# Robust Stability of Nonlinear Diffusion Fuzzy Neural Networks with Parameter Uncertainties and Time Delays

Authors: Ruofeng Rao; Gaozhi Tang; Jiuqi Gong; Xiaoyan Wan; Guanghong Wu; Qiao Zhang; Shouming Zhong
Journal: Complexity
 (2018)

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## Abstract

In this paper, a class of nonlinearp-Laplace diffusion BAM Cohen-Grossberg neural networks (BAM CGNNs) with time delays is investigated. In the case of p>1 with p≠2, the authors construct novel Lyapunov functional to overcome the mathematical difficulties of nonlinear p-Laplace diffusion time-delay model with parameter uncertainties, deriving the LMI-based robust stability criterion applicable to computer MATLAB LMI toolbox and deleting the boundedness of the amplification functions. And in the case of p=2, LMI-based sufficient conditions are also inferred for robust input-to-state stability of reaction-diffusion Markovian jumping BAM CGNNs with the event-triggered control, which is different from those of many previous related literature. In particular, the role of diffusion can be reflected in newly acquired criteria. Finally, numerical examples verify the effectiveness of the proposed methods.


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## Body

## 1. Introduction

In recent decades, reaction-diffusion neural networks have been the subject of research due to the fact that electrons have diffusion behaviors in an inhomogeneous magnetic field, and the role of diffusion items have always been investigated and discussed in many existing results ([1–4]). Since the conduction velocity of electrons and components is limited, the phenomenon of time delays inevitably appears in various practical projects. Thereby, time-delay reaction-diffusion systems are relatively common objects of study. For example, in [5], the following time-delay reaction-diffusion Cohen-Grossberg neural networks (CGNNs) with impulse was studied (see [7, (7)]),
(1)∂u∂t=∇⋅ℛ∘∇u−AuBu−Cfu+Dgut−τt,x+J,t≥0,t≠tk,utk+,x=Mutk−,x+NHutk−−τt,x,k=1,2,…,where ℛ∘∇u is Hadamard product of matrix ℛ and vector gradient ∇u (see [6] for details).In [7], the stability of the following BAM Cohen-Grossberg neural networks (BAM CGNNs) with distributed delays was discussed.
(2)dxdt=−AxtBxt−Cfyt−τt+M∫−∞tKt−sfxsds+σt,xt,yt−τtdwt,dydt=−A~ytB~yt−C~gxt−τ~t+M~∫−∞tK~t−sgxsds+σ~t,xt−τ~t,ytdwt.The Cohen-Grossberg-type BAM neural network model was initially proposed by Cohen and Grossberg [8] in 1983. The model not only generalizes the single-layer autoassociative Hebbian correlator to a two-layer pattern matched heteroassociative circuit but also possesses Cohen-Grossberg dynamics, and it has promising application potentials for tasks of classification, parallel computation, associative memory, and nonlinear optimization problems. Since then, a lot of research has been done on BAM CGNNs models ([7, 9–11]). Besides, owing to biological engineering backgrounds and population dynamics, economics, physical engineering, and other reasons, the stability of nonlinear diffusion systems have received widespread attention [11–17]. For example, in [11], the author studied the following nonlinear diffusion fuzzy system, involved to time-delay BAM Cohen-Grossberg neural networks.
(3)∂uit,x∂t=∑j=1m∂∂xjDijt,x,u∇uit,xp−2∂ui∂xj−aiuit,xbiuit,x−∑j=1nm~ijfjvjt,x−⋀j=1nm^ijfjvjt−τjt,x−⋁j=1nmˇijfjvjt−τjt,x,t≥0,x∈Ω,∂ujt,x∂t=∑i=1m∂∂xiDjit,x,u∇ujt,xp−2∂uj∂xi−cjvjt,xdjvjt,x−∑i=1nn~jigiuit,x−⋀i=1nn^jigiuit−ρit,x−⋁i=1nnˇjigiuit−ρit,x,t≥0,x∈Ω,uθ,x=ϕθ,x,vθ,x=υθ,x,θ,x∈−∞,0×Ω,ut,x=0∈Rn,vt,x=0∈Rn,t,x∈R×∂Ω.Under the complex conditions(4)λ1p¯D+p¯λminA¯B>p−1p¯pnm~+m^+mˇλmaxA¯λmaxF+p¯pnn~+n^+nˇ1−τλmaxC¯λmaxG,λ1p¯D+p¯λminC¯D>p−1p¯pnn~+n^+nˇλmaxC¯λmaxG+p¯pnm~+m^+mˇ1−τλmaxA¯λmaxF,and other conditions, a stability result ([11, Theorem3.2]) was given, where
(5)D=minjkinft,x,uDjkt,x,u,D=minjiinft,x,vDjit,x,v.In recent years, some methods and ideas of related literature ([5–45]) inspire our current work. In this paper, we shall discuss the robust stability of nonlinear p-Laplacian diffusion Takagi-Sugeno (T-S) fuzzy system with discrete delays and distributed delays. Actually, T-S fuzzy models provide a successful method to describe certain complex nonlinear system using some local linear subsystems ([31, 32, 46]). Besides, there exist parameter errors unavoidable in factual systems due to aging of electronic components, external disturbance, and parameter perturbations. Therefore, the robustness of the system stability should be investigated, too. Our main objectives are as follows:
(1)
Changing (4) into linear matrix inequalities (LMIs) applicable to computer MATLAB LMI toolbox, which can be adapted to large-scale calculation in practical engineering.
(2)
Ensure that the nonlinear diffusion term plays a role in the LMI-based stability criterion while in some existing results ([6, Theorem 3.1], [18, Theorem 3.1], [19, Theorem 3.1],), the role of the nonlinear diffusion term was neglected in their LMI-based criteria.
(3)
Deleting the boundedness of amplification functionai· in some existing results (see, e.g., [7, 9, 19, 21]).For these purposes, we need to achieve the following works:(i)
Improve [11, Lemma3.1] and make it adopted to LMI-based criterion, in which the nonlinear diffusion can play roles.
(ii)
Construct a novel Lyapunov functional and design comprehensive applications of variational method, Young inequality, and LMI technique so that LMI-based criterion can be derived for the nonlinear diffusion fuzzy system with parameter uncertainties, discrete delays, and distributed delays.
(iii)
Relax the restrictions of amplification functionai· so that the boundedness of ai· is not necessary. At the same time, employing LMI technique guarantees structuring LMI-based criterion.
(iv)
Explore the input-to-state stability of reaction-diffusion Markovian jumping BAM CGNNs with time delays and the event-triggered controlFor convenience’s sake, we still need to introduce some standard notations:(i)
A=aijn×n≥0≤0: a nonnegative (nonpositive) matrix, that is, aij≥0≤0 for all i,j=1,2,…,n.
(ii)
A≥B≤B: represents the matrix C = (A − B) satisfying C ≥ 0 (≤4 0).
(iii)
A=aijn×n>0<0: a positive (negative) definite matrix.
(iv)
A=aijn×n≥0≤0: a nonnegative (nonpositive) definite matrix.
(v)
A1≥A2A1≤A2: this means A1−A2 is a nonnegative (nonpositive) definite matrix.
(vi)
A1>A2A1<A2: this means A1−A2 is a positive (negative) definite matrix.
(vii)
λmaxΦ and λmaxΦ,λminΦdenote the largest and smallest eigenvalue of matrix Φ, respectively.
(viii)
DenoteC=cijn×n for any matrix C=cijn×n;
(ix)
ut,x=u1t,x,u2t,x,…,unt,xT for any vector ut,x=u1t,x,u2t,x,…,unt,xT.
(x)
ut,x=u1t,x,u2t,x,…,unt,xT≥0≤0 implies uit,x≥0≤0 for all i=1,2,…,n.
(xi)
ut,x≥vt,x≤vt,x implies uit,x≥vit,x≤vit,x for all i=1,2,…,n, where ut,x=u1t,x,u2t,x,…,unt,xT and vt,x=v1t,x,v2t,x,…,vnt,xT
(xii)
I: identity matrix with compatible dimension.
(xiii)
The Sobolev spaceW1,pΩ=u∈Lp:Du∈Lp (see [28] for details).
(xiv)
Denote byλ1 the lowest positive eigenvalue of the boundary value problem (see [28] for details)
(6)−Δpςt,x=λςt,x,x∈Ω,ςt,x=0,x∈∂Ω.


## 2. Preliminaries

Consider the following Takagi-Sugeno fuzzyp-Laplace partial differential equations with distributed delay.Fuzzy rulej:Ifω1t is µj1 and ⋯ωs∗t is µjs∗ then
(7)∂u∂t=∇⋅Dt,x,u∘∇pu−Aut,xBut,x−Cj=ΔCjtfvt−τt,x+Mj+ΔMjt∫t−ρttfvs,xds,∂v∂t=∇⋅Dt,x,v∘∇pv−A~vt,xB~vt,x−C~j+ΔC~jtgut−τ~t,x+M~j+ΔM~jt∫t−ρ~ttgus,xds,uθ,x=ϕθ,x,vθ,x=υθ,x,θ,x∈−τ∗,0×Ω,ut,x=0∈Rn,vt,x=0∈Rn,t,x∈R×∂Ω,where ωktk=1,2,…,s∗ is the premise variable and µjkj=1,2,…,r;k=1,2,…,s∗ is the fuzzy set that is characterized by membership function. r is the number of the if-then rules, and s is the number of the premise variables. Dt,x,u=diagD1t,x,u,D2t,x,u,…,Dnt,x,u and Dt,x,v=diagD1t,x,v,D2t,x,v,…,Dnt,x,v are diffusion coefficients matrices. D∘∇pu is Hadamard product of matrix D and ∇pu (see e.g., [13] for details) and so is Dt,x,v∘∇pv. Let p>1 be a given scalar, and Ω⊂Rn be a bounded domain with a smooth boundary ∂Ω of class C2 by Ω. Denote ut,x=u1t,x,u2t,x,…,unt,xT∈Rn and vt,x=v1t,x,v2t,x,…,vnt,xT∈Rn. For any given i∈N≜1,2,…,n, uit,xis the state variable of the ith neuron at time t in space variable x and so is vit,x. fvt−τt,x=f1v1t−τit,x,…,fivit−τit,x,…,fnvnt−τit,xT, and gut−τ~t,x=g1u1t−τ~1t,x,…,giuit−τ~it,x,…,gnunt−τ~nt,xT, in which fivit−τit,x is the neuron activation function of the ith unit of time t−τit in space variable x and so is giuit−τ~it,x. Both τit and τ~it are discrete time delays with 0≤τit≤τ and 0≤τ~it≤τ~,∀i∈N. And distributed delays ρt and ρ~twith 0≤ρt≤ρ and 0≤ρ~t≤ρ~. In addition, the positive scalar τ∗=maxτ,τ~,ρ,ρ~. Here, τ,τ~, and τ∗ all may be +∞. Besides, there is a positive scalar l0< 1 such that τi′t≤l0 and τ~i′t≤l0 for all i∈N. Aut,x=diaga1u1t,x,…,aiuit,x,…,anunt,x, and A~vt,x=diaga~1v1t,x,…,a~ivit,x,…,a~nvnt,x, in which aiuit,x represents an amplification function and so does a~ivit,x. But,x=b1u1t,x,…,biuit,x,…,bnunt,xT, and B~ut,x=b~1v1t,x,…,b~ivit,x,…,b~nvnt,xT, in which both biuit,x and b~ivit,x are appropriately behavior functions. Cj,C~j,Mj and M~j are connection weight strength coefficient matrices, and ΔCjt,ΔC~jt,ΔMjt and ΔM~jt are real-valued matrix functions which represent time-varying parameter uncertainties.By means of a standard fuzzy inference method, (7) can be inferred as follows,
(8)∂u∂t=∇⋅Dt,x,u∘∇pu−Aut,xBut,x−∑j=1r0hjwtCj+ΔCjtfvt−τt,x+Mj+ΔMjt∫t−ρttfvs,xds,∂u∂t=∇⋅Dt,x,v∘∇pv−A~vt,xB~vt,x−∑j=1r0hjwtC~j+ΔC~jtgut−τ~t,x+M~j+ΔM~jt∫t−ρ~ttgus,xds,uθ,x=ϕθ,x,vθ,x=υθ,x,θ,x∈−τ∗,0×Ω,vt,x=0∈Rn,vt,x=0∈Rn,t,x∈R×∂Ω,where wt=w1t,w2t,…,ws∗tT, hjwt=wjwt/∑k=1rwkwt, and wjwt:Rs∗→0,1j=1,2,…,r0 is the membership function of the system with respect to the fuzzy rule j. hj can be regarded as the normalized weight of each if-then rule, satisfying hjωt≥0 and ∑j=1r0hjωt=1.Particularly in the case ofp=2, the system (8) is the so-called reaction-diffusion impulsive Markovian jumping BAM Cohen-Grossberg neural networks (BAM CGNNs). Inspired by some methods and conclusions of some related literature ([47–51]), we shall discuss the input-to-state stability reaction-diffusion BAM CGNNs with the event-triggered control in Section 4, for seldom existing literature involved to such complex model with feedback control.Lemma 2.1.
aq−1b≤q−1/qaq+bq/q, ∀a,b∈0,+∞, and q>1.
Note that Lemma2.1 is the particular case of the famous Young inequality.Lemma 2.2
(Schur complement [52]) Given matrices Qt, St, and Rt with appropriate dimensions, where Qt=QtT and ℛt=ℛtT, then
(9)QtStSTtℛt>0,if and only if
(10)ℛt>0,Qt−Stℛ−1tSTt>0,or
(11)Qt>0,ℛt−STtQ−1tSTt>0,where Qt, St, and ℛt are dependent on t.


## 3. Robust Stability on Nonlinearp-Laplacian Diffusion System in the Case of p≠2

Throughout this paper, we assume thatDt,x,u=diagD1,…,Dj…Dn and Dt,x,v=D1,…,Dj,…,Dn, where we denote Dj=Djt,x,u and Dj=Djt,x,v for short. In addition, we always denote ut,x=u1t,x,u2t,x,…,unt,xT and vt,x=v1t,x,v2t,x,…,vnt,xT. Denote ut,x by u and uit,x by ui and so do v and vj.Lemma 3.1.
Letp>1 be a positive real number, and Q=diagq1,q2,…,qn a positive definite matrix. Let u and v be a solution of (8). Then we have
(12)∫Ω∑j=1nqjui∑k=1n∂∂xkDj∇ujp−2∂uj∂xkdx≤∫Ωu1p/2,u2p/2,…,unp/2−λ1QDu1p/2u2p/2⋮unp/2dx,∫Ω∑j=1nqjvi∑i=1n∂∂xiDj∇vjp−2∂vj∂xidx≤∫Ωv1p/2,v2p/2,…,vnp/2−λ1QDv1p/2v2p/2⋮vnp/2dx,Proof.
Sinceu is a solution of (8), it follows by Gauss formula and the Dirichlet zero boundary condition that
(13)∫Ω∑j=1nqiuj∑k=1n∂∂xkDj∇ujp/2∂uj∂xkdx=−∑k=1n∑j=1n∫ΩqjDj∇ujp−2∂uj∂xk2dx≤∫Ωu1p/2,u2p/2,…,unp/2−λ1QDu1p/2u2p/2⋮unp/2dx.Another inequality can be similarly proved. And so the proof is completed.Remark 1.
Lemma3.1 improves [11, Lemma 3.1] and [18, Lemma 2.3] for the first time, which makes a contribution to the final LMI criterion.Remark 2.
In the case ofΩ=0,a⊂R1 or W01,p0,a, the first eigenvalue
(14)λ1=2a∫0p−11/pdt1−tP/p−11/pp
(see, e.g., [28]).Remark 3.
IfΩ=x1,x2T:0<x1<a,0<x2<b⊂R2 and p=2, the first eigenvalue λ1=π/a2+π/b2 (see, e.g., [26]).In this section, we suppose(H1)
There exist positive definite matricesA¯=diaga¯1,a¯2,…,a¯n, A¯=diaga¯1,a¯2,…,a¯n, A¯~=diaga¯~1,a¯~2,…,a¯~n, and A~¯=diaga~¯1,a~¯2,…,a~¯n such that
(15)0<a¯i≤aissp−2≤a¯i,0<a¯~i≤a~issp−2≤a~¯i,0≠s∈R,i=1,2,…,n;where Au=diaga1u1,a2u2,…,anun, and A~u=diaga~1u1,a~2u2,…,a~nun.
(H2)
There exists positive definite matricesB=diagb1,b2,…,bn and B~=diagb~1,b~2,…,b~n such that bi0=0=b~i0 and bis/s≥bi,b~i/s≥b~i,0≠s∈R,i=1,2,…,n, where Bu=b1u1,b2u2,…,bnunT and B~u=b~1u1,b~2u2,…,b~nunT.
(H3)
There are positive definite matricesF=diagF1,F2,…,Fn and G=diagG1,G2,…,Gn such that fis≤Fis,gis≤Gis,∀s∈R,i=1,2,…,n, where fv=f1v1,…,fnvnTand gv=g1v1,…,gnvnT.Remark 4.
The condition (H1) implies that the boundedness of amplification functionsai and a~i are unnecessary in the case of p>1 with p≠2, for we may take ais=a¯isp−2, which is actually unbounded for s∈−∞,+∞. Below, we denote for convenience
(16)Cjt=Cj+ΔCjtC~jt=C~j+ΔCjt,Mjt=Mj+ΔMjtM~jt=M~j+ΔMjt,where Cjt=cijktn×n, C~jt=c~ijktn×n, Mjt=mijktn×n, and M~jt=m~ijktn×n are diagonal matrices.Theorem 3.2.
Suppose that the conditions (H1)–(H3) hold andp≜p1/p2>1 with p1 being an even number and p2 being an odd number. Besides, there are four nonnegative matrices C∗, C~∗, M∗, and M~∗ such that
(17)−C∗≤ΔCjt≤C∗−C~∗≤ΔC~jt≤C~∗,−M∗≤ΔMjt≤M∗,−M~∗≤ΔM~jt≤M~∗.
Assume, in addition,(18)K≜B−∑j=1r0Cj+C∗G+ρMj+M∗G−∑j=1r0C~j+C~∗G+ρ~M~j+M~∗GB~−1≥0,and there is a positive definite matrix Q=diagq1,q2,…,qn such that
(19)λ1QD+QA¯B−∑j=1r0p−1pQA¯Cj+C∗F+ρp−1pQA¯Mj+M∗F+1p1−l0QA~¯C~j+C~∗G+ρ~pQA~¯M~j+M~∗G>0,and
(20)λ1QD+QA¯~B~−∑j=1r0p−1pQA~¯C~j+C~∗G+ρ~p−1pQA~¯M~j+M~∗G+1p1−l0QA¯Cj+C∗F+ρpQA¯Mj+M∗F>0,then there exists the globally asymptotically robust stable unique equilibrium point for (8).Remark 5.
Condition (4) does not complete the matrix form. However, (19)–(20) are complete linear matrices inequalities, which have even gotten better at dealing with the calculation of the large operations involved in the practical engineering by way of computer MATLAB programming.Proof.
There are three steps to the proof.Step 1.
We claim that the null solution is the unique equilibrium point for (8).In fact, we know from (H2)–(H3) thatbi0=b~i0=fi0=f~i0=gi0=g~i0=0, and hence u=0 and v=0 are the equilibrium solution of (8).Moreover, we prove that the equilibrium point is unique. Indeed, it follows from (H1) thatais>0. Let (21) be an equilibrium point for (8)
(21)uv,then we get
(22)00=Bu−∑j=1r0hjwtCjtfv+Mjt∫t−ρttfvdsB~v−∑j=1r0hjwtC~jtgu+M~jt∫t−ρ~ttguds=Bu−∑j=1r0hjwtCjt+MjtρtfvB~v−∑j=1r0hjwtC~jt+M~jtρ~tgu.If (23) is another equilibrium point of (8)
(23)u⌣v⌣,we can actually deduce from (22) that
(24)Bu−Bu⌣≤∑j=1r0Cjtfv−fv⌣+Mjt∫t−ρtfv−fv⌣ds≤∑j=1r0Cjtfv−fv⌣+Mjt∫t−ρtfv−fv⌣ds,and then
(25)Bu−u⌣≤∑j=1r0CjtFv−v⌣+Mjt∫t−ρtFv−v⌣ds=∑j=1r0CjtF+ρMjtFv−v⌣=∑j=1r0Cjt+C∗F+ρMj+M∗Fv−v⌣.Similarly,(26)B~v−v⌣≤∑j=1r0C~jtG+ρ~MjtGu−u~=∑j=1r0C~j+C~∗G+ρM~j+M~∗Gv−v⌣.Combining (25) and (26) implies
(27)B−∑j=1r0Cj+C∗F+ρMj+M∗F−∑j=1r0C~j+C~∗G+ρ~M~j+M~∗GB~u−u⌣v−v⌣≤0∈R2n,and (18) yields
(28)u−u⌣v−v⌣=0∈R2n,or
(29)uv=u⌣v⌣.Thereby, the null solution is the unique equilibrium point for (8).Remark 6.
In ordinary differential systems, the uniqueness of the equilibrium solution can be determined by the existence of the equilibrium solution and its global asymptotic stability. However, (8) is a partial differential system, including two different variables: t and x. Since the existence of the equilibrium solution and its global asymptotic stability only determines the equilibrium solution which is unique about variable t, but it may be not unique on variable x. Hence, it is necessary to verify the uniqueness of the equilibrium solution.Step 2.
To derive LMI-based criterion in which the nonlinear diffusion terms can play roles, we need to construct new Lyapunov-Krasovskii functionals as follows:(30)Vt=V1t+V2t+V3t+V4t+V5t+V6t,where
(31)v1t=∫ΩuTt,xQut,xdx=∑i=1n∫Ωqiui2dx,v2t=∫ΩvTt,xQvt,xdx=∑j=1n∫Ωqivj2dx,(32)v3t=21−l0∑i=1n∑j=1r0∑k=1na¯iqicijk+c∗ijkFk∫Ω∫t−τkttvks,xppdsdx,v4t=21−l0∑i=1n∑j=1r0∑k=1na~¯iqic~ijk+c~∗ijkGk∫Ω∫t−τ~kttuks,xppdsdx,(33)v5t=2p∑j=1r0∫Ω∫−ρ0dσ∫t+σtv1s,xp/2,…,vns,xp/2QA¯Mj+M∗Fv1s,xp/2v2s,xp/2⋮vns,xp/2dsdx,(34)v6t=2p∑j=1r0∫Ω∫−ρ~0dσ∫t+σtu1s,xp/2,…,uns,xp/2QA~¯M~j+M~∗Gu1s,xp/2u2s,xp/2⋮uns,xp/2dsdx,Remark 7.
The uncertainty of parameters brings a difficulty to design the Lyapunov functions. If imitating the previous Lyapunov functions in existing literature, for example, let(35)v3t=21−l0∑i=1n∑j=1r0∑k=1na¯iqicijktFk∫Ω∫t−τkttvks,xppdsdx,
one can find it impossible that the sufficient conditions of stability criterion can be derived. In addition, Lyapunov functions (33) and (34) help us to derive the complete linear matrix inequality condition for the stability criterion of nonlinear diffusion system (8).Step 3.
We claim that the null solution is globally asymptotically robust stable.Evaluating the time derivation ofV1t along the trajectory of the (8), we can derive from Lemma 3.1
(36)v1′t≤∫Ω2−λ1UTt,xQDUt,x−uTQAuBudx+2∫Ω∑j=1r0hjwtuTQAuCjtfvt−τt,xdx+2∫Ω∑j=1r0hjwtuTQAuMjt∫t−ρttfvs,xdsdx,where we simply denote
(37)Ut,x=u1t,xP/2u2t,xP/2⋮unt,xP/2,vt,x=v1t,xp/2v2t,xp/2⋮vnt,xp/2,and
(38)Ut−τt,x=u1t−τ1t,xP/2u2t−τ2t,xP/2⋮unt−τnt,xP/2,Vt−τ~t,x=v1t−τ~1t,xP/2v2t−τ~2t,xP/2⋮vnt−τ~nt,xP/2.It follows by (H1), (H2), and the conditions on the parameterp that
(39)saisbis≥aibisp,∀s∈R.So combining (H1), (H2), and (39) results in
(40)2∫ΩuTQAuBudx=2∫Ω∑i=1nuiqiaiuibiuidx≥2∫Ω∑i=1nqia¯ibiuipdx=∫ΩUTt,x2QA¯BUt,xdx,From Lemma2.1, (H1), and (H3), we get
(41)2∫Ω∑j=1r0hjwtuTQAuCjtfvt−τt,xdx≤2∑j=1r0∫ΩuTQAuCjtfvt−τt,xdx≤2∑i=1n∑j=1r0∑k=1na¯iqicijktFk∫Ωp−1puip+vkt−τkt,xppdx=∑j=1r0∫ΩUTt,x2p−1pQA¯CjtFUt,x+∑j=1r0∫ΩVTt−τt,x2pQA¯CjtFVt−τt,xdx≤∑j=1r0∫ΩUTt,x2p−1pQA¯CjtFUt,x+∑j=1r0∫ΩVTt−τt,x2pQA¯Cj+C∗FVt−τt,xdx,where Cjt=cijktn×n.Besides, we can conclude from (H2), (H3), and Lemma2.1 that
(42)2∫Ω∑j=1r0hjwtuTQAuMjt∫t−ρttfvs,xdsdx≤2∑j=1r0∫Ω∫t−ρttuTQAuMjtfvs,xdsdx≤2∑i=1n∑j=1r0∑k=1na¯iqimijktFk∫Ω∫t−ρtuip−1vks,xdsdx≤∑j=1r0∫ΩUTt,x2ρp−1pQA¯MjtFUt,xdsdx+∑j=1r0∫Ω∫t−ρtVTs,x2pQA¯Mj+M∗FVs,xdsdx,where Mjt=mijktn×n.So we have(43)v1′t≤∫ΩUTt,x−2λ1PD−2QA¯B+∑j=1r02p−1pQA¯CjtF+2ρp−1pQA¯MjtFUt,x+∑j=1r0∫ΩVTt−τt,x2pQA¯Cj+C∗FVt−τt,xdx+∑j=1r0∫Ω∫t−ρtVTs,x2pQA¯Mj+M∗FVs,xdsdx.Besides, we get by (32)
(44)v3′t≤21−l0∑i=1n∑j=1r0∑k=1na¯iqicijk+c∗ijkFk∫Ωvkt,xppdx−2∑i=1n∑j=1r0∑k=1na¯iqicijk+c∗ijkFk∫Ωvkt−τkt,xppdx=∑j=1r0∫ΩVTt,x2p1−l0QA¯Cj+C∗FVt,xdx−∑j=1r0∫ΩVTt−τt,x2pQA¯Cj+C∗FVt−τt,xdx,One can deduce from (33) that
(45)v5′t=∑j=1r0∫Ω∫−ρ0VTt,x2pQA¯Mj+M∗FVt,xdx−∑j=1r0∫Ω∫−ρ0VTt+s,x2pQA¯Mj+M∗FVt+s,xdsdx=∑j=1r0∫ΩVTt,x2pQA¯Mj+M∗FVs,xdx−∑j=1r0∫Ω∫t−ρtVTs,x2pQA¯Mj+M∗FVs,xdsdx.Hence,(46)v1′t+v3′t+v5′t≤2∫ΩUTt,x−λ1QD−QA¯B+∑j=1r0p−1pQA¯CjtF+ρp−1pQA¯MjtFUt,x+2∫ΩVTt,x∑j=1r01p1−l0QA¯Cj+C∗F+ρpQA¯Mj+M∗FVt,xdx.Similarly, we can deduce fromV2, V4, and V6 that
(47)V2′t+V4′t+V6′t≤2∫ΩVTt,x−λ1QD−QA~¯B~+∑j=1r0p−1pQA~¯C~jtG+ρ~p−1pQA~¯M~jtGVt,x+2∫ΩUTt,x∑j=1r01p1−l0QA~¯C~j+C~∗G+ρ~pQA~¯M~j+M~∗GUt,xdx.Therefore, (17), (19), and (20) yield
(48)V′t≤2∫ΩUT−λ1QD−QA¯B+∑j=1r0p−1pQA¯CjtF+ρp−1pQA¯MjtF+1p1−l0QA~¯C~j+C~∗G+ρ~pQA~¯M~j+M~∗GU+2∫ΩVT−λ1QD−QA~¯B~+∑j=1r0p−1pQA~¯C~jtG+ρ~p−1pQA~¯M~jtG+1p1−l0QA¯Cj+C~∗F+ρpQA¯Mj+M∗FV≤0.It follows by the standard Lyapunov functional theory that the null solution of (8) is globally asymptotically robust stable. And the proof is completed.Remark 8.
There have been some other approaches removing boundedness of amplification functions. For example, in [53], an appropriate Lyapunov-Krasovskii functional is set up to derive the LMI-based μ-stability for discrete time-delay system. This is really a good result. However, in this paper, our system model (8) is the continuous system different from the discrete system ([54, (1)]). Of course, the main results of this paper are inspired by some methods and ideas of these documents.Remark 9.
The Lyapunov functionals (33) and (34) are similar to the quadric form different from those of [11, 13, 14, 20]. Actually, the quadric form and matrix form help us to derive the LMI-based criterion.Remark 10.
The boundedness of amplification functionsai and a~j may be unbounded while amplification functions are always proposed to be bounded in many existing results (see, e.g., [7, 9, 18–21]).If the diffusion phenomena are ignored, (8) degenerates into the following BAM CGNNs with discrete and distributed time-varying delays:
(49)dxdt=−AxtBxt−∑j=1r0hiwtCj+ΔCjtfyt−τt+Mj+ΔMjt∫t−ρttfysds,dydt=−A~ytB~yt−∑j=1r0hiwtC~j+ΔC~jtgxt−τ~t+M~j+ΔM~jt∫t−ρ~ttgxsds,xs=ϕs,ys=υs,s∈−τ∗,0.Since in ordinary differential systems, the uniqueness of the equilibrium solution can be determined by the existence of the equilibrium solution and its global asymptotic stability, and the diffusion items disappear, we can directly deduce the following corollary from Theorem3.2:Corollary 3.3.
Suppose that the conditions (H1)–(H3) hold. Besides, there are four nonnegative matricesC∗,C~∗,M∗, and M~∗ such that
(50)−C∗≤ΔCjt≤C∗,−C~∗≤ΔC~jt≤C~∗,−M∗≤ΔMjt≤M∗,−M~∗≤ΔM~jt≤M~∗,and there is a positive definite matrix Q=diagq1,q2,…,qn such that
(51)QA¯B−∑j=1r0p−1pQA¯CjF+ρp−1pQA¯MjF+1p1−l0QA~¯C~jG+ρ~¯pQA~¯M~jG>0,and
(52)QA¯~B~−∑j=1r0p−1pQA~¯C~jG+ρ~p−1pQA~¯M~jG+1p1−l0QA¯CjF+ρpQA~MjF>0,then there exists the unique globally asymptotically robust stable equilibrium point for (49).Remark 11.
For the BAM CGNNs (53), Corollary 3.3 deletes the boundedness of amplification functions ai and a~j, improving related results (see, e.g., [7, 9, 18, 19, 21]). This is also shown below (Table 1).Table 1
Comparisons of amplification functionajujt,x in related results.
Theorems and examples
Boundedness conditions ofaj
Theorem3.2, (Corollary 3.3), Example 5.1
No requirements
[9, Theorem 2.1, Theorem 3.1, and Example 1]
0<a¯<ajr<a¯,∀r∈R
[2, Theorem 1 and Example 1]
0<a¯j≤ajr≤a¯j,∀r∈R
[3, Theorem 3.1 and Example 4.1]
0<a¯j≤ajr<∞withaj′rr≥0,∀r∈R
[3, Theorem 3.1 and Example 4.1]
0<a¯j≤ajr,∀r∈R
[5, Theorem 4 and Example 1]
0<a¯j≤ajr≤a¯j,∀r∈R
[6, Theorem 3.1 and Example 4.1]
0<a¯j≤ajr≤a¯j,∀r∈R


## 4. Input-to-State Stability of Markovian Jumping Reaction-Diffusion BAM CGNNs with Event-Triggered Control in the Case ofp=2

In this section, we consider the following Markovian jumping reaction-diffusion BAM CGNNs with event-triggered control under Dirichlet zero-boundary value.(53)∂u∂t=∇⋅Dt,x,u∘∇u−Aut,xBut,x−ζv~t,x−∑j=1r0hiwtCrj+ΔCrjtfvt−τt,x+Mrjφv^t,x,∂u∂t=∇⋅Dt,x,v∘∇v−A~vt,xB~vt,x−ζu~t,x−∑j=1r0hiwtC~rj+ΔC~rjtgut−τ~t,x+M~rjφu^t,x,for all t>0, the initial value is uθ,x=ϕθ,x, vθ,x=ψθ,x, and ∀θ,x∈−τ∗,0×Ω, where u~,v~ represent feedback, and u~,v~ represent the unknown exogenous disturbance of the neuron. For any k=0,1,2,…, the time tk is the triggering time or update time. Between the triggering times tk and tk+1, the feedback control is designed as
(54)u^t,x=ξutk,x,v^t,x=ηvtk,x,t∈tk,tk+1,where t0=0,u^t,x=u^1t,x,…,u^nt,xT, v^t,x=v^1t,x,…,v^nt,xT, ξ=diagξ1,…,ξn, and η=diagη1,…,ηn. Here ξi and ηi are constants for all i.LetΩ⌣,ϒ,ℙ be the given probability space where Ω⌣ is the sample space, ϒ is σ, the algebra of subset of the sample space, and ℙ is the probability measure defined on ϒ. Let S=1,2,…,n0 and the random form process rt:0,+∞→S be a homogeneous, finite-state Markovian process with right continuous trajectories with generator Π=γijn0×n0 and transition probability from mode i at time t to mode j at time t+∆t, i, and j∈S.
(55)ℙrt+δ=jrt=i=γijδ+oδ,j≠i,1+γijδ+oδ,j≠i,where γij≥0 is transition probability rate from i to jj≠i and γii=−∑j=1,j≠in0γij,δ>0 and limδ→0oδ/δ=0.Letet,x=e1t,x,…,ent,xT and e~t,x=e~1t,x,…,e~nt,xT be the error signal defined by
(56)et,x=utk,x−ut,x,e~t,x=vtk,x−vt,x,t∈tk,tk,+1,then we actually get
(57)u^t,x=ξut,x+et,x,v^t,x=ηvt,x+e~t,x,t≥0.Define the event-triggering mechanism by(58)tk+1=inft>tk:θu~L22+v~L22+εuL22+vL22−ut−τ~tL22+vt−τtL22−WeL22+e~L22<0,where θ>0, ε∈0,1, and W>0.Remark 12.
Such thattk+1 is always defined well on many occasions. For example, let initial value uθ,x=ϕθ,x≡0, vθ,x=ψθ,x≡0, and ∀θ,x∈−τ∗,0×Ω, then we must get t1>0.In this section, we assume that the conditions (H1)–(H3) hold still in the case ofp=2.Besides, suppose that(H4)
ζv~=ζ1v~1,ζ2v~2,…,ζnv~nT with a positive definite matrix L=diagl1,l2,…,ln such that
(59)ζis−ζit≤lis−t,∀s,t∈Rn,i=1,2,…,n.
(H5)
φv^=φ1v^1,φ2v^2,…,φnv^nT with a positive definite matrix L^=diagl^1,l^2,…,l^n such that
(60)φis−φit≤l^is−t,∀s,t∈Rn,i=1,2,…,n.For any moder∈S,
(61)Crjt=Crj+ΔCrjt,C~rjt=C~rj+ΔCrjt,which do not have to be diagonal matrices or other special matrices.In addition, we assume that(62)f0=g0=0=ζ0=φ0,which can guarantee that u=0, and v=0 is a trivial solution of (53).Besides, there are nonnegative matricesCr∗ and C~r∗ such that
(63)−Cr∗≤ΔCrjt≤Cr∗,−C~r∗≤ΔC~rjt≤C~r∗.Before giving the man result of this section, we need the following lemma:Lemma 4.1
([54]) Let x∈Rn, y∈Rn, and ε>0. Then we have
(64)xTy+yTx≤εxTx+ε−1yTyDefinition 4.2.
System (53) is called robust stochastic input-to-state in mean square stable for all admissible uncertainties satisfying (63), if for t>0, there exist function β∈Kℒ and γ∈K such that
(65)EuL22+vL22≤βt,Eu0,xL22+v0,xL22+γu~L22+∥v~∥L22,where K=γ⋅∣γ:R+→R+ is continuous strictly increasing with γ0=0 with R+=0,∞, Kℒ=β⋅,⋅∣βt,⋅∈K for each fixed t,βt,y is decreasing for fixed y and limt→∞βt,y=0.Theorem 4.3.
Assume the conditions (H1)–(H5) hold. Suppose that there is a sequence of positive definition matricesPrr∈S and positive scalars α1, α2, α3, and α4 such that the following LMI conditions hold for any mode r∈S:
(66)QrPrA¯Cr1+Cr∗F⋯PrA¯Crr0+Cr∗F1α2PrA¯Mr1⋯1α2PrA¯Mrr0∗−I⋯00⋯0∗∗⋱00…0∗∗⋯−I0⋯0∗∗⋯∗−I⋯0∗∗∗∗∗⋱0∗∗∗∗∗⋯−I>0,Q~rPrA~¯C~r1+C~r∗G⋯PrA~¯C~rr0+C~r∗G1α2PrA~¯Mr1⋯1α2PrA~¯Mrr0∗−I⋯00⋯0∗∗⋱00…0∗∗⋯−I0⋯0∗∗⋯∗−I⋯0∗∗∗∗∗⋱0∗∗∗∗∗⋯−I>0,where I represents the identity matrix with suitable dimension under different cases for convenience.
(67)Qr=2λ1PrD+2PrA¯B−∑j=1n0γrjPj−1α1I−2r0α4ξ2L^2−εr0I,Q~r=2λ1PrD+2PrA¯~B~−∑j=1n0γrjPj−1α3I−2r0α2η2L^2−εr0I.
If, in addition,(68)maxλmax2r0α2η2L^2,λmax2r0α4ξ2L^2≤r0W,then (53) is a robust stochastic input-to-state stable in mean square.Proof.
Construct the Lyapunov-Krasovskii functionals as follows:(69)Vt,r=V1t,r+V2t,r,r∈S,V1t,r=∫ΩuTt,xPrut,xdx,V2t,r=∫ΩvTt,xPrvt,xdx,where each Prr∈S is positive definition diagonal matrix.Due to
(70)v^t,x=ηvt,x+e~t,x,we get
(71)v^t,xTv^t,x≤2vTη2v+e~t,xTη2e~t,x,and
(72)2∫ΩuTPrAu∑j=1r0Mrjφv^t,xdx≤2∑j=1r0∫ΩuTPrA¯MrjL^v^t,xdx≤∑j=1r0∫Ω1α2uTPrA¯MrjMrjTA¯Pru+α2v^t,xTL^2v^t,xdx≤∑j=1r0∫Ω1α2uTPrA¯MrjMrjTA¯Pru+2α2vTη2L^2v+2α2e~t,xTη2L^2e~t,xdx,where u=ut,x and v=vt,x.
Similarly, we get(73)2∫ΩuTPrLv~t,xdx≤∫Ω1α1uTu+α1v~t,xTLPrPrLv~t,xdx.
Next,(74)2∑j=1r0∫ΩuTPrAuCrjtfvt−τt,xdx≤2∑j=1r0∫ΩuTPrA¯CrjtFvt−τt,xdx≤∑j=1r0∫ΩuTPrA¯CrjtFFCrjtTA¯Prudx+r0∫Ωvt−τt,xTvt−τt,xdx.
LetL be the weak infinitesimal operator, then we get
(75)ℒV1t,x≤∫Ωut,xT−2λ1PrD−2PrA¯B+∑j=1n0γrjPj+1α2I+∑j=1r0PrA¯CrjtFFCrjtTA¯Pr+∑j=1r01α2PrA¯MrjMrjTA¯Prut,xdx+∫Ωv~t,xTα1LPrPrLv~t,xdx+r0∫Ωvt−τ,xTvt−τt,xdx+∫Ωvt,xT2r0α2η2L^2vt,x+∫Ωe~t,xT2r0α2η2L^2e~t,xdx
Similarly,(76)ℒV2t,x≤∫Ωut,xT−2λ1PrD−2PrA¯~B~+∑j=1n0γrjPj+1α3I+∑j=1r0PrA~¯C~rjtGGC~rjtTA~¯Pr+∑j=1r01α4PrA~¯M~rjM~rjTA~¯Prvt,xdx+∫Ωu~t,xTα3LPrPrLu~t,xdx+r0∫Ωut−τ~t,xTut−τ~t,xdx+∫Ωut,xT2r0α4ξ2L^2ut,x+∫Ωet,xT2r0α4ξ2L^2et,xdx.
Hence, we get(77)ℒVt,r≤∫Ωut,xT−2λ1PrD−2PrA¯B+∑j=1r0γrjPj+1α1I+∑j=1r0PrA¯CrjtFFCrjtTA¯Pr+∑j=1r01α2PrA¯MrjMrjTA¯Pr+2r0α4ξ2L^2ut,xdx+∫Ωvt,xT−2λ1PrD−2PrA¯~B~+∑j=1n0γrjPj+1α3I+∑j=1r0PrA~¯C~rjtGGC~rjtTA~¯Pr+∑j=1r01α4PrA~¯M~rjM~rjTA~¯Pr+2r0α2η2L^2vt,xdx+∫Ωu~t,xTα3LPrPrLu~t,xdx+∫Ωv~t,xTα1LPrPrLv~t,xdx+r0∫Ωut−τ~t,xTut−τ~t,xdx+r0∫Ωvt−τt,xTvt−τt,xdx+∫Ωet,xT2r0α4ξ2L^2et,xdx+∫Ωe~t,xT2r0α2η2L^2e~t,xdx.
That means(78)ℒVt,x≤−ΘruL23+vL23+r0ut−τ~tL22+vt−τtL22+ΨeL23+e~L23+Φru~L23+v~L23,where
(79)Θr=minλminQr−∑j=1r0PrA¯Crj+Cr∗FFCrj+Cr∗TA¯Pr−∑j=1r01α2PrA¯MrjMrjTA¯Pr,λminQ~r−∑j=1r0PrA~¯C~rj+C~r∗GGC~rj+C~r∗TA~¯Pr−∑j=1r01α4PrA~¯M~rjM~rjTA~¯Pr,Ψ=maxλmax2r0α2η2L^2,λmax2r0α4ξ2L^2≤r0W,and
(80)Φr=maxλmaxα3LPrPrL,λmaxα1LPrPrL.
In addition, for anyt∈tk,tk+1, the definition of tk+1 derives
(81)θu~L22+v~L22+εuL22+vL22−ut−τ~tL22+ut−τtL22−WeL22+e~L22≥0.
So we get(82)ℒVt,r≤−Θr−εr0uL22+uL22+Φr+θr0u~L22+v~L22,or
(83)ℒVt,r≤−β1uL22+vL22+β2u~L22+v~L22,where (66) and Schur complement lemma yield that Θr−εr0>0 and hence β1>0 with β1=minr∈SΘr−εr0 and β2=maxr∈SΦr+θr0>0.
Furthermore, Dynkin’s formula yields(84)ddtEVt,r=ELVt,r≤−β1uL22+vL22+β2u~L22+v~L22≤−β1λmaxPrEVt,r+β2u~L22+v~L22≤−β3EVt,r+β2u~L22+v~L22,where
(85)β3=minr∈Sβ1λmaxPr>0.
Applications of the Comparison principle to (84) reaches
(86)EVt,r≤EV0,re−β3t−β2u~L22+v~L22β3e−β3t−1≤EV0,re−β3t+β2u~L22+v~L22β3≤λmaxPru0,xL22+v0,xL22e−β3t+β2u~L22+v~L22β3,which derives
(87)uL22+vL22≤λmaxPrλmaxPru0,xL22+v0,xL22e−β3t+β2u~L22+v~L22β3λminPr,which together with Definition 4.2 implies that (53) is robust stochastic input-to-state stable in mean square.Remark 13.
Theorem4.3 provides a new stability criterion which is different from the existing criteria of [55–59]. In addition, to the best of our knowledge, it is the first time to investigate input-to-state stability of reaction-diffusion time-delay system with event-triggered control. Especially, the diffusion items play roles in the criterion.


## 5. Numerical Examples

Example 5.1.
Consider (8) with the following parameters: p=8/3,u=u1t,x,u2t,xT, v=v1t,x,v2t,xT∈R2, x∈Ω=0,π, and then the first eigenvalue
(88)λ1=2π∫08/3−11/8/3dt1−t8/3/8/31/8/38/3=9558
(see Remark2).Leta1u1=0.1u1231+cos2u1+1, a2u2=0.2u223, a~1v1=0.2u123, a~2v2=0.1u2231+sin4v2−10, b1u1=2u11+sin2u1, b2u2=2.5u2, d1u1=2v11+cos2u1, d2u2=2.7u2, f1v1=0.16v1sinv1, f2v2=0.166v2, g1u1=0.15u1sinu1, g2u2=0.17u2, τ1t=τ2t=t/2=τ~1t=τ~2t, ρ1t=ρ2t=ρ~1t=5=ρ~2t, and l0=0.5, ρ=ρ~=5,
(89)A¯=0.1000.2,A¯=0.2000.2=A~¯,A¯~=0.2000.1,B=2002.5,B~=2002.7,Dt,x,u=0.003000.006,Dt,x,u=0.0033000.0057,F=0.16000.166,G=0.15000.17.Letr0=2, and
(90)C1=−0.0530.00110.00180.085,C2=0.0860.0009−0.00110.0085,C~1=0.0360.001−0.00110.085,C~2=0.0350.0011−0.00090.088,M1=−0.0230.00130.00080.072,M2=0.0760.0003−0.00020.0072,M~1=0.0360.0003−0.00090.036,M~2=0.0320.0003−0.00020.077,C∗=0.0330.0011−0.00180.0063,C~∗=0.00630.00130.00120.00036,M∗=0.00330.0061−0.00190.0013,M~∗=0.00330.00110.00390.0066.So we can use MATLAB software to compute (18), obtaining
(91)K=2.00000−0.1078−0.069302.5000−0.0644−0.0813−0.0685−0.10812.00000−0.0734−0.125102.7000−1=0.50140.00180.02710.01290.00092.40120.01300.01210.01720.02170.50160.00110.01370.01860.00130.3713≥0.Moreover, utilizing MATLAB LMI toolbox to solve LMIs (19)–(20) reaches the feasibility data as follows:
(92)Q=46.61340046.9921.Now, one can conclude from Theorem3.2 that there exists the globally asymptotically robust stable unique equilibrium point for (8).Remark 14.
From Table1, we know, our new results (Theorem 3.2 and Corollary 3.3) is novel because the boundedness of amplification functions becomes unnecessary.Remark 15.
From Table2, we know, our Theorem 3.2 is novel, different from those of existing results.Example 5.2.
Consider (63) with the following parameters: Ω=0,10×0,10 and u=u1,u2T, v=v1,v2T∈R2, and x=x1,x2∈Ω⊂R2. And so λ1=0.02π2=0.1974 (see Remark 3).Table 2
Comparisons of our results with other results related to reaction-diffusion.
Related results
Value ofp
Diffusion plays a role
Applicable to MATLAB LMI toolbox
Our Theorem3.2
p>1
Yes
Completely applicable
[19, Theorem 3.1]
p>2
No
Not
[8, Theorem 3.1]
p=2
Yes
[11, Theorem3.2]
p>1
No
Not
[16, Theorem 1–3]
p=2
Yes
Not
[18, Theorem 3.1]
p>1
No
YesLeta1u1=0.1u1231+cos2u1+1, a2u2=0.2u223, a~1v2=0.2v123, a~2v2=0.1v2231+sin4v2−10, b1u1=2u11+sin2u1, b2u2=2.5u2, d1v1=2v11+cos2v1, d2v2=2.7v2, f1v1=0.16v1sinv1, f2v2=0.166v2, g1u1=0.15u1sinu1, g2u2=0.17u2, τ1t=τ2t=t/2=τ~1t=τ~2t,
(93)A¯=0.1000.2,A¯=0.2000.2=A~¯,A¯~=0.2000.1B=2002.5,B~=2002.7,Dt,x,u=0.003000.006,Dt,x,u=0.0033000.0057,F=0.16000.166,G=0.15000.17.Letr0=2=n0, and
(94)C11=−0.0530.00110.00180.085,C12=0.0860.0009−0.00110.0058,C~11=0.0360.001−0.00110.016,C~12=0.0350.0011−0.00090.088,M11=−0.0230.00130.00080.072,M12=0.0760.0003−0.00020.0027,M~11=0.0360.0003−0.00090.036,M~12=0.0320.0003−0.00020.077,C1∗=0.0330.0011−0.00180.0063,C~1∗=0.00630.00130.00120.00036,C~2∗=0.00580.00130.00120.00053,C21=−0.0490.00140.00180.085,C22=0.0890.0009−0.00180.0058,C~21=0.0540.001−0.00110.016,C~22=0.0430.0011−0.00090.088,M21=−0.0320.00130.00080.072,M22=0.0830.0003−0.00020.0027,M~21=0.0430.0003−0.00090.036,M~22=0.0390.0003−0.00020.077,C2∗=0.00430.0011−0.00180.0063,ξ=0.0010000.0015,η=0.0021000.0033,ζv~=ζ1v~1,ζ2v~2T=0.01sinv~1,0.02sin2v~2φu^=φ1u^1,φ2u^2T=0.01sin3u^1,0.02u^2.Letθ=0.01, ε=0.001, and W=0.5 and then we can compute and verify that (68) is satisfied. Now using computer MATLAB LMI-toolbox to solve LMI (66) gives the feasibility data as follows:
(95)P1=7.7856006.899,P2=6.6189006.9973,α1=1.5346,α2=1.5986,α3=1.1323,α4=0.9869.According to Theorem4.3, (53) is robust stochastic input-to-state stable in mean square.Remark 16.
This paper is inspired by the methods and conclusions of the previous literature [55–59]. But the sufficient conditions of Theorem 4.3 is easier to be verified than those of existing results.


## 6. Conclusions

In this paper, we mainly provided two novel conclusions forp-Laplace diffusion BAM CGNNs. In the case of p>1 with p≠2, the authors construct novel Lyapunov functional to overcome the mathematical difficulties of nonlinear p-Laplace diffusion time-delay model with parameter uncertainties, deriving the LMI-based robust stability criterion applicable to computer MATLAB LMI toolbox, deleting the boundedness of the amplification functions. On the other hand, when p=2, LMI-based sufficient conditions are also inferred for robust input-to-state stability of reaction-diffusion Markovian jumping BAM CGNNs with the event-triggered control, which is different from those of many previous related literature. As far as we are concerned, seldom literature involved a reaction-diffusion stochastic system with time delays and the event-triggered control. It is the first time to explore the method for the stability analysis of this system. Finally, numerical examples illustrate the effectiveness and feasibility via computer MATLAB LMI toolbox.



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*Source: 6263931-2018-07-16.xml*