# Existence, Uniqueness, and Exponential Stability of Uncertain Delayed Neural Networks with Inertial Term: Nonreduced Order Case

Authors: M. Iswarya; R. Raja; Q. Zhu; M. Niezabitowski; J. Alzabut; C. Maharajan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this work, we mainly focus on uncertain delayed neural network system with inertial term. Here, the existence, uniqueness, and exponential stability of inertial neural networks are derived without shifting the second order differential system into first order through substituting variables. Initially, we construct a proper Lyapunov–Krasovskii functional to investigate the stability of novel uncertain delayed inertial neural networks, which is different from the classical Lyapunov functional approach. By utilizing the Kirchhoff’s matrix tree theorem, Cauchy–Schwartz inequality, homeomorphism theorem, and some inequality techniques, the necessary and sufficient conditions are derived for the designed framework. Subsequently, to exhibit the strength of this outcome, we framed a quantitative example.


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## Body

## 1. Introduction

In recent years, because of the fruitful applications in various domains such as gesture recognition [1], image quality enhancement [2], secure communication [3], face detection [4], image compression [5] and medical image processing [6], the dynamical behaviour of different kinds of neural networks (NNs), namely, Cohen-Grossberg neural networks (CGNNs) [7], recurrent neural networks (RNNs) [8, 9], Hopfield neural networks [10], bidirectional associative memory neural networks (BAMNNs) [11], and chaotic neural networks (CNNs) [12], have been studied widely. In addition to that, time delays of a dynamical framework play a supreme role in various disciplines. Because of the finite propagation velocity, time delays are unavoidable in signal transmission of neural networks, which may lead the unwanted dynamical responds such as chaotic and bifurcation. Marcus and Westervelt [13] firstly proposed the stability analog of neural networks with time delays in 1989.(1)xi.t=−aixit+∑j=1nbijfjxjt+∑j=1ncijfjxjt−τ+Ii,where xit∈ℝn indicates the position of the ith neuron, ai>0 is the positive constant weight, bijn×n,cijn×n indicates the transmission weight matrices from ith neuron to jth neuron, and fj:ℝn⟶ℝn characterize the transfer function, the constant delay τ>0, and Ii denotes the external input. In the nervous system, the time-delays occur due to the propagation time of neurotransmitters forming presynaptic neurons to postsynaptic neurons. The time delays are occurring not only in constant manner [14] but also according to the number of parallel pathways with different lengths and sizes of the axon; it may be classified into various kinds such as discrete [15, 16], distributed, and mixed delays [17–19].(2)x˙it=−aixit+∑j=1nbijfjxjt+∑j=1ncij∫t−σttfjxjsds+Ii,where the term σt signifies the distributed time-varying delay. Besides the aforementioned delay kinds, there is another kind of delay called proportional delay, which explains that, δt=t−qijt, where qij>0, for i,j=1,2,…,n, pantograph delay factor with 0<qij<1. This kind of delay is also know as discrete unbounded delay since the delay term δt⟶+∞ as t⟶+∞. Due to the constrains of the continuously distributed delay, the proportional delay is less conservative.(3)x˙t=−aixit+∑j=1nbijfjxjt+∑j=1ncijfjxjqijt+Ii.Moreover, this type of delay is used in web quality of service (QoS) routing decision [5]. Due to its applications in the areas such as wireless LAN [20], network application servers [21], and integro differential equations [22], the study on proportional delay increases the research interest of researchers; in [23], the synchronization of CGNNs with proportional delays was studied on the basis of Lyapunov theory, the exponential synchronization of INNs with proportional delays and reaction diffusion was investigated under Lyapunov–-Krasovskii functional and Wirtinger inequality in [24]; in [25]. the exponential stability of CNNs with proportional delay was investigated by utilizing the matrix theory and Lyapunov theory. Furthermore, in recent days, the stability analysis of dynamical systems is a hot research topic [26, 27].In recent years, most of the studies are mainly focused on first order derivative of the NNs, although it is great significant to introduce the inertial term in NNs because of the reason that the influence of the inductance, which was firstly investigated in electronic neural networks and found that the coupling of the neurons is in inertial way; the dynamics can be complex [28, 29]. In 1997, Wheeler and Schieve initially proposed the inertial neural networks [30].(4)x¨it=−aix˙it−bixit+∑j=1ncijfjxjt+Ii.Here,x¨it denotes the inertial term and bi>0 is a positive constant. In recent years, the convergence analysis of INNs has been focused by lot of research scientists, for instance by utilizing the Lyapunov functional method, inequality techniques, and analytical method, the global Lagrange stability of INNs with discrete and distributed time-varying delays [31, 32]. In [33], Huang and Cao investigated the stability of inertial CGNNs with the method of model transformation, differential mean value theorem, Lyapunov stability theory, and Linear Matrix Inequality (LMI) techniques. Maharajan et al. [34] investigated the Lagrange stability of inertial BAMNNs on the basis of Lyapunov–Krasovskiii Functional (LKF), integral inequality, and LMI approach. Tang et al. [35] proposed the exponential stability of complex-valued INNs using LKF and LMI. Kong et al. [36] studied the novel fixed-time stability of fuzzy INNs by using the inclusion theory and LKF approach.Due to various reasons such as aging of device, existence of error in modeling, and exterior noise in the course of the hardware implementation, the parameter uncertainty occurs, which is unavoidable. Moreover, the system variables of NNs are affected by some improper dynamical responds such as instability, bifurcation, and oscillation. Recent days, many scholars did their research works in the domain of uncertain inertial neural networks [37–39].(5)x¨t=−ai+Δaix˙it−bi+Δbixit+∑j=1ncij+Δcijfjxjt+Ii.Moreover, in the previous studies of INNs, most of the authors used the reduced order approach, which means that they reduced the order of the system from second order into first order by using the variable transformation. Evidently, the variable transformation leads an increase in dimension of the system and few more parameters have to be introduced. This will initiate the computational complexity of the work.Furthermore, Lyapunov function [40] which is a scalar function defined in the phase space is an effective tool in obtaining the stability of dynamical systems. Comparing to the linear system to verify the stability of nonlinear system is a difficult task; to overcome this problem, the Lyapunov function was proposed by A. M. Lyapunov. Now a days, various kinds of Lyapunov functionals are applied in the stability of neural networks system, for instances [41] Kong et al. studied the synchronization INNs on the basis of indefinite Lyapunov–Krasovskii functional method, by utilizing the new augmented Lyapunov–Krasovskii functional. Gao et al. [42] investigated the stability problem of NNs. The stability of impulsive system is proposed by using a novel Lyapunov-like functional approach by Shao et al. [27].From this, it is well known that, in stability theory, the Lyapunov method plays a major role, but the construction of a suitable Lyapunov function is quite harder. In order to overcome this situation, a novel technique was explored in the base of Lyapunov method and graph theory. In this approach, we have incorporated the concepts of graph theory with Lyapunov theory. In particular, in this approach, the topological structure of the NNs is observed as a digraph accompanied by single neuron as a node and interaction between them as a directed arcs.Compared with the aforementioned works, the aim of this work is as listed below:(i)
In this work, we investigate the exponential stability results of inertial neural networks along with the existence of uncertainty and distributed and proportional time-varying delay.(ii)
Without using the traditional variable transformation approach, in this work, we directly design the second order system on the basis of novel Lyapunov functional which contains the first order state variable and the cofactor of theith diagonal element of the Laplacian matrix in graph theory.(iii)
By utilizing the homeomorphism theorem, the sufficient conditions for the existence of unique equilibrium point are derived for the designed framework.(iv)
It is pointed out that the necessary criteria for the stability results are presented under the Lyapunov–Krasovskii functional and the concepts of graph theory in direct approach, which is totally different from the traditional reduced order approach.(v)
In the end of this work, we give a numerical example and simulation results to show the effectiveness of the work.We systemized the article content in the following way: Section2, describes the mathematical framework of the accompanied by proportional and distributed time-varying delays. Afterwards, the basic concepts on this subject are given. The solution of existence of an unique equilibrium point is discussed in Section 3, and in Section 4, the exponential stability is discussed. In Section 5, a numerical illustration is derived and the conclusion of this study is given in Section 6.


## 2. Basic Concepts and Model Description

In the present study, the existence and exponential stability of unique equilibrium point for uncertain delayed inertial neural networks (UDINN) with proportional delays are investigated.For the entire study, we consider the following: we examineM=1,2,…,n, ℝ=−∞,+∞ represents the set of all real numbers, ℝ+=0,+∞, ℕ represents the set of all natural numbers, the n-dimensional Euclidean space is ℝn and the set of all n×m real matrices is ℝn×m, and the Euclidean norm for any vector y and the trace norm for any matrix A are denoted y and traceATA, respectively. A digraph G=V,E is weighted, if we allocate a positive weight mab for every arcs a,b. The Laplacian matrix of G is defined as(6)ℒp=−mab,ifa≠b,∑h≠jmah,a=b.For the Lyapunov function,vit,yi∈C1,2ℝ+×ℝn;ℝ+,i∈M, which is differentiable and continuous at t and twice differentiable at yi.Lemma 1 [43]).
The weighted digraphG,Awithn≥2of vertices andA=aijn×n. Let the collection of all spanning unicyclic ofG,AbeUandithdiagonal element of cofactor of theLpis denoted asci, then the following identity holds:(7)∑i,j=1nciaiFijt,zit,zjt≤∑U∈UWU∑k,h∈ΕℭUFkht,zkt,zht,where, for k,h∈M, an arbitrary function Fkht,zkt,zht, the set of all spanning unicyclic graph U, WU, and ℭU denotes, respectively, the weight and dicycle of U. Additionally, ci>0 if G,A is strongly connected for i=1,2,…,n.Lemma 2 (Homeomorphism theorem44).
Ifℋis a continuous function fromℝntoℝnwhich satisfies the following conditions,(1)
ℋu is one-one function on ℝn,(2)
ℋu⟶∞,asu⟶∞,
thenℋu is homeomorphism of ℝn.Lemma 3 (Cauchy–Schwarz inequality45).
Letα1,α2,…,αnandα1,α2,…,αnbe two sequences of real numbers, one has(8)∑i=1nαi2∑i=1nβi2≥∑i=1nαiβi2,the inequality turns into equality, whenever the real sequences α1,α2,…,αn and α1,α2,…,αn are proportional.Compared with the aforementioned discussions, we describe the following uncertain neural networks with distributed, proportional delay and inertial term as(9)y¨it=−αi+Δαiy˙it−βi+Δβiyit+∑j=1nγij+Δγijfjyjt+∑j=1nδij+Δδijgjyjpit+∑j=1nζij+Δζij∫t−τitthjyjsds+Ii,where yit∈ℝn denotes the ith node of neurons, the positive parameters αi,βi,γij,δij,ζij∈ℝ indicate, respectively, connection weights of ith node and the connection weight of ith node to jth node. The unknown weighted matrices Δαi,Δβi,Δγij,Δδij,Δζij with the conditions Δαi≤λ1i;Δβi≤λ2i;Δγij≤λ3i;Δδij≤λ4i;Δζij≤λ5i. fj,gj,hj denote, respectively, the activation function, activation function of the proportional delay pit, and the activation function of the distributed delay τit with the condition 0≤τ˙it<τi.The initial value of the given system (9) is(10)yis=ϕis,dyisds=ψis,−τ≤s≤0.Remark 1.
The proportional delaypitis also known as unbounded time-varying delay, since the delay factor,1−pit=σt. Here,σtis a continuous function which fulfils the conditionσt⟶∞ast⟶∞. More precisely, suppose that in system (9), ifpi=1and there is no distributed delay and also with the absence of an uncertainty, equation (9) can be rewritten as(11)y¨it=−αiy˙it−βiyit+∑j=1nγ˜ijf˜jyjt,which is generalized INNs.


## 3. Existence of a Unique Equilibrium Point

Definition 1.
Letyt=y1t,y2t,…,yntbe the solution of (9), satisfying the initial conditions. If there exist two positive constantsrandMϕ,ψsuch that(12)y˙it≤Mϕ,ψe−rt,yit≤Mϕ,ψe−rt,∀t∈0,+∞,for allt≥t0, then the system (9) is globally exponentially stable.Assumption 1.
ℛ1 For any i=1,2,…,n, the transfer functions fj⋅,gj⋅ and hj⋅ which is from ℝn to ℝn satisfies the Lipschitz condition. That is, there exist Lipschitz constants Fj,Gj,andHj such that for any u,v∈ℝ,(13)fju−fjv≤Fju−v,gju−gjv≤Gju−v,hju−hjv≤Hju−v.Assumption 2.
ℛ2 For each i∈M, there exist some positive constants ai,bi, and ci such that(14)Air<0,4Airℬir>Cir2,∀i∈n.Assumption 3.
ℛ3 For each i∈M, there exist some positive constants ai and ci such that(15)ai2βi+λ2i+aiciαi+λ1i−ci2>0;Air<0,ℬir<0,∀i∈M.
For the sake of simplicity, we present the upcoming denotations:(16)Nij=Fjai2γij+λ3i+aiciγij+λ3i+pi0ai2δij+λi4+aiciδij+λi4e1−piGj+pi×ai2ζij+λi5+aiciζij+λi5Hje2rτiτi,Air=aici−ai2αi+λ1i+12∑j=1nai2Fjγij+λ3i+Gjδij+λ4i+Hjζij+λ5i,ℬir=−aiciβi+λ2i+12∑j=1naj2Fjγij+λ3i+Gjδij+λ4i+Hjζij+λ5i+12∑j=1n×Fiγji+λ3j+Giδji+λ4j+Hjζji+λ5jajcj+12∑j=1nFjγij+λ3i+Gjδij+λ4i+Hjζij+λ5iaici,Cir=bi+ci2−aiciαi+λ1i−ai2βi+λ2i,where ai,bi, and ci are positive constants, for all i∈M.Theorem 1.
Suppose that there is a constantℬir<0which is defined in denotation, and assumptionℛ1holds, then there exists a unique equilibrium point for system (9).Proof.
Let us begin with a mapℋyi=ℋ1y1,ℋ2y2,…,ℋnyn∈ℝn. Here,(17)ℋyi=−βi+λ2iyi+∑j=1nγi+λ3ifjyj+∑j=1nδi+λ4igjyj+∑j=1nζi+λ5ihjyj+Ii.
Next, by using Lemma2, we will show that ℋyi is homeomorphism for all i∈M. Initially, we show that the map ℋyi is a one-one map on ℝn. Suppose if not, then there exist ui,vi∈ℝn and ui≠vi such that ℋui=ℋvi,(18)βi+λ2iui−vi=∑j=1nγij+λ3ifjuj−fjvj+∑j=1nδij+λ4igjuj−gjvj+∑j=1nζij+λ5ihjuj−hjvj.
Then,(19)∑i=1naiciβi+λ2iui−vi2=∑i=1n∑j=1naiciui−viγij+λ3ifjuj−fjvj+∑i=1n∑j=1naici×ui−viδij+λ4igjuj−gjvj+∑i=1n∑j=1naiciui−vi×ζij+λ5ihjuj−hjvj≤∑i=1n∑j=1naiciγij+λ3iFj+δij+λ4iGj+ζij+λ5iHj×ui−viuj−vj≤∑i=1n∑j=1naici2γij+λ3iFj+δij+λ4iGj+ζij+λ5iHjui−vi2+∑i=1n∑j=1najcj2γji+λ3jFi+δji+λ4jGi+ζji+λ5jHiui−vi2≤∑i=1naiciβi+λ2i−12∑j=1n|γij+λ3i|Fj+|δij+λ4i|Gj+ζij+λ5iHjaici−12∑j=1nγji+λ3jFi+δji+λ4jGi+ζji+λ5jHiajcjui−vi2≤0.
Since,ℬir<0, it shows that ui=vi, ∀i∈M; this implies the contradiction to ui≠vi. Hence, ℋyi is one-one function. Our next aim is to show that ℋyip⟶+∞asyip⟶+∞.(20)∑i=1nyiaiciℋyi−ℋ0=−∑i=1nβi+λ2iaiciyi2+∑i=1n∑j=1naiciγij+λ3iyifjyj−fj0+∑i=1n∑j=1naiciδij+λ4iyigjyj−gj0+∑i=1n∑j=1naiciζij+λ5iyihjyj−hj0≤−∑i=1nβi+λ2iaiciyi2−∑j=1naiciγij+λ3iyiFjyj−∑j=1naiciδij+λ4iyiGjyj−∑j=1naiciζij+λ5iyiHjyj≤−∑i=1naiciβi+λ2i−12∑j=1nγij+λ3iFj+δij+λ4iGj+|ζij+λ5i|Hjaici−12∑j=1nγji+λ3jFi+δji+λ4jGi+ζji+λ5j×Hiajcjyi2≤−ξi∑i=1nyi2.
Here,(21)ξi=min1≤i≤naiciβi+λ2i−12∑j=1nγij+λ3iFj+δij+λ4iGj+ζij+λ5iHjaici−12×∑j=1nγji+λ3jFi+δji+λ4jGi+ζji+λ5jHiajcj,and ξi>0, since ℬir<0,∀i∈M.(22)∑i=1nyiaiciℋyi−ℋ0≤−ξi∑i=1nyi2.
By using Cauchy–Schwartz inequality in Lemma3,(23)∑i=1nyi2≤1ξi∑i=1naiciyiℋyi−ℋ0≤aξi∑i=1nyiℋyi−ℋ0.where a=max1≤i≤naici. It follows that(24)y2≤aξℋiy−ℋ02≤aξℋiy2−ℋ02,which reveals that, ℋyi2⟶+∞ as yi⟶+∞. By Lemma 2, ℋyi is a homeomorphism on ℝn which implies that system (9) has a unique equilibrium point yi∗=y1∗,y2∗,…,yn∗T.Corollary 1.
Suppose that the assumptionsℛ1andℛ3hold, then (9) has a unique equilibrium point, which is asymptotically stable.Remark 2.
Suppose that the value of uncertain parameters is equal to zero; there occurs a change of an unbounded time-varying delay (proportional delay) into discrete time-varying delay, and then system (9) is converted into INNs with discrete and distributed model and it is described as follows:(25)y¨it=−αiy˙it−βiyit+∑j=1nγijfjyjt+∑j=1nδijgjyjt−ηt+∑j=1nζij∫t−τitthjyjs×ds+Ii.


## 4. Exponential Stability

Theorem 2.
Under the assumptionsℛ1andℛ2, the system (9) is exponentially stable. Proof. Let us defined the global Lyapunov functional as follows(26)Vt=∑i=1nςivit.Where, ςi denotes the cofactor of ith diagonal element of ℒp of digraph G,Aand(27)vit=vi1t+vi2t+vi3t+vi4t.
Here,(28)vi1t=12∑i=1nbiyi2te2rtvi2t=12∑i=1naiy˙it+ciyit2e2rtvi3t=12∑i=1n∑j=1n1piGjai2δij+λi4+aiciδij+λi4e1−pi∫pitte2rsf2yjsds,vi4t=12∑i=1n∑j=1n1Hjai2ζij+λi5+aiciζij+λi5e2rτi∫−τit0∫t+θθe2rsf2yjsdsdθ.Along the solution (9) we derive vit as follows(29)v˙i1t=12∑i=1nbie2rt2y˙ityit+2ryi2t=∑i=1nbie2rty˙ityit+12∑i=1n2rbie2rtyi2t,(30)v˙i2t=12∑i=1n2re2rtaiy˙it+ciyit2+2e2rtaiy˙it+ciyitaiy¨it+ciy˙it=12∑i=1ne2rt2rai2+2aiciy˙i2t+12∑i=1ne2rt2ai2y˙it+2aiciyity¨it+12∑i=1ne2rt×4raici+2ci2yity˙it+12∑i=1ne2rt2rci2yi2t,(31)v˙i3t=12∑i=1n∑j=1n1piai2δij+λi4+aiciδij+λi4e1−piGje2rtyj2t−e2rpityj2pitpi=12∑i=1n∑j=1ne2rtGj1piai2δij+λi4+aiciδij+λi4e1−piyj2t−ai2δij+λi4+aiciδij+λi4yj2pit,(32)v˙i4t=12∑i=1n∑j=1nai2ζij+λi5+aiciζij+λi5Hje2rτi+tτiyj2t−e2rτi∫t−τitte2rtyj2sds.Substitute (29)–(32) in (27) we get,(33)v˙it=∑i=1ne2rtbi+2raici+ci2y˙ityit+rai2+aiciy˙i2t+rbi+rci2yi2t+2ai2×y˙it+2aiciyity¨it+12∑j=1n1piai2δij+λi4+aiciδij+λi4e1−piGj+ai2×ζij+λi5+aiciζij+λi5Hje2rτiτiyj2t−12∑j=1nai2δij+λi4+aiciδij+λi4×Gjyi2pit−12∑i=1n∑j=1nai2ζij+λi5+aiciζij+λi5Hje2rτi∫t−τitte2rtyj2sds≤∑i=1ne2rtbi+2raici+ci2y˙ityit+rai2+aiciy˙i2t+rbi+rci2yi2t−ai2αi+λ1iy˙i2t−aiciαi+λ1iy˙ityit−2ai2βi+λ2iy˙ityit−2aiciβi+λ2iyi2t+∑j=1nai2y˙it+aiciyitγij+Δγijfjyjt+∑j=1nai2y˙it+aiciyiδij+Δδij×gjyjpit+∑j=1nai2y˙it+aiciyitζij+Δζij∫t−τitthjyjsds+12∑j=1n1piai2×δij+λi4+aiciδij+λi4e1−piGj+ai2ζij+λi5+aiciζij+λi5Hje2rτiτi×yj2t−12∑j=1nai2δij+λi4+aiciδij+λi4Gjyi2pit−12∑i=1n∑j=1nai2ζij+λi5+aici×ζij+λi5Hje2rτi∫t−τitte2rtyj2sds.By using the assumptionℛ1 and the inequality uv≤1/2u2+v2 we get,(34)ai2y˙it+aiciyitγij+Δγijfjyjt=ai2γij+Δγijy˙itfjyjt+aiciγij+Δγijyit×fjyjt≤12Fjγij+λ3iai2y˙i2t+yj2t+aiciyi2t+yj2t.Similarly,(35)ai2y˙it+aiciyitδij+Δδijgjyjpit=ai2δij+Δδijy˙itgjyjpit+aiciδij+Δδij×yitgjyjt≤12Gjδij+λ4iai2y˙i2t+yj2pit+aiciyi2t+yj2pit,and(36)ai2y˙it+aiciyitζij+Δζij∫t−τitthjyjsds=ai2ζij+Δζijy˙it∫t−τitthjyjsds+aici×ζij+Δζijyit∫t−τitthjyjsds≤12Hjζij+λ5iai2y˙i2t+∫t−τittyj2sds+aiciyi2t+∫t−τittyj2sds.Substitute in (34)–(36) in (33), we have,(37)v˙it≤∑i=1ne2rtbi+2raici+ci2−aiciαi+λ1i−2ai2βi+λ2iy˙ityit+rai2+aici−ai2αi+λ1i+∑j=1nFjγij+λ3i2ai2+∑j=1nGjδij+λ4i2ai2+∑j=1nHjζij+λ5i2ai2×y˙i2t+rbi+rci2−2aiciβi+λ2i+∑j=1nFjγij+λ3i2aici+∑j=1nGjδij+λ4i2×aici+∑j=1nHjζij+λ5i2aiciyi2t+12∑j=1n1piai2δij+λi4+aiciδij+λi4e1−pi×Gj+ai2ζij+λi5+aiciζij+λi5Hje2rτiτi+∑j=1nFjγij+λ3i2ai2+∑j=1nFjγij+λ3i2×aiciyj2t−12∑j=1nai2δij+λi4+aiciδij+λi4Gjyj2pit+∑j=1nGjδij+λ4i2ai2×yj2pit+12∑j=1nGjδij+λ4iaiciyjpit−12∑i=1n∑j=1nai2ζij+λi5+aiciζij+λi5×Hje2rτi∫t−τitte2rtyj2sds+∑j=1nHjζij+λ5i2ai2∫t−τittyj2sds+∑j=1nHjζij+λ5i2aici×∫t−τittyj2sdsv˙it≤∑i=1ne2rtbi+2raici+ci2−aiciαi+λ1i−2ai2βi+λ2iy˙ityit+rai2+aici−ai2αi+λ1i+∑j=1nFjγij+λ3i2ai2+∑j=1nGjδij+λ4i2ai2+∑j=1nHjζij+λ5i2ai2y˙i2t+rbi+rci2−2aiciβi+λ2i+∑j=1nFjγij+λ3i2aici+∑j=1nGjδij+λ4i2aici+∑j=1nHjζij+λ5i2aiciyi2t+12∑j=1n1piai2δij+λi4+aiciδij+λi4e1−piGj+ai2ζij+λi5+aiciζij+λi5Hje2rτiτi+12∑j=1nFjγij+λ3iai2+aiciyj2tv˙it≤∑i=1ne2rtbi+2raici+ci2−aiciαi+λ1i−2ai2βi+λ2iy˙ityit+rai2+aici−ai2αi+λ1i+∑j=1nFjγij+λ3i2ai2+∑j=1nGjδij+λ4i2ai2+∑j=1nHjζij+λ5i2ai2y˙i2t+rbi+rci2−2aiciβi+λ2i+∑j=1nFjγij+λ3i2aici+∑j=1nGjδij+λ4i2aici+∑j=1n×Hjζij+λ5i2aiciyi2t+12∑j=1n1pipiFjai2γij+λ3i+aiciγij+λ3i+ai2δij+λi4+aiciδij+λi4e1−piGj+piai2ζij+λi5+aiciζij+λi5Hje2rτiτiyj2t−12∑i=1n1pj×pjFiaj2γji+λ3j+ajcjγji+λ3j+aj2δji+λj4+ajcjδji+λj4e1−pjGi+pjaj2ζji+λj5+aiciζji+λj5Hie2rτjτjyi2t,(38)=e2rt∑i=1nAiry˙i2t+Ciryity˙it+ℬiryi2t+∑i=1n∑j=1nNijyj2t−yi2t.Here, we need to show that,1/2∑i=1n∑j=1nNijyj2t−yi2t≤0. The obtained results has shown under Lemma 1, that(39)12∑i=1n∑j=1nNijyj2t−yi2t=12∑Q∈ℚWQ∑s,r∈ECQyr2t−ys2t.Along the directed cycleCQ, we indicate the set ECQ by,(40)ECQ=im,im+1|m=1,2,…,n−1,in=i1.According to (40), we get(41)∑s,r∈ECQyr2t−ys2t=yi12t−yi22t+yi22t−yi32t+⋯+yin−12t−yin2t+yin2t−yi12t=0.From (39) and (41) we get,(42)12∑i=1n∑j=1nNijyj2t−yi2t≤0.Then, by (38) implies that,(43)v˙i≤e2rt∑i=1nAiry˙i2t+Ciryity˙it+ℬiryi2t=e2rt∑i=1nAiry˙i2t+CirAiryity˙it+ℬiryi2t=e2rt∑i=1nAiry˙i2t+2Cir2Airyity˙it+Cir2Air2yi2t+ℬir−Cir24Air2yi2t=e2rt∑i=1nAiry˙i2t+Cir2Airyit2+ℬir−Cir24Air2yi2t=0,∀t∈0,∞,that is,(44)V˙t≤0.This implies thatVt≤V0,∀t∈0,+∞ and(45)12e2rt∑i=1nbiyi2t+aiy˙it+ciyit2≤V0.Note that,(46)e2rtaiy˙it+ciyit2=aierty˙it+ciertyit2.That is(47)aiy˙itert=aierty˙it+ciertyit−ciyit.Then, there exist an constantMϕ,ψ>0 such that,(48)y˙it≤Mϕ,ψe−rt,yit≤Mϕ,ψe−rt,∀t∈0,+∞.Hence from Definition1 the system (9) is exponentially stable.Corollary 2.
Assume that the system (25) is exponentially stable if the assumption ℛ1 and following conditions holds,(49)Nij=Fjai2γij+aiciγij+ηiai2δij+δijaicie1−ηiGj+ηiai2ζij+aiciζijHje2rτiτi0>aici−ai2αi+12∑j=1nai2Fjγij+Gjδij+Hjζij0>−aiciβi+12∑j=1naj2Fjγij+Gjδij+Hjζij+12+Giδji+Hjζjiajcj+12∑j=1nFj×∑j=1nFiγjiγij+Gjδij+Hjζijaici,Where ai,bi and ci are positive constants, for all i∈M.Remark 3.
In the stability theory, a Lyapunov functional plays a vital role. Basically, there are two approaches to fulfill the criteria of attaining the system stability with the Lyapunov functional (i.e. classical and traditional). In classical approach the construction of Lyapunov functional concerns only with the state variable of the proposed system, whereas in the case of traditional approach a Lyapunov functional that is not only contains the state variables but also the involvement of the derivatives of those state variables helps in attain the system stability much earlier. Moreover, the mathematical framework of the inertial neural networks involves with the second order differential equations which builts the computational complexity while manipulating, hence in order to avoid this burden, we utilized a traditional Lyapunov functional which is the great variation compared with the classical one.


## 5. Numerical Simulation

Example 1.
Let us consider the following inertial neural networks with proportional delay, distributed delay and uncertain parameters as:(50)y¨t=−α+Δαty˙t−β+Δβyt+γ+Δγfyt+δ+Δδgypt+ζ+Δζ∫t−σtthysds+Ii.Where(51)yt=y1t,y2tT;fj=gj=hj=1.5x−1;α=2001.75;β=8002.83;γ=−11.01−3.8932.89;δ=−2.81−7.21−2.521.81;ζ=−30.810.07−0.02;Δαt=−2.81−7.21−2.521.81;Δβt=−2.81−7.21−2.521.81;Δγt=−2.81−7.21−2.521.81;Δδt=−2.81−7.21−2.521.81;Δζt=−2.81−7.21−2.521.81;Fj=Gj=Hj=1.5;r=0.2;a1=0.75,a2=0.25;b1=2.25,b2=2.75;c1=0.02,c2=1.75;we obtain(52)A1r=−2.12;A2r=−29.4;B1r=−2.27;B2r=−31.22;C1r=−2.86;C2r=3.92.Which implies that(53)Air<0;4AirBir>Cir2,fori=1,2.
Clearly, all the conditions in Theorems1 and 2 are satisfied, hence (50) is globally exponentially stable and has unique fixed point. Which was shown in Figures 1 and 2 with distinct initial condition and external inputs.Figure 1
Exponential stability of (50) with Ii=21.
(a)(b)Figure 2
Exponential stability of (50) with Ii=sint.
(a)(b)Remark 4.
Figure1 demonstrates the state trajectory of the system (9) in first and second order with constant external input and Figure 2 indicates the state trajectory of (9) with variable external input.


## 6. Conclusion

In this proposed work, we studied the existence, uniqueness and exponential stability of INNs with uncertain, proportional and distributed delays. Without shifting the order of the differential equation, under the Lyapunov functional method and a novel graph theoretic approach the necessary and the sufficient condition for stability of INNs was obtained. And by using the homeomorphism theorem the existence of an unique equilibrium point was derived. In the end of this work, a numerical simulations was presented to show the exactness of this proposed work.In future, we will discuss the synchronization of inertial neural networks and its applications to real life applications which were mentioned in the introductory section of our manuscript.



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