# Exponential Lagrange Stability for Markovian Jump Uncertain Neural Networks with Leakage Delay and Mixed Time-Varying Delays via Impulsive Control

Authors: J. Yogambigai; M. Syed Ali; Quanxin Zhu; Jingwei Cai
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The problem of exponential Lagrange stability analysis of Markovian jump neural networks with leakage delay and mixed time-varying delays is studied in this paper. By utilizing the Lyapunov functional method, employing free-weighting matrix approach and inequality techniques in matrix form, we establish several novel stability criteria such that, for all admissible parameter uncertainties, the suggested neural network is exponentially stable in Lagrange sense. The derived criteria are expressed in terms of linear matrix inequalities (LMIs). A numerical example is provided to manifest the validity of the proposed results.


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## Body

## 1. Introduction

It is known that stability is an important research topic in the mathematical field [1–13]. As a special class of mathematical models, neural networks are similar to the brain synapses link structure; neural networks possess multiple dynamic behaviors [14]. For these reasons, neural frameworks have received considerable attention as a result of their intensive applications in determination of some optimization issue, associative memory, classification of patterns, and other areas [15–18]. Since axonal signal transmission time delays often occur in various neural networks and may also cause undesirable dynamic network behaviors such as oscillation and instability, thus, it is important to study the stability of neural networks. Many practical network systems, such as image processing, communication, fault diagnosis, fixed-point computations, parallel computations, and industrial automation, can be modeled as neural networks (NNs) with time delays [19, 20].Time-varying delays are used to indicate the promoted speed of signals is finite and uncertain in systems [21–28]. Noting that while signal propagation is sometimes instantaneous and can be modeled with discrete delays, it may also be distributed during a certain time period so that distributed delays are incorporated into the model [29]. Thus, the issue of stability analysis for neural framework with time-varying delays is investigated via LMI technique by many authors [30–34]. Very recently, a leakage delay, which is the time delay in leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, has been applied to use in studying the problem of stability for neural networks [35–37]. In fact, the leakage term has also a great impact on the dynamical behavior of neural networks. The authors in [38] investigated the stability problem of neural networks with leakage delays and impulses: a piecewise delay method.When one models real nervous systems, random disturbance and parameter uncertainties are a unit ineluctable to be thought about. As a result, the connection weights of the neuron depend upon bound resistance and capacitance values that embody uncertainties [34]. Therefore, it is of great importance to consider the random effect on the stability of neural frameworks with parameter uncertainties [39–41]. On the other hand, Markovian jump neural networks can be regarded as a special class of hybrid systems, which can model dynamic systems whose structures are subject to random abrupt parameter changes resulting from component or interconnection failures, sudden environment changes, changing subsystem interconnections, and so forth [42–45]. A neural network has limited modes, which may jump from one to another at various periods. Thus, neural networks with such a jumping character may transform to each other, according to a Markovian chain [44, 46].Applications of this kind of neural networks can be found in modeling production systems, economic systems, and other practical systems. But in practice applications, for example, the state of electronic networks is often subject to instantaneous changes at certain instants, this is impulsive phenomenon. Impulsive neural network model belongs to new category of dynamical systems, which is neither continuous nor discrete ones. Examples of impulsive phenomena can also be found in other fields of automatic control system, artificial intelligence, robotics, etc. As we know, neural networks could be stabilized or destabilized by impulsive phenomena [44, 47]. The presence of impulse means that the state trajectory does not preserve the basic properties. One of the most desirable properties of neural framework is the Lyapunov global stability. From a dynamical system point of view, globally stable networks in Lyapunov sense are monostable systems, which have a unique equilibrium attracting all trajectories asymptotically.However, the equilibrium sometimes does not exist in many real physical systems. When a neural network is used as an associative memory storage or for pattern recognition, the existence of many equilibriums is also necessary. On the other hand, monostable neural networks have been found to be computationally restrictive and multistable dynamics are essential to deal with important neural computations desired. For example, only the neuron with the strongest input should remain active in a winner-take-all network depending on the external input [48, 49]. This is possible only if there are multiple equilibria with some being unstable. As we know, Lagrange stability is one of the most important properties in multistability analysis of the total system which does not require the information of equilibriums [50–54]. The boundedness of solutions and the existence of globally attractive sets lead to a total system concept of stability: (asymptotic) Lagrange stability [55, 56]. Hence we concentrate on studying Lagrange stability of the neural networks with mixed time-varying delays and leakage delay in the presence of uncertainties.Motivated by the above discussions, it is necessary to study the stability properties in Lagrange sense for impulsive neural networks with parameter uncertainties. To the best of our knowledge, there are no published papers on the Lagrange stability analysis of uncertain Markovian jump neural networks with mixed time-varying delays and leakage delay. To fill this gap, we try to perform an exponential stability in Lagrange sense analysis of conceded neural frameworks with mixed time-varying delays and parameter uncertainties. By using novel Lyapunov-Krasovskii functionals together with the zero function, we establish the Lagrange stability of the neural networks with parameter uncertainties. In particular, these sufficient conditions are communicated as far as LMIs that can be solved numerically. Finally, a numerical illustration is given to demonstrate the adequacy of the obtained results.Notation. R n denotes the n-dimensional Euclidean space and Rm×n is the set of all m×n real matrices. The superscript ′T′ denotes matrix transposition and A≥B (respectively, A<B) where A and B are symmetric matrices (respectively, positive definite); · denotes the Euclidean norm in Rn. If Q is a square matrix, denoted by λmax⁡(Q) (respectively, λmin⁡(Q)) it means the largest (respectively, smallest) eigenvalue of Q. Moreover, let (Ω,F,Ftt≥0,P) be a complete probability space with a filtration Ftt≥0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right-continuous). The asterisk ∗ in a symmetric matrix is used to denote term that is induced by symmetry; diag(·) stands for the diagonal matrix; I and 0 denote the identity matrix and zero matrix of appropriate dimensions, respectively.


## 2. Problem Formulation and Preliminaries

Given a complete probability space{Ω,F,Ftt≥0,P} with a natural filtration Ftt≥0 satisfying the usual conditions, where Ω is the sample space, F is the algebra of events, and P is the probability measure defined on F, let rt,t≥0 be a right-continuous Markovian chain on the probability space (Ω,F,Ftt≥0,P) taking values in the finite space S = 1,2,…,m with generator Φ=ϕkjm×m(k,j∈S) given by(1)Prt+Δ=j∣rt=k=ϕkjΔ+0Δ,ifk≠j,1+ϕkkΔ+0Δ,ifk=j.Here Δ>0 and ϕkj≥0 is the transition rate from k to j if j≠k, while ϕkk=-∑j≠kϕkj.We consider the Markovian jump uncertain neural networks with both time-varying discrete delays and distributed delays as well as leakage delay and impulsive perturbations:(2)η˙t=-Brtηt-ρ+D0rtfηt+D1rtfηt-τt+D2rt∫t-dttfηsds+J,t≠thΔηth=-Hhrtηth--Brt∫th-ρthηsds,t=th,h∈Z+where η(t)∈Rn stands for the neuron state vector of the system; f·∈Rn is the nonlinear activation function, respectively, J=J1,J2,…,JnT∈Rn is an external input vector; B(rt)=diag(b1(rt),b2(rt),…,bn(rt)) is self-feedback parameter matrix of the neurons, where bi(rt)>0; D0(rt)=d0rtn×n∈Rn×n,D1(rt)=[d1(rt)]n×n∈Rn×n,D2(rt)=d2rtn×n∈Rn×n are connection weight matrices. Also, Hh(rt) is the impulses gain matrix at the moment of time th. The discrete set th satisfies 0=t0<t1<⋯<th<⋯,limh→∞th.  η(th-) denotes the left-hand limits at th. Similarly, η(th+) denotes the right-hand limits at th. ρ is the leakage delay, τ(t) is the discrete time-varying delay, and d(t) is the distributed time-varying delay.The initial condition associated with model (2) is η(s)=ψ(s), where s∈[-β,0], and ψ is differentiable on [-β,0], and β=max⁡ρ,τ,d.The time-varying delaysτ(t) and d(t) satisfy (3)0≤τt≤τ,0≤dt≤d,where τ and d are constants.HereB(rt)=B(rt)+ΔB(rt),D0(rt)=D0(rt)+ΔD0(rt),D1(rt)=D1(rt)+ΔD1(rt) and D2(rt)=D2(rt)+ΔD2(rt).  ΔB(rt),ΔD0(rt),ΔD1(rt), and ΔD2(rt) are real-valued unknown matrices representing time-varying parameter uncertainties and are assumed to be of the form(4)ΔBrtΔD0rtΔD1rtΔD2rt=MrtFrtNbrtN0rtN1rtN2rtwhere M(rt),Nb(rt),N0(rt),N1(rt), and N2(rt) are known real constant matrices for all rt∈S and F(rt) is the uncertain time-varying matrix satisfying FT(rt)F(rt)=I,∀rt∈S.Assumption (A).There is a positive diagonal matrix L=diagl1,l2,…,ln such that f(0)=0 and (5)fβ1-fβ2≤Lβ1-β2for β1≠β2.For convenience, each possible value ofrt is denoted by k, k∈S in the sequel. Then we have Bk=B(rt),D0k=D0(rt),D1k=D1(rt),D2k=D0(rt),Hhk=Hhrt.Definition 1 (see [57]).
The neural network (2) is said to be uniformly stable in Lagrange sense, if, for any δ>0, there is a positive constant H=H(δ)>0 such that ηt,ψ<H for any ψ∈Rδ=ψ∈C-β,0,Rn:ψ<H,t≥0.Definition 2 (see [57]).
If there exists a radially unbounded and positive definite functionV(·), a nonnegative continuous function H(·), and two positive constants γ and ε such that, for any solution η(t) of neural network (2), V(η(t))>γ implies V(η(t))-γ≤H(ψ)e-εt for any t≥0 and ψ∈Rδ, then the neural network (2) is said to be globally exponentially attractive (GEA) with respect to V(η(t)), and the compact set Ω={η(t)∈Rn:V(η(t))≤γ} is said to be a GEA set of (2).Definition 3 (see [57]).
The neural network (2) is globally exponentially stable (GES) in Lagrange sense, if it is both uniformly stable in Lagrange sense and GEA. If there is a need to emphasize the Lyapunov-like functions, the neural network will be called GES in Lagrange sense with respect to V.Definition 4 (see [44]).
The functionV:[t0,∞)×Rn×S→R+ belongs to class ψ0 if (a) the function V is continuous on each of the sets [th-1,th)×Rn×S and for all t≥t0, V(t,0,i)≡0, i∈S; (b)V(t,x,i) is locally Lipschitzian in x∈Rn, i∈S; (c) for each h=1,2,… and i,j∈S, there exist finite limits (6)limt,z,j→th-,x,i⁡Vth-,x,iandlimt,z,j→th+,x,j⁡Vth+,x,jwith V(th+,x,j)=V(th,x,j) satisfied.Lemma 5 (see [58]).
Letη,μ∈Rn,R be a positive definite matrix; then the following inequality holds: (7)ηTμ+μTη≤ηTR-1η+μTRμ.Lemma 6 (see [59]).
For any constant matrixR>0, any scalars α and β with α<β, and a vector function η(t):[α,β]→R such that the integrals concerned are well defined, the following inequality holds: (8)∫αβηsdsTR∫αβηsds≤β-α∫αβηTsRηsds.Lemma 7 (see [59]).
The LMIR=R11R12R12TR22<0 with R11=R11T, R22=R22T is equivalent to one of the following conditions: (9)iR22<0,R11-R12R22-1R12T<0;iiR11<0,R22-R12TR11-1R12<0.Lemma 8 (see [57]).
LetV(t)∈C0,+∞,R, and there are two positive constants δ and ρ such that (10)D+Vt≤-δVt+ρ,t≥0,and then(11)Vt-δρ≤V0-δρe-δt,t≥0.In particular, if V(t)≥δ/ρ for t≥0, then V(t) exponentially approaches ρ/δ as t increases.Lemma 9 (see [60]).
LetD,E, and F(t) be real matrices of appropriate dimensions, and F(t) satisfies FT(t)F(t)=I. Then, the following inequality holds for any constant ε>0: (12)DFtE+ETFTtDT=εDDT+ε-1ETE.


## 3. Main Results

The following theorem presents a Lagrange stability condition for the Markovian jump neural networks (2) with uncertainties (i.e., ΔB(rt)≠0,ΔD0(rt)≠0,ΔD1(rt)≠0,ΔD2(rt)≠0).Theorem 10.
Under assumption (A), for given constantsτ,d,δ>0, there exist positive definite matrices W1,W2,W3,W4,R1k,R2,R3, positive diagonal matrices Q,S, and any appropriate dimensional matrices P1,P2,X11,X12,X13,X22,X23,X33 such that the following linear matrix inequalities (LMIs) hold:(13)X=X11X12X13∗X22X23∗∗X33>0,(14)R1kI-HhkTR1l∗R1l≥0,h∈Z+,hererth=l(15)Ψ=Ψr,s<0r,s=1,2,…,14,where (16)Ψ1,1=-R1kBk-BkR1k+δR1k+∑j=1mϕkjR1j+δR2+τX11+X13+X13T+LQL+ε2NbkTNbk,Ψ1,2=τX12-X13+X23T,Ψ1,3=R1kD0k-ε2NbkTN0k,Ψ1,4=R1kD1k-ε2NbkTN1k,Ψ1,5=R1kD2k-ε2NbkTN2k,Ψ1,6=BkR1kBk-δR1kBk-∑j=1mϕkjR1jBk,Ψ1,10=R1kTMk,Ψ1,11=R1k,Ψ2,2=τX22-X23-X23T+LSL,Ψ3,3=dR3-Q+ε1N0kTN0k+ε2N0kTN0k,Ψ3,4=ε1N0kTN1k+ε2N0kTN1k,Ψ3,5=ε1N0kTN2k+ε2N0kTN2k,Ψ3,6=-D0kTR1kBk,Ψ3,7=D0kTP1T,Ψ3,8=D0kTP2T-ε1N0kTNbk,Ψ4,4=-S+ε1N1kTN1k+ε2N1kTN1k,Ψ4,5=ε1N1kTN2k+ε2N1kTN2k,Ψ4,6=-D1kTR1kBk,Ψ4,7=D1kTP1T,Ψ4,8=D1kTP2T-ε1N1kTNbk,Ψ5,5=-e-δddR3+ε1N2kTN2k+ε2N2kTN2k,Ψ5,6=-D2kTR1kBk,Ψ5,7=D2kTP1T,Ψ5,8=D2kTP2T-ε1N2kTNbk,Ψ6,6=δBkR1kBk+∑j=1mϕkjBkR1jBk-e-δρρR2,Ψ6,10=-R1kTMkBk,Ψ6,12=BkR1k,Ψ7,7=τeδτX33-P1-P1T,Ψ7,8=-P1Bk-P2T,Ψ7,9=P1TMk,Ψ7,13=P1,Ψ8,8=-P2Bk-BkP2T+ε1NbkTNbk,Ψ8,9=P2TMk,Ψ8,14=P2,Ψ9,9=-ε1I,Ψ10,10=-ε2I,Ψ11,11=-W1,Ψ12,12=-W2,Ψ13,13=-W3,Ψ14,14=-W4,
and otherΨ(r,s) are equal to zero.
Then the Markovian jump uncertain neural network (2) is globally exponentially stable in Lagrange sense. Moreover, the set(17)Ω=ηt∈Rn:ηt≤JTW1+W2+W3+W4JδλminR11/2eρBkis an estimation of globally exponentially attractive set of (2).Proof.
Consider the following Lyapunov-Krasovskii functional candidate for model (2) as(18)Vrt,t=∑a=15Vart,t,where(19)V1rt,t=ηt-Bk∫t-ρtηsdsTR1kηt-Bk∫t-ρtηsds,V2rt,t=∫-ρ0∫t+θteδs-tηTsR2ηsdsdθ,V3rt,t=∫-d0∫t+θteδs-tfTηsR3fηsdsdθ,V4rt,t=∫-τ0∫t+θteδτ+s-tη˙TsX33η˙sdsdθ,V5rt,t=∫0t∫θ-τθθeδs-tϑTθ,sXϑθ,sdsdθ,and ϑ(θ,s)=ηTθηTθ-τθη˙TsT.
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.Calculating the time derivative of Vb(rt,t)(b=2,3,4,5), we have(21)V˙2rt,t=-δ∫-ρ0∫t+θteδs-tηTsR2ηsdsdθ+ρηTtR2ηt-∫t-ρteδs-tηTsR2ηsds≤-δV2rt,t+ρηTtR2ηt-e-δρ∫t-ρtηTsR2ηsds≤-δV2rt,t+ρηTtR2ηt-e-δρρ∫t-ρtηsdsTR2∫t-ρtηsds,(22)V˙3rt,t≤-δV3rt,t+dfTηtR3fηt-e-δdd∫t-dttηsdsTR3∫t-dttηsds,(23)V˙4rt,t=-δ∫-τ0∫t+θteδτ+s-tη˙TsX33η˙sdsdθ+τeδτη˙TtX33η˙t-∫t-τteδτ+s-tη˙TsX33η˙sds≤-δV4rt,t+τeδτη˙TtX33η˙t-∫t-τtη˙TsX33η˙sds,(24)V˙5rt,t=∫t-τtteδs-tϑTt,sXϑt,sds-δ∫0t∫θ-τθθeδs-tϑTθ,sXϑθ,sdsdθ≤-δV5rt,t+∫t-τttϑTt,sXϑt,sds=-δV5rt,t+τtηTtX11ηt+ηTtX12ηt-τt+ηTt-τtX12Tηt+ηTt-τtX22ηt-τt+ηTtX13+X13Tηt+ηTt-X13+X23Tηt-τt+ηTt-τt-X13T+X23ηt+ηTt-τtX23T+X23ηt-τt+∫t-τtη˙TsX33η˙sds.From assumption (A), it follows that(25)0≤ηTtLQLηt-fTηtQfηt,(26)0≤ηTt-τtLSLηt-τt-fTηt-τtSfηt-τt.Left-multiplying two sides of (2) by η˙T(t)P1, we obtain(27)η˙TtP1η˙t=-η˙TtP1Bkηt-ρ+η˙TtP1D0kfηt+η˙TtP1D1kfηt-τt+η˙TtP1D2k∫t-dttfηsds+η˙TtP1J.Taking the transpose on two sides of (27), we have(28)η˙TtP1Tη˙t=-ηTt-ρBkP1Tη˙t+fTηtD0kTP1Tη˙t+fTηt-τtD1kTP1Tη˙t+∫t-dttfηsdsTD2kTP1Tη˙t+JTP1Tη˙t.Adding (27) to (28) and using Lemma 5, we get(29)0≤η˙Tt-P1-P1T+P1W3-1P1η˙t-η˙TtP1Bkηt-ρ-ηTt-ρBkP1Tη˙t+η˙TtP1×D0kfηt+fTηtD0kTP1Tη˙t+η˙TtP1D1kfηt-τt+fTηt-τtD1kTP1Tη˙t+η˙TtP1D2k∫t-dttfηsds+∫t-dttfηsdsTD2kTP1Tη˙t+JTtW3JSimilarly, we obtain the zero inequality from (2) by multiplying ηT(t-ρ)P2 and using Lemma 5 as follows:(30)0=-ηTt-ρP2η˙t-η˙TtP2Tηt-ρ-ηTt-ρP2Bk+BkP2Tηt-ρ+ηTt-ρP2D0kfηt+fTηtD0kTP2Tηt-ρ+ηTt-ρP2D1kfηt-τt+fTηt-τtD1kTP2Tηt-ρ+ηTt-ρP2D2k∫t-dttfηsds+∫t-dttfηsdsTD2kTP2Tηt-ρ+ηTt-ρP2J+JTtP2Tηt-ρ≤-ηTt-ρP2η˙t-η˙TtP2Tηt-ρ+ηTt-ρ-P2Bk-BkP2T+P2W4-1P2ηt-ρ+ηTt-ρP2D0kfηt+fTηtD0kTP2Tηt-ρ+ηTt-ρP2D1kfηt-τt+fTηt-τtD1kTP2Tηt-ρ+ηTt-ρP2D2k∫t-dttfηsds+∫t-dttfηsdsTD2kTP2Tηt-ρ+JTtW4J.Combining (20)-(26), (29), and (30), we obtain(31)V˙rt,t≤-δVrt,t+ξTtΘξt+JTtW1+W2+W3+W4J,where ξT=ηTtηTt-τtfTηtfT(ηt-τt∫t-dttfηsdsT∫t-ρtηsdsTη˙TtηTt-ρT, and (32)Θ=Θij8×8,in which Θ1,1=-R1kBk-BkR1k+δR1k+R1kW1-1R1k+∑j=1mϕkjR1j+δR2+τX11+X13+X13T+LQL, Θ1,2=τX12-X13+X23T, Θ1,3=R1kD0k, Θ1,4=R1kD1k, Θ1,5=R1kD2k, Θ1,6=BkR1kBk-δR1kBk-∑j=1mϕkjR1jBk, Θ2,2=τX22-X23-X23T+LSL, Θ3,3=dR3-Q, Θ3,6=-D0kTR1kBk=D0kTP1T, Θ3,8=D0kTP2T, Θ4,4=-S, Θ4,6=-D1kTR1kBk, Θ4,7=D1kTP1T, Θ4,8=D1kTP2T, Θ5,5=-e-δd/dR3, Θ5,6=-D2kTR1kBk, Θ5,7=D2kTP1T, Θ5,8=D2kTP2T, Θ6,6=δBkR1kBk+BkR1kW2-1R1kBk+∑j=1mϕkjBkR1jBk-e-δρ/ρR2, Θ7,7=τeδτX33-P1-P1T+P1W3-1P1, Θ7,8=-P1Bk-P2T, Θ8,8=-P2Bk-BkP2T+P2W4-1P2, and the rest of Θij are zero.
Now we consider the change ofV(rt,t) at impulse time t=th,h∈Z+. By (2), we have(33)ηth-Bk∫th-ρthηsds=ηth--Hhkηth--Bk∫th-ρthηsds-Bk∫th-ρthηsds=I-Hhkηth--Bk∫th-ρthηsds.Moreover, it follows from (14) that(34)R1kI-HhkTR1l∗R1l≥0⟺I-I-HhkT∗IR1kI-HhkTR1l∗R1lI0-I-HhkI≥0⟺R1k-I-HhkTR1lI-Hhk0∗R1l≥0⟺R1k-I-HhkTR1lI-Hhk≥0Combining (33) and (34), we obtain(35)V1rth,th=ηth-Bk∫th-ρthηsdsTR1kηth-Bk∫th-ρthηsds=ηth--Bk∫th-ρthηsdsTI-HhkTR1kI-Hhkηth--Bk∫th-ρthηsds≤ηth--Bk∫th-ρthηsdsTR1kηth--Bk∫th-ρthηsds=V1rth-,th-.This yields(36)Vrth,th≤Vrth-,th-,h∈Z+.Obviously, by employing Lemma 6, we see that Ψ<0 in (15) is equivalent to Θ<0. Thus, it follows from (31) that(37)V˙rt,t≤-δVrt,t+JTtW1+W2+W3+W4J.Combining Lemma 7 and (37), we get(38)Vrt,t-JTtW1+W2+W3+W4Jδ≤Vr0,0-JTtW1+W2+W3+W4Jδe-δt.On the other hand, we obtain(39)Vr0,0=η0-Bk∫-ρ0ηsdsTR1kη0-Bk∫-ρ0ηsds+∫-ρ0∫θ0eδsηTsR2ηsdsdθ+∫-d0∫θ0eδsfTηsR3fηsdsdθ+∫-τ0∫θ0eδτ+sη˙TsX33η˙sdsdθ≤1+ρBk2R1k+ρδR2+dδR3max1≤i≤nli2sups∈-h,0ψs2+τδeδτX33sups∈-h,0ψ˙s2.Let(40)H=1+ρBk2R1k+ρδR2+dδR3max1≤i≤nli2sups∈-h,0ψs2+τδeδτX33sups∈-h,0ψ˙s2.From (38), we have(41)Vrt,t-JTtW1+W2+W3+W4Jδ≤He-δt.Hence, the Markovian jump neural network is globally exponentially attractive.
From definition ofV(rt,t), we have(42)Vrt,t≥λmin⁡R1kηt-Bk∫t-ρtηsds2.It follows from (41) and (42) that(43)ηt-Bk∫t-ρtηsds≤Hλmin⁡R1k+JTtW1+W2+W3+W4Jδλmin⁡R1k1/2.Furthermore,(44)ηt≤Hλmin⁡R1k+JTtW1+W2+W3+W4Jδλmin⁡R1k1/2+Bk∫t-ρtηsds.From the well-known Gronwall inequality, we get(45)ηt≤Hλmin⁡R1k+JTtW1+W2+W3+W4Jδλmin⁡R1k1/2eρBk.Therefore, the Markovian jump neural network (2) is uniformly stable in Lagrange sense. So, the Markovian jump neural network (2) is globally exponentially stable in Lagrange sense. Moreover, we know from (41) and (42) that a globally exponentially attractive set of (2) is as follows:(46)Ω=ηt∈Rn∣Vrt,t≤JTtW1+W2+W3+W4Jδ⊆ηt∈Rn∣ηt≤JTtW1+W2+W3+W4Jδλmin⁡R1k1/2eρBk.The proof is completed.Theorem10 provides a Lagrange stability criteria for Markovian jump neural networks with uncertainties (i.e., ΔB(rt)≠0,ΔD0(rt)≠0,ΔD1(rt)≠0,ΔD2(rt)≠0). In the following results we derived the Lagrange stability conditions for the Markovian jump neural networks without uncertainties (i.e., ΔB(rt)=ΔD0(rt)=ΔD1(rt)=ΔD2(rt)=0).Theorem 11.
Under assumption(A), for given constants τ,d,δ>0, there exist positive definite matrices W1,W2,W3,W4,R1k,R2,R3, positive diagonal matrices Q,S, and any appropriate dimensional matrices P1,P2,X11,X12,X13,X22,X23,X33 such that the following linear matrix inequalities (LMIs) hold:(47)X=X11X12X13∗X22X23∗∗X33>0,(48)R1kI-HhkTR1l∗R1l≥0,h∈Z+,hererth=l(49)Ψ˘=Ψ˘r,s<0r,s=1,2,…,12,where(50)Ψ˘1,1=-R1kBk-BkR1k+δR1k+∑j=1mϕkjR1j+δR2+τX11+X13+X13T+LQL,Ψ˘1,2=τX12-X13+X23T,Ψ˘1,3=R1kD0k,Ψ˘1,4=R1kD1k,Ψ˘1,5=R1kD2k,Ψ˘1,6=BkR1kBk-δR1kBk-∑j=1mϕkjR1jBk,Ψ˘1,9=R1k,Ψ˘2,2=τX22-X23-X23T+LSL,Ψ˘3,3=dR3-Q,Ψ˘3,6=-D0kTR1kBkΨ˘3,7=D0kTP1T,Ψ˘3,8=D0kTP2T,Ψ˘4,4=-S,Ψ˘4,6=-D1kTR1kBk,Ψ˘4,7=D1kTP1T,Ψ˘4,8=D1kTP2T,Ψ˘5,5=-e-δddR3,Ψ˘5,6=-D2kTR1kBk,Ψ˘5,7=D2kTP1T,Ψ˘5,8=D2kTP2T,Ψ˘6,6=δBkR1kBk+∑j=1mϕkjBkR1jBk-e-δρρR2,Ψ˘6,10=BkR1k,Ψ˘7,7=τeδτX33-P1-P1T,Ψ˘7,8=-P1Bk-P2T,Ψ˘7,11=P1,Ψ˘8,8=-P2Bk-BkP2T,Ψ˘8,12=P2,Ψ˘9,9=-W1,Ψ˘10,10=-W2,Ψ˘11,11=-W3,Ψ˘12,12=-W4,and  otherΨ˘r,sare  zero.Then, the Markovian jump neural network (2) (with ΔB(rt)=ΔD0(rt)=ΔD1(rt)=ΔD2(rt)=0) is globally exponentially stable in Lagrange sense. Moreover, the set (51)Ω=ηt∈Rn∣ηt≤JTW1+W2+W3+W4JδλminR11/2eρBk is an estimation of globally exponentially attractive set of (2).Proof.
Define the Lyapunov-Krasovskii functional candidate asV(rt,t) as defined in Theorem 10. By employing the same method in Theorem 10, we can easily prove the desired result.Remark 12.
Now, we consider the following neural network, as a special case of neural network (2) that reduces to a delayed neural network without Markovian jump parameters described by(52)η˙t=-Bkηt-ρ+D0kfηt+D1kfηt-τt+D2k∫t-dttfηsds+J,t≠thΔηth=-Hhkηth--Bk∫th-ρthηsds,t=th,κ∈Z+Using the same method in Theorem 10, we can get the following resultsCorollary 13.
Under assumption(A), for given constants τ,d,δ>0, there exist positive definite matrices W1,W2,W3,W4,R1k,R2,R3, positive diagonal matrices Q,S, and any appropriate dimensional matrices P1,P2,X11,X12,X13,X22,X23,X33 such that the following linear matrix inequalities (LMIs) hold:(53)X=X11X12X13∗X22X23∗∗X33>0,(54)R1kI-HhkTR1k∗R1k≥0,h∈Z+,(55)Ψ^=Ψ^r,s<0r,s=1,2,…,14,where(56)Ψ^1,1=-R1kBk-BkR1k+δR1k+δR2+τX11+X13+X13T+LQL+ε2NbkTNbk,Ψ^1,2=τX12-X13+X23T,Ψ^1,3=R1kD0k-ε2NbkTN0k,Ψ^1,4=R1kD1k-ε2NbkTN1k,Ψ^1,5=R1kD2k-ε2NbkTN2k,Ψ^1,6=BkR1kBk-δR1kBk,Ψ^1,10=R1kTMk,Ψ^1,11=R1k,Ψ^2,2=τX22-X23-X23T+LSL,Ψ^3,3=dR3-Q+ε1N0kTN0k+ε2N0kTN0k,Ψ^3,4=ε1N0kTN1k+ε2N0kTN1k,Ψ^3,5=ε1N0kTN2k+ε2N0kTN2k,Ψ^3,6=-D0kTR1kBk,Ψ^3,7=D0kTP1T,Ψ^3,8=D0kTP2T-ε1N0kTNbk,Ψ^4,4=-S+ε1N1kTN1k+ε2N1kTN1k,Ψ^4,5=ε1N1kTN2k+ε2N1kTN2k,Ψ^4,6=-D1kTR1kBk,Ψ^4,7=D1kTP1T,Ψ^4,8=D1kTP2T-ε1N1kTNbk,Ψ^5,5=-e-δddR3+ε1N2kTN2k+ε2N2kTN2k,Ψ^5,6=-D2kTR1kBk,Ψ^5,7=D2kTP1T,Ψ^5,8=D2kTP2T-ε1N2kTNbk,Ψ^6,6=δBkR1kBk-e-δρρR2,Ψ^6,10=-R1kTMkBk,Ψ^6,12=BkR1k,Ψ^7,7=τeδτX33-P1-P1T,Ψ^7,8=-P1Bk-P2T,Ψ^7,9=P1TMk,Ψ^7,13=P1,Ψ^8,8=-P2Bk-BkP2T+ε1NbkTNbk,Ψ^8,9=P2TMk,Ψ^8,14=P2,Ψ^9,9=-ε1I,Ψ^10,10=-ε2I,Ψ^11,11=-W1,Ψ^12,12=-W2,Ψ^13,13=-W3,Ψ^14,14=-W4,and  otherΨ^r×sare  zero.Then, the uncertain neural network (2) is globally exponentially stable in Lagrange sense. Moreover, the set (57)Ω=ηt∈Rn∣ηt≤JTW1+W2+W3+W4JδλminR1k1/2eρBkis an estimation of globally exponentially attractive set of (2).Remark 14.
In this paper we studied exponential Lagrange stability for Markovian jump uncertain neural networks with leakage delay and mixed time-varying delays via impulsive control. The stability of neural networks was studied by applying the differential inequality and Lyapunov method [30–34]. And the authors considered Lyapunov stability, where the Lyapunov stability of equilibrium point can be regarded as a special case of the Lagrange stability. In addition we considered model uncertainties, Markovian jumping parameters, and leakage delay in the concerned networks since it will affect the stability of the systems. Thus our results are more general than the results in the literature.


## 4. Numerical Example

This section provides numerical result to demonstrate the effectiveness of the presented strategy.Example 1.
Consider the following delayed uncertain Markovian jump neural networks with 2 modes:(58)η˙t=-Bkηt-ρ+D0kfηt+D1kfηt-τt+D2k∫t-dttfηsds+J,t≠thΔηth=-Hhkηth--Bk∫th-ρthηsds,t=th,h∈Z+(59)B1=1001,D01=0.2-10-3,D11=-0.2-0.20.2-0.3,D21=-0.3-0.80.4-0.1,B2=1001,D02=21-10.2,D12=-0.3-0.40.4-0.5,D22=0.4-0.2-0.20.4,Hh1=Hh2=0.4000.4,M1=0.2000.2,Nb1=N01=N11=N21=0.1000.1,M2=0.5000.5,Nb2=N02=N12=N22=0.2000.2,J=0.50.2,L=diag0.5,0.5.In order to obtain Lagrange stability, we take τ=0.8, d=0.5, δ=0.5, and ρ=0.5 and apply Theorem 10; we get feasible solutions to LMIs (14) and (15) as follows:(60)R11=0.1899-0.1535-0.15350.2065,R12=0.60080.04960.04960.6771,R2=1.2472-0.8560-0.85601.2001,R3=8.2198-0.7129-0.71296.6828,W1=514.86500.07850.0785514.9600,W2=514.66680.02660.0266514.6908,W3=514.8894-0.0147-0.0147514.8409,W4=515.2687-0.0025-0.0025515.1238,Q=8.4321008.4321,S=6.0806006.0806,P1=1.16080.05060.05060.7860,P2=1.16080.05060.05060.7860,X11=2.2685-0.0457-0.04572.3587,X12=-3.5666-0.0549-0.0549-3.6697,X13=-2.1441-0.0331-0.0331-2.2075,X22=2.59250.05420.05422.6978,X23=2.14270.03260.03262.2061,X33=0.36260.02630.02630.4137.From Theorem 10, we know that the uncertain neural networks concerned are globally exponentially stable in Lagrange sense, and (61)Ω=ηt∈Rn∣ηt≤12.0212 is an estimation of globally exponentially attractive set of (2).


## 5. Conclusion

The problem of Lagrange stability analysis has been investigated in this paper for uncertain Markovian jump neural networks with mixed time-varying delays and leakage delay. By choosing a Lyapunov-Krasovskii functional, the improved delay-dependent criteria are based on the Lagrange stability approach with attractive set. The Lagrange stability criterion has been obtained by solving a set of LMIs, which can guarantee the global exponential stability of the concerned system. A numerical example is given to illustrate the validity and feasibility of the obtained theoretical results.



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