The robust exponential stability problem for a class of uncertain impulsive stochastic neural networks of neutral-type with Markovian parameters and mixed time-varying delays is investigated. By constructing a proper exponential-type Lyapunov-Krasovskii functional and employing Jensen integral inequality, free-weight matrix method, some novel delay-dependent stability criteria that ensure the robust exponential stability in mean square of the trivial solution of the considered networks are established in the form of linear matrix inequalities (LMIs). The proposed results do not require the derivatives of discrete and distributed time-varying delays to be 0 or smaller than 1. Moreover, the main contribution of the proposed approach compared with related methods lies in the use of three types of impulses. Finally, two numerical examples are worked out to verify the effectiveness and less conservativeness of our theoretical results over existing literature.
1. Introduction
Up to now, the stability analysis of neural networks is an important research field in modern cybernetic area, since most of the successful applications of neural networks significantly depend on the stability of the equilibrium point of neural networks. Many papers related to this problem have been published in the literature; see [1] for a survey.
During implementation of artificial neural networks, time-varying delays [2–4] are unavoidable due to finite switching speeds of the amplifiers, and the neural signal propagation is often distributed in a certain time period with the presence of an amount of parallel pathways with a variety of axon sizes and lengths. Therefore, it is necessary to consider mixed time-varying delays (discrete time-varying delay and distributed time-varying delay) to design the neural networks models. There are many works focusing on the mixed time-varying delays [5–8], among which delay-dependent criteria are generally less conservative than delay-independent ones when the sizes of time-delays are small, and the maximum allowable delay bound is the main performance index of delay-dependent stability analysis [9]. In addition, as a special type of time delayed neural networks, neutral-type neural networks precisely describe that the past state of the networks will affect the current state. Therefore, the problems of stability and synchronization for such a class of neural networks have been studied in many references; see [10–22].
It is well known that the other three sources which may lead to instability and poor performances in neural networks are stochastic perturbation, impulsive perturbations, and parametric uncertainties. Most of this viewpoint is attributable to the following three reasons: (1) A neural network can be stabilized or destabilized by certain stochastic inputs [23–26]. (2) In the real world, many evolutionary processes are characterized by abrupt changes at time. These changes are called impulsive phenomena, which have been found in various fields, such as physics, optimal control, and biological mathematics [27]. (3) The effects of parametric uncertainties cannot be ignored in many applications [28–30]. Hence, stochastic perturbation, impulsive perturbations, and parametric uncertainties also should be taken into consideration when dealing with the stability issue of neural networks.
On the other hand, Markovian jumping systems [31] can be seen as a special class of hybrid systems with two different states, which involve both time-evolving and event-driven mechanisms. So such systems would be used to model the abrupt phenomena such as random failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. Thus, many relevant analysis results for Markovian jumping neural networks with impulses have been reported; see [32–38] and the references therein.
Recently, by using the concept of the minimum impulsive interval, Bao and Cao [11], Zhang et al. [12], and Gao et al. [13] derived some sufficient conditions to ensure exponential stability in mean square for neutral-type impulsive stochastic neural networks with Markovian jumping parameters and mixed time delays. However, in [11–13], the authors ignored parametric uncertainties. And in these three papers, the derivatives of time-varying delays need to be zero or smaller than one. So far, there are few results on the study of robust exponential stability of neutral-type impulsive stochastic neural networks with Markovian jumping parameters, mixed time-varying delays, and parametric uncertainties. More importantly, the impulses can be divided into three types to discuss the following: the impulses are stabilizing; the impulses are neutral-type (i.e., they are neither helpful for stability of neural networks nor destabilizing); and the impulses are destabilizing. Some interesting results for analyzing and synthesizing impulsive nonlinear systems that divide impulses into three types can be seen in [39–46]. In [39–41, 43], the authors studied the stability problem of impulsive neural networks with discrete time-varying delay by using the Lyapunov-Razumikhin method; several criteria for global exponential stability of the discrete-time or continuous-time neural networks are established in terms of matrix inequalities. In [42, 44–46], combining the impulsive comparison theory and triangle inequality, some important results about three-type impulses for different neural networks have been obtained. However, distributed time-varying delay has not been taken into account in all abovementioned references; how to deal with the stability problem of Markovian jumping impulsive stochastic neural networks with mixed delays is also a meaningful direction. Motivated by above discussion, based on the concepts of three-type impulses, this paper focuses on the robust exponential stability in mean square of impulsive stochastic neural networks with Markovian jumping parameters, mixed time-varying delays, and parametric uncertainties. By constructing a proper exponential-type Lyapunov-Krasovskii functional, linear matrix inequality (LMI) technique, Jensen integral inequality and free-weight matrix method, several novel sufficient conditions in terms of linear matrix inequalities (LMIs) are derived to guarantee the robust exponential stability in mean square of the trivial solution of the considered model. Compared with references [11–13], the constructed model renders more practical factors since the parametric uncertainties have been taken into account, and the derivatives of discrete and distributed time-varying delays need to be 0 or smaller than 1. Moreover, the main contribution of the proposed approach compared with related methods lies in the use of three types of impulses.
The organization of this paper is as follows. In Section 2, the robust exponential stability problem of impulsive stochastic neural networks with Markovian jumping parameters, mixed time-varying delays, and parametric uncertainties is described and some necessary definitions and lemmas are given. Some new robust exponential stability criteria are obtained in Section 3. In Section 4, two illustrative examples are given to show the effectiveness and less conservatism of the proposed method. Finally, conclusions are given in Section 5.
Notation. Let R denote the set of real numbers, let R+ denote the set of all nonnegative real numbers, let Rn and Rn×m denote the n-dimensional and n×m dimensional real spaces equipped with the Euclidean norm, and let · refer to the Euclidean vector norm and the induced matrix norm. N+ denotes the set of positive integers. For any matrix X∈Rn×n, X>0 denotes that X is a symmetric and positive definite matrix. If X1, X2 are symmetric matrices, then X1≤X2 means that X1-X2 is a negative semidefinite matrix. XT and X-1 mean the transpose of X and the inverse of a square matrix. I denotes the identity matrix with appropriate dimensions. Let τ>0 and C([-τ,0];Rn) denote the family of all continuous Rn-valued functions ξ(θ) on [-τ,0] with the norm ξ=sup-τ≤θ≤0ξ(θ). Let ω(t)=[ω1(t),ω2(t),…,ωn(t)]T be an n-dimensional Brownian motion defined on a complete probability space (Ω,F,P) with a natural filtration {Ft}t≥0 (i.e., Ft=σ{ω(s):0≤s≤t}), which satisfies Edω(t)=0 and E[dω(t)]2=dt. LFtp([-τ,0];Rn)(t≥0) denote the family of all Ft measurable bounded C([-τ,0];Rn)-valued random variables ξ={ξ(θ):-τ≤θ≤0} such that ∫-τ0Eξspds<∞, where E{·} stands for the correspondent expectation operator with respect to the given probability measure P. The notation ⋆ always denotes the symmetric block in one symmetric matrix. Matrix dimensions, if not explicitly stated, are assumed to be compatible for operations.
2. Model Description and Preliminaries
Let {r(t),t≥0} be a right continuous Markov chain in a complete probability space (Ω,F,P) taking values in a finite state space S={1,2,…,N} with generator Π=(πij)N×N given by (1)Prt+Δt=j∣rt=i=πijΔt+oΔt,ifi≠j,1+πiiΔt+oΔt,ifi=j,where Δt>0 and limΔt→0(o(Δt/Δt)=0. Here πij≥0(i≠j) is the transition rate from mode i to mode j while πii=-∑j≠iπij is the transition rate from mode i to mode i.
Consider a class of impulsive stochastic neural networks of neural-type with Markovian jumping parameters, mixed time-varying delays, and parametric uncertainties, which can be presented by the following impulsive integrodifferential equation:(2)dut-Drtut-τ3t=-Crtut+Art+ΔArtfut+Brt+ΔBrtfut-τ1t+Ert+ΔErt∫t-τ2ttfusds+Jdt+σt,rt,ut,ut-τ1t,ut-τ2t,ut-τ3tdωt,t≠tk,Dutk+=WkrtDutk-,t=tk,k∈N+,for t>0, where u(t)=(u1(t),u2(t),…,un(t))T∈Rn is the state vector associated with n neurons at time t. In the continuous part of system (2), C(r(t))=diag{c1(r(t)),c2(r(t)),…,cn(r(t))} is a diagonal matrix with positive entries ci(r(t))>0(i=1,2,…,n); the matrices A(r(t))=(aij(r(t)))n×n, B(r(t))=(bij(r(t)))n×n, and E(r(t))=(eij(r(t)))n×n are the connection weight matrix, the discrete time-varying delay connection weight matrix, and the distributed-delay connection weight matrix, respectively; ΔA(r(t)), ΔB(r(t)), and ΔD(r(t)) are the time-varying parametric uncertainties; f(u(t))=(f1(u1(t)),f2(u2(t)),…,fn(un(t)))T∈Rn is the nonlinear neuron activation function which describes the behavior in which the neurons respond to each other; J=[J1,J2,…,Jn]T∈Rn is a constant external input vector; τ1(t), τ2(t), and τ3(t) are, namely, the discrete time-varying delay, distributed time-varying delay, and neutral time-varying delay, which satisfy 0≤h1≤τ1(t)≤h2, τ˙1(t)≤μ1, 0≤τ2(t)≤τ2, τ˙2(t)≤μ2, 0≤τ3(t)≤τ3, and τ˙3(t)≤μ3; the noise perturbation (or the diffusion coefficient) σ(t,r(t),u(t),u(t-τ1(t)),u(t-τ2(t)),u(t-τ3(t))):R+×S×Rn×Rn×Rn→Rn×n is a Borel measurable function. In the discrete part of system (2), Du(tk+)=Wk(r(t))Du(tk-), k∈N+ is the impulse at the moment of time tk of an operator defined as Du(t)=u(t)-D(r(t))u(t-τ3(t)); Wk(r(t))∈Rn×n is the impulse gain matrix at the moment of time tk; the discrete instant set {tk} satisfies 0=t0<t1<t2<⋯<tk<⋯, limk→∞tk=∞; Du(tk-) and Du(tk+) are the left-hand and right-hand limits of operator Du(t) at tk, respectively; as usual, we always assume that Du(tk+)=Du(tk).
Remark 1.
In the continuous part of system (2), the evolution of state vector u(t) is driven by the evolution of the operator Du(t)=u(t)-D(r(t))u(t-τ3(t)). Consequently, we consider state jumping of the operator Du(t) at impulsive time in the discrete part of system (2). In system (2) of [13], ω˙(t) has been used to build the main model, which is wrong since Brown motion is nowhere differentiable with probability 1 [47].
For convenience, we denote r(t)=i, i∈S; then the matrices D(r(t)), C(r(t)), A(r(t)), B(r(t)), E(r(t)), ΔA(r(t)), ΔB(r(t)), and ΔE(r(t)) will be written as Di, Ci, Ai, Bi, Ei, ΔAi, ΔBi, and ΔEi, respectively. Therefore, system (2) can be rewritten as follows: (3)dut-Diut-τ3t=-Ciut+Ai+ΔAifut+Bi+ΔBifut-τ1t+Ei+ΔEi∫t-τ2ttfusds+Jdt+σt,i,ut,ut-τ1t,ut-τ2t,ut-τ3tdωt,t≠tk,Dutk+=WikDutk-,t=tk,k∈N+.
The initial condition of system (3) is given in the following form: (4)us=φs,s∈-τ,0,r0=i0,τ=maxh2+τ3,τ2+τ3,for any φ(s)∈LF02([-τ,0];Rn).
To prove our main results, the following hypotheses are needed:
All the eigenvalues of matrix Di,i∈S, are inside the unit circle, which guarantees the stability of difference system u(t)-Diu(t-τ3(t))=0.
Each neuron activation function fj is continuous [48], and there exist scalars lj- and lj+ such that (5)lj-≤fja-fjba-b≤lj+,
for any a,b∈R, a≠b, j=1,2,…,n, where lj+ and lj- can be positive, negative, or zero. And we set (6)L1=diagl1-,l2-,…,ln-,L2=diagl1+,l2+,…,ln+.
The noise matrix σ(t,i,·,·,·,·) is local Lipschitz continuous and satisfies the linear growth condition as well, and σ(0,i,0,0,0,0)=0. Moreover, there exist positive definite matrices H1i, H2i, H3i, and H4i(i∈S) such that (7)traceσTt,i,z1,z2,z3,z4σt,i,z1,z2,z3,z4≤z1TH1iz1+z2TH2iz2+z3TH3iz3+z4TH4iz4,
for all z1, z2, z3, z4∈Rn, t∈R+, and i∈S.
The time-varying admissible parametric uncertainties ΔAi(t), ΔBi(t), ΔEi(t), i∈S, are in terms of (8)ΔAitΔBitΔEit=ZiFitHiJiKi,
where Zi, Hi, Ji, and Ki are known real constant matrices with appropriate dimensions and Fi(t) is the uncertain time-varying matrix-valued function satisfying(9)FiTtFit≤I,∀t≥0.
In this paper, we always assume that some conditions are satisfied so that system (3) has a unique equilibrium point. Let u∗=(u1∗,u2∗,…,un∗)∈Rn be the equilibrium point of system (3). For simplicity, we can shift the equilibrium u∗ to the origin by letting x(t)=u(t)-u∗. Then system (3) can be transformed into the following one: (10)dxt-Dixt-τ3t=-Cixt+Ai+ΔAigxt+Bi+ΔBigxt-τ1t+Ei+ΔEi∫t-τ2ttgxsdsdt+σt,i,xt,xt-τ1t,xt-τ2t,xt-τ3tdωt,t≠tk,Dxtk+=WikDxtk-,t=tk,k∈N+,where g(x(·))=f(u(·)+u∗)-f(u∗). The initial condition of system (10) is given in terms of (11)xs=ψs=φs-u∗,s∈-τ,0,r0=i0,τ=maxh2+τ3,τ2+τ3.
Noting that g(0)=0 and σ(0,i,0,0,0,0)=0, we know that the trivial solution of system (10) exists. Thus, the stability problem of u∗ of system (3) converts to the stability problem of the trivial solution of system (10). On the other hand, from hypothesis (H1), we get (12)lj-≤gja-gjba-b≤lj+,for any a,b∈R, a≠b, j=1,2,…,n.
Next, let x(t;ξ) denote the state trajectory from the initial data x(θ)=ξ(θ) on -τ≤θ≤0 in LFt2([-τ,0];Rn). Based on above discussion, system (10) has a trivial solution x(t;0)≡0 corresponding to the initial condition ξ=0. For simplicity, we write x(t;ξ)=x(t).
The following definition and lemmas are useful for developing our main results.
Definition 2 (see [49]).
The trivial solution of system (10) is said to be exponentially stable in mean square if for every ξ∈LF02([-τ,0];Rn), there exist constants γ>0 and M>0 such that the following inequality holds: (13)Ext;ξ2≤Me-γtsup-τ≤θ≤0Eξθ2,where γ is called the exponential convergence rate.
Lemma 3 (Jensen integral inequality; see Gu [50]).
For any constant matrix M>0, any scalars s1 and s2 with s1<s2, and a vector function η(t):[a,b]→R such that the integrals concerned are well defined, then the following inequality holds: (14)∫s1s2ηsdsTM∫s1s2ηsds≤s2-s1∫s1s2ηsMηsds.
Lemma 4 (Wang et al. [51]).
For given matrices E, F, and G with FTF≤I and scalar ε>0, the following inequality holds: (15)GFE+ETFTGT≤εGGT+ε-1ETE.
Remark 5.
Some inequalities have been widely used to derive less conservative conditions to analyze and synthesize problems of time-delay systems, for example, Gronwall-Bellman inequality [52], Halanay inequality [53], Jensen integral inequality, Wirtinger integral [54], and reciprocally convex approach [55] in which Jensen integral inequality is the most used, and Lemma 4 also holds if s1=s2.
Remark 6.
Similar to [8], we further investigate the substantial influence of the three-type impulses for the exponential stability issue of stochastic neural networks of neutral-type with both Markovian jump parameters and mixed time delays.
3. Main Results
In this section, the robust exponential stability in mean square of the trivial solution for system (10) is studied under hypotheses (H1) to (H4).
Before proceeding, by using the model transformation technique, we rewritten system (10) as (16)dxt-Dixt-τ3t=ztdt+σtdωt,t≠tk,k∈N+,where(17)zt=-Cixt+Ai+ΔAigxt+Bi+ΔBigxt-τ1t+Ei+ΔEi∫t-τ2ttgxsds,σt=σt,i,xt,xt-τ1t,xt-τ2t,xt-τ3t.
Theorem 7.
Assume that hypotheses (H1)–(H4) hold. For given scalars h1, h2, τ2, τ3, and μ1, μ2, μ3, the trivial solution of system (10) is robustly exponentially stable in mean square if there exist positive scalars λi, αi≥-1 (αi≠0), α=max{1+αi}(i∈S), κ1, κ2, γ, positive definite matrices Pi(i∈S), Q1, Q2, Q3, Q4, Q5, Q6, positive diagonal matrices Ri, Si(i∈S), and any real matrices Nq(q=1,2,…,10) of appropriate dimensions such that (18)Pi≤λiI,(19)WikTPlWik-Pi≤αiPiherertk=l,(20)Φi=Φi′Γ1iΓ2i∗-κ1I0∗∗-κ2I<0,where (21)Φi′=ϕimn′13×13,m=1,2,…,13,n=1,2,…,13,Γ1i=PiZi012n×n,Γ2i=DiTPiZi012n×n,ϕi1,1′=-PiCi-CiTPi+γPi+λiH1i+∑j=1NπijPj+e-γh2Q1+∑j=23e-γτjQj+h2-h1γeγh2-eγh1Q5+τ3γeγτ3-1Q6-2L1RiL2+N1+N1T+N6+N6T,ϕi1,2′=-N1+N2T,ϕi1,3′=-N6+N7T,ϕi1,4′=-∑jNπijPjDi-γPiDi+CiTPiDi-N1Di-N6Di+N3T+N8T,ϕi1,5′=N1Di+N4T,ϕi1,6′=N6Di+N9T,ϕi1,7′=PiAi+L1+L2Ri,ϕi1,8′=PiBi,ϕi1,9′=PiEi,ϕi1,10′=-N1+N5T,ϕi1,11′=-N6+N10T,ϕi2,2′=λiH2i-1-μ1hμ1Q1-2L1SiL2-N2-N2T,ϕi2,4′=-N2Di-N3T,ϕi2,5′=N2Di-N4T,ϕi2,8′=L1+L2Si,ϕi2,10′=-N2-N5T,ϕi3,3′=λiH3i-1-μ2hμ2Q2-N7-N7T,ϕi3,4′=-N7Di-N8T,ϕi3,6′=N7Di-N9T,ϕi3,11′=-N7-N10T,ϕi4,4′=γDiTPiDi+DiT∑j=1NπijPjDi+λiH4i-1-μ3hμ3Q3-N3Di-DiTN3T-N8Di-DiTN8T,ϕi4,5′=N3Di-DiTN4T,ϕi4,6′=N8Di-DiTN9T,ϕi4,7′=-DiTPiAi,ϕi4,8′=-DiTPiBi,ϕi4,9′=-DiTPiEi,ϕi4,10′=-N3-DiTN5T,ϕi4,11′=-N8-DiTN10T,ϕi5,5′=N4Di+DiTN4T,ϕ5,10′=-N4+DiTN5T,ϕi6,6′=N9Di+DiTN9T,ϕi6,11′=-N9+DiTN10T,ϕi7,7′=τγeγτ-1Q4-2Ri+κ1HiTHi+κ2HiTHi,ϕi8,8′=-2Si+κ1JiTJi+κ2JiTJi,ϕi9,9′=-Q4+κ1KiTKi+κ2KiTKi,ϕi10,10′=-N5-N5T,ϕi11,11′=-N10-N10T,ϕi12,12′=-Q5,ϕi13,13′=-Q6,and the function h(u)∈R+, u∈R, is defined as(22)hu=1,u>1,e-2γτ,u≤1and for αi>0, -γ+lnα/inf{tk-tk-1}<0, k∈N+, other elements of Φi′ are all equal to 0.
Proof.
Let xt=x(t+s), s∈[-τ,0]. As discussed in [56–59], {xt,r(t),t≥0} is a C([-τ,0];Rn)×S-valued Markov process. Construct the following stochastic Lyapunov-Krasovskii functional candidate for system (10): (23)Vt,i,xt=V1t,i,xt+V2t,i,xt+V3t,i,xt,where(24)V1t,i,xt=eγtxt-Dixt-τ3tTPixt-Dixt-τ3t,V2t,i,xt=∫t-τ1tteγs-h2xTsQ1xsds+∫t-τ2tteγs-τ2xTsQ2xsds+∫t-τ3tteγs-τ3xTsQ3xsds,V3t,i,xt=τ∫-τ0∫t+βteγs-βgTxsQ4gxsdsdβ+h2-h1∫-h2-h1∫t+βteγs-βxTsQ5xsdsdβ+τ3∫-τ30∫t+βteγs-βxTsQ6xsdsdβ.
For t∈[tk-1,tk), k∈N+, denote L to be the weak infinitesimal operator of the random process xt=x(t+s), s∈[-τ,0]; then along the trajectory of system (10) we have (25)LVt,i,xt=LV1t,i,xt+LV2t,i,xt+LV3t,i,xt,where(26)LV1t,i,xt=γeγtxt-Dixt-τ3tTPixt-Dixt-τ3t+2eγtxt-Dixt-τ3tTPi-Cixt+Ai+ΔAigxt+Bi+ΔBigxt-τ1t+Ei+ΔEi∫t-τ2ttgxsds+eγttraceσTtPiσt+eγtxt-Dixt-τ3tT∑j=1NπijPjxt-Dixt-τ3t,(27)LV2t,i,xt=eγt-h2xTtQ1xt-1-τ˙1teγt-τ1t-h2xTt-τ1tQ1xt-τ1t+eγt-τ2xTtQ2xt-1-τ˙2teγt-τ2t-τ2xTt-τ2tQ2xt-τ2t+eγt-τ3xTtQ3xt-1-τ˙3teγt-τ3t-τ3xTt-τ3tQ3xt-τ3t,(28)LV3t,i,xt=τ∫-τ0eγt-βgTxtQ4gxtdβ-τ∫-τ0eγtgTxt+βQ4gxt+βdβ+h2-h1∫-h2-h1eγt-βxTtQ5xtdβ-h2-h1∫-h2-h1eγtxTt+βQ5xt+βdβ+τ3∫-τ30eγt-βxTtQ6xtdβ-τ3∫-τ30eγtxTt+βQ6xt+βdβ=τγeγτ-1eγtgTxtQ4gxt-τeγt∫t-τtgTxsQ4gxsds+h2-h1γeγh2-eγh1eγtxTtQ5xt-h2-h1eγt∫t-h2t-h1xTsQ5xsds+τ3γeγτ3-1eγtxTtQ6xt-τ3eγt∫t-τ3txTsQ6xsds≤τγeγτ-1eγtgTxtQ4gxt-τ2teγt∫t-τ2ttgTxsQ4gxsds+h2-h1γeγh2-eγh1eγtxTtQ5xt-h2-h1eγt∫t-h2t-h1xTsQ5xsds+τ3γeγτ3-1eγtxTtQ6xt-τ3eγt∫t-τ3txTsQ6xsds.From hypotheses (H3) and (18), we have(29)traceσTtPiσt≤λitraceσTtσt≤λixTtH1ixt+xTt-τ1tH2ixt-τ1t+xTt-τ2tH3ixt-τ2t+xTt-τ3tH4ixt-τ3t.Combining (20) and (27) together yields (30)LV2t,i,xt≤eγtxTte-γh2Q1+e-γτ2Q2+e-γτ3Q3xt-1-μ1hμ1xTt-τ1tQ1xt-τ1t-1-μ2hμ2xTt-τ2tQ2xt-τ2t-1-μ3hμ3xTt-τ3tQ3xt-τ3t.
If τ2(t)>0, h2>h1, based on (28) and Lemma 3, it is easy to derive that (31)LV3t,i,xt≤τγeγτ-1eγtgTxtQ4gxt-eγt∫t-τ2ttgxsdsTQ4∫t-τ2ttgxsds+h2-h1γeγh2-eγh1eγtxTtQ5xt-eγt∫t-h2t-h1xsdsTQ5∫t-h2t-h1xsds+τ3γeγτ3-1eγtxTtQ6xt-eγt∫t-τ3txsdsTQ6∫t-τ3txsds.Note that inequality (31) still holds if τ2(t)=0 and h2=h1 since (32)∫t-τ2ttgTxsQ4gxsds=∫t-τ2ttgxsdsTQ4∫t-τ2ttgxsds=0,∫t-h2t-h1xTsQ5xsds∫t-h2t-h1xsdsTQ5∫t-h2t-h1xsds=0,∫t-τ3txTsQ6xsds=∫t-τ3txsdsTQ6∫t-τ3txsds=0.
On the other hand, by hypothesis (H2), one can get that there exist positive diagonal matrices Ri=diagr1i,r2i,…,rni, Si=diag{s1i,s2i,…,sni},i∈S, such that the following inequalities hold (33)0≤2eγt∑j=1nrjigjxjt-lj-xjtlj+xjt-gjxjt=2eγtxTtL1+L2Rigxt-xTtL1RiL2xt-gTxtRigxt,0≤2eγt∑j=1nsjigjxjt-τ1t-lj-xjt-τ1tlj+xjt-τ1t-gjxjt-τ1t=2eγtxTt-τ1tL1+L2Sigxt-τ1t-xTt-τ1tL1SiL2xt-τ1t-gTxt-τ1tSigxt-τ1t.
Moreover, by utilizing the well-known Newton-Leibniz formulae and (16), it can be deduced that for any matrices Nq, q=1,2,…,10, with appropriate dimensions, the following equalities also hold(34)0=2eγtxTtN1+xTt-τ1tN2+xTt-τ3tN3+xTt-τ1t-τ3t-τ1tN4+∫t-τ1ttzsdsTN5xt-Dixt-τ3t-xt-τ1t-Dixt-τ1t-τ3t-τ1t-∫t-τ1ttzsds-∫t-τ1ttσsdωs,0=2eγtxTtN6+xTt-τ2tN7+xTt-τ3tN8+xTt-τ2t-τ3t-τ2tN9+∫t-τ2ttzsdsTN10xt-Dixt-τ3t-xt-τ2t-Dixt-τ2t-τ3t-τ2t-∫t-τ2ttzsds-∫t-τ2ttσsdωs.
Considering hypothesis (H4), substituting (26)–(34) and Edω(t)=0 into (25) yields that for t∈[tk-1,tk), k∈N+, (35)ELVt,i,xt≤eγtEχTtΦi′′χt,where(36)χTt=xTtxTt-τ1txTt-τ2txTt-τ3t∫t-τ2ttgxsdsT∫t-τ1ttzsdsT∫t-τ2ttzsdsT∫t-h2t-h1xsdsT∫t-τ3txsdsThhhhhxTt-τ1t-τ3t-τ1thhhhhxTt-τ2t-τ3t-τ2tgTxthhhhhgTxt-τ1t∫t-τ2ttgxsdsThhhhh∫t-τ1ttzsdsT∫t-τ2ttzsdsThhhhh∫t-h2t-h1xsdsT∫t-τ3txsdsT,Φi′′=Φi′κ1=0,κ2=0+PiZi012n×nFit06n×nHiTJiTKiT04n×nT+06n×nHiTJiTKiT04n×nFiTtPiZi012n×nT-DiTPiZi012n×nFit06n×nHiTJiTKiT04n×nT-06n×nHiTJiTKiT04n×nFiTtDiTPiZi012n×nT.
Combining Lemma 4 and (35) together yields that there exist two positive scalars κ1 and κ2 such that(37)Φi′′≤Ξi=Φi′κ1>0,κ2>0+κ1-1PiZi012n×nPiZi012n×nT+κ2-1DiTPiZi012n×nDiTPiZi012n×nT.Applying the Schur complement equivalence [60] to (20) yields Ξi<0. Therefore, Φi′′<0, which means (38)ELVt,i,xt≤0,t∈tk-1,tk,k∈N+.
For t=tk, k∈N+, according to (19) and (23) and Edω(t)=0, we have (39)EVtk,l,xtk=EVtk-,i,xtk-+EeγtkDxTtk-WikTPlWik-PiDxtk-≤EVtk-,i,xtk-+αiEV1tk-,i,xtk-;if -1≤αi<0, then (40)EVtk,l,xtk≤EVtk-,i,xtk-;if αi>0, then (41)EVtk,l,xtk≤1+αiEVtk-,i,xtk-≤αEVtk-,i,xtk-.So, from inequalities (38) and (40), for all i∈S, t≥0, it is true through the mathematical induction that (42)EVt,i,xt≤EV0,r0,x0,-1≤αi<0.Similarly, based on inequalities (38) and (41), for all i∈S, t∈[tk-1,tk), k∈N+, it is true through the mathematical induction that (43)EVt,i,xt≤αk-1EV0,r0,x0=EV0,r0,x0ek-1lnα≤EV0,r0,x0etk-1/inftk-tk-1lnα≤EV0,r0,x0elnα/inftk-tk-1t,αi>0.From (23), (42), and (43), the following inequalities are, namely, hold (44)EDxtTDxt≤EV0,r0,x0mini∈SλminPie-γt,-1≤αi<0,t≥0,(45)EDxtTDxt≤EV0,r0,x0mini∈SλminPie-γ+lnα/inftk-tk-1t,αi>0,t∈tk-1,tk,k∈N+.On the other hand, defined L=diag{l1,l2,…,ln} within lj=max{lj-,lj+},j=1,2,…,n, from Lemma 4; it is easy to obtain that there exists a positive scalar ϵ such that (46)EV0,r0,x0=Ex0-Dix0-τ30TPr0x0-Dix0-τ30+∫-τ100Eeγs-h2xTsQ1xsds+∑j=23∫-τj00Eeγs-τjxTsQjxsds+τ∫-τ0∫β0Eeγs-βgTxsQ4gxsdsdβ+h2-h1∫-h2-h1∫β0Eeγs-βxTsQ5xsdsdβ+τ3∫-τ30∫β0Eeγs-βxTsQ6xsdsdβ≤1+ϵxT0Pr0x0+1+ϵ-1xT-τ30DiTPr0Dix-τ30+λmaxQ1e-γh2∫-h20eγsds+∑j=23λmaxQje-γτj∫-τj0eγsds+τλmaxLTQ4L∫-τ0∫β0eγs-βdsdβ+h2-h1λmaxQ5∫-h2-h1∫β0eγs-βdsdβ+τ3λmaxQ6∫-τ30∫β0eγs-βdsdβsup-τ≤θ≤0Eξθ2≤1+ϵ+1+ϵ-1DiTDimaxi∈Sλi+λmaxQ1e-γh2γ1-e-γh2+∑j=23λmaxQje-γτjγ1-e-γτj+τλmaxLTQ4Lγeγτ-1γ-τ+h2-h1λmaxQ5γeγh2-eγh1γ-h2-h1+τ3λmaxQ6γeγτ3-1γ-τ3sup-τ≤θ≤0Eξθ2=M1sup-τ≤θ≤0Eξθ2,where (47)M1=1+ϵ+1+ϵ-1DiTDimaxi∈Sλi+λmaxQ1e-γh2γ1-e-γh2+∑j=23λmaxQje-γτjγ1-e-γτj+τλmaxLTQ4Lγeγτ-1γ-τ+h2-h1λmaxQ5γeγh2-eγh1γ-h2-h1+τ3λmaxQ6γeγτ3-1γ-τ3.In addition, one can see that (48)ExTtxt=EDxt+Dixt-τ3tTDxt+Dixt-τ3t=EDxtTDxt+2DxtTDixt-τ3t+xTt-τ3tDiTDixt-τ3t=EDxtTDxt+2EDxtTDixt-τ3t+ExTt-τ3tDiTDixt-τ3t.By utilizing Lemma 4 and (48), a positive scalar ε can be found, such that (49)ExTtxt≤1+εEDxtTDxt+1+ε-1λmaxDiTDiExTt-τ3txt-τ3t,(50)1+ε-1λmaxDiTDieγτ3<1.If -1≤αi<0, by using (44) and (49), for any t∗≥0, we can get the following result by the same derivation in [22]:(51)sup0≤t≤t∗ExTtxteγt≤1+εEV0,r0,x0mini∈SλminPi+1+ε-1λmaxDiTDieγτ3sup0≤t≤t∗ExTt-τ3txt-τ3teγt-τ3t≤1+εEV0,r0,x0mini∈SλminPi+1+ε-1λmaxDiTDieγτ3sup-τ3t≤θ<0Eξθ2+sup0≤t≤t∗ExTtxteγt≤1+ε-1λmaxDiTDieγτ3sup-τ≤θ≤0Eξθ2+sup0≤t≤t∗ExTtxteγt+1+εEV0,r0,x0mini∈SλminPi.Because (46) and (50) hold, we have (52)sup0≤t≤t∗ExTtxteγt≤Msup-τ≤θ≤0Eξθ2,where(53)M=1+εM1/mini∈SλminPi+1+ε-1λmaxDiTDieγτ31-1+ε-1λmaxDiTDieγτ3.Letting t∗→∞ yields (54)supt∈0,∞ExTtxteγt≤Msup-τ≤θ≤0Eξθ2.Obviously, for -1≤αi<0, t≥0, (55)ExTtxt≤Me-γtsup-τ≤θ≤0Eξθ2.Next, along the same line of (55), it can be deduced that for αi>0, t∈[tk-1,tk), k∈N+, (56)ExTtxt≤M′e-γ+lnα/inftk-tk-1tsup-τ≤θ≤0Eξθ2,where (57)M′=1+εM1/mini∈SλminPi+1+ε-1λmaxDiTDieγ-lnα/inftk-tk-1τ31-1+ε-1λmaxDiTDieγ-lnα/inftk-tk-1τ3.Hence, for αi≥-1 (αi≠0), by Definition 2 and (55) and (56), it can be seen that the trivial solution of system (10) is robustly exponentially stable in mean square. Moreover, the exponential convergence rate is (58)γ,if-1≤αi<0,γ-lnαinftk-tk-1,ifαi>0.This completes the proof of Theorem 7.
Remark 8.
In fact, exponential convergence rate of the trivial solution of system (10) is the inherent essence. The constructed exponential-type Lyapunov-Krasovskii functional in the proof of Theorem 7 is aimed at estimating a closely approximate exponential convergence rate of the trivial solution of system (10) mathematically.
Remark 9.
When -1≤αi<0, the impulses are stabilizing; when αi>0, the impulses are destabilizing; and when Wik=I, the impulses are neutral-type (i.e., they are neither helpful for stability of system (10) nor destabilizing). αi≠0 is necessary since the Markovian jumping would occur at the impulsive time instants; that is, Pi is changing with the mode’s change, and there always exist scalars αi>0 such that Pl≤(1+αi)Pi. To the best of authors’ knowledge, there is no result about dividing the impulses into three types for robust global exponential stability for impulsive stochastic neural networks of neutral-type with Markovian parameters, mixed time delays, and parametric uncertainties. Moreover, because the stability analysis for the case of neutral-type impulses is similar to that of destabilizing impulses, the robust exponential stability in mean square of system (10) has been classified into two categories: -1≤αi<0 and αi>0.
Remark 10.
As shown in (58), the effects of the three types of impulses for the exponential convergence rate of the trivial solution of system (10) have been explicitly presented, which further verifies the characteristics of the different impulses.
When system (10) is without parametric uncertainties, by constructing the same Lyapunov-Krasovskii functional, from Theorem 7, the following corollary can be deduced to guarantee the exponential stability in mean square of the trivial solution of system (10).
Corollary 11.
Assume that hypotheses (H1)–(H3) hold. For given scalars h1, h2, τ2, τ3, and μ1, μ2, μ3, the trivial solution of system (10) is exponentially stable in mean square if there exist positive scalars λi, αi≥-1(αi≠0), α=max{1+αi}(i∈S), γ, positive definite matrices Pi(i∈S), Q1, Q2, Q3, Q4, Q5, Q6, positive diagonal matrices Ri, Si(i∈S), and any real matrices Nq(q=1,2,…,10) of appropriate dimensions such that (59)Pi≤λiI,WikTPlWik-Pi≤αiPiherertk=l,Φi′<0,where (60)Φi′=ϕimn′13×13,m=1,2,…,13,n=1,2,…,13,ϕi1,1′=-PiCi-CiTPi+γPi+λiH1i+∑j=1NπijPj+e-γh2Q1+∑j=23e-γτjQj+h2-h1γeγh2-eγh1Q5+τ3γeγτ3-1Q6-2L1RiL2+N1+N1T+N6+N6T,ϕi1,2′=-N1+N2T,ϕi1,3′=-N6+N7T,ϕi1,4′=-∑jNπijPjDi-γPiDi+CiTPiDi-N1Di-N6Di+N3T+N8T,ϕi1,5′=N1Di+N4T,ϕi1,6′=N6Di+N9T,ϕi1,7′=PiAi+L1+L2Ri,ϕi1,8′=PiBi,ϕi1,9′=PiEi,ϕi1,10′=-N1+N5T,ϕi1,11′=-N6+N10T,ϕi2,2′=λiH2i-1-μ1hμ1Q1-2L1SiL2-N2-N2T,ϕi2,4′=-N2Di-N3T,ϕi2,5′=N2Di-N4T,ϕi2,8′=L1+L2Si,ϕi2,10′=-N2-N5T,ϕi3,3′=λiH3i-1-μ2hμ2Q2-N7-N7T,ϕi3,4′=-N7Di-N8T,ϕi3,6′=N7Di-N9T,ϕi3,11′=-N7-N10T,ϕi4,4′=γDiTPiDi+DiT∑j=1NπijPjDi+λiH4i-1-μ3hμ3Q3-N3Di-DiTN3T-N8Di-DiTN8T,ϕi4,5′=N3Di-DiTN4T,ϕi4,6′=N8Di-DiTN9T,ϕi4,7′=-DiTPiAi,ϕi4,8′=-DiTPiBi,ϕi4,9′=-DiTPiEi,ϕi4,10′=-N3-DiTN5T,ϕi4,11′=-N8-DiTN10T,ϕi5,5′=N4Di+DiTN4T,ϕ5,10′=-N4+DiTN5T,ϕi6,6′=N9Di+DiTN9T,ϕi6,11′=-N9+DiTN10T,ϕi7,7′=τγeγτ-1Q4-2Ri,ϕi8,8′=-2Si,ϕi9,9′=-Q4,ϕi10,10′=-N5-N5T,ϕi11,11′=-N10-N10T,ϕi12,12′=-Q5,ϕi13,13′=-Q6,and the function h(u)∈R+, u∈R, is defined as (61)hu=1,u>1,e-2γτ,u≤1.And for αi>0, -γ+lnα/inf{tk-tk-1}<0, k∈N+, other elements of Φi′ are all equal to 0.
When system (10) is without Markovian jumping parameters, parametric uncertainties, distributed time-varying delay, impulses, and stochastic perturbation, then system (10) can be written as (62)dxt-Dxt-τ3t=-Cxt+Agxt+Bgxt-τ1tdt.Construct a Lyapunov-Krasovskii functional as follows: (63)Vt,xt=eγtxt-Dxt-τ3tTPxt-Dxt-τ3t+∫t-τ1tteγs-h2xTsQ1xsds+∫t-τ3tteγs-τ3xTsQ2xsds+h2-h1∫-h2-h1∫t+βteγs-βxTsQ3xsdsdβ+τ3∫-τ30∫t+βteγs-βxTsQ4xsdsdβ.From Theorem 7, the following corollary can be deduced to guarantee the exponential stability of the trivial solution of system (62).
Corollary 12.
Assume that hypotheses (H1)-(H2) hold. For given scalars h1, h2, τ3, and μ1, μ3, the trivial solution of system (62) is exponentially stable if there exist positive scalar γ, positive definite matrices P, Q1, Q2, Q3, Q4, positive diagonal matrices R, S, and any real matrices Nq(q=1,2,…,5) of appropriate dimensions such that (64)Φ′<0,where (65)Φ′=ϕimn′9×9,m=1,2,…,9,n=1,2,…,9,ϕ1,1′=-PC-CTP+γP+e-γh2Q1+e-γτ3Q2+h2-h1γeγh2-eγh1Q3+τ3γeγτ3-1Q4-2L1RL2+N1+N1T,ϕ1,2′=-N1+N2T,ϕ1,3′=-γPD+CTPD-N1D+N3T,ϕ1,4′=N1D+N4T,ϕ1,5′=PA+L1+L2R,ϕ1,6′=PB,ϕ1,7′=-N1+N5T,ϕ2,2′=-1-μ1hμ1Q1-2L1SL2-N2-N2T,ϕ2,3′=-N2D-N3T,ϕ2,4′=N2D-N4T,ϕ2,6′=L1+L2S,ϕ2,7′=-N2-N5T,ϕ3,3′=γDTPD-1-μ3hμ3Q2-N3D-DTN3T,ϕ3,4′=N3D-DTN4T,ϕ3,5′=-DTPA,ϕ3,6′=-DTPB,ϕ3,7′=-N3-DTN5T,ϕ4,4′=N4D+DTN4T,ϕ4,7′=-N4+DTN5T,ϕ5,5′=-2R,ϕ6,6′=-2S,ϕ7,7′=-N5-N5T,ϕ8,8′=-Q3,ϕ9,9′=-Q4,and the function h(u)∈R+, u∈R, is defined as (66)hu=1,u>1,e-2γτ,u≤1.And other elements of Φ′ are all equal to 0.
4. Numerical Results
In this section, two numerical examples are presented to illustrate the effectiveness of the obtained results.
Example 13 (see [13]).
Let the state space of Markov chain {r(t),t≥0} be S={1,2} with generator (67)Π=-0.450.450.5-0.5.Consider 2D delayed impulsive stochastic neural networks of neutral-type (10) with Markovian switching and parametric uncertainties: (68)C1=2.9002.8,C2=2.5002.6,A1=0.20.180.30.19,A2=0.300.40,B1=0.80.20.20.3,B2=2.51.512.5,D1=0.2000.2,D2=0.3000.3,E1=40.040.144,E2=41.514,Z1=0.1-0.20.70.2,Z2=-0.1-0.2-0.10.2,H1=-0.30.1-0.20.1,H2=0.3-0.40.7-0.1,J1=-0.5-0.40.2-0.2,J2=-0.1-0.40.40.3,K1=-0.20.20.10.8,K2=0.10.3-0.4-0.3,F1t=sint00cost,F2t=cost00sint,gxt=tanhxt,τ1t=0.6+0.6sin2t,τ2t=0.25+0.25cos4t,τ3t=1.5+1.5cost,σ1t=σ2t=0.3x1t000.2x2t-τ1t+0.3x2t000.2x2t-τ2t+0.3x1t-τ2t000.2x2t-τ3t.Then system (10) satisfies hypotheses (H1)–(H3) with (69)h1=0,h2=1.2,τ2=0.5,τ3=3,μ1=1.2,μ2=1,μ3=1.5,τ=4.2,H11=H12=0.18I,H21=H22=0.08I,H31=H32=0.18I,H41=H42=0.08I,L1=0,L2=I,L=I.
Case of the Stabilizing Impulses. Study the following impulsive gain matrices:(70)W1k=0.9000.9,W2k=0.9000.9,k∈N+.By choosing α1=-0.1, α2=-0.1, then the impulses are the stabilizing impulses. We set tk=0.5+tk-1, k∈N+, Δt=0.001. The 2-state Markov chain with r(0)=1 is shown in Figure 1, among which the right continuous Markov chain {r(t),t≥0} is denoted by the solid blue line, and the Markov chain of the impulsive time instants {r(tk),k∈N+} is denoted by the red point, and the black point is used to judge whether the Markovian jumping occurs at the impulsive time instants, that is, r(tk)-r(tk-Δt). From Figure 1, we can conclude that the Markovian jumping does not occur at the impulsive time instants when tk=0.5+tk-1, k∈N+, Δt=0.001.
By using the LMI toolbox in MATLAB, we search the maximum exponential convergence rate which is 5.4297 subject to the LMIs (18)–(20). Let γ=0.5; we can obtain the following feasible solutions to the LMIs (18)–(20) in Theorem 7: (71)P1=0.0019-0.0001-0.00010.0010,P2=0.0018-0.0010-0.00100.0023,Q1=0.0229-0.0002-0.00020.0214,Q2=0.0192-0.0002-0.00020.0177,Q3=0.0089000.0091,Q4=0.00200.00010.00010.0027,Q5=0.0231-0.0003-0.00030.0213,Q6=0.0023000.0021,R1=0.1752000.1752,R2=0.1703000.1703,S1=0.1399000.1399,S2=0.1356000.1356,N1=-0.2415-0.0016-0.0016-0.2542,N2=0.27730.00140.00160.2885,N3=0.09340.00050.00050.0965,N4=-0.0946-0.0002-0.0002-0.0968,N5=0.25950.00060.00050.2638,N6=-0.2368-0.0016-0.0016-0.2497,N7=0.23950.00150.00150.2517,N8=0.09230.00070.00070.0964,N9=-0.0947-0.0003-0.0003-0.0974,N10=0.25810.00070.00070.2635,λ1=0.0677,λ2=0.0816,κ1=0.0015,κ2=0.0015.
Set the simulation step size h=0.05 and r(0)=1, Δt=0.001. The dynamic behavior of system (10) with the stabilizing impulses in Example 13 is presented in Figure 2, with the initial condition of every state uniformly randomly selected from [-0.1;0.1], s∈[-4.2,0]. Therefore, it can be verified that system (10) with the stabilizing impulses is robustly exponentially stable in mean square with exponential convergence rate 0.5.
Case of the Destabilizing Impulses. Study the following impulsive gain matrices:(72)W1k=1.08001.08,W2k=1.08001.08,k∈N+.By choosing α1=0.5, α2=0.5, then the impulses are the destabilizing impulses. In order to find the maximum exponential convergence rate, we first assume that the Markovian jumping may occur at the impulsive time instants. By using the LMI toolbox in MATLAB, we search the maximum exponential convergence rate which is 5.4020 subject to the LMIs (18)–(20), and inf{tk-tk-1}>ln(1.5)/5.4020=0.0751. Then set tk=0.08+tk-1, k∈N+, Δt=0.01. The 2-state Markov chain with r(0)=1 is shown in Figure 3, among which the right continuous Markov chain {r(t),t≥0} is denoted by the solid blue line, and the Markov chain of the impulsive time instants {r(tk),k∈N+} is denoted by the red point, and the black circle is used to judge whether the Markovian jumping occurs at the impulsive time instants, that is, r(tk)-r(tk-Δt). From Figure 3, we can conclude that the Markovian jumping would occur at the impulsive time instants when tk=0.08+tk-1, k∈N+, Δt=0.01, which further verify the correctness of the assumption.
Set the simulation step size h=0.04 and r(0)=1, Δt=0.01. The dynamic behavior of system (10) with the destabilizing impulses in Example 13 is presented in Figure 4, with the initial condition of every state uniformly randomly selected from [-0.1;0.1], s∈[-4.2,0]. Therefore, it can be verified that system (10) with the destabilizing impulses is robustly exponentially stable in mean square.
Case of the Neural-Type Impulses. Study the following impulsive gain matrices:(73)W1k=1001,W2k=1001,k∈N+.By choosing α1=1, α2=1, then the impulses are the neutral-type impulses. In order to find the maximum exponential convergence rate, we first assume that the Markovian jumping may occur at the impulsive time instants. By using the LMI toolbox in MATLAB, we search the maximum exponential convergence rate which is 5.4039 subject to the LMIs (19)-(20), and inf{tk-tk-1}>ln(2)/5.4039=0.1283. Then we set tk=0.15+tk-1, k∈N+, Δt=0.01. The 2-state Markov chain with r(0)=1 is shown in Figure 5, among which the right continuous Markov chain {r(t),t≥0} is denoted by the solid blue line, and the Markov chain of the impulsive time instants {r(tk),k∈N+} is denoted by the red point, and the black circle is used to judge whether the Markovian jumping occurs at the impulsive time instants, that is, r(tk)-r(tk-Δt). From Figure 5, we can conclude that the Markovian jumping would occur at the impulsive time instants when tk=0.15+tk-1, k∈N+, Δt=0.01, which further verify the correctness of the assumption.
Set the simulation step size h=0.05 and r(0)=1, Δt=0.01. The dynamic behavior of system (10) with the neutral-type impulses in Example 13 is presented in Figure 6, with the initial condition of every state uniformly randomly selected from [-0.1;0.1], s∈[-4.2,0]. Therefore, it can be verified that system (10) with the neutral-type impulses is robustly exponentially stable in mean square.
The 2-state Markov chain with tk=0.5+tk-1, k∈N+, Δt=0.001 in Example 13.
The dynamic behavior of system (10) with the stabilizing impulses, with the initial condition of every state uniformly randomly selected from [-0.1;0.1], s∈[-4.2,0] in Example 13.
The 2-state Markov chain with tk=0.08+tk-1, k∈N+, Δt=0.01 in Example 13.
The dynamic behavior of system (10) with the destabilizing impulses, with the initial condition of every state uniformly randomly selected from [-0.1;0.1], s∈[-4.2,0] in Example 13.
The 2-state Markov chain with tk=0.15+tk-1, k∈N+, Δt=0.01 in Example 13.
The dynamic behavior of system (10) with the neutral-type impulses, with the initial condition of every state uniformly randomly selected from [-0.1;0.1], s∈[-4.2,0] in Example 13.
Example 14 (see [16]).
Consider 2D delayed neural networks of neutral-type (62): (74)C=5005,A=111-1,B=111-1,D=-0.500-0.5,gxt=0.25tanhx1t,0.25tanhx2tT,τ1t=0.5τ′+0.5τ′cos1τ′t,τ′>0,τ3t=1.Then system (64) satisfies hypotheses (H1)-(H2) with(75)h1=0,h2=τ′,τ3=1,μ1=0.5,μ3=0,τ=τ′+1,L1=0,L2=diag0.25,0.25,L=diag0.25,0.25.By using the LMI toolbox in MATLAB, we search for the fact that the LMI (64) in Corollary 12 is feasible for any γ≤12.5883 and τ′≤2.0000. A comparison of the maximum upper delay bound (MADB) h2 for different values of γ that guarantee the exponential stability of system (62) is made in Table 1 from which we can see that for this system of Example 14, the results in this paper are less conservative than that in [16].
The maximum allowable delay bound (MADB) h2 for different values of γ.
Methods
γ
0.2000
1.3800
Example 1 in [16]
h2
22.2000
1.0000
Corollary 12
h2
88.1009
15.6047
5. Conclusion
In this paper, delay-dependent robust exponential stability criteria for a class of uncertain impulsive stochastic neural networks of neutral-type with Markovian parameters and mixed time-varying delays have been derived by the use of the Lyapunov-Krasovskii functional method, Jensen integral inequality, free-weight matrix method, and the LMI framework. The proposed results do not require the derivatives of discrete and distributed time-varying delays to be 0 or smaller than 1. Moreover, the main contribution of the proposed approach compared with related methods lies in the use of three types of impulses. Finally, two numerical examples are worked out to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature. One of our future research directions is to apply the proposed method to study the synchronization problem for Markovian jumping chaotic delayed neural networks of neutral-type via impulsive control.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-Destruction Detecting and Engineering Computing under Grant no. 2014QZJ01 and Grant no. 2015QYJ01 and National Natural Science Foundation of China under Grant 61573010 and Grant 11501391.
ZhangH.WangZ.LiuD.A comprehensive review of stability analysis of continuous-time recurrent neural networks20142571229126210.1109/TNNLS.2014.23178802-s2.0-84903269771CaoJ.RakkiyappanR.MaheswariK.Exponential H∞ filtering analysis for discrete-time switched neural networks with random delays using sojourn probabilities2016593387402RakkiyappanR.DharaniS.CaoJ.Synchronization of neural networks with control packet loss and time-varying delay via stochastic sampled-data controller201526123215322610.1109/tnnls.2015.24258812-s2.0-84929008863CaoJ.SivasamyR.RakkiyappanR.Sampled-data H∞ synchronization of chaotic Lur′e systems with time delay201635811835ZhuQ.CaoJ.RakkiyappanR.Exponential input-to-state stability of stochastic Cohen-Grossberg neural networks with mixed delays20157921085109810.1007/s11071-014-1725-2MR33027552-s2.0-84925514252ZhuQ.RakkiyappanR.ChandrasekarA.Stochastic stability of Markovian jump BAM neural networks with leakage delays and impulse control201413613615110.1016/j.neucom.2014.01.0182-s2.0-84897961888ZhuQ.CaoJ.Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays201141234135310.1109/tsmcb.2010.20533542-s2.0-79952006906ZhuQ.CaoJ.Stability analysis for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays20107313–152671268010.1016/j.neucom.2010.05.0022-s2.0-77955329498KwonO. M.ParkJ. H.LeeS. M.ChaE. J.Analysis on delay-dependent stability for neural networks with time-varying delays2013103211412010.1016/j.neucom.2012.09.0122-s2.0-84870389532ZhangG.LinX.ZhangX.Exponential stabilization of neutral-type neural networks with mixed interval time-varying delays by intermittent control: a CCL approach201433237139110.1007/s00034-013-9651-y2-s2.0-84897724897BaoH.CaoJ.Stochastic global exponential stability for neutral-type impulsive neural networks with mixed time-delays and Markovian jumping parameters20111693786379110.1016/j.cnsns.2010.12.027MR2787823ZBL1227.340792-s2.0-79953680155ZhangH.DongM.WangY.SunN.Stochastic stability analysis of neutral-type impulsive neural networks with mixed time-varying delays and Markovian jumping20107313–152689269510.1016/j.neucom.2010.04.0162-s2.0-77955310225GaoY.ZhouW.JiC.TongD.FangJ.Globally exponential stability of stochastic neutral-type delayed neural networks with impulsive perturbations and Markovian switching20127032107211610.1007/s11071-012-0603-z2-s2.0-84870881717ZhengC.-D.GuY.LiangW.WangZ.Novel delay-dependent stability criteria for switched Hopfield neural networks of neutral type201515811712610.1016/j.neucom.2015.01.0612-s2.0-84926524370XiaJ.ParkJ. H.ZengH.Improved delay-dependent robust stability analysis for neutral-type uncertain neural networks with Markovian jumping parameters and time-varying delays20151491198120510.1016/j.neucom.2014.09.0082-s2.0-84912103152LienC.-H.YuK.-W.LinY.-F.ChungY.-J.ChungL.-Y.Global exponential stability for uncertain delayed neural networks of neutral type with mixed time delays200838370972010.1109/TSMCB.2008.9185642-s2.0-44849117844ZhangH.LiuZ.HuangG.-B.Novel delay-dependent robust stability analysis for switched neutral-type neural networks with time-varying delays via SC technique20104061480149110.1109/tsmcb.2010.20402742-s2.0-78149330319LakshmananS.ParkJ. H.JungH. Y.KwonO. M.RakkiyappanR.A delay partitioning approach to delay-dependent stability analysis for neutral type neural networks with discrete and distributed delays20131116818910.1016/j.neucom.2012.12.0162-s2.0-84876710672ZhouW.ZhuQ.ShiP.SuH.FangJ.ZhouL.Adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching parameters201444122848286010.1109/TCYB.2014.23172362-s2.0-84913537576ZhuQ.ZhouW.ZhouL.WuM.TongD.Mode-dependent projective synchronization for neutral-type neural networks with distributed time-delays20141409710310.1016/j.neucom.2014.03.0322-s2.0-84901486301LiX.Global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type2010215124370438410.1016/j.amc.2009.12.068MR25961142-s2.0-76849111658ZhangY.GuD.-W.XuS.Global exponential adaptive synchronization of complex dynamical networks with neutral-type neural network nodes and stochastic disturbances201360102709271810.1109/TCSI.2013.22491512-s2.0-84884909583DengF.LuoQ.MaoX.Stochastic stabilization of hybrid differential equations20124892321232810.1016/j.automatica.2012.06.044MR29569142-s2.0-84864991419MaoX.YinG. G.YuanC.Stabilization and destabilization of hybrid systems of stochastic differential equations200743226427310.1016/j.automatica.2006.09.006ZBL1111.930822-s2.0-33845950815BlytheS.MaoX.LiaoX.Stability of stochastic delay neural networks2001338448149510.1016/S0016-0032(01)00016-3ZBL0991.931202-s2.0-0035400754ZhuQ.CaoJ.Mean-square exponential input-to-state stability of stochastic delayed neural networks201413115716310.1016/j.neucom.2013.10.0292-s2.0-84894040421LakshmikanthamV.BainovD. D.SimeonovP. S.1989SingaporeWorld Scientific10.1142/0906ZhangH.MaT.HuangG.-B.WangZ.Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control201040383184410.1109/TSMCB.2009.20305062-s2.0-77952582245LiH.ChenB.ZhouQ.FangS.Robust exponential stability for uncertain stochastic neural networks with discrete and distributed time-varying delays2008372193385339410.1016/j.physleta.2008.01.0602-s2.0-41949098489HuangT.LiC.DuanS.StarzykJ. A.Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects201223686687510.1109/TNNLS.2012.21921352-s2.0-84868212651KrasovskiiN. N.LidskiiA.Analytical design of controllers in systems with random attributes. I. Statement of the problem, method of solving19612210211025MR0152395ZhengC.-D.WangY.WangZ.New stability results of neutral-type neural networks with continuously distributed delays and impulses20149191880189610.1080/00207160.2013.865727ZBL1329.341182-s2.0-84907594317ZhuQ.CaoJ.Stability of Markovian jump neural networks with impulse control and time varying delays20121352259227010.1016/j.nonrwa.2012.01.021MR29119132-s2.0-84859111285ZhengC.-D.WangY.WangZ.Stability analysis of stochastic fuzzy Markovian jumping neural networks with leakage delay under impulsive perturbations201435131728175510.1016/j.jfranklin.2013.12.0132-s2.0-84894064818DongM.ZhangH.WangY.Dynamics analysis of impulsive stochastic Cohen-Grossberg neural networks with Markovian jumping and mixed time delays2009727–91999200410.1016/j.neucom.2008.12.0072-s2.0-61849133380RakkiyappanR.BalasubramaniamP.Dynamic analysis of Markovian jumping impulsive stochastic Cohen-Grossberg neural networks with discrete interval and distributed time-varying delays20093440841710.1016/j.nahs.2009.02.008MR25616582-s2.0-68749109339JiangH.LiuJ.Dynamics analysis of impulsive stochastic high-order BAM neural networks with Markovian jumping and mixed delays20114214917010.1142/s1793524511001398MR28181802-s2.0-84863121664ZhuQ.CaoJ.Stability analysis of Markovian jump stochastic BAM Neural networks with impulse control and mixed time delays201223346747910.1109/tnnls.2011.21826592-s2.0-84865321328WuS.-L.LiK.-L.ZhangJ.-S.Exponential stability of discrete-time neural networks with delay and impulses2012218126972698610.1016/j.amc.2011.12.0792-s2.0-84862804797WuS. L.LiK.-L.HuangT. Z.Exponential stability of static neural networks with time delay and impulses20115894395110.1049/iet-cta.2010.03292-s2.0-79960613846WuS.-L.LiK.-L.HuangT.-Z.Global exponential stability of static neural networks with delay and impulses: discrete-time case201217103947396010.1016/j.cnsns.2012.02.0132-s2.0-84862793381LuJ.HoD. W. C.CaoJ.A unified synchronization criterion for impulsive dynamical networks20104671215122110.1016/j.automatica.2010.04.0052-s2.0-78049289886ChenW.-H.ZhengW. X.Global exponential stability of impulsive neural networks with variable delay: an LMI approach20095661248125910.1109/tcsi.2008.2006210MR27246712-s2.0-67650377042ZhangW.TangY.WuX.FangJ.-A.Synchronization of nonlinear dynamical networks with heterogeneous impulses20146141220122810.1109/TCSI.2013.22860272-s2.0-84897965396ZhangW.TangY.FangJ.-A.WuX.Stability of delayed neural networks with time-varying impulses201236596310.1016/j.neunet.2012.08.014ZBL1258.341662-s2.0-84866853414WongW. K.ZhangW.TangY.WuX.Stochastic synchronization of complex networks with mixed impulses201360102657266710.1109/TCSI.2013.22443302-s2.0-84884900555HighamD. J.An algorithmic introduction to numerical simulation of stochastic differential equations200143352554610.1137/s0036144500378302MR18723872-s2.0-0035439412KhalilH. K.1996Upper Saddle River, NJ, USAPrentice HallZhuQ.CaoJ.Robust exponential stability of markovian jump impulsive stochastic Cohen-Grossberg neural networks with mixed time delays20102181314132510.1109/TNN.2010.20541082-s2.0-77955510720GuK.An integral inequality in the stability problem of time-delay systemsProceedings of the 39th IEEE Confernce on Decision and ControlDecember 2000Sydney, Australia280528102-s2.0-0034439996WangY.XieL.de SouzaC. E.Robust control of a class of uncertain nonlinear systems199219213914910.1016/0167-6911(92)90097-C2-s2.0-0026910256GronwallT. H.Note on the derivatives with respect to a parameter of the solutions of a system of differential equations191920429229610.2307/1967124MR1502565HalanayA.YorkeJ. A.Some new results and problems in the theory of differential-delay equations197113558010.1137/1013004MR0284675ZBL0216.11902SeuretA.GouaisbautF.Wirtinger-based integral inequality: application to time-delay systems20134992860286610.1016/j.automatica.2013.05.030MR30844752-s2.0-84881481825ParkP. G.KoJ. W.JeongC.Reciprocally convex approach to stability of systems with time-varying delays201147123523810.1016/j.automatica.2010.10.0142-s2.0-78650807125WangZ.LiuY.LiuX.Exponential stabilization of a class of stochastic system with markovian jump parameters and mode-dependent mixed time-delays20105571656166210.1109/TAC.2010.20461142-s2.0-77954546505HuangH.HuangT.ChenX.A mode-dependent approach to state estimation of recurrent neural networks with Markovian jumping parameters and mixed delays201346506110.1016/j.neunet.2013.04.014ZBL1296.931822-s2.0-84878262539YangX.CaoJ.LuJ.Synchronization of randomly coupled neural networks with markovian jumping and time-delay201360236337610.1109/TCSI.2012.22158042-s2.0-84873413526SkorohodA. V.2009American Mathematical SocietyBoydS.El GhaouiL.FeronE.1994Philadelphia, Pa, USASIAM