate dimensions and positive scalars εi such that the following LMIs hold for all i,j∈S, i6= j:





 Θ13,13

√

τMM₁ √

τ2M₂ √

τ2M₃ Θ₁^T εiΘ₂^T

∗ −Z1 0 0 0 0

∗ ∗ −Z1−Z2 0 0 0

∗ ∗ ∗ −Z2 0 0

∗ ∗ ∗ ∗ −εi 0

∗ ∗ ∗ ∗ ∗ −ε_i







<0, (26)

where Θ₁ =

E_i^TM₅₁^T 0E_i^TM₅₂^T 0E_i^TM₅₃^T 0n,8n , Θ2= [N_1i0_n,7nN_2iN_3i0_n,3n],

P_i=





P_11i P_12i P_13i

∗ P_22i P_23i

∗ ∗ P33i



, Q_p=

Q_p1 Q_p2

∗ Q_p3

,

u=1,2,3, π¯=

N

∑

j=1

πi j,

Θ1,1=π¯P11j−P11i+Q₁₁+ (τ₂+1)Q₂₁+Q31+τ_m²R1

+τ_MR₂+τ₂R₃−F₁D₁+2M₅₁(A_i−I) +2M₁₁, Θ1,2=π¯P₁₂_j−P_12i+M₃₁−M₄₁, Θ1,3=M₄₁−M₁₁

+M₁₂^T +M₂₁−M₃₁+M₅₂^T(A_i−I)^T, Θ1,4=π¯P₁₃_j−P_13i−M₂₁,

Θ1,5=π¯P₁₁_j+Q₁₂+ (τ₂+1)Q₂₂+Q₃₂−M₅₁ +M₅₂^T(A_i−I)^T+M₁₃^T,

Θ1,6=π¯P₁₂_j, Θ1,8=π¯P₁₃_j,Θ1,9=F₂D₁+M₅₁B_i, Θ1,10=M₅₁C_i, Θ1,11=M₄₁, Θ1,12=−M₄₁, Θ1,13=M₄₁, Θ2,2=π¯P₂₂_j−P_22i−Q₁₁,