# Delay-Dependent Guaranteed Cost Controller Design for Uncertain Neural Networks with Interval Time-Varying Delay

Authors: M. Rajchakit; P. Niamsup; T. Rojsiraphisal; G. Rajchakit
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper studies the problem of guaranteed cost control for a class of uncertain
delayed neural networks. The time delay is a continuous function belonging to a given
interval but not necessary to be differentiable. A cost function is considered as a
nonlinear performance measure for the closed-loop system. The stabilizing controllers
to be designed must satisfy some exponential stability constraints on the closed-loop
poles. By constructing a set of augmented Lyapunov-Krasovskii functionals combined
with Newton-Leibniz formula, a guaranteed cost controller is designed via memoryless
state feedback control, and new sufficient conditions for the existence of the guaranteed
cost state feedback for the system are given in terms of linear matrix inequalities
(LMIs).  Numerical examples are given to illustrate the effectiveness of the obtained
result.


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## Body

## 1. Introduction

The last few decades have witnessed the use of artificial neural networks (ANNs) in many real-world applications and have offered an attractive paradigm for a broad range of adaptive complex systems. In recent years, ANNs have enjoyed a great deal of success and have proven useful in wide variety pattern recognition feature-extraction tasks. Examples include optical character recognition, speech recognition, and adaptive control, to name a few. To keep the pace with the huge demand in diversified application areas, many different kinds of ANN architecture and learning types have been proposed to meet varying needs as robustness and stability. Stability and control of neural networks with time delay have attracted considerable attention in recent years [1–8]. In many practical systems, it is desirable to design neural networks which are not only asymptotically or exponentially stable but can also guarantee an adequate level of system performance. In the area of control, signal processing, pattern recognition, and image processing, delayed neural networks have many useful applications. Some of these applications require that the equilibrium points of the designed network be stable. In both biological and artificial neural systems, time delays due to integration and communication are ubiquitous and often become a source of instability. The time delays in electronic neural networks are usually time varying, and sometimes vary violently with respect to time due to the finite switching speed of amplifiers and faults in the electrical circuitry. Guaranteed cost control problem [9–12] has the advantage of providing an upper bound on a given system performance index and thus the system performance degradation incurred by the uncertainties or time delays is guaranteed to be less than this bound. The Lyapunov-Krasovskii functional technique has been among the popular and effective tool in the design of guaranteed cost controls for neural networks with time delay. Nevertheless, despite such diversity of results available, most existing work either assumed that the time delays are constant or differentiable [13–16]. Although, in some cases, delay-dependent guaranteed cost control for systems with time-varying delays was considered in [12, 13, 15], the approach used there can not be applied to systems with interval, nondifferentiable time-varying delays. To the best of our knowledge, the guaranteed cost control and state feedback stabilization for uncertain neural networks with interval, non-differentiable time-varying delays have not been fully studied yet (see, e.g., [4–26] and the references therein), which are important in both theories and applications. This motivates our research.In this paper, we investigate the guaranteed cost control for uncertain delayed neural networks problem. The novel features here are that the delayed neural network under consideration is with various globally Lipschitz continuous activation functions, and the time-varying delay function is interval, non-differentiable. A nonlinear cost function is considered as a performance measure for the closed-loop system. The stabilizing controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles. Based on constructing a set of augmented Lyapunov-Krasovskii functionals combined with Newton-Leibniz formula, new delay-dependent criteria for guaranteed cost control via memoryless feedback control are established in terms of LMIs, which allow simultaneous computation of two bounds that characterize the exponential stability rate of the solution and can be easily determined by utilizing Matlabs LMI control toolbox.The outline of the paper is as follows. Section2 presents definitions and some well-known technical propositions needed for the proof of the main result. LMI delay-dependent criteria for guaraneed cost control and a numerical examples showing the effectiveness of the result are presented in Section 3. The paper ends with conclusions and cited references.


## 2. Preliminaries

The following notation will be used in this paper.ℝ+ denotes the set of all real nonnegative numbers; ℝn denotes the n-dimensional space with the scalar product 〈x,y〉 or xTy of two vectors x,y, and the vector norm ∥·∥; Mn×r denotes the space of all matrices of (n×r)-dimensions. AT denotes the transpose of matrix A; A is symmetric if A=AT; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λmax(A)=max{Reλ;λ∈λ(A)}.xt:={x(t+s):s∈[-h,0]}, ∥xt∥=sups∈[-h,0]∥x(t+s)∥; C1([0,t],ℝn) denotes the set of all ℝn-valued continuously differentiable functions on [0,t]; L2([0,t],ℝm) denotes the set of all the ℝm-valued square integrable functions on [0,t].MatrixA is called semipositive definite (A≥0) if 〈Ax,x〉≥0, for all x∈ℝn;A is positive definite (A>0) if 〈Ax,x〉>0 for all x≠0;A>B means A-B>0. The notation diag{⋯} stands for a block-diagonal matrix. The symmetric term in a matrix is denoted by *.Consider the following uncertain neural networks with interval time-varying delay:(2.1)x˙(t)=-(A+ΔA(t))x(t)+(W0+ΔW0(t))W0f(x(t))+(W1+ΔW1(t))g(x(t-h(t)))+Bu(t),t≥0,x(t)=ϕ(t),t∈[-h1,0],
where x(t)=[x1(t),x2(t),…,xn(t)]T∈ℝn is the state of the neural; u(·)∈L2([0,t],ℝm) is the control; n is the number of neurals, and
(2.2)f(x(t))=[f1(x1(t)),f2(x2(t)),…,fn(xn(t))]T,g(x(t-h(t)))=[g1(x1(t-h(t))(t)),g2(x2(t-h(t))(t)),…,gn(xn(t-h(t)))]T,
are the activation functions; A=diag(a¯1,a¯2,…,a¯n),a¯i>0 represents the self-feedback term; B∈Rn×m is control input matrix; W0,W1 denote the connection weights, the discretely delayed connection weights and the distributively delayed connection weight, respectively; the time-varying uncertain matrices ΔA(t),ΔW0(t), and ΔW1(t) are defined by
(2.3)ΔA(t)=EaFa(t)Ha,ΔW0(t)=Ew0Fw0(t)Hw0,ΔW1(t)=Ew1Fw1(t)Hw1,
where Ea,Ew0,Ew1,Ha,Hw0,andHw1 are known constant real matrices with appropriate dimensions. Fa(t),Fw0(t),andFw1(t) are unknown uncertain matrices satisfying
(2.4)FaT(t)Fa(t)≤I,Fw0T(t)Fw0(t)≤I,Fw1T(t)Fw1(t)≤I,t∈R+.
The time-varying delay function h(t) satisfies the condition
(2.5)0≤h0≤h(t)≤h1.
The initial functions ϕ(t)∈C1([-h1,0],Rn), with the norm
(2.6)∥ϕ∥=supt∈[-h1,0]∥ϕ(t)∥2+∥ϕ˙(t)∥2.
In this paper we consider various activation functions and assume that the activation functions f(·),g(·) are Lipschitzian with the Lipschitz constants fi,ei>0:
(2.7)|fi(ξ1)-fi(ξ2)|≤fi|ξ1-ξ2|,i=1,2,…,n,∀ξ1,ξ2∈ℝ,|gi(ξ1)-gi(ξ2)|≤ei|ξ1-ξ2|,i=1,2,…,n,∀ξ1,ξ2∈ℝ.
The performance index associated with the system (2.1) is the following function:
(2.8)J=∫0∞‍f0(t,x(t),x(t-h(t)),u(t))dt,
where f0(t,x(t),x(t-h(t)),u(t)):R+×Rn×Rn×Rm→R+ is a nonlinear cost function that satisfies
(2.9)∃Q1,Q2,R:f0(t,x,y,u)≤〈Q1x,x〉+〈Q2y,y〉+〈Ru,u〉,
for all (t,x,u)∈R+×Rn×Rm and Q1,Q2∈Rn×n,R∈Rm×m are given symmetric positive definite matrices. The objective of this paper is to design a memoryless state feedback controller u(t)=Kx(t) for system (2.1) and the cost function (2.8) such that the resulting closed-loop system
(2.10)x˙(t)=-[(A+EaFa(t)Ha)-BK]x(t)+(W0+Ew0Fw0(t)Hw0)f(x(t))+(W1+Ew1Fw1(t)Hw1)g(x(t-h(t)))
is exponentially stable and the closed-loop value of the cost function (2.10) is minimized.Definition 2.1.
Givenα>0. The zero solution of closed-loop system (2.8) is α-exponentially stabilizable if there exists a positive number N>0 such that every solution x(t,ϕ) satisfies the following condition:
(2.11)∥x(t,ϕ)∥≤Ne-αt∥ϕ∥,∀t≥0.Definition 2.2.
Consider the control system (2.1). If there exists a memoryless state feedback control law u*(t)=Kx(t) and a positive number J* such that the zero solution of the closed-loop system (2.10) is exponentially stable and the cost function (2.8) satisfies J≤J*, then the value J* is a guaranteed constant and u*(t) is a guaranteed cost control law of the system and its corresponding cost function.We introduce the following technical well-known propositions, which will be used in the proof of our results.Proposition 2.3 (Schur complement lemma [27]).
Given constant matricesX,Y, and Z with appropriate dimensions satisfying X=XT,Y=YT>0, then X+ZTY-1Z<0 if and only if
(2.12)(XZTZ-Y)<0.Proposition 2.4 (integral matrix inequality [28]).
For any symmetric positive definite matrixM>0, scalar γ>0 and vector function ω:[0,γ]→Rn such that the integrations concerned are well defined, the following inequality holds
(2.13)(∫0γ‍ω(s)ds)TM(∫0γ‍ω(s)ds)≤γ(∫0γ‍ωT(s)Mω(s)ds).


## 3. Design of Guaranteed Cost Controller

In this section, we give a design of memoryless guaranteed feedback cost control for uncertain neural networks (2.1). Let us set
(3.1)W11=-PAT-AP-2αP+0.25BRBT+∑i=01‍Gi+2ϵ1EaTEa+6ϵ1-1PHaTHaP+4ϵ2-1PFHw0THw0FP,W12=P+AP-0.5BBT,W13=e-2αh0H0+0.5BBT+AP,W14=e-2αh1H1+0.5BBT+AP,W15=P+0.5BBT+AP,W22=∑i=01‍WiDiWiT+∑i=01‍hi2Hi+(h1-h0)U-2P-BBT+ϵ1EaTEa+ϵ2Ew0TEw0+ϵ3Ew1TEw1,W23=P,W24=P,W25=P,W33=-e-2αh0G0-e-2αh0H0-e-2αh1U+∑i=01‍WiDiWiT+ϵ1EaTEa+ϵ2Ew0TEw0+ϵ3Ew1TEw1,W34=0,W35=e-2αh1U,W44=∑i=01‍WiDiWiT-e-2αh1U-e-2αh1G1-e-2αh1H1+ϵ1EaTEa+ϵ2Ew0TEw0+ϵ3Ew1TEw1,W45=e-2αh1U,W55=-e-2αh1U+W0D0W0T+4ϵ3-1PEHw1THw1EP+ϵ1EaTEa+ϵ2Ew0TEw0+ϵ3Ew1TEw1,E=diag{ei,i=1,…,n},F=diag{fi,i=1,…,n},λ1=λmin(P-1),λ2=λmax(P-1)+h0λmax[P-1(∑i=01‍Gi)P-1]+h12λmax[P-1(∑i=01‍Hi)P-1]+(h1-h0)λmax(P-1UP-1).Theorem 3.1.
Consider control system (2.1) and the cost function (2.8). If there exist symmetric positive definite matrices P,U,G0,G1,H0,and H1, and diagonal positive definite matrices Di,i=0,1, and ϵi>0,i=1,2,3 satisfying the following LMIs
(3.2)ℰ=[W11W12W13W14W15*W22W23W24W25**W33W34W35***W44W45****W55]<0,(3.3)𝒮1=[-PA-ATP-∑i=01‍e-2αhiHi2PFPQ1*-D00**-Q1-1]<0,(3.4)𝒮2=[W1D1W1T-e-2αh1U2PEPQ2*-D10**-Q2-1]<0,
then
(3.5)u(t)=-12BTP-1x(t),t≥0
is a guaranteed cost control and the guaranteed cost value is given by
(3.6)J*=λ2∥ϕ∥2.
Moreover, the solution x(t,ϕ) of the system satisfies
(3.7)∥x(t,ϕ)∥≤λ2λ1e-αt∥ϕ∥,∀t≥0.Proof.
LetY=P-1,y(t)=Yx(t). Using the feedback control (2.8) we consider the following Lyapunov-Krasovskii functional:
(3.8)V(t,xt)=∑i=16‍Vi(t,xt),V1=xT(t)Yx(t),V2=∫t-h0t‍e2α(s-t)xT(s)YG0Yx(s)ds,V3=∫t-h1t‍e2α(s-t)xT(s)YG1Yx(s)ds,V4=h0∫-h00‍∫t+st‍e2α(τ-t)x˙T(τ)YH0Yx˙(τ)dτds,V5=h1∫-h10‍∫t+st‍e2α(τ-t)x˙T(τ)YH1Yx˙(τ)dτds,V6=(h1-h0)∫t-h1t-h0‍∫t+st‍e2α(τ-t)x˙T(τ)YUYx˙(τ)dτds.
It is easy to check that
(3.9)λ1∥x(t)∥2≤V(t,xt)≤λ2∥xt∥2,∀t≥0.
Taking the derivative of V1 we have
(3.10)V˙1=2xT(t)Yx˙(t)=yT(t)[-P(A+EaFa(t)Ha)T-(A+EaFa(t)Ha)P]y(t)-yT(t)BBTy(t)+2yT(t)(W0+Ew0Fw0(t)Hw0)f(·)y(t)+2yT(t)(W1+Ew1Fw1(t)Hw1)g(·)y(t),V˙2=yT(t)G0y(t)-e-2αh0yT(t-h0)G0y(t-h0)-2αV2,V˙3=yT(t)G1y(t)-e-2αh1yT(t-h1)G1y(t-h1)-2αV3,V˙4=h02y˙T(t)H0y˙(t)-h1e-2αh0∫t-h0t‍x˙T(s)H0x˙(s)ds-2αV4,V˙5=h12y˙T(t)H1y˙(t)-h1e-2αh1∫t-h1t‍y˙T(s)H1y˙(s)ds-2αV4,V˙6=(h1-h0)2y˙T(t)Uy˙(t)-(h1-h0)e-2αh1∫t-h1t-h0‍y˙T(s)Uy˙(s)ds-2αV6,
Applying Proposition 2.4 and the Leibniz-Newton formula
(3.11)∫st‍y˙(τ)dτ=y(t)-y(s).
We have for j=1,2,i=0,1(3.12)-hi∫t-hit‍y˙T(s)Hjy˙(s)ds≤-[∫t-hit‍y˙(s)ds]THj[∫t-hit‍y˙(s)ds]≤-[y(t)-y(t-h(t))]THj[y(t)-y(t-h(t))]=-yT(t)Hiy(t)+2xT(t)Hjy(t-h(t))-yT(t-hi)Hjy(t-hi).
Note that
(3.13)∫t-h1t-h0‍y˙T(s)Uy˙(s)ds=∫t-h1t-h(t)‍y˙T(s)Uy˙(s)ds+∫t-h(t)t-h0‍y˙T(s)Uy˙(s)ds.
Applying Proposition 2.4 gives
(3.14)[h1-h(t)]∫t-h1t-h(t)‍y˙T(s)Uy˙(s)ds≥[∫t-h1t-h(t)‍y˙(s)ds]TU[∫t-h1t-h(t)‍y˙(s)ds]≥[y(t-h(t)-y(t-h1))]TU[y(t-h(t)-y(t-h1))].
Since h1-h(t)≤h1-h0, we have
(3.15)[h1-h0]∫t-h1t-h(t)‍y˙T(s)Uy˙(s)ds≥[y(t-h(t)-y(t-h1))]TU[y(t-h(t)-y(t-h1))],
then
(3.16)-[h1-h0]∫t-h1t-h(t)‍y˙T(s)Uy˙(s)ds≤-[y(t-h(t)-y(t-h1))]TU[y(t-h(t)-y(t-h1))].
Similarly, we have
(3.17)-(h1-h0)∫t-h(t)t-h0‍y˙T(s)Uy˙(s)ds≤-[y(t-h0)-y(t-h(t))]TU[y(t-h0)-y(t-h(t))].
Then, we have
(3.18)V˙(·)+2αV(·)≤yT(t)[-P(A+EaFa(t)Ha)T-(A+EaFa(t)Ha)P]y(t)-yT(t)BBTy(t)+2yT(t)(W0+Ew0Fw0(t)Hw0)f(·)+2yT(t)(W1+Ew1Fw1(t)Hw1)g(·)+yT(t)(∑i=01‍Gi)y(t)+2α〈Py(t),y(t)〉+y˙T(t)(∑i=01‍hi2Hi)y˙(t)+(h1-h0)y˙T(t)Uy˙(t)-∑i=01‍e-2αhiyT(t-hi)Giy(t-hi)-e-2αh0[y(t)-y(t-h0)]TH0[y(t)-y(t-h0)]-e-2αh1[y(t)-y(t-h1)]TH1[y(t)-y(t-h1)]-e-2αh1[y(t-h(t))-y(t-h1)]TU[y(t-h(t))-y(t-h1)]-e-2αh1[y(t-h0)-y(t-h(t))]TU[y(t-h0)-y(t-h(t))].
Using (2.8)
(3.19)Py˙(t)+(A+EaFa(t)Ha)Py(t)-(W0+Ew0Fw0(t)Hw0)f(·)-(W1+Ew1Fw1(t)Hw1)g(·)+0.5BBTy(t)=0,
and multiplying both sides with [2y(t),-2y˙(t),2y(t-h0),2y(t-h1),2y(t-h(t))]T, we have
(3.20)2yT(t)Py˙(t)+2yT(t)(A+EaFa(t)Ha)Py(t)-2yT(t)(W0+Ew0Fw0(t)Hw0)f(·)-2yT(t)(W1+Ew1Fw1(t)Hw1)g(·)+yT(t)BBTy(t)=0,-2y˙T(t)Py˙(t)-2y˙T(t)(A+EaFa(t)Ha)Py(t)+2y˙T(t)(W0+Ew0Fw0(t)Hw0)f(·)+2y˙T(t)(W1+Ew1Fw1(t)Hw1)g(·)-y˙T(t)BBTy(t)=0,2yT(t-h0)Py˙(t)+2yT(t-h0)(A+EaFa(t)Ha)Py(t)-2yT(t-h0)(W0+Ew0Fw0(t)Hw0)×f(·)-2yT(t-h0)(W1+Ew1Fw1(t)Hw1)g(·)+yT(t-h0)BBTy(t)=0,2yT(t-h1)Py˙(t)+2yT(t-h1)(A+EaFa(t)Ha)Py(t)-2yT(t-h1)(W0+Ew0Fw0(t)Hw0)×f(·)-2yT(t-h1)(W1+Ew1Fw1(t)Hw1)g(·)+yT(t-h1)BBTy(t)=0,2yT(t-h(t))Py˙(t)+2yT(t-h(t))(A+EaFa(t)Ha)Py(t)-2yT(t-h(t))×(W0+Ew0Fw0(t)Hw0)f(·)-2yT(t-h(t))(W1+Ew1Fw1(t)Hw1)g(·)+yT(t-h(t))BBTy(t)=0.
Adding all the zero items of (3.20) and f0(t,x(t),x(t-h(t)),u(t))-f0(t,x(t),x(t-h(t)),u(t))=0, respectively, into (3.18) and using the condition (2.7) for the following estimations:
(3.21)f0(t,x(t),x(t-h(t)),u(t))≤〈Q1x(t),x(t)〉+〈Q2x(t-h(t)),x(t-h(t))〉+〈Ru(t),u(t)〉=〈PQ1Py(t),y(t)〉+〈PQ2Py(t-h(t)),y(t-h(t))〉+0.25〈BRBTy(t),y(t)〉,2〈W0f(x),y〉≤〈W0D0W0Ty,y〉+〈D0-1f(x),f(x)〉,2〈W1g(z),y〉≤〈W1D1W1Ty,y〉+〈D1-1g(z),g(z)〉,2〈D0-1f(x),f(x)〉≤〈FD0-1Fx,x〉,2〈D1-1g(z),g(z)〉≤〈ED1-1Ez,z〉,2〈EaFa(t)HaPy,y〉≤〈ϵ1EaTEay,y〉+〈ϵ1-1PHaTHaPy,y〉,ϵ1>0,2〈Ew0Fw0(t)Hw0Pf(x),y〉≤〈ϵ2Ew0TEw0y,y〉+〈ϵ2-1PD0Hw0THw0D0Py,y〉,ϵ2>0,2〈Ew1Fw1(t)Hw1Pg(z),y〉≤〈ϵ3Ew1TEw1y,y〉+〈ϵ3-1PD1Hw1THw1D1Pz,z〉,ϵ3>0,
we obtain
(3.22)V˙(·)+2αV(·)≤ζT(t)ℰζ(t)+yT(t)S1y(t)+yT(t-h(t))S2y(t-h(t))-f0(t,x(t),x(t-h(t)),u(t)),
where ζ(t)=[y(t),y˙(t),y(t-h0),y(t-h1),y(t-h(t))], and
(3.23)ℰ=[W11W12W13W14W15*W22W23W24W25**W33W34W35***W44W45****W55],S1=-PA-ATP-∑i=01‍e-2αhiHi+4PFD0-1FP+PQ1P,S2=W1D1W1T-e-2αh2U+4PED1-1EP+PQ2P.
Note that by the Schur complement lemma, Proposition 2.3, the conditions S1<0 and S2<0 are equivalent to the conditions (3.3) and (3.4), respectively. Therefore, by condition (3.2), (3.3), and (3.4), we obtain from (3.22) that
(3.24)V˙(t,xt)≤-2αV(t,xt),∀t≥0.
Integrating both sides of (3.24) from 0 to t, we obtain
(3.25)V(t,xt)≤V(ϕ)e-2αt,∀t≥0.
Furthermore, taking condition (3.9) into account, we have
(3.26)λ1∥x(t,ϕ)∥2≤V(xt)≤V(ϕ)e-2αt≤λ2e-2αt∥ϕ∥2,
then
(3.27)∥x(t,ϕ)∥≤λ2λ1e-αt∥ϕ∥,t≥0,
which concludes the exponential stability of the closed-loop system (2.8). To prove the optimal level of the cost function (2.4), we derive from (3.22) and (3.2)–(3.4) that
(3.28)V˙(t,zt)≤-f0(t,x(t),x(t-h(t)),u(t)),t≥0.
Integrating both sides of (3.28) from 0 to t leads to
(3.29)∫0t‍f0(t,x(t),x(t-h(t)),u(t))dt≤V(0,z0)-V(t,zt)≤V(0,z0),
dute to V(t,zt)≥0. Hence, letting t→+∞, we have
(3.30)J=∫0∞‍f0(t,x(t),x(t-h(t)),u(t))dt≤V(0,z0)≤λ2∥ϕ∥2=J*.
This completes the proof of the theorem.Remark 3.2.
Note thath(t) is non-differentiable and interval time-varying delay; therefore, the stability criteria proposed in [5–8, 12, 15–26] are not applicable to this system.Example 3.3.
Consider the uncertain neural networks with interval time-varying delays (2.1), where
(3.31)A=[0.1000.3],W0=[0.10.10.20.3],W1=[0.20.20.10.4],B=[0.10.2],E=[0.3000.4],F=[0.2000.3],Q1=[0.20.10.10.4],Q2=[0.30.20.20.5],R=[0.10.10.10.3],Ea=[0.10.10.10.3],Ew0=[0.10.10.10.2],Ew1=[0.20.10.10.3],Ha=[0.30.20.20.2],Hw0=[0.20.10.10.2],Hw1=[0.30.10.10.3],h(t)=0.1+1.3sin2tift∈ℐ=⋃k≥0[2kπ,(2k+1)π]h(t)=0ift∈R+∖ℐ.
Note that h(t) is non-differentiable; therefore, the stability criteria proposed in [4–8, 12, 15–26] are not applicable to this system. Given α=0.1,h0=0.1,andh1=1.4, by using the Matlab LMI toolbox, we can solve for P,U,G0,G1,H0,H1,D0, and D1 which satisfy the conditions (3.2)–(3.4) in Theorem 3.1. A set of solutions are ϵ1=0.0017,ϵ2=0.0013,ϵ3=0.0012,
(3.32)P=[1.1578-0.1128-0.11281.0597],U=[2.3269-0.3820-0.38202.6681],G0=[1.45960.13970.13971.2369],G1=[2.26940.81140.81141.0125],H0=[0.64550.04520.04520.5104],H1=[0.30050.02330.02330.2306],D0=[0.0011000.0011],D1=[0.7809000.7809].
Then
(3.33)u(t)=-0.2292x1(t)-0.1816x2(t),t≥0
is a guaranteed cost control law and the cost given by
(3.34)J*=5.4631∥ϕ∥2.
Moreover, the solution x(t,ϕ) of the system satisfies
(3.35)∥x(t,ϕ)∥≤3.6984e-0.1t∥ϕ∥,∀t≥0.The exponential convergence dynamics of the network (2.1) are shown in Figure 1.Figure 1
The simulation of the solutionsx1(t) and x2(t) with the initial condition ϕ(t[105]T,t∈[-0.4,0].Example 3.4.
Consider the uncertain neural networks with interval time-varying delays (2.1), where
(3.36)A=[1002],W0=[1123],W1=[2214],B=[12],E=[3004],F=[2003],Q1=[2114],Q2=[3225],R=[1113],Ea=[1113],Ew0=[1112],Ew1=[2113],Ha=[3222],Hw0=[2112],Hw1=[3113],h(t)=0.1+0.7sin2tift∈ℐ=⋃k≥0[2kπ,(2k+1)π]h(t)=0ift∈R+∖ℐ.
Note that h(t) is non-differentiable; therefore, the stability criteria proposed in [5–8, 12, 15–26] are not applicable to this system. Given α=0.3,h0=0.1,h1=0.8, by using the Matlab LMI toolbox, we can solve for P,U,G0,G1,H0,H1,D0, and D1 which satisfy the conditions (3.2)–(3.4) in Theorem 3.1. A set of solutions are ϵ1=0.9,ϵ2=0.8,ϵ3=0.7,
(3.37)P=[0.7832-0.0213-0.02130.0011],U=[0.1297-0.0019-0.00190.0197],G0=[0.17950.01370.01370.2211],G1=[1.21970.96480.96480.7391],H0=[0.89310.11830.11830.7197],H1=[0.68510.12970.12970.5726],D0=[0.1397000.2278],D1=[0.6812000.6813].
Then
(3.38)u(t)=-0.7314x1(t)-0.0196x2(t),t≥0,
is a guaranteed cost control law and the cost given by
(3.39)J*=24.3219∥ϕ∥2.
Moreover, the solution x(t,ϕ) of the system satisfies
(3.40)∥x(t,ϕ)∥≤12.3690e-0.3t∥ϕ∥,∀t≥0.The exponential convergence dynamics of the network (2.1) are shown in Figure 2.Figure 2
The simulation of the solutionsx1(t) and x2(t) with the initial condition ϕ(t)=[1-1]T,t∈[-1,0.8].


## 4. Conclusions

In this paper, the problem of guaranteed cost control for uncertain neural networks with interval nondifferentiable time-varying delay has been studied. A nonlinear quadratic cost function is considered as a performance measure for the closed-loop system. The stabilizing controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles. By constructing a set of time-varying Lyapunov-Krasovskii functionals combined with Newton-Leibniz formula, a memoryless state feedback guaranteed cost controller design has been presented, and sufficient conditions for the existence of the guaranteed cost state-feedback for the system have been derived in terms of LMIs.



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*Source: 587426-2012-11-07.xml*