# LMI Approach to Stability Analysis of Cohen-Grossberg Neural Networks withp-Laplace Diffusion

Authors: Xiongrui Wang; Ruofeng Rao; Shouming Zhong
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

The nonlinearp-Laplace diffusion (p>1) was considered in the Cohen-Grossberg neural network (CGNN), and a new linear matrix inequalities (LMI) criterion is obtained, which ensures the equilibrium of CGNN is stochastically exponentially stable. Note that, if p=2, p-Laplace diffusion is just the conventional Laplace diffusion in many previous literatures. And it is worth mentioning that even if p=2, the new criterion improves some recent ones due to computational efficiency. In addition, the resulting criterion has advantages over some previous ones in that both the impulsive assumption and diffusion simulation are more natural than those of some recent literatures.


---

## Body

## 1. Introduction and Preparation

It is well known that Cohen-Grossbeg neural network (CGNN) was proposed by Cohen and Grossberg [1] in 1983. Since then there have been a lot of interested results obtained in many literatures (see [2–9]) due to its general applications, such as pattern recognition, image and signal processing, optimization automatic control, and artificial intelligence. Usually, there exist the impulsive effect and time-varying delays phenomenon in various neural networks [3, 5–7, 10–14]. Besides, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields [15–18]. However, diffusion disturbance was always simulated simply by linear Laplace diffusion [15–18]. Few papers involved the nonlinear reaction-diffusion [19]. So in this paper, we investigate the stability of the following stochastic CGNN with nonlinear p-Laplace diffusion (p>1):
(1.1)du(t,x)={∇·(D~(t,x,u)∘∇pu)-A(u(t,x))2×[B(u(t,x))-Cf(u(t,x))+Dg(u(t-τ(t),x))](D~(t,x,u)∘∇pu)}dt+σ(u(t,x))dw(t),t∈[tk,tk+1)u(tk+,x)=Mu(t-,x),t=tku(t0+θ,x)=φ(θ,x),(θ,x)∈[-τ,0]×∂Ω∂ui(t,x)∂ν=0,(t,x)∈[-τ,+∞)×∂Ω,i=1,2,…,n,
where Ω is a bounded subset in Rm with smooth boundary ∂Ω, and ∂ui(t,x)/∂ν=((∂ui(t,x)/∂x1),(∂ui(t,x)/∂x2),…,(∂ui(t,x)/∂xm))T denotes the outward normal derivative on ∂Ω.Remark 1.1.
Ifp=2, system (1.1) was studied by [5] though there is a little difference between Dirichlet boundary condition and Neumann boundary condition. However, our impulsive assumption u(tk+,x)=Mu(tk-,x) is more natural than that of [5], which will result in some difference in methods.Here,M is a diagonal matrix, Ω∈Rm is a bounded compact set with smooth boundary, u=(u1,u2,…,un)T∈Rn, and w(t)=(w1(t),w2(t),…,wn(t))T is a n-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,P) with the natural filtration {ℱt}t≥0 generated by the process {w(s):0≤s≤t}. We associate Ω with the canonical space generated by all {wi(t)} and denoted by ℱ the associated σ-algebra generated by w(t) with the probability measure p. A(u(t,x)) presents an amplification function, B(u(t,x)) is an appropriately behavior function, and f and g denote the activation function. τ(t)(0≤τ(t)≤τ) corresponds to the transmission delays at time, and tk is called the impulsive moment with 0<t1<t2<⋯<tk<⋯ and limk→∞tk=+∞. We always assume u(tk+,x)=u(tk,x). ∇pu=(∇pu1,…,∇pun)T, ∇pui=(|∇ui|p-2(∂ui/∂x1),…,|∇ui|p-2(∂ui/∂xm))T, D~∘∇pu=(D~ik|∇ui|p-2(∂ui/∂xk))n×m is Hadamard product of matrix D~ and ∇pu [20]. Here, the diffusion parameters matrix D~(t,x,u) is denoted simply as D~=(D~ik)n×m. Let Yi=(yi1,…,yim)T,i=1,2,…,n, and matrix 𝕐=(Y1,…,Yn)T, and we denote ∇·Yi=∑k=1m‍∂yik/∂xk,∇·𝕐=(∇·Y1,∇·Y2,…,∇·Yn)T. Particularly, ∇pu=∇u for the case of p=2.Remark 1.2.
Diffusion effects always occur in the neural networks when electrons are moving in asymmetric electromagnetic fields [15–18], and diffusion behavior is so complicated that it cannot always be simulated by linear Laplace diffusion. So in this paper, the nonlinear p-Laplace diffusion is considered in System (1.1).Assume, in addition, the following.(H1)
A(u(t,x)) is a bounded, positive, and continuous diagonal matrix, that is, there exist two positive diagonal matrices A_ and A¯ such that 0<A_≤A(u(t,x))≤A¯.(H2)
B(u(t,x))=(b1(u1(t,x)),b2(u2(t,x)),…,bn(un(t,x)))T such that there exists a positive diagonal matrix B=diag(B1,B2,…,Bn) satisfying (bi(ui(t,x))/ui(t,x))≥Bi for all i.(H3)
There exist two positive diagonal matricesF=diag(F1,F2,…,Fn) and G=diag(G1,G2,…,Gn) such that
(1.2)0≤fi(r)r≤Fi,0≤gi(r)r≤Gi,∀i.(H4)
The null solution is the equilibrium point of system (1.1), that is, the following conditions hold:
(1.3)B(0)-Cf(0)-Dg(0)=0,trace[σT(u(t,x))σ(u(t,x))]≤uT(t,x)Qu(t,x),where the symmetrical matrix Q>0.For convenience's sake, we introduce some standard notations(i)
L2(R)(R×Ω): the space of real Lebesgue measurable functions of R×Ω, it is a Banach space for the 2-norm ∥u(t)∥2=(∑i=1n‍∥ui(t)∥)1/2 with ∥ui(t)∥=(∫Ω‍|ui(t,x)|2dx)1/2, where |ui(t,x)| is Euclid norm.(ii)
LF02([-τ,0]×Ω;Rn): the family of all F0-measurable C([-τ,0]×Ω;Rn-) value random variable ξ={ξ(θ,x):-τ≤θ≤0,x∈Ω} such that sup-τ≤θ≤0E∥ξ(θ)∥22<∞, where E{·} stands for the mathematical expectation operator with respect to the given probability measure p.(iii)
Q=(qij)n×n>0 (<0): a positive (negative) definite symmetrical matrix, that is, yTQy>0 (<0) for any 0≠y∈Rn.(iv)
Q=(qij)n×n≥0 (≤0): a semipositive (semi-negative) definite symmetrical matrix, that is, yTQy≥0 (≤0) for any y∈Rn.(v)
Q≥Q~(Q≤Q~): this means Q-Q~ is a semi-positive (semi-negative) symmetrical definite matrix.(vi)
Q>Q~(Q<Q~): this means Q-Q~ is a positive (negative) symmetrical definite matrix.(vii)
λmax(Φ),λmin(Φ) denotes the largest and smallest eigenvalue of symmetrical matrix Φ, respectively.(viii)
I: identity matrix with compatible dimension.(ix)
Denote|C|=(|cij|)n×n for any matrix Cn×n; |u(t,x)|=(|u1|,|u2|,…,|un|) for any u∈Rn.Letu(t,x;φ) denote the state trajectory from the initial data u(t0+θ,x;φ)=φ(θ,x) on -τ≤θ≤0 in LF02([-τ,0]×Ω;Rn).Definition 1.3.
The null solution of impulsive system (2.2) is globally stochastically exponentially stable in the mean square if for every φ∈LF02([-τ,0]×Ω;Rn), there exists scalars β>0 and γ>0 such that
(1.4)E(∥u(t,φ)∥22)≤γe-βtsup-τ≤θ≤0E(∥φ(θ)∥22).Lemma 1.4 (see [11]).
LetU,P be any matrices, ε>0 is a positive number and matrix H=HT>0, then
(1.5)PTU+UTP≤εPTHP+ε-1UTH-1U.Lemma 1.5 (Schur complement [3]).
The LMI(1.6)(Q(t)S(t)ST(t)R(t))>0,
where matrix S(t) and symmetrical matrices Q(t) and  R(t) depend on t, is equivalent to any one of the following conditions: (L1)
R(t)>0,Q(t)-S(t)R-1(t)ST(t)>0;(L2)
Q(t)>0,R(t)-ST(t)Q-1(t)ST(t)>0.Lemma 1.6 (see [21]).
Consider the following differential inequality:(1.7)D+v(t)≤-av(t)+b[v(t)]τ,t≠tkv(tk)≤akv(tk-)+bk[v(tk-)]τ,
where v(t)≥0,[v(tk)]τ=supt-τ≤s≤tv(s),[v(tk-)]τ=supt-τ≤s≤tv(s) and v(t) is continuous except tk,k=1,2,…, where it has jump discontinuities. The sequence tk satisfies 0=t0<t1<t2<⋯<tk<tk+1<⋯, and limk→∞tk=∞. Suppose that(1)
a>b≥0;(2)
tk-tk-1>δτ, where δ>1, and there exist constants γ>0,M>0 such that
(1.8)ρ1ρ2⋯ρk+1≤Meγtk,where ρi=max{1,ai+bieλτ}, λ>0 is the unique solution of equation λ=a-beλτ then
(1.9)v(t)≤M[v(0)]τe-(λ-γ)t.
In addition, if θ=supk∈Z{1,ak+bkeλτ}, then
(1.10)v(t)=θ[v(0)]τe-(λ-(ln(θeλτ)/δτ))t,t≥0.Computer simulation is shown in Figures1, 2, 3, and 4.Figure 1
Computer simulation of the stateu1(t,x).Figure 2
Sectional curve of the state variableu1(t,x).Figure 3
Computer simulation of the stateu2(t,x).Figure 4
Sectional curve of the state variableu2(t,x).


## 2. Main Results

Theorem 2.1.
If assumptions (H1)–(H4) hold, in addition, the following conditions are satisfied:(C1)
there exists diagonal matricesP1=diag(p11,p12,…,p1n)>0 and P2>0 such that
(2.1)(-2P1A_B+F2+P1Q+P2P1A¯|C|P1A¯|D||CT|A¯P1-I0|DT|A¯P10-I)<0;(C2)
min{(λminΘ/λmaxP1),(1-μ)}>(λmaxG2/λminP2)≥0, where Θ=2P1A_B-P1A¯|C||C|TA¯P1-P1A¯|D||D|TA¯P1-F2-P1Q-P2, where τ′(t)≤μ<1 for all t;(C3)
there exists a constantδ>1 such that infk∈Z(tk-tk-1)>δτ, δ2τ>ln(ρeλτ) and λ-(ln(ρeλτ)/δτ)>0, where λ>0 is the unique solution of the equation λ=a-beλτ, and ρ=max{1,(λmax(MP1M)/λminP1)+eλτ}, a=min{(λminΘ/λmaxP1),(1-μ)},b=λmaxG2/λminP2, then the null solution of system (1.1) is stochastically exponentially stable with convergence rate (1/2)(λ-(ln(ρeλτ)/δτ)).Proof.
First, we can get by Guass formula (see [20, Lemma 2.3])
(2.2)∫Ω‍uTP1(∇·(D~(t,x,u)∘∇pu)dx=∫Ω‍uTP1(∑k=1m‍∂∂xk(D~1k|∇u1|p-2∂u1∂xk),…,∑k=1m‍∂∂xk(D~nk|∇un|p-2∂un∂xk))Tdx=∫Ω‍∑j=1n‍p1juj∑k=1m‍∂∂xk(D~jk|∇uj|p-2∂uj∂xk)dx=-∑k=1m‍∑j=1n‍∫Ω‍p1jD~jk|∇uj|p-2(∂uj∂xk)2dx.
Construct the Lyapunov functional as follows:(2.3)V=V1+V2,
where
(2.4)V1=∫Ω‍uT(t,x)P1u(t,x)dx=∫Ω‍|uT(t,x)|P1|u(t,x)|dxV2=∫Ω‍∫t-τ(t)t‍uT(s,x)P2u(s,x)dsdx.
Then(2.5)LV1=∫Ω‍-∑k=1m‍∑j=1n‍2p1jD~jk|∇uj|p-2(∂uj∂xk)2dx-2∫Ω‍uT(t,x)P1A(u(t,x))B(u(t,x))dx+2∫Ω‍uT(t,x)P1A(u(t,x))Cf(u(t,x))dx+2∫Ω‍uT(t,x)P1A(u(t,x))Dg(u(t-τ(t),x))dx+∫Ω‍trace[σT(u(t,x))P1σ(u(t,x))]dx≤-2∫Ω‍|uT(t,x)|P1A_B|u(t,x)|dx+∫Ω‍[|uT(t,x)|P1A¯|C||C|TA¯P1|u(t,x)|+|fT(u(t,x))||f(u(t,x))|]dx+∫Ω‍[|uT(t,x)|P1A¯|D||D|TA¯P|u(t,x)|+gT(u(t-τ(t),x))g(u(t-τ(t),x))]dx+∫Ω‍uT(t,x)P1Qu(t,x)dx≤-∫Ω‍|uT(t,x)|[2P1A_B-P1A¯|C||C|TA¯P1-P1A¯|D||D|TA¯P1-F2-P1Q]|u(t,x)|dx+∫Ω‍uT(t-τ(t),x)G2u(t-τ(t),x)dxLV2=∫Ω‍[uT(t,x)P2u(t,x)-(1-τ′(t))uT(t-τ(t))P2u(t-τ(t))]dx≤∫Ω‍|uT(t,x)|P2|u(t,x)|dx+(μ-1)∫Ω‍uT(t-τ(t))P2u(t-τ(t))dx.
And then we have
(2.6)LV≤-∫Ω‍|uT(t,x)|Θ|u(t,x)|dx+∫Ω‍uT(t-τ(t),x)[G2+(μ-1)P2]u(t-τ(t),x)dx.
Next, we use the method similar as that of [22]. Since u(t,x) is the solution of system, and V(u(t,x))∈C2[Rm,R+] for all t, we can get by Ito^ formula
(2.7)V(t)=V(tk)+∫tkt‍LV(u(s,x))ds+∫tkt‍∂V∂uσ(u(s,x))dw(s).
Then we have(2.8)EV(u(t,x))=EV(u(tk,x))+∫tkt‍ELV(s,x)ds,t∈[tk,tk+1).
Thus, for small enoughΔt>0, we have
(2.9)EV(u(t+Δt,x))=EV(u(tk,x))+∫tkt+Δt‍ELV(s,x)ds,t∈[tk,tk+1),
and then
(2.10)EV(u(t+Δt,x))-EV(u(t,x))=∫tt+Δt‍ELV(s,x)ds≤E∫tt+Δt‍[-∫Ω‍|uT(s,x)|Θ|u(s,x)|dxE∫tt+Δt‍2+∫Ω‍uT(s-τ(s),x)[G2+(μ-1)P2]u(s-τ(s),x)dx]ds,t∈[tk,tk+1).
Since
(2.11)∫Ω‍|uT(t,x)|Θ|u(t,x)|dx+∫Ω‍uT(t-τ(t),x)(1-μ)P2u(t-τ(t),x)dx≥λminΘλmaxP1∫Ω‍|uT(t,x)|P1|u(t,x)|dx+(1-μ)∫Ω‍uT(t-τ(t),x)P2u(t-τ(t),x)dx≥min{λminΘλmaxP1,(1-μ)}V∫Ω‍uT(t-τ(t),x)G2u(t-τ(t),x)dx≤λmaxG2λminP2∫Ω‍uT(t-τ(t),x)P2u(t-τ(t),x)dx,
Then,(2.12)D+EV(u(t,x))≤-min{λminΘλmaxP1,(1-μ)}EV(t)+λmaxG2λminP2[EV(t)]τ,t∈[tk,tk+1).
Next, we have
(2.13)V(tk)=∫Ω‍uT(tk,x)P1u(tk,x)dx+∫Ω‍uT(tk-τ(tk),x)P2u(tk-τ(tk),x)dx=∫Ω‍[uT(tk-,x)MP1Mu(tk-,x)dx+∫Ω‍uT(tk-τ(tk),x)P2u(tk-τ(tk),x)dx≤λmax(MP1M)λminP1V(tk-)+[V(tk-)]τ.
Now the conditions (C1)–(C3) and Lemma1.6 deduce
(2.14)EV(t)≤ρ[V(0)]τe-(λ-(ln(ρeλτ)/δτ))t
or
(2.15)E∥u(t,x)∥2≤(ρλmaxPλminPsup-τ≤s≤0E∥ϕ(s)∥22)e-(λ-(ln(ρeλτ)/δτ))t
which together with Definition 1.3 implies the accomplishment of the proof.Remark 2.2.
The nonlinearp-Laplace diffusion (p>1) brings a great difficulties in judging the stability. However, even if p=2, Theorem 2.1 has more computational efficiency than [15, Theorem 3.1] due to LMI criterion.


## 3. Examples

Consider the following impulsive CGNN:(3.1)du(t,x)={∇·((0.0005000.0005)∘∇pu)-(1.65+0.5sinu1001.58+0.5sinu2)2×[(5.19u1-0.02u1cosu15.18u2-0.01u2cosu2)-(0.1-0.03-0.030.1)f(u(t,x))2221-(0.1-0.03-0.030.1)g(u(t-τ(t),x))]}dt+σ(u(t,x))dw(t),t∈[tk,tk+1)u(tk+)=(1.350001.350)u(t-,x),t=tku(t0+θ,x)=ϕ(θ,x),(θ,x)∈[-0.65,0]×∂Ω∂ui(t,x)∂ν=0,(t,x)∈[-τ,+∞)×∂Ω,i=1,2,
where Ω={(x1,x2)T∈R2|-2≤x1,x2≤2}, and the corresponding matrices
(3.2)F=(0.100000.100)=G,Q=(0.0001000.0001).
We might as well assume that t0=0,tk-tk-1=0.525,τ(t)=0.65,τ′(t)≤μ=0.99 for all t≥t0, and
(3.3)f(u)=g(u)=(|u1+1|-|u1-1|20|u2+1|-|u2-1|20),ϕ(s,x)=(x(1-cos(5πx))cos189(x-0.25)e-100s(1-x)sin2(4πx)cos201(x-0.55)e-100s),-0.65≤s≤0,A_=(1001),A¯=(2002),B=(5.160005.160).
By way of MATLAB LMI Control Toolbox, we can solve the LMI condition in (C1) and get
(3.4)P1=(0.2392000.2392),P2=(1.2291001.2291).
Next, we will prove that such P1 and P2 make (C2) and (C3) hold. Indeed, by computing directly, we can obtain a=0.01,b=0.0081, and then (C2) holds. Moreover, we might as well assume δ=82, and then we have λ=0.0019,ρ=2.8235, thus (C3) is satisfied. Now from Theorem 2.1 we can compute the convergence 9.3809*10-0.004.


## 4. Conclusions

In this paper, we investigate the influence of impulse, time-delays and diffusion behaviors on the stability of stochastic Cohen-Grossberg neural network (CGNN). The LMI conditions of stochastic exponential stability of impulsive CGNN withp-Laplace reaction-diffusion terms was given, and an illustrate example was also given to show the effectiveness of the obtained result. Besides, the result obtained in this paper is also valid to the Laplace reaction-diffusion (in the case of p=2) and has more computational efficiency due to the LMI approach even if p=2 (Remark 2.2).



---

*Source: 523812-2012-12-24.xml*