1 Introduction

In recent years,neural networks(especially switched neural networks,recurrent neural networks,Hop field neural networks,and cellular neural networks)have been successfully applied in many areas such as pattern recognition,associative memory,image processing,fault diagnosis,and combinatorial optimization.Many researchers focused on studying the existence,uniqueness,and global robust asymptotic stability of the equilibrium point in the presence of time delays and parameter uncertainties for various classes of nonlinear neural networks(see[1–6]).When a neural network incorporates abrupt changes in its structure,a Markovian jump system is very appropriate to describe its dynamics.Thus,the problem of stochastic robust stability for uncertain delayed neural networks with Markovian jumping parameters is investigated via LMI technique in[7–13].

The linear matrix inequality approach is one of the most extensively used in recent publications.For instance,in[14–18],a class of switched Hop field neural networks with time-varying delay by integrating the theory of switched systems with neural networks with time-varying delay are reported,global exponential stability conditions for switched Hop field neural networks with time-varying delay are addressed on the basis of the Lyapunov-Krasovskii functional approach,and a delay-dependent robust stability criteria are presented by employing LMIs and free weighting matrices methods.

In practical implementations,uncertainties are inevitable in neural networks because of the existence of modeling errors and external disturbance.It is important to ensure the neural networks system is stable under these uncertainties(see[19–21]).Both time delays and uncertainties can destroy the stability of neural networks in an electronic implementation.Therefore,it is of great theoretical and practical importance to investigate the robust stability for delayed neural networks with uncertainties(see[22–26]).

In the recent years,H∞ concept was proposed to reduce the effect of the disturbance input on the regulated output to within a prescribed level.Hence,H∞ finite time boundness for switched neural networks with time varying delays takes considerable attention[27–33].Usually,some performance indexes,that is,H∞ and L∞,are employed to deal with external disturbances.Many results are developed with these performance indexes(see[34–39]references therein).

To the best of authors knowledge,there is no results available on the H∞control for Markovian jumping neutral-type neural networks(MJNNs)time varying delays.Motivated by this,in this article,we analyze the adaptive finite time stability for the MJNNs.LMI-matrixbased criteria for determining finite time stability for the MJNNNs are developed.

This article is organized as follows.In Section 2,we presents problem formulation,notations,definitions,and technical lemmas.In Section 3,stability conditions are proposed for delayed neural network systems;a reliable H∞controller is derived to guarantee the exponential stability with uncertainties of the resulting closed-loop neural network systems.Section 4 illustrative examples and comparison results are given to show the conservativeness and effectiveness of the proposed results.

2 Notations

Throughout this article,we will use the notation A＞0 to denote that the matrix A is a symmetric and positive definite matrix.Let(⊗,F,{Ft}t≥0,P)be a complete probability space with a filtration{Ft}t≥0satisfying the usual conditions(it is right continuous and F0contains all P-null sets);be the family of all bounded,F0-measurable,C([−τ,0];Rn)-valued random variables ξ={ξ(θ): −τ≤ θ ≤ 0}such thatThe mathematical expectation operator with respect to the given probability measure P is denoted by E{·}.The shorthand diag{···}denotes the block diagonal matrix. ‖ ·‖ stands for the Euclidean norm.Moreover,the notation∗always denotes the symmetric block in one symmetric matrix.Let{ηt,t≥ 0}be a homogeneous, finite-state Markovian process with right continuous trajectories and taking values in finite set S={1,2,···,s}with a given probability space(⊗,F,P)and the initial model η0. Π =[πij]s×s,i,j ∈ S,which denotes the transition rate matrix with transition probability

where Δt ＞ 0,and πij ≥ 0 is the transition rate from mode i to mode j satisfying πij ≥ 0 for i/=j with

2.1 Problem formulation and preliminaries

Consider the following Markovian jumping neural networks of neutral type with distributed time varying delays

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where x(t)=[x1(t),x2(t),···,xn(t)]T ∈ Rnis the state,u(t) ∈ Rlis the control input,w(t)∈ Rnis the disturbance input which belongs to L2[0,∞),and z(t)∈ Rqis the controlled output.f(x(t))is the neuron activation function,and φ(θ)is a continuous vectorvalued initial function.For eachin which Diare a positive diagonal matrices,Ai,Bi,Ci,Di,Ei,F1i,F2i,Gi,J1i,J2i∈ Rn×nare the weight connection matrices with appropriate dimensions,and ΔAi(t),ΔBi(t),ΔCi(t),ΔDi(t),ΔJ1i(t)are uncertain real-valued matrices.The variables τ(t),σ(t),and ρ(t)represent respectively the time varying delay,distributive,and neutral delays satisfyingandand d2are positive constants,and

Assumption 1 Assume that the uncertainties are norm-bounded and of the form

Definition 2.4([37]Finite-time H∞performance) For a given time constant Tf,neural networks(2.1)is said to be robust finite-time H∞performance with respect toR,η(t))if the networks is finite-time bounded and the following inequality holds

针铁矿对Cu2+、Pb2+的吸附能力随着pH值的升高而变强；针铁矿表面的羟基是其吸附和固化金属离子的原因；利用酸碱滴定法和格氏图处理方法，能够计算出针铁矿表面活性位点密度，针铁矿表面活性位点密度为9.107 site/nm2。

Assumption 2 For a given time constant Tf,the external disturbance w(t)satisfies

Assumption 3 The activation functions satisfy the following condition,for any i=1,2,···,n,there exist constantssuch that

For presentation convenience,we denote

Definition 2.1([8]Finite-time stability) For a given time constant τ＞0,neural networks(2.1)is said to be stochastically finite-time stable with respect toif

where c2 ＞ c1 ＞ 0,R is a positive definite matrix,and η(t)is a switching signal.

Remark 2.2 Consider neural network(2.1)with u(t)≡0 and w(t)≡0,the neural networks is said to be uniformly finite-time stable with respect toif(2.4)holds.

Definition 2.3([37]Finite-time boundedness) For a given time constant Tf,neural networks(2.1)with u(t)≡ 0 is said to be finite-time bounded with respect toif condition(2.4)holds,where c2＞ c1＞ 0,R is a positive definite matrix,η(t)is a switching signal,and w(t)satisfy(2.3).

共直流母线分布式系统具有稳定性更高,容量更大,为分布式发电提供了很好的模式,有很好的发展前景。因此其控制方法[2-3]、能量管理策略[4-5]等正成为研究热点。

where M1i,M2i,M3i,M4i,M5iare known real-valued constant matrices with appropriate dimensions,Ωiare unknown and possibly time-varying matrix with Lebesgue measurable elements satisfying

Definition 2.5([33]Robust finite-time H∞control) For a given time constant Tf,neural networks(2.1)is said to be robust finite-time stabilizable with H∞disturbance attenuation level γ,if there exists a controller u(t)=Kη(t)x(t),t∈ (0,Tf],such that(i)The corresponding closed-loop neural network is finite-time bounded;(ii)Under zero initial condition,inequality(2.5)holds for any w(t)satisfying Assumption 2.

Lemma 2.6([25]) For any constant matrix M ∈ Rn×n,M=MT ＞ 0,scalar η2＞ η1≥ 0,and vector function w:[η1,η2] → Rnsuch that the integrations concerned are well defined,then we have

Lemma 2.7([40]) Let U,V,W,and X be real matrices of appropriate dimensions with X satisfying X=XT,then for all VTV≤I,X+UV W+UTVTWT＜0,if and only if there exists a scalar δ＞ 0 such that X+ δUUT+ δ−1WTW ＜ 0.

3 Main Results

3.1 Finite-time boundedness analysis

In this section,we consider the problem of finite time boundedness for the following neural networks

Theorem3.1Consider the neural networks(3.1),and letIf there exist positive scalars α,λk,(k=1,2,···,7),positive definite matrices Pi,Q1,Q2,Q3,S1,S2,T,and positive diagonal matrices U1,U2with appropriate dimensions,such that the following LMI holds,

with

andand g=max(X).Then,the network is finite-time bounded with respect to

Proof Choose the following Lyapunov-Krasovskii functional

where

Taking the derivative of the V(t)along the trajectory of(3.1),we have

Using Lemma 2.6,we obtain

By Assumption 3,it is obtained that

can be compactly written as

Then,for any positive diagonal matrices U1and U2,the following inequalities hold true

Combining and adding(3.4)–(3.11),we obtain

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here

with

Using Schur complement lemma,we have

pre-multiply and post multiply by the term of(3.14)by diagand then by using fact that,and replacingwe get Λ＜0.From(3.2),we have

Then,

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Then,under the controllerthe systems is robustly finite time stable with H∞performance indexwhich satisfies

On the other hand,

We obtain

then it holds that

here

By Definition 2.1,we have

According to Definition 2.3,we know that neural networks(3.1)is finite-time bounded with respect toThus,the proof is completed.

In this section,we present numerical examples to illustrate the effectiveness and advantage of the obtained theoretical results.

Example 2 Consider the neural networks(2.1)with parameters as follows:

Proof The proof is similar to that of Theorem 3.1,so it is omitted here.

3.2 Finite time H∞control

Consider the neural networks(2.1),under the controller u(t)=Kη(t)x(t),t∈ (0,Tf],the corresponding neural network is given by

Theorem 3.3Consider the neural networks(3.23)–(3.25),and letIf there exist positive scalars α,τ,σ,ρ,δ1,δ2,positive definite symmetric matrices Pi,Q1,Q2,Q3,S1,S2with appropriate dimensions and positive diagonal matrices U1,U2,such that the following inequalities hold,

where

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Noting that λwe have λmax(Pi)＜1,then

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ProofReplacing Di,Ai,Bi,Ci,Giin the left side of(3.26)withJ2iKi,and using Schur complement lemma,we can obtain

Letthen it can be obtained that

By Assumption 1,we have

with

By Lemma 2.7,we can get

Using Schur complement lemma,from(3.26)we obtain Ψi＜ 0.Thus,the proof is completed.

4 Numerical Example

Corollary 3.2Consider neural networks(3.1)withThen,the networks is finite-time bounded,if there exist positive scalars α,λk,k=(1,2,···,7),positive definite matrices Pi,Q1,Q2,Q3,S1,S2,and positive diagonal matrices U1,U2with appropriate dimensions,such that the following LMI holds.

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Example 1 Consider neural networks(3.1)with parameters as follows

X1=diag{0,0},X2=diag{−2,2}.For the given values of Tf=0.7,τ=0.3,=0.5,d1=0.6,d2=0.4,=0.8,α=1.5,c1=0.01,d=0.2,by Theorem 3.1 we know that the optimal value of depends on parameter α.By solving the matrix inequalities(3.2)–(3.3),we can get the optimal bound of with different value of α in each subsystems.The smallest bound can be obtained as=0.4315 when α=1.5.

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M1i=M2i=M3i=M4i=M5i=0.3I,X1=diag{0,0},X2=diag{−2,2}.For the given values of Tf=5,¯τ=0.5,¯ρ=0.2,d1=0.9,d2=0.3,¯σ=1,α=4,c2=4,d=2,µ=2,R=I and taking γ2as optimized variable with a fixed α,by solving the optimal value problem for each subsystem according to Theorem 3.4,we get the following conclusion.For subsystem 1,when α ∈ [1.618,11.512],the LMI in Theorem 3.3 has feasible solution,and γmin=0.6000.

By solving the matrix equalities in Theorem 3.3,we obtain

Example 3 Consider the class of Markovian jumping neural networks[41]

with parameters as follows:

X1=diag{0,0},X2=diag{1,1}.For the given values of Tf=5,=0.26,d1=0.2,=1,c1=0.5,d=2,R=I,by solving the optimal value problem for each subsystem according to Corollary 3.2,we get the minimal value of c2=1.8101.It should be mentioned that the minimal c2in[41]is 5.4296.Thus delayed Markovian jumping neural networks is stochastically finite-time bounded with respect to(c1,c2,Tf)with minimal c2smaller than that in[41].Therefore,the stability conditions proposed in this article are less conservative than[41].

Example 4 Consider the class of Markovian jumping neural networks[42],

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with parameters as follows:

the activation function X1=diag{0.2,0.2,0.2},X2=diag{0.4,0.4,0.4}.For the given values of Tf=5,d1=0.2,c1=0.5,c2=0.3,d=2,R=I,by solving the problem for each subsystem according to Corollary 3.2,we get the upper bound of¯τ=4.450,whereas for the upper bound of¯τ=2.5 in[42],it shows that delayed Markovian jumping neural networks is stochastically finite-time bounded with respect to(c1,c2,T).Hence the results in this article are less conservative than that in[42].

Conclusion

In this article,we investigate the problem of robust finite-time H∞control for a class of uncertain neutral-type neural networks with distributed time varying delays.The finite-time boundedness analysis and finite-time H∞controller design for neural networks system with H∞ disturbance attenuation level γ are studied.Numerical examples are provided to show the effectiveness of the proposed method.The results are compared with the existing results to show the conservativeness.

语法测试是检验教学效果及学生学习效果的重要手段之一。但语法教学现状的研究仅仅对中考、高考、大学英语四/六级、英语专业四/八级等有所研究，对于国外几大英语考试如剑桥商务英语、托业、托福、雅思、美国研究生入学资格考试等的研究不足。除此以外，对于校本测试（如期末考试）等的研究也较少。这些研究上的不足将会影响到教学手段及教学方法的调整。

池州市位于长江中下游冲积平原与丘陵接合部位，总体地形南高北低。本区自晚第三季以来，由于遭受强烈剥蚀与侵蚀作用，山丘多低矮，构成剥蚀低山丘陵地貌，分布于本区东南面，最高峰为太朴山，标高+723.0 m。局部为可溶岩出露区，岩溶微地貌发育。向北地势渐低，北部为长江冲积平原及湖盆，地表平坦而开阔。

试验所用模型和模拟的模型相同，拉形高度15mm,加工深度15mm,成形角度40°。自行设计工装，加工工装示意图如图5所示，包括上支撑板、中间板、下支撑板、支撑模以及提供动力的千斤顶，工装主要功能是板料实现预拉形。所用材料为工业纯铝1060，板料厚度为1mm。

Figure 1 State trajectories of the system in Example 1

Figure 2 State trajectories of the system in Example 2

Figure 3 State trajectories of the system in Example 3

Figure 4 State trajectories of the system in Example 4

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