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

Department of Mathematics, Alagappa University, Karaikudi 630 003, India Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi 630 003, India School of Mathematics and Statistics, Hunan Normal University, Hunan 410 081, China School of Information Science and Engineering, Chengdu University, Chengdu, 610106, China Faculty of Automatic Control, Electronics and Computer Science, Department of Automatic Control, and Robotics, Silesian University of Technology, Akademicka 16, Gliwice 44-100, Poland Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia Group of Mathematics, Faculty of Engineering, Ostim Technical University, 06374 Ankara, Turkey Department of Mathematics, V.S.B Engineering College, Karur, India


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) where x i (t) ∈ R n indicates the position of the ith neuron, a i > 0 is the positive constant weight, (b ij ) n×n , (c ij ) n×n indicates the transmission weight matrices from ith neuron to jth neuron, and f j : R n ⟶ R n characterize the transfer function, the constant delay τ > 0, and I i 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. e 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][18][19].
f j x j (s) ds + I i , (2) where the term σ(t) signifies the distributed time-varying delay.
Besides the aforementioned delay kinds, there is another kind of delay called proportional delay, which explains that, δ(t) � t − q ij t, where q ij > 0, for i, j � 1, 2, . . . , n, pantograph delay factor with 0 < q ij < 1. is kind of delay is also know as discrete unbounded delay since the delay term δ(t) ⟶ + ∞ as t ⟶ + ∞. Due to the constrains of the continuously distributed delay, the proportional delay is less conservative.
c ij f j x j q ij t + I i . (3) 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, € x i (t) denotes the inertial term and b i > 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][38][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. is 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. e 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 ith 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. e 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.

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 represents the set of all real numbers, R + � (0, +∞), N represents the set of all natural numbers, the n-dimensional Euclidean space is R n and the set of all n × m real matrices is R n×m , and the Euclidean norm for any vector y and the trace norm for any matrix A are denoted |y| and weight m ab for every arcs (a, b). e Laplacian matrix of G is defined as which is differentiable and continuous at t and twice differentiable at y i .

Lemma 1 [43]).
e weighted digraph(G, A)withn ≥ 2of vertices andA � (a ij ) n×n . Let the collection of all spanning unicyclic of(G, A)beUandithdiagonal element of cofactor of theL p is denoted asc i , then the following identity holds: where, for k, h ∈ M, an arbitrary function F kh (t, z k (t), z h (t)), the set of all spanning unicyclic graph U, W(U), and C U denotes, respectively, the weight and dicycle of U. Addi- Lemma 2 (Homeomorphism theorem 44). IfHis a continuous function fromR n toR n which satisfies the following conditions, Lemma 3 (Cauchy-Schwarz inequality 45). Let (α 1 , α 2 , . . . , α n )and(α 1 , α 2 , . . . , α n )be two sequences of real numbers, one has the inequality turns into equality, whenever the real sequences Compared with the aforementioned discussions, we describe the following uncertain neural networks with distributed, proportional delay and inertial term as Mathematical Problems in Engineering where y i (t) ∈ R n denotes the ith node of neurons, the positive parameters α i , β i , c ij , δ ij , ζ ij ∈ R indicate, respectively, connection weights of ith node and the connection weight of ith node to jth node. e unknown weighted matrices Δα i , Δβ i , Δc ij , Δδ ij , Δζ ij with the conditions f j , g j , h j denote, respectively, the activation function, activation function of the proportional delay p i t, and the activation function of the distributed delay τ i (t) with the condition 0 ≤ _ τ i (t) < τ i . e initial value of the given system (9) is Remark 1. e proportional delay p i t is also known as unbounded time-varying delay, since the delay factor, Here, σ(t) is a continuous function which fulfils the condition σ(t) ⟶ ∞ as t ⟶ ∞ . More precisely, suppose that in system (9), ifp i � 1and there is no distributed delay and also with the absence of an uncertainty, equation (9) can be rewritten as which is generalized INNs.

Existence of a Unique Equilibrium Point
Definition 1. Let y(t) � (y 1 (t), y 2 (t), . . . , y n (t)) be the solution of (9), satisfying the initial conditions. If there exist two positive constantsrandM ϕ,ψ such that for allt ≥ t 0 , then the system (9) is globally exponentially stable.
. . , n, the transfer functions f j (·), g j (·) and h j (·) which is from R n to R n satisfies the Lipschitz condition. at is, there exist Lipschitz constants F j , G j , an d H j such that for any u, v ∈ R, Assumption 3. (R 3 ) For each i ∈ M, there exist some positive constants a i and c i such that For the sake of simplicity, we present the upcoming denotations: where a i , b i , and c i are positive constants, for all i ∈ M.

Theorem 1.
Suppose that there is a constant B r i < 0 which is defined in denotation, and assumption R 1 holds, then there exists a unique equilibrium point for system (9).
Proof. Let us begin with a map H(y i ) � (H 1 (y 1 ), H 2 (y 2 ), . . . , H n (y n )) ∈ R n . Here, Next, by using Lemma 2, we will show that H(y i ) is homeomorphism for all i ∈ M. Initially, we show that the

Mathematical Problems in Engineering
Since, B r i < 0, it shows that u i � v i , ∀i ∈ M; this implies the contradiction to u i ≠ v i . Hence, H(y i ) is one-one function. Our next aim is to show that Here, and By using Cauchy-Schwartz inequality in Lemma 3, where a � max 1≤i≤n |a i c i | . It follows that which reveals that, ‖H(y i )‖ 2 ⟶ + ∞ as ‖y i ‖ ⟶ + ∞. By Lemma 2, H(y i ) is a homeomorphism on R n which implies that system (9) has a unique equilibrium point y * i � (y * 1 , y * 2 , . . . , y * n ) T . □ Corollary 1. Suppose that the assumptions R 1 and R 3 hold, then (9) has a unique equilibrium point, which is asymptotically stable.

Exponential Stability
Theorem 2. Under the assumptions (R 1 ) and (R 2 ) , the system (9) is exponentially stable. Proof. Let us defined the global Lyapunov functional as follows Where, ς i denotes the cofactor of ith diagonal element of L p of digraph (G, A)and Here, θ t+θ e 2rs f 2 y j (s) dsdθ.

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Here, we need to show that, (1/2) n i�1 n j�1 N ij (y 2 j (t) − y 2 i (t)) ≤ 0. e obtained results has shown under Lemma 1, that Along the directed cycle C Q , we indicate the set E(C Q ) by, According to (40), we get From (39) and (41) we get, en, by (38) implies that, that is, is implies that V(t) ≤ V(0), ∀t ∈ [0, +∞) and Note that, at is en, there exist an constant M ϕ,ψ > 0 such that, Hence from Definition 1 the system (9) is exponentially stable.

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Corollary 2. Assume that the system (25) is exponentially stable if the assumption R 1 and following conditions holds,

Numerical Simulation
Example 1. Let us consider the following inertial neural networks with proportional delay, distributed delay and uncertain parameters as: € y (t) � − (α + Δα(t)) _ y(t) − (β + Δβ)y(t) +(c + Δc)f(y(t)) +(δ + Δδ)g(y(pt)) h(y(s))ds + I i . Where y(t) � y 1 (t), y 2 (t) T ; Clearly, all the conditions in eorems 1 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 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.

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.

Mathematical Problems in Engineering
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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.