#### 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.

#### 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.where indicates the position of the neuron, is the positive constant weight, indicates the transmission weight matrices from neuron to neuron, and characterize the transfer function, the constant delay , and 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].where the term signifies the distributed time-varying delay. Besides the aforementioned delay kinds, there is another kind of delay called proportional delay, which explains that, , where , for , pantograph delay factor with . This kind of delay is also know as discrete unbounded delay since the delay term as . Due to the constrains of the continuously distributed delay, the proportional delay is less conservative.

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].

Here, denotes the inertial term and 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].

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 the 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: Section 2, 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 examine , represents the set of all real numbers, , represents the set of all natural numbers, the n-dimensional Euclidean space is and the set of all real matrices is , and the Euclidean norm for any vector and the trace norm for any matrix are denoted and , respectively. A digraph is weighted, if we allocate a positive weight for every arcs . The Laplacian matrix of is defined as

For the Lyapunov function, , which is differentiable and continuous at and twice differentiable at .

Lemma 1 [43]). *The weighted digraph withof vertices and. Let the collection of all spanning unicyclic ofbeanddiagonal element of cofactor of theis denoted as, then the following identity holds:where, for , an arbitrary function , the set of all spanning unicyclic graph , , and denotes, respectively, the weight and dicycle of . Additionally, if is strongly connected for .*

Lemma 2 (Homeomorphism theorem 44). *Ifis a continuous function fromtowhich satisfies the following conditions,*(1)* is one-one function on ,*(2)*,**then is homeomorphism of .*

Lemma 3 (Cauchy–Schwarz inequality 45). *Let andbe two sequences of real numbers, one hasthe inequality turns into equality, whenever the real sequences and are proportional.*

Compared with the aforementioned discussions, we describe the following uncertain neural networks with distributed, proportional delay and inertial term aswhere denotes the node of neurons, the positive parameters indicate, respectively, connection weights of node and the connection weight of node to node. The unknown weighted matrices with the conditions . denote, respectively, the activation function, activation function of the proportional delay , and the activation function of the distributed delay with the condition .

The initial value of the given system (9) is

*Remark 1. **The proportional delay ** is also known as unbounded time-varying delay, since the delay factor, ** . Here, ** is a continuous function which fulfils the condition ** as ** . More precisely, suppose that in system* (9), *if**and there is no distributed delay and also with the absence of an uncertainty, equation* (9) *can be rewritten as*which is generalized INNs.

#### 3. Existence of a Unique Equilibrium Point

*Definition 1. **Let ** be the solution of* (9), *satisfying the initial conditions. If there exist two positive constants**and**such that**for all**, then the system* (9) *is globally exponentially stable.*

*Assumption 1. * For any , the transfer functions and which is from to satisfies the Lipschitz condition. That is, there exist Lipschitz constants such that for any ,

*Assumption 2. * For each , there exist some positive constants , and such that

*Assumption 3. * For each , there exist some positive constants and such thatFor the sake of simplicity, we present the upcoming denotations:where , and are positive constants, for all .

Theorem 1. *Suppose that there is a constant ** which is defined in denotation, and assumption ** holds, then there exists a unique equilibrium point for system* (9)*.*

*Proof. **Let us begin with a map ** . Here,*Next, by using Lemma 2, we will show that is homeomorphism for all . Initially, we show that the map is a one-one map on . Suppose if not, then there exist and such that ,Then,Since, , it shows that , ; this implies the contradiction to . Hence, is one-one function. Our next aim is to show that .Here,and , since .By using Cauchy–Schwartz inequality in Lemma 3,where . It follows thatwhich reveals that, as . By Lemma 2, is a homeomorphism on which implies that system (9) has a unique equilibrium point .

Corollary 1. *Suppose that the assumptions ** and ** hold, 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:*

#### 4. Exponential Stability

Theorem 2. *Under the assumptions ** and ** , the system* (9) *is exponentially stable.* Proof. *Let us defined the global Lyapunov functional as follows*Where, denotes the cofactor of diagonal element of of digraph and*Here,*

Along the solution (9) we derive as follows

Substitute (29)–(32) in (27) we get,

By using the assumption and the inequality we get,

Similarly,and