# New Results on Stability of Delayed Cohen–Grossberg Neural Networks of Neutral Type

Authors: Ozlem Faydasicok
Journal: Complexity
 (2020)

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## Abstract

This research work conducts an investigation of the stability issues of neutral-type Cohen–Grossberg neural network models possessing discrete time delays in states and discrete neutral delays in time derivatives of neuron states. By setting a new generalized appropriate Lyapunov functional candidate, some novel sufficient conditions are proposed for global asymptotic stability for the considered neural networks of neutral type. This paper exploits some basic properties of matrices in the derivation of the results that establish a set of algebraic mathematical relationships between network parameters of this neural system. A key feature of the obtained stability criteria is to be independent from time and neutral delays. Therefore, the derived results can be easily tested. Moreover, a constructive numerical example is studied to check the verification of presented global stability conditions.


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## Body

## 1. Introduction

In the past few decades, a variety of neural network models including Hopfield neural networks (HNNs), cellular neural networks (CNNs), Cohen–Grossberg neural networks (CGNNs), and bidirectional associative memory neural networks (BAMNNs) have been utilized for solving some typical engineering problems associated with pattern recognitions, signal processing, associative memories, and optimization related problems [1–9]. In these typical engineering applications, it is usually desired that the dynamics of the employed neural network must exhibit some certain behaviors depending on the characteristics of the problem to be solved. For instance, if one needs to solve an optimization problem, then the aimed designed neural network may require to possess a unique and globally asymptotically stable equilibrium point for every fixed input value. In this aspect, it becomes an important requirement to analyze stability behaviors of dynamical neural systems. On the contrary, neural networks have also been electronically implemented for real time applications of various classes of engineering problems. It is known that, in the process of electronically implementing a neural network, because of the finite switching speed of operational amplifiers and signal transmission times of neurons due to the communications of neurons, delay parameters encounter. The presence of the time delay parameters may lead to various complex nonlinear dynamics including instability, periodic solutions, and chaos. Therefore, one needs to consider the possible effects of these time delays on the stability properties of neural systems. In the recent literature, the stability issues for delayed neural networks have been addressed by a variety of researchers, and various sets of novel sufficient results on global asymptotic stability of the equilibrium point for different neural network models have been published [10–25]. It should mention that stability analysis of neural networks whose mathematical model with only time delays may not be appropriate to address the complete characteristics of dynamics for these types of neural network models. The reason for this fact is that, in many cases, beside the states involving time delays, the time derivative of the states may also have some different types of delays. In this sense, we need to consider the neural networks having delays both in states and in time derivative of states. A neural network that is modelled in this way is called neutral-type neural network. A widely studied neural network of this class is that of Cohen–Grossberg neural networks possessing discrete time and neutral delay parameters. This neural network model is defined by the nonlinear dynamical equations:(1)x˙it=dixit−cixit+∑j=1naijfjxjt+∑j=1nbijfjxjt−τj+ui+∑j=1neijx˙jt−ζj.where xit is representing states for the neurons, cixit is some behaved functions, and dixit is representing the amplification functions. The constant elements aij and bij are representing interconnection weights among the neurons. τj1≤j≤n is representing time delays and ζj1≤j≤n is representing neutral delays. The element eij is denoting the weights of time derivative of states including delays. fj⋅ is denoting neuronal activation functions, and ui is constant input of ith neuron. In system (1), we state some general assumptions. Let τ=maxτj, ζ=maxζj, 1≤j≤n, and δ=maxτ,ζ. Under these assumptions, neural network model (1) keeps the initial values stated by xit=φit and x˙it=ϑit∈C−δ,0,R. We note that C−δ,0,R represents real-valued functions that are described on the interval −δ,0 to R.We can make some remarks on system (1) to address the role of this system. If we make some simple changes in the mathematical model of system (1), we can easily have some other forms of neural network models. If we let eij=0,∀i,j, then system (5) becomes a delayed Cohen–Grossberg network. If we let eij=0,∀i,j and dixit=1 and cixit=xit, then, system (1) will define the class of Hopfield neural networks. If we let eij=0,∀i,j, dixit=1ci, cixit=xit, and fj⋅ be a specific activation function with the binary output values, then neural network (1) turns into the cellular neural network. Thus, the stability analysis of (1) will also address the stability of many different neural network models.In stability analysis of the neutral-type network system whose dynamical activities are governed by (1), the primary question to be addressed is the determination of mathematical relationships in the neuron states and the functions dixit, cixit, and fixit. The well-known basic assumptions on these nonlinear functions are given below.A1: the function dixit has the following property:(2)0<μi≤dixit≤ρi,∀i,∀xit∈R,where μi and ρi are positive valued real constants.A2: cixit have the following property:(3)0<γi≤cixit−ciyitxit−yit=cixit−ciyitxit−yit≤ψi,∀i,∀xit,yit∈R,xit≠yt,where γi and ψi are positive valued real constants.A3: the function dixit has the following property:(4)fixit−fiyit≤ℓixit−yit,∀i,∀xit,yit∈R,xit≠yit,where ℓi is positive-valued real constant.If neutral-type neural networks possess discrete delays, then the mathematical models of such neural systems can be stated in the forms of vectors and matrices. Then, we can study the stability of such neural networks by exploiting the linear matrix inequality approach using the other appropriate mathematical methods. In [26–34], the stability of neutral-type neural networks have been studied, and by constructing some classes of suitable Lyapunov functionals together with setting some useful lemmas and new mathematical techniques, different novel stability results on the considered neutral-type neural networks of various types of linear matrix inequalities have been presented. In [35–40], novel global stability conditions for neutral-type neural networks in the forms of different linear matrix inequality formulations have been proposed by employing various proper Lyapunov functionals with the triple or four integral terms. In [41, 42] various stability problems for neutral-type neural networks have been analyzed, and by setting the semifree weighting matrix techniques and an augmented Lyapunov functional, some less conservative and restrictive global stability conditions of linear matrix inequalities have been proposed. In [42], the stability for neural networks of neutral-type possessing discrete delays has been suitable conducted, and by employing a proper Lyapunov functional that makes a combination of the descriptor model transformation, a novel stability criterion has been formulated in linear matrix inequalities. In [16], stability of neural systems has been addressed, and by proposing an appropriate Lyapunov functionals utilizing auxiliary function-type integral inequalities and reciprocally convex method, various sets of stability results via linear matrix inequalities have been obtained. In [43], the Lagrange stability issue of neutral-type neural systems having mixed delays has been analyzed, and by using the suitable Lyapunov functionals and applying some appropriate linear matrix inequality techniques, various sufficient criteria have been obtained to ensure Lagrange stability of neural networks of neutral type. In [44], the issues associated with stability of neutral-type singular neural systems involving different delay parameters have been studied, and by setting a novel adequate Lyapunov functional and some rarely integral inequalities, a new global asymptotic stability condition via linear matrix inequality has been derived. In [45], dynamical issues of neural networks of neutral-type possessing some various delay parameters have been analyzed, and different stability results have been derived employing linear matrix inequality combining with Razumikhin-like approaches.We should point out that the results of [16, 26–45] employ some various classes of linear matrix inequality techniques to obtain different sets of stability conditions for neutral-type neural networks. However, the stability results derived via the linear matrix inequality method are required to test some negative definite properties of high-dimensional matrices whose elements are formed by the system parameters of neural networks. Due to these complex calculation problems, it becomes a necessity to propose different stability conditions for neutral-type neural networks, which are not stated in linear matrix inequality forms. In this concept, the current paper will focus on the dynamical analysis of neural system (1) to derive some easily verifiable algebraic stability conditions.


## 2. Stability Analysis

The basic contribution of this section will be deriving some stability conditions implying the stability of neutral-type Cohen–Grossberg neural system whose model is given by (1). We now proceed with a first step to provide a simpler procedure with the proofs of the stability conditions. This step needs to transform the equilibrium points x^=x^1,x^2,…,x^nT of Cohen–Grossberg neural system represented by equation (1) to the origin. This will be achieved by utilizing the simple formula zit=xit−x^i, which turns neutral-type neural network (1) to an equivalent neutral-type neural network which has a set of differential equations that govern the dynamics of this system:(5)z˙it=αizit−βizit+∑j=1naijgjzjt+∑j=1nbijgjzjt−τj+∑j=1neijz˙jt−ζj.(5) can be represented in the vectors and matrices forms as given below:(6)z˙t=αzt−βzt+Agzt+Bgzt−τ+Ez˙t−ζ,where the system matrices are A=aijn×n,B=bijn×n,E=eijn×n, and(7)zt=z1t,z2t,…,zntT,zt−τ=z1t−τ1,z2t−τ2,…,znt−τnT,gzt=g1z1t,g2z2t,…,gnzntT,αzt=diagα1z1t,α2z2t,…,αnznt,βzt=β1z1t,β2z2t,…,βnzntT,gzt−τ=g1z1t−τ1,g2z2t−τ2,…,gnznt−τnT,z˙t−ζ=z˙1t−ζ1,z˙2t−ζ2,…,z˙nt−ζnT.After transforming neutral system (1) into neutral system (5), we have new transformed functions in system (5). The function αizit are of the form(8)αizit=dizit+x^i.The functionβizit are of the form(9)βizit=cizit+x^i−cix^i.The functiongizit are of the form(10)gizit=fizit+x^i−fix^i.According the properties byA1, A2, and A3, these new transformed functions possess the following properties:(11)0<μi≤αizit≤ρi,∀i,γizi2t≤zitβizit≤ψizi2t,∀i,gizit≤ℓizit,∀i.Fact 1.
Consider a real matrixA=aijn×n and a real vector x=x1,x2,…,xnT. We can state the following inequality:(12)xTATAx≤∑i=1n∑j=1n∑k=1nakiakjxi2.Fact 2.
Consider a real matrixA=aijn×n and a real vector x=x1,x2,…,xnT. We can state the following inequality:(13)xTATAx≤A22x22.
A combination offacts 1 and 2 can be expressed by the following fact.Fact 3.
Consider a real matrixA=aijn×n and a real vector x=x1,x2,…,xnT. We can state the following inequality:(14)xTATAx≤k1∑i=1n∑j=1n∑k=1nakiakjxi2+k2A22∑i=1nxi2,where k1 and k2 are the binary constants such that k1+k2=1 and k1k2=0.Fact 4.
LetA=aijn×n be a real matrix, D=diagdi>0 be a positive diagonal matrix, and x=x1,x2,…,xnT be a real vector. The following inequality can be stated:(15)xTATDDAx≤∑i=1n∑j=1n∑k=1ndk2akjakjxi2.Fact 5.
Consider any two real vectorsx=x1,x2,…,xnT and y=y1,y2,…,ynT. The following inequality can be stated:(16)2xTy≤kxTx+1kyTy,where k can be chosen as any arbitrary positive real number.
The key contribution of this paper can now be presented by the theorem stated below.Theorem 1.
Suppose the conditions given byA1, A2, and A3 hold. Let κ and ξ be positive real-valued numbers. Then, the origin of Cohen–Grossberg neural system of neutral type expressed by (5) is globally asymptotically stable if the system parameters of (5) satisfy the conditions:(17)εi=γi2ℓi2−2+κp1∑j=1n∑k=1nakiakj+p2A22−2+ξq1∑j=1n∑k=1nbkibkj+q2B22>0,εi=r11ρi2−1+1κ+1ξ∑j=1n∑k=1n1μk2ekiekj+r21ρM2−1+1κ+1ξ1μm2E22>0,where μm=min1≤i≤nμi, ρM=max1≤i≤nρi, p1, p2, q1, q2, r1, and r2 are the binary constants such that p1+p2=1, p1p2=0, q1+q2=1, q1q2=0, r1+r2=1, and r1r2=0.Proof.
Construct a suitable Lyapunov functional candidate given by(18)Vt=∑i=1n∫0zit2βisαisds+∑i=1n∫t−ζit1αi2zisz˙i2sds+k∑i=1n∫t−τitzi2sds,+2+ξ∑i=1n∫t−τitℓi2q1∑j=1n∑k=1nbkibkj+q2B22zi2sds,where k is a real-valued positive number whose appropriate value will be specified in the process of the proof. If we take the time derivative of the Lyapunov functional Vt along the trajectories of Cohen–Grossberg neural network model defined by (5), we will derive the equation:(19)V˙t=∑i=1n2βizitαizitz˙it+2+ξ∑i=1nℓi2q1∑j=1n∑k=1nbkibkj+q2B22zi2t−2+ξ∑i=1nℓi2q1∑j=1n∑k=1nbkibkj+q2B22zi2t−τi+∑i=1n1αi2zitz˙i2t−1αi2zit−ζiz˙i2t−ζi+k∑i=1nzi2t−k∑i=1nzi2t−τi.
(19) may be rearranged as(20)V˙t=2βTztα−1ztz˙t+∑i=1n−1αi2zit−ζiz˙i2t−ζi+kzi2t−kzi2t−τi+α−1ztz˙tTα−1ztz˙t+2+ξ∑i=1n∫t−τitℓi2q1∑j=1n∑k=1nbkibkj+q2B22zi2t−2+ξ∑i=1n∫t−τitℓi2q1∑j=1n∑k=1nbkibkj+q2B22zi2t−τi.
Note the inequalities(21)2βTztα−1ztz˙t=2βTzt−βzt+Agzt+Bgzt−τ+2βTztα−1ztEz˙t−ζ(22)=α−1ztz˙tTα−1ztz˙t−βzt+Agzt+Bgzt−τ+α−1ztEz˙t−ζT−βzt+Agzt+Bgzt−τ+α−1ztEz˙t−ζ.
Combining (22) with (23) leads to(23)2βTzt+α−1ztz˙tTα−1ztz˙t−βTztβzt+gTztATAgzt+gTzt−τBTBgzt−τ+2gTztATBgzt−τ+2gTztATα−1ztEz˙t−ζ+2gTzt−τBTα−1ztEz˙t−ζ+z˙Tt−ζETα−1ztα−1ztEz˙t−ζ.
By the virtue of fact 5, the following inequalities can be written as(24)2gTztATBgzt−τ≤gTztATAgzt+gTzt−τBTBgzt−τ,(25)2gTztATα−1ztEz˙t−ζ≤κgTztATAgzt+1κz˙Tt−ζETα−2ztEz˙t−ζ,(26)2gTzt−τBTα−1ztEz˙t−ζ≤ξgTzt−τBTBgzt−τ+1ξz˙Tt−ζETα−2ztEz˙t−ζ.
Using (24)–(26) in (23) results in(27)2βTzt+α−1ztz˙tTα−1ztz˙t−βTztβzt+2+κgTztATAgzt+2+ξgTzt−τBTBgzt−τ+1+1κ+1ξz˙Tt−ζETα−2ztEz˙t−ζ
We first note the following equality:(28)−βTztβzt=−∑i=1nβi2zit.
By fact 3, we express the inequalities:(29)gTztATAgzt≤∑i=1np1∑j=1n∑k=1nakiakjgi2zit+p2A22gi2zt,gTzt−τBTBgzt−τ≤q1∑i=1n∑j=1n∑k=1nbkibkjgi2zit−τi+q2B22∑i=1ngi2zit−τi.
Using the property offact 4, we express the inequality:(30)z˙Tt−ζETα−2ztEz˙t−ζ≤∑i=1nr1∑j=1n∑k=1n1αk2zktekiekjz˙i2t−ζi+r2α−2zt22E22z˙i2t−ζi.
Using (28)–(30) in (27) yields(31)≤2βTzt+α−1ztz˙tTα−1ztz˙t+2+ξ∑i=1nq1∑j=1n∑k=1nbkibkjgi2zit−τi+q2B22gi2zit−τi−∑i=1nβi2zit+2+κp1∑j=1n∑k=1nakiakjgi2zit+p2A22gi2zit+1+1κ+1ξr1∑i=1n∑j=1n∑k=1n1αk2zktekiekjz˙i2t−ζi+r2α−2zt22E22∑i=1nz˙i2t−ζi.
A1 implies the following inequalities:(32)1αk2zkt≤1μk2(33)α−2zt22≤1μm2.
A2 implies that(34)γi2zi2t≤βi2zit.
A3 implies that(35)gi2zit≤ℓi2zi2t(36)gi2zit−τi≤ℓi2zi2t−τi.
Using (32)–(36) in (37) leads to(37)≤2βTzt+α−1ztz˙tTα−1ztz˙t−∑i=1nγi2zi2t+2+κ∑i=1nℓi2p1∑j=1n∑k=1nakiakj+p2A22zi2t+2+ξ∑i=1nℓi2q1∑j=1n∑k=1nbkibkj+q2B22zi2t−τi+1+1κ+1ξ∑i=1nr1∑j=1n∑k=1n1μk2ekiekj+r21μm2E22z˙i2t−ζi.
A1 implies the following inequality:(38)1ρM2≤1ρi2≤1αi2zit−ζi.
Then, by (38), we can write(39)−∑i=1n1αi2zit−ζiz˙i2t−ζi≤−r1∑i=1n1ρi2z˙i2t−ζi−r21ρM2∑i=1nz˙i2t−ζi.
Using (37) and (39) in (20) results in(40)V˙t≤∑i=1n−γi2zi2t+2+κℓi2p1∑j=1n∑k=1nakiakj+p2A22zi2t+2+ξ∑i=1nℓi2q1∑j=1n∑k=1nbkibkj+q2B22zi2t+1+1κ+1ξ∑i=1nr1∑j=1n∑k=1n1μk2ekiekj+r21μm2E22z˙i2t−ζi−r1∑i=1n1ρi2z˙i2t−ζi−r21ρM2∑i=1nz˙i2t−ζi+k∑i=1nzi2t−zi2t−τi=−∑i=1nℓi2γi2ℓi2−2+κp1∑j=1n∑k=1nakiakj+p2A22−2+ξq1∑j=1n∑k=1nbkibkj+q2B22zi2t−∑i=1nr11ρi2−1+1κ+1ξ∑j=1n∑k=1n1μk2ekiekjz˙i2t−ζi−∑i=1nr21ρM2−1+1κ+1ξ1μm2E22z˙i2t−ζi+k∑i=1nzi2t−zi2t−τi=∑i=1n−εiℓi2zi2t−εiz˙i2t−ζi+k∑i=1nzi2t−zi2t−τi≤∑i=1n−εmℓm2zi2t−εmz˙i2t−ζi+k∑i=1nzi2t−zi2t−τi=−εmℓm2−kzt22−εmz˙t−ζ22−kzt−τ22,where εm=min1≤i≤nεi, εm=min1≤i≤nεi, and ℓm=min1≤i≤nℓi.
From (40), one can obtain the inequality:(41)V˙t≤−εmℓm2−kzt22.
In (41), the choice k<εmℓm2 will make it possible for V˙t to be negative definite and every transformed states zt≠0.
In (40), taking zt≠0 will directly yield the result:(42)V˙t≤−kzt−τ22.
(42) directly implies that V˙t will have negative values for every transformed delayed state zt−τ≠0.
Let the transformed statezt=0 together with delayed transformed state zt−τ=0. Then, we immediately get from (40) that(43)V˙t≤−εmz˙t−ζ22.
(43) directly yields that time derivative of this studied Lyapunov functional V˙t will be negative for every z˙t−ζ≠0.
Letzt=0, zt−τ=0, and z˙t−ζ=0. This case leads to the fact that z˙t=0. Hence, from (19), we get that V˙t=0. Hence, we note that V˙t=0 at the equilibrium point which is the origin of system (5) and V˙t<0 except for the equilibrium point. Hence, this Lyapunov functional analysis ensures that the origin of system (5) is asymptotically stable. In addition, the Lyapunov functional given by (18) is radially bounded, meaning that Vt⟶∞ when zt⟶∞. The radially unboundedness of this Lyapunov functional guarantees that the origin of neutral-type Cohen–Grossberg neural network (5) is globally asymptotically stable. Q.E.D.


## 3. An Instructive Example

This section gives an instructive example for the sake of indicating the applicability of results expressed by the conditions of Theorem1.Example: consider a case of neutral-type neural system (1) of four neurons, which has the system matrices given as follows:(44)A=12aaaa−a−aaa−aa−aaa−a−aa,B=12bbbb−b−bbb−bb−bbb−b−bb,E=12eeeeeeeeeeeeeeee,where a, b, and e are being some positive constants. For this example, we also make the choices for the parameters μ1=μ2=μ3=μ4=1, ρ1=ρ2=ρ3=ρ4=1, γ1=γ2=γ3=γ4=2, ψ1=ψ2=ψ3=ψ4=2, and ℓ1=ℓ2=ℓ3=ℓ4=1. For the system matrices A, B, and E, one may calculate(45)ATA=a20000a20000a20000a2,BTB=b20000b20000b20000b2,ETE=e2e2e2e2e2e2e2e2e2e2e2e2e2e2e2e2.Then, we can obtain the following:A22=a2,B22=b2,E22=4e2, and(46)∑j=14∑k=14akiakj=a2,∑j=14∑k=14bkibkj=b2,∑j=14∑k=14ekiekj=4e2,i=1,2,3,4.According to Theorem1, this example establishes the following conditions:(47)εi=4−2+κp1+p2a2−2+ξq1+q2b2=4−2+κa2−2+ξb2,εi=r11−1+1κ+1ξ4e2+r21−1+1κ+1ξ4e2=1−1+1κ+1ξ4e2.Letκ=2 and ξ=2. Then, εi=41−a2−b2 and ϵi=1−8e2,i=1,2,3,4. Clearly, the conditions a2+b2<1 and e<1/22 establish the global stability of system (5).


## 4. Conclusions

This research work has been conducted as an investigation of the stability issues for neutral-type Cohen–Grossberg neural network models possessing discrete time delays in states and discrete neutral delays in time derivatives of neuron states. By setting a novel generalized appropriate Lyapunov functional candidate, some new sufficient conditions have been proposed for global asymptotic stability for the considered delayed neural networks of neutral type. This paper has exploited some basic properties of matrices in the derivation of the results that established a set of algebraic mathematical relationships between network parameters of the neural system. The obtained stability criteria proved to be independent from the time and neutral delays. Therefore, the proposed results can be easily verified. A constructive numerical example has also been presented to check the applicability of the presented global stability conditions.



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