# State Feedback Stabilization for Neutral-Type Neural Networks with Time-Varying Discrete and Unbounded Distributed Delays

Authors: Yantao Wang; Xue Lin; Xian Zhang
Journal: Journal of Control Science and Engineering
 (2012)

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## Abstract

The problem of stabilization for a class of neutral-type neural networks with discrete and unbounded distributed delays is investigated. By introducing an appropriate Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative, delay-range-dependent and rate-dependent stabilization criteria are presented in the form of LMIs with nonlinear constraints. In order to solve the nonlinear problem, a cone complementarity linearization (CCL) algorithm is offered. In addition, several numerical examples are provided to illustrate the applicability of the proposed approach.


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## Body

## 1. Introduction

In 1943, McCulloch and Pitts [1] proposed the concept of artificial neurons and proved that a single neuron can perform logic functions. This is usually viewed as the beginning of era of artificial neural networks, which is called neural networks for short. Recently, neural networks have received considerable attention due to their wide applications in solving some optimization problems, pattern recognition, image processing, and signal processing. Since time delay is frequently a source of instability and/or oscillation of many practical systems, several approaches have been proposed for analysis and synthesis of delayed neural networks (see [2–8] and the references therein).A neutral-type time-delay system contains delays in the state and its derivatives. Such system can be found in population ecology, lossless transmission lines, heat exchangers, and so forth. Because of its wider applications, the stability for the class of neutral-type systems has received considerable attention in the last several decades (see, e.g., [9–13]). Corresponding to the class of neutral-type time-delay systems, we will also get a class of neutral-type neural networks. Stability criteria for neutral-type neural networks have been proposed in [3, 14–21] by constructing appropriate Lyapunov-Krasovskii functional and applying the linear matrix inequality (LMI) approach. Recently, the stability conditions for the neutral-type neural networks with discrete and unbounded distributed delays were provided in [17–21]. Specifically, Rakkiyappan and Balasubramaniam [21] studied the global asymptotic stability for a class of neutral-type neural networks with two unbounded distributed delays by the so-called LMI technique. Lu [22] designed a state feedback controller stabilizing the neutral-type neural networks with discrete and bounded distributed delays. However, there are no results about the stabilization for neutral-type neural networks with unbounded distributed delay(s), this motivates our research.This paper deals with the stabilization problem of a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays. New sufficient conditions for the existence of the state feedback controller are proposed by introducing a Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative. Since these sufficient conditions are presented in the form of LMIs with nonlinear constraints, the so-called CCL technique is employed to deal with the nonlinear constraints. This allows us to obtain a state feedback gain by LMI Control Toolbox of MATLAB. Several numerical examples are given to illustrate the effectiveness of the proposed approach.


## 2. Problem Formulation

Consider the class of neutral-type neural networks with time-varying discrete and unbounded distributed delays described by the integro differential equation:(1)x˙i(t)=-aixi(t)+∑j=1n‍bijfj(xj(t))+∑j=1n‍cijfj(xj(t-τ(t)))+∑j=1n‍eij∫-∞t‍zj(t-s)fj(xj(s))ds+∑j=1n‍dijx˙j(t-h(t))+∑j=1n‍fij∫-∞t‍zj(t-s)x˙j(s)ds+ℓi+Ii(t),(2)xi(θ)=ϕi(θ),∀θ∈(-∞,0],
where i=1,2,…,n,xi(t) is the state variable of the ith neuron, ai>0 is a constant, bij, cij, dij, eij, and fij are connection weight coefficients of the neurons, fj is the neuron activation function, ℓi is the external bias vector element, Ii(t) is the external control input vector, ϕi(θ) denotes the initial condition, τ(t) and h(t) are time-varying delays satisfying
(3)0≤τ1≤τ(t)≤τ2,τ˙(t)≤τ3,0≤h1≤h(t)≤h2,h˙(t)≤h3,
and zj(t) is a real value nonnegative continuous function on [0,+∞) and satisfies
(4)∫0∞‍zj(s)ds=1.
When Ii(t)≡0 for all i, the global asymptotic stability of the system (1) has been considered in [21]. Here, the control input item Ii(t) in (1) is borrowed from [22].Throughout this paper, we assume that the neuron activation functionsfj (j=1,2,…,n) satisfy the following condition:
(5)|fj(ζ1)-fj(ζ2)|≤Lj|ζ1-ζ2|∀ζ1,ζ2∈ℝ,ζ1≠ζ2,
where Lj (j=1,2,…,n) are known positive constants.Letx*=(x1*,x2*,…,xn*) be an equilibrium point of the system (1). Set yi=xi-xi*,   i=1,…,n. Then the system (1) can be written as
(6)y˙(t)=-Ay(t)+Bg(y(t))+Cg(y(t-τ(t)))+Dy˙(t-h(t))+E∫-∞t‍Z(t-s)g(y(s))ds+F∫-∞t‍Z(t-s)y˙(s)ds+I(t),y(θ)=ϕ(θ),∀θ∈(-∞,0],
where y(t)=[y1(t)y2(t)⋯yn(t)]T, A=diag(a1,a2,…,an), B=[bij], C=[cij], D=[dij], E=[eij], F=[fij], Z(t)=diag(z1(t),z2(t),…,zn(t)), I(t)=[I1(t)⋯In(t)]T, g(y(t))=[g1(y1(t))g2(y2(t))⋯gn(yn(t))]T with gj(yj(t))=fj(yj(t)+xj*)-fj(xj*), and ϕ(θ)=[ϕ1(θ)-x1*⋯ϕn(θ)-xn*]. From (5), we have gj2(ζ)≤Lj2ζ2, j=1,2,…,n, for all ζ∈ℝ.The main purpose of this paper is to develop a delay-range-dependent and rate-dependent condition for the existence of a state feedback controller(7)I(t)=Ky(t),
which stabilizes the neutral-type neural network (6) with time-varying discrete and unbounded distributed delays.A delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained based on a Lyapunov-Krasovskii functional approach accompanied with a CCL technique, a delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained. Set yt(θ)=y(t+θ), θ∈(-∞,0]. Assume that there exists a scalar V>0 such that ∫-∞0‍∥yt(s)∥2ds≤V and ∫-∞0‍∥y˙t(s)∥2ds≤V. This guarantees that the so-called Lyapunov-Krasovskii functional theory can be applied to the considered stabilization problem.


## 3. Main Results

In this section, we will investigate a delay-range-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6). This requires the following several lemmas.Lemma 1 (Jensen inequality [23]).
Given a real symmetry positive-definite matrixP∈ℝn×n, a pair of scalars a and b satisfying b≥a≥0. If a vector-valued function  x:[a,b]→ℝn is derivable on [a,b], then
(8)(b-a)∫ab‍x˙T(w)Px˙(w)dw≥[x(b)-x(a)]TP[x(b)-x(a)].Lemma 2 (Schur complementary lemma [24]).
For a given matrixS:=[S11S12S12TS22] with S11=S11T and S22=S22T, the following conditions are equivalent.(i)
S<0.(ii)
S22<0 and S11-S12S22-1S12T<0.(iii)
S11<0 and S22-S12TS11-1S12<0.Lemma 3 (Cauchy-Schwarz inequality [25]).
If the functionsf and g are integral on [a,b], then
(9)(∫ab‍f(s)g(s)ds)2≤(∫ab‍f2(s)ds)(∫ab‍g2(s)ds).Based on the above three lemmas, the following delay-range-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6) can be investigated by constructing an appropriate Lyapunov-Krasovskii functional and applying the CCL technique.Theorem 4.
For given scalarsh1, h2, h3, τ1, τ2, and τ3 such that h2>h1>0 and τ2>τ1>0, the system (6) subject to (3) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices P,P~,Ti,T~i,Qi(i=1,…,6),Q~4,Q~5,Q~6, Rj(j=1,…,5),R~4,R~5, M:=diag(m1,…,mn), N:=diag(n1,…,nn), N~:=diag(n~1,…,n~n), a matrix K~,  and positive scalars ε1 and ε2,  such that
(10)[Π~A~TA~TA~TA~TA~TA~TA~-T100000A~0-T20000A~00-T3000A~000-T400A~0000-T50A~00000-T6]<0,(11)[-T~1P~P~-Q~4]≤0,[-T~2P~P~-τ12-1R~4]≤0,[-T~3P~P~-τ2-1R~5]≤0,(12)[-T~4P~P~-h12-1Q~5]≤0,[-T~5P~P~-h2-1Q~6]≤0,[-T~6P~P~-N~]≤0,(13)PP~=I,TiT~i=I(i=1,…,6),R4R~4=I,R5R~5=I,Q4Q~4=I,Q5Q~5=I,Q6Q~6=I,NN~=I,where(14)Π~≔e1T(∑i=13‍Ri+∑i=13‍Qi+ε1LTL)e1-e2T((1-τ3)R3-ε2LTL)e2+A~Te1+e1TA~-(1-h3)e5TQ3e5-h12-1(e5-e6)TQ6(e5-e6)-τ12-1(e2-e4)TR5(e2-e4)+e9T(M-ε1I)e9-e11TMe11-e12TNe12-ε2e10Te10-(1-h3)e8TQ4e8-τ12-1(e2-e4)TR4(e2-e4)-τ12-1(e2-e3)TR4(e2-e3)-e3TR1e3-e4TR2e4-τ12-1(e2-e3)TR5(e2-e3)-τ1-1(e1-e3)TR5(e1-e3)-e6TQ1e6-e7TQ2e7-h12-1(e5-e6)TQ5(e5-e6)-h12-1(e5-e7)TQ6(e5-e7)-h12-1(e5-e7)TQ5(e5-e7)-h1-1(e1-e6)TQ6(e1-e6),A~=[-PA+K~000000PDPBPCPEPF],L=diag(L1,…,Ln),ei=[0n×(i-1)In0n×(12-i)](i=1,…,12),h12=h2-h1,τ12=τ2-τ1.Furthermore, when the LMIs (10)–(12) with the constraint (13) are feasible, a desired state feedback gain is given by K=P-1K~.Proof.
Choose a Lyapunov-Krasovskii functional as follows:(15)V(yt)=∑i=16‍Vi(yt),
with
(16)V1(yt)=yT(t)Py(t),(17)V2(yt)=∫t-h(t)t‍[yT(s)Q3y(s)+y˙T(s)Q4y˙(s)]ds+∑i=12‍∫t-hit‍yT(s)Qiy(s)ds+∫t-τ(t)t‍yT(s)R3y(s)ds+∑i=12‍∫t-τit‍yT(s)Riy(s)ds,V3(yt)=∫-τ2-τ1‍∫t+θt‍y˙T(s)R4y˙(s)dsdθ+∫-τ20‍∫t+θt‍y˙T(s)R5y˙(s)dsdθ,V4(yt)=∫-h2-h1‍∫t+θt‍y˙T(s)Q5y˙(s)dsdθ+∫-h20‍∫t+θt‍y˙T(s)Q6y˙(s)dsdθ,V5(yt)=∑j=1n‍mj∫0∞‍zj(δ)∫t-δt‍gj2(yj(ξ))dξdδ,V6(yt)=∑j=1n‍nj∫0∞‍zj(δ)∫t-δt‍y˙j2(ξ)dξdδ.
Set∥yt∥w2=∥y(t)∥2+∫-∞0‍∥yt(θ)∥2dθ+∫-∞0‍∥y˙t(θ)∥2dθ. By some derivation, we have
(18)λmin(P)∥y(t)∥2≤V(yt)≤(a1+a2+a3)∥yt∥w2,
where
(19)a1=λmax(P)+λmax(Q1)+λmax(Q2)+λmax(Q3)+λmax(Q4),a2=λmax(R1)+λmax(R2)+λmax(R3),a3=τ12λmax(R4)+τ2λmax(R5)+h12λmax(Q5)+h2λmax(Q6)+λmax(MLTL)+λmax(N),
which implies that the functional V(yt) is a legitimate due to [26]. Next we deal with the derivatives of Vi(yt)(i=1,…,6).
Set(20)A~1=[-A+K000000DBCEF],ξ(t)=[{∫-∞t‍}yT(t),yT(t-τ(t)),yT(t-τ1),yT(t-τ2),yT(t-h(t)),yT(t-h1),yT(t-h2),y˙T(t-h(t)),gT(y(t)),gT(y(t-τ(t))),(∫-∞t‍Z(t-s)g(y(s))ds)T,(∫-∞t‍Z(t-s)y˙(s)ds)T]T.
When the controller (7) is applied to the system (6), the resultant closed-loop system is obtained as follows:
(21)y˙(t)=A~1ξ(t).
Due to Lemma 1, (3), the derivatives of Vi(yt) (i=1,…,4) along the trajectory of (21) are
(22)V˙1(yt)=2yT(t)Py˙(t)=2yT(t)PA~1ξ(t)=ξT(t)(e1TPA~1+A~1TPe1)ξ(t),(23)V˙2(yt)=yT(t)(∑i=13‍Qi+∑i=13‍Ri)y(t)-(1-h˙(t))yT(t-h(t))Q3y(t-h(t))-(1-h˙(t))y˙T(t-h(t))Q4y˙(t-h(t))-yT(t-h1)Q1y(t-h1)-yT(t-h2)Q2y(t-h2)-(1-τ˙(t))yT(t-τ(t))R3y(t-τ(t))-yT(t-τ1)R1y(t-τ1)-yT(t-τ2)R2y(t-τ2)+y˙T(t)Q4y˙(t)≤ξT(t)e1T(∑i=13‍Qi+∑i=13‍Ri)e1ξ(t)+ξT(t)A~1TQ4A~1ξ(t)-ξT(t)e7TQ2e7ξ(t)-(1-h3)ξT(t)e8TQ4e8ξ(t)-ξT(t)e6TQ1e6ξ(t)-(1-h3)ξT(t)e5TQ3e5ξ(t)-(1-τ3)ξT(t)e2TR3e2ξ(t)-ξT(t)e3TR1e3ξ(t)-ξT(t)e4TR2e4ξ(t),(24)V˙3(yt)=y˙T(t)(τ12R4+τ2R5)y˙(t)-∫t-τ2t-τ1‍y˙T(s)R4y˙(s)ds-∫t-τ2t‍y˙T(s)R5y˙(s)ds=y˙T(t)(τ12R4+τ2R5)y˙(t)-∫t-τ2t-τ(t)‍y˙T(s)R4y˙(s)ds-∫t-τ(t)t-τ1‍y˙T(s)R4y˙(s)ds-∫t-τ2t-τ(t)‍y˙T(s)R5y˙(s)ds-∫t-τ(t)t-τ1‍y˙T(s)R5y˙(s)ds-∫t-τ1t‍y˙T(s)R5y˙(s)ds≤ξT(t)A~1T(τ12R4+τ2R5)A~1ξ(t)-τ12-1ξT(t)(e2-e4)TR4(e2-e4)ξ(t)-τ12-1ξT(t)(e2-e3)TR4(e2-e3)ξ(t)-τ12-1ξT(t)(e2-e4)TR5(e2-e4)ξ(t)-τ12-1ξT(t)(e2-e3)TR5(e2-e3)ξ(t)-τ1-1ξT(t)(e1-e3)TR5(e1-e3)ξ(t),(25)V˙4(yt)=y˙T(t)(h12Q5+h2Q6)y˙(t)-∫t-h2t-h1‍y˙T(s)Q5y˙(s)ds-∫t-h2t‍y˙T(s)Q6y˙(s)ds=y˙T(t)(h12Q5+h2Q6)y˙(t)-∫t-h2t-h(t)‍y˙T(s)Q5y˙(s)ds-∫t-h(t)t-h1‍y˙T(s)Q5y˙(s)ds-∫t-h2t-h(t)‍y˙T(s)Q6y˙(s)ds-∫t-h(t)t-h1‍y˙T(s)Q6y˙(s)ds-∫t-h1t‍y˙T(s)Q6y˙(s)ds≤ξT(t)A~1T(h12Q5+h2Q6)A~1ξ(t)-h12-1ξT(t)(e5-e7)TQ5(e5-e7)ξ(t)-h12-1ξT(t)(e5-e6)TQ5(e5-e6)ξ(t)-h12-1ξT(t)(e5-e7)TQ6(e5-e7)ξ(t)-h12-1ξT(t)(e5-e6)TQ6(e5-e6)ξ(t)-h1-1ξT(t)(e1-e6)TQ6(e1-e6)ξ(t).
By Lemma3 and (4), we can obtain the derivatives of V5(yt) and V6(yt) as follows:
(26)V˙5(yt)=∑j=1n‍mj∫0∞‍zj(δ)gj2(yj(t))dδ-∑j=1n‍mj∫0∞‍zj(δ)gj2(yj(t-δ))dδ=∑j=1n‍mjgj2(yj(t))∫0∞‍zj(δ)dδ-∑j=1n‍mj∫0∞‍zj(δ)dδ×∫0∞‍zj(δ)gj2(yj(t-δ))dδ≤gT(y(t))Mg(y(t))-∑j=1n‍mj(∫0∞‍zj(δ)gj(yj(t-δ))dδ)2=gT(y(t))Mg(y(t))-(∫-∞0‍Z(t-s)g(y(s))ds)T×M(∫-∞0‍Z(t-s)g(y(s))ds)=ξT(t)e9TMe9ξ(t)-ξT(t)e11TMe11ξ(t),(27)V˙6(yt)≤y˙T(t)Ny˙(t)-∑j=1n‍nj(∫0∞‍zj(δ)y˙j(t-δ)dδ)2=ξT(t)A~1TNA~1ξ(t)-ξT(t)e12TNe12ξ(t).
For the functions gj, j=1,2,…,n, using gj2(ζj)≤Lj2ζj2, we get that
(28)0≤-ε1ξT(t)e9Te9ξ(t)+ε1ξT(t)e1TLTLe1ξ(t),0≤-ε2ξT(t)e10Te10ξ(t)+ε2ξT(t)e2TLTLe2ξ(t),
where L=diag(L1,…,Ln). Noting that K=P-1K~, we have A~=PA~1. The combination of (22)–(28) gives
(29)V˙(yt)=∑i=16‍V˙i(yt)≤ξT(t)Π¯ξ(t),
where
(30)Π¯=Π~+A~1T(Q4+τ12R4+τ2R5+h12Q5+h2Q6+N)A~1.
By Lemma2 and (10)–(13), we can get that
(31)[Π~A~1TPA~1TPA~1TPA~1TPA~1TPA~1TPPA~1Π2200000PA~10Π330000PA~100Π44000PA~1000Π5500PA~10000Π660PA~100000Π77]<0,
where
(32)Π22=-PQ4-1P,Π33=-τ12-1PR4-1P,Π44=-τ2-1PR5-1P,Π55=-h12-1PQ5-1P,Π66=-h2-1PQ6-1P,Π77=-PN-1P.
Thus it is easy to see thatV˙(yt)<0, which, together with (7) and [26, Theorem  3.1.6], implies that the system (6) is asymptotically stabilizable. The proof is completed.Remark 5.
A delay-range-dependent and rate-dependent condition under which the system (6) is asymptotically stabilizable is investigated in Theorem 4. In [7, 8], the stability of a class of the neural networks with unbounded distributed delays has been studied; however, the neutral term was not considered in their models. In [18], only a unbounded distributed delay is taken into account. But in this paper the neutral-type neural networks with two unbounded distributed delays are studied. Therefore, Theorem 4 in this paper can be viewed as an extension of the corresponding results in [7, 8, 18].Remark 6.
In order to apply [26, Theorem 3.1.6] to conclude that the system (6) is asymptotically stabilizable, we denote a new norm ∥·∥w2 on a space of functions.
Noting that the stabilization criterion proposed in Theorem4 includes the nonlinear constraint (13), we cannot obtain the desired gain K by using the LMI Controller Toolbox of MATLAB. In order to deal with the nonlinear constraint (13), we offer the following CCL algorithm. The algorithm is available to determine the maximum value of h2 for given h1, h3, τ1, τ2, and τ3 and the corresponding gain K.Algorithm 7 (CCL algorithm).
Step1. Choose a sufficiently small initial h2>h1, such that there exists a feasible solution to (10)–(12) and
(33)[PIIP~]≥0,[NIIN~]≥0,[R4IIR~4]≥0,[R5IIR~5]≥0,[TiIIT~i]≥0(i=1,…,6),[QjIIQ~j]≥0(j=4,5,6).
Set hmax=h2.Step2. Find a feasible set of P0, P~0, N0, N~0, Ti0, T~i0(i=1,…,6), Rl0, R~l0(l=4,5), Qj0, and Q~j0(j=4,5,6) satisfying (10)–(12), (33). Set k=0.Step3. Solve the following LMI problem for the variables K~, P, P~, N, N~, Ti, T~i(i=1,…,6), Rl, R~l(l=4,5), Qj, and Q~j(j=4,5,6):
(34)minsubjectto(8)-(10),(29)trΘk,
where
(35)Θk=PkP~+PP~k+NkN~+NN~k+∑i=16‍(TiT~ik+TikT~i)+∑l=45‍(RlR~lk+RlkR~l)+∑j=46‍(QjQ~jk+QjkQ~j).SetPk+1=P, P~k+1=P~, Nk+1=N, N~k+1=N~, Tik+1=Ti, T~ik+1=T~i(i=1,…,6), Rlk+1=Rl, R~lk+1=R~l(l=4,5), Qjk+1=Qj and Q~jk+1=Q~j(j=4,5,6).Step4. If the LMI. (36)Π^:=[Π~A~1TQ4A~1TR4A~1TR5A~1TQ5A~1TQ6A~1TNQ4A~1-Q400000R4A~10-τ12-1R40000R5A~100-τ2-1R5000Q5A~1000-h12-1Q500Q6A~10000-h2-1Q60NA~100000-N]<0  is feasible for the variables N, Ti(i=1,…,6), Rl(l=4,5), and Qj(j=4,5,6) and the matrices K~ and P obtained in Step 3, then set K=P-1K~ and stop. If the LMI (36) is infeasible within a specified number of iterations, then stop; otherwise, set k=k+1 and go to Step 3.Remark 8.
The idea of the above algorithm is taken from [27]. Compared with the CCL algorithms proposed in [28, 29], the merit of the above algorithm is that more freedoms are provided to test the iteration stop conditions (see [27, Remark  4] for details).Next, we consider the following neutral-type neural network (37), which is a special case of (6). A stability criterion for the neural network (37) has been presented in [21]. The following corollary can be immediately obtained from Theorem 4, which gives a stabilization criterion for the neutral-type neural network (37):
(37)y˙(t)=-Ay(t)+Bg(y(t))+Cg(y(t-h(t)))+Dy˙(t-h(t))+E∫-∞t‍Z(t-s)g(y(s))ds+F∫-∞t‍Z(t-s)y˙(s)ds+I(t),
where
(38)0≤h(t)≤h2,h˙(t)≤h3.Corollary 9.
For given scalarsh2 and h3 such that h2>0, the neutral-type neural network (37) subject to (38) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices P,P~,Qi(i=1,…,4),Q~2,Q~4,Tj,T~j(j=1,2,3),M:=diag(m1,…,mn),N:=diag(n1,…,nn), and N~:=diag(n~1,…,n~n), a matrix K~ and positive scalars ε1 and ε2, such that
(39)Σ:=[Σ~𝒜T𝒜T𝒜T𝒜-T100𝒜0-T20𝒜00-T3]<0,[-T~1P~P~-Q~2]≤0,[-T~2P~P~-h2-1Q~4]≤0,[-T~3P~P~-N~]≤0,(40)PP~=I,TjT~j=I(j=1,2,3),Q2Q~2=I,Q4Q~4=I,NN~=I,
where
(41)Σ~:=e^1T(Q1+Q3+ε1LTL)e^1-e^2((1-h3)Q1-ε2LTL)e^2+𝒜Te^1+e^1T𝒜-(1-h3)e^4TQ2e^4-e^3TQ3e^3+e^5T(M-ε1I)e^5-e^7TMe^7-h2-1(e^2-e^3)TQ4(e^2-e^3)-h2-1(e^1-e^2)TQ4(e^1-e^2)-e^8TNe^8-ε2e^6Te^6,𝒜=[-PA+K~00PDPBPCPEPF],e^i=[0n×(i-1)In0n×(8-i)],i=1,…,8.
Furthermore, when the LMIs (39) with the constraint (40) are feasible, a desired state feedback gain is given by K=P-1K~.Proof.
If we use the followingξ^(t) and V^(yt) instead of ξ(t) and V(yt) in the proof of Theorem 4, respectively, then the proof can be easily completed:
(42)V^(yt)=∑i=15‍V^i(yt),
with
(43)V^1(yt)=yT(t)Py(t),V^2(yt)=∫t-h(t)t‍[yT(s)Q1y(s)+y˙T(s)Q2y˙(s)]ds+∫t-h2t‍yT(s)Q3y(s)ds,V^3(yt)=∫-h20‍∫t+θt‍y˙T(s)Q4y˙(s)dsdθ,V^4(yt)=∑j=1n‍mj∫0∞‍Zj(δ)∫t-δt‍gj2(yj(ξ))dξdδ,V^5(yt)=∑j=1n‍nj∫0∞‍Zj(δ)∫t-δt‍y˙j2(ξ)dξdδ.ξ^(t)=[{∫-∞t‍}yT(t),yT(t-h(t)),yT(t-h2),y˙T(t-h(t)),gT(y(t)),gT(y(t-h(t))),(∫-∞t‍Z(t-s)g(y(s))ds)T,(∫-∞t‍Z(t-s)y˙(s)ds)T]T.The following system model is a practical partial element equivalent circuit (PEEC) that is described in [30, Figure 1] by Bellen et al.:
(44)y˙(t)=Ay(t)+By(t-τ)+Cy˙(t-τ)+I(t),(45)y(θ)=ϕ(θ),∀θ∈[-τ,0],
where τ is a positive scalar representing the system delay.It should be emphasized that the matrixA in (44) is not necessarily diagonal, which is different from the one in (6). However, one can easily find that the proof of Theorem 4 is always available whether the matrix A is diagonal or not. For this reason, we can derive the following Corollaries 10 and 12 by a method similar to obtaining Theorem 4.Corollary 10.
For a given scalarτ>0, the neutral-type system (44) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices P, P~,  Qi(i=1,…,3), Q~2, Q~3, Tj, and T~j(j=1,2) and a matrix K~, such that
(46)Ξ:=[Ξ~AˇTAˇTAˇ-T10Aˇ0-T2]<0,(47)[-T~1P~P~-Q~2]≤0,[-T~2P~P~-τ-1Q~3]≤0,(48)PP~=I,TjT~j=I(j=1,2),Q2Q~2=I,Q3Q~3=I,
where
(49)Ξ~:=eˇ1TQ1eˇ1-eˇ2TQ1eˇ2+AˇTeˇ1+eˇ1TAˇ-eˇ3TQ2eˇ3-τ-1(eˇ1-eˇ2)TQ3(eˇ1-eˇ2),Aˇ=[PA+K~PBPC],eˇi=[0n×(i-1)In0n×(3-i)],i=1,2,3.
Furthermore, when the LMIs (46) and (47) with the nonlinear constraint (48) are feasible, a desired state feedback gain is given by K=P-1K~.Remark 11.
Similar to Algorithm7, one can solve the stabilization criteria proposed in Corollaries 9 and 10.Corollary 12.
For a given scalarτ>0, the unforced system of (44) is asymptotically stable if there exist real symmetry positive-definite matrices P and Qi(i=1,2,3) such that
(50)Θ:=eˇ1TQ1eˇ1-eˇ2TQ1eˇ2+A-TPeˇ1+eˇ1TPA--eˇ3TQ2eˇ3-τ-1(eˇ1-eˇ2)TQ3(eˇ1-eˇ2)+A-T(Q2+τQ3)A-<0,
where
(51)A-=[ABC],
and eˇi is defined as in Corollary 10.


## 4. Numerical Examples

In this section, we will illustrate our approach by several numerical examples.Example 13.
Consider the neutral-type neural network (6), where
(52)A=[2003],B=[-0.340-0.10.1],C=[0.10.2-0.15-0.18],D=[0.1000.1],E=[0.41-0.50.690.31],F=[0.3-0.150.5-0.2],Lj=1,j=1,2,τ3=0.5,τ2=4.8,τ1=4.1,h3=0.5,h1=4.5,h2=7.5,g(y(s))=tanh(y(s)2).
Solving the LMIs (10)–(12) with nonlinear constraint (13) by Algorithm 7 and the LMI Control Toolbox of MATLAB, we can get a desired state feedback gain stabilizing the system as follows:
(53)K=[0.7763-0.1070-0.10701.7242].
The state response curves of the closed-loop system are given in Figures1 and 2, where the initial functions in Figures 1 and 2 are, respectively, chosen as ϕ(θ)=[-0.1-20], θ∈(-∞,0] and ϕ(θ)=[eθe2θ], θ∈(-∞,0].State responses of the closed-loop system forz1(s)=z2(s)=1-|s-1|, 0≤s≤2.
(a)
(b)State responses of the closed-loop system forz1(s)=z2(s)=(2/π)e-(1/2)s2, s∈(-∞,0].
(a)
(b)Example 14.
Consider the system (37) with the following parameter matrices:
(54)A=[3.6003.6],B=[1.1980.10.11.198],C=[0.10.160.050.1],D=[0.2000.2],E=[0.4-0.20.30.2],F=[0.3-0.150.5-0.2],Lj=1,j=1,2,ϕ(θ)=[-0.1-20],θ∈(-∞,0],g(y(s))=tanh(y(s)2),z(s)=1-|s-1|(0≤s≤2),h2=8.
When h3=0.15 by the LMI Control Toolbox of MATLAB, it can be verified that the LMI in [21, Theorem 3.1] is not feasible, so it cannot guarantee that the unforced system is stable. Now we check the stabilization condition proposed in Corollary 9 of this paper and find that it is feasible. So, by Corollary 9 the system is asymptotically stabilizable by state feedback controller, and a desired state feedback gain is given by
(55)K=[1.6655-0.3168-0.31681.8069],
and, further, the state response curves of the closed-loop system are given in Figure 3.
Whenh3=0, by the LMI Control Toolbox of MATLAB, it can be verified that the LMI in [21, Theorem 3.2] is not feasible, which cannot guarantee the unforced system is stable. However, the stabilization condition proposed in Corollary 9 can be satisfied. Hence, by Corollary 9, the neutral-type neural network is asymptotically stabilizable via state feedback controller, and a desired state feedback controller gain is given by
(56)K=[1.6847-0.3110-0.31101.8198],
and, further, the state response curves of the closed-loop system are given in Figure 4.State responses of the closed-loop system whenh3=0.15.
(a)
(b)State responses of the closed-loop system whenh3=0.
(a)
(b)Example 15.
Consider the PEEC system (44) with the following parameter matrices:
(57)A=100×[β123-9012-6],B=100×[10-3-0.5-0.5-1-0.5-1.50],C=172[-152403-241].
Applying Corollaries12 and 10, the maximum delay bounds τM for the various β are presented in Table 1, which clearly shows the effectiveness of our approach. Through Table 1, we can get that the stability criterion proposed in this paper is less conservative than the ones in [31–33].
The following example is provided for applying the approach proposed in this paper to a realistic problem which is motivated by the small PEEC model in [30, Figure 1].Table 1
Allowable upper bound ofτM for different β.
β
-
2.121
-
2.122
-
2.123
-
2.124
-
2.125
[31, Theorem 3.1]
220.2
220.1
201.0101
202.5203
163.2513
[32, Theorem  1]
401.5
401.4764
401.4507
401.4251
401.3979
[33, Theorem  1]
2202.57
2200.34
2198.331
2214.12
2229.1
Corollary12  in this paper
3000.1
3000.11
3000.1
3000.1
3000.11
Corollary10  in this paper
11000
9000
6000
6000
8000Example 16.
Consider the PEEC system (44) with
(58)A=100×[-7123-9012-6],B=100×[10-3-0.5-0.5-1-0.5-1.50],C=172[-152403-241],τ=1.
When I(t)≡0, Bellen et al. [30] gave sufficient conditions for the asymptotic stability of the zero solution to (44) by utilizing a suitable reformulation of the system.
By applying Algorithm7 and the LMI Control Toolbox of MATLAB to solve LMIs (46) and (47) with the nonlinear constraint (48), a required state feedback gain is obtained as follows:
(59)K=[66.1662-283.9848-0.7157-487.718545.6016-31.1715-74.6213-146.45001.7714].
The simulation results for the PEEC system are shown by Figures 5 and 6 with the initial value ϕ(θ)=[sin(t)sin(2t)sin(3t)]T. Figure 5 represents the state responses of the PEEC system when I(t)≡0, and Figure 6 represents the state responses of the resultant closed-loop system. From Figures 5 and 6, we can see that both the open-loop and closed-loop systems are stable, however, the speed of convergence towards the null point in the closed-loop system is faster than one in the open-loop system.Figure 5
State responses of the open-loop system.Figure 6
State responses of the closed-loop system.


## 5. Conclusions

In this paper, the delay-range-dependent and rate-dependent stabilization criteria for a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays are established. The criteria are derived by constructing an appropriate Lyapunov-Krasovskii functional and using certain matrix technique. A CCL algorithm is developed to obtain the state feedback gainK. Numerical examples are provided to show that our results are more suitable than some existing ones, which illustrate the merits of the proposed approach.



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*Source: 517157-2012-09-10.xml*