Global asymptotic stability and projective lag synchronization for uncertain inertial competitive neural networks

In this paper, the global asymptotic stability and projective lag synchronization of second‐order competitive neural networks with mixed time‐varying delays and uncertainties are studied without converting the original system into the usual first‐order system. Firstly, according to the Lyapunov functional method, inequality technique and the designed adaptive control strategy, the algebraic criteria of stability, and projective lag synchronization are derived by adjusting the control gain parameters in the controller. The obtained sufficient conditions are simple and easy to verify. Different from the traditional feedback controller, the adaptive controller can adjust its characteristics according to the system model, and external disturbance makes the system have better control performance. Besides, unlike the existing ones, this stability and synchronization problem is directly analyzed by constructing some new Lyapunov functionals with state variables and state variable derivatives. Finally, the effectiveness and practicability of the results are verified by numerical examples.


INTRODUCTION
The neural network (NN) is a parallel distributed system formed by a large number of processing elements (called neurons) with storage functions connected in a specific network topology.Although the function of individual neurons is simple, and the information processing speed is slow.The NN has high information processing speed by constantly adjusting the connection weights among internal neurons.Common NN models include bidirectional associative memory NNs, memristive NNs, and inertial NNs (INNs).Competitive NNs were first proposed by Meyer-Base et al. [1].This NN has two specific state variables: short-term memory (STM) and long-term memory (LTM).In this model, STM describes the fast neural activity, while LTM describes slow and unsupervised synaptic modifications.In recent years, many researchers have paid extensive attention to the dynamic properties of competitive NNs.For example, Balasundaram et al. [2] provided sufficient conditions for exponential stability of competitive NNs with impulsive effects.Lou and Cui [3] studied the exponential synchronization problem for a class of competitive NNs.Zhu and Shi [4] investigated synchronization of memristive competitive NNs with different time scales.For other theoretical results about competitive networks, we can refer to previous studies [5][6][7][8][9][10][11][12][13][14].
In 1986, Babcock and Westervelt [15] proposed a NN model described by second-order differential equations, namely, INNs.Its essence is to imitate the inertial properties in practical problems by introducing inductance into the neural circuit.Compared to the traditional first-order capacitance-resistance model, the second-order system has more complex dynamic characteristics and broad biological background.In addition, early research shows that inertia terms are powerful tools for generating bifurcation [16] and chaos [17].Therefore, introducing the inertia term into NNs is essential.The INN has many applications, such as associative memory [18] and pattern encryption [19].So far, there have been many achievements in INNs [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].In this existing research [22][23][24][25][26][27][28][29][30][31], a common method transforms the original second-order NN (SNN) into the first-order NN by selecting the appropriate variable transformation.Obviously, variable transformation leads to an increase in system dimensions and computational complexity.In previous studies [20,[32][33][34], the stability of SNNs with time delay is investigated by the non-reduction method.The global exponential stability of inertial competitive NN (ICNNs) with discrete time delays is discussed [20].Li et al. [32] studied stability and synchronization of INNs with discrete delay by non-reduced-order method.The exponential synchronization of inertial complex-valued NNs is discussed by the non-reduced-order method [33].Chen and Kong [34] investigated the exponential stability of fuzzy INNs by the technique of not reducing the order.It can be seen that there are few kinds of literature that introduce inertial terms into competitive NNs.Therefore, it is necessary to introduce the inertia term into the competitive NN and use the non-reduced-order method to study it.
On the one hand, due to equipment aging, modeling errors, external noise in hardware implementation, and other reasons, parameter uncertainty is inevitable, usually encountered in the NNs.In recent years, many scholars have done research in the field of uncertain INNs.At present, many scholars have considered the influence of uncertain terms on INNs [35][36][37][38][39]. Iswarya et al. [40] studied the exponential stability of uncertain INNs.Fixed time synchronization of uncertain fuzzy INNs is discussed in Kong et al. [41].
On the other hand, the NNs synchronous attracted people's attention because of their wide applications in many fields, such as secure communication, image encryption, and information science.So far, a wide range of synchronization phenomena have been studied, such as finite time synchronization [42], projective synchronization [43], exponential synchronization [44], and lag synchronization [45].In practical application, it is not easy to directly obtain the exact parameters of the master-slave system.Therefore, adaptive synchronization [46] may be a good choice in this situation.It is worth mentioning that due to the limited signal transmission and switching time, the time delay is inevitable, so it is difficult to achieve the complete synchronization of NNs, and it is very important to consider lag synchronization.Compared with other types of synchronization, projective synchronization has typical advantages because of the unpredictability of proportional constant, and it can enhance communication security.The combination of projective synchronization and lag synchronization can better improve the signal transmission and application value.At present, Feng et al. [47] studied a general method for projective lag synchronization of heterogeneous chaotic maps with different dimensions.Pratap et al. [48] discussed global projection lag synchronization of mixed time-varying delay BAM NN with fractional memristors.Other results on projective lag synchronization can be found in previous studies [49][50][51][52][53].
Therefore, it is necessary to introduce uncertain terms and inertial terms into competitive NNs.
As far as we know, few articles study the stability and projective lag synchronization of NNs with time-varying delays by non-reduced-order methods.All these have led to the motivation of this research.The main contributions of this paper are: 1.The inertial term, uncertain term, and discrete and infinite distributed delays are all taken into account in the dynamic model of competitive NNs, which makes the discussed model more versatile.2. Different from the previous state feedback control methods [45,54,55] used to realize the projective lag synchronization of NNs.This paper adopts adaptive control to realize projective lag synchronization.3. Based on inequality technique, the considered NN can achieve projective lag synchronization by adjusting the controller, thus showing good robustness.
The rest of the paper is structured as follows: Section 2 gives the description of the model and some preliminary knowledge.The main results of asymptotic stabilization and projective lag synchronization are revealed in Sections 3 and 4.
Two simulation examples are given in Section 5 to illustrate the effectiveness of the results.Ultimately, the conclusion is drawn in Section 6.

MODEL DESCRIPTION AND PRELIMINARIES
In this paper, a type of ICNNs with mixed delays and uncertain parameters are described as where x i (t) is the level of neuronal current activity,   (t) and   (t − (t)) are activation function, w il (t) is the synaptic efficiency,  l is an external stimulus intensity, D i represents a connection weight between the i and  neurons, c i is an external stimulus, Di and Di represent a delayed feedback synaptic weight, I i (t) is external input, and q is constant.Without loss of generality, the input stimulus  is assumed to be normalized with unit magnitude |||| 2 = 1; then, the above networks are simplified as the initial conditions associated with system (2) are given by (3) If Di = 0 and uncertain is not considered, then (2) is transformed into If Di = 0, s i (t) = 0 and uncertain is not considered, then (2) is transformed into Remark 2. Obviously, the system considered in Shi et al. [20] is a special case of (1), and system ( 6) is model ( 1) in Shi and Zeng [56].Thus, model ( 1) is more general.
Let (2) be the drive system, and the response system is described as follows: , ςi (t) are continuous and bounded, and u i (t), v i (t) are controllers.Now, we define the error term as where  is the lag delay and k i is a scaling constant.
Proof.Firstly, consider a generalized Lyapunov function as where . . .
According to (27) Take the limit on both sides of ( 28) From Lemma 1, one has So this system can achieve global asymptotic stability.The proof is completed.□

PROJECTIVE LAG SYNCHRONIZATION
In this section, we consider the projective lag synchronization between the drive system (2) and response system (7).First, from the definition of e i (t),  i (t), the error dynamical system can be written as The synchronization error system is converted into the following form: Theorem 2. Under Assumptions 1-6, then ICNNs will achieve PLS under the following adaptive controllers: where r i ,  i ,  i , ri , δi , ψi are positive constants.
Proof.Firstly, consider a generalized Lyapunov function as where Together with ( 33)-( 38), the derivative function of V(t) is given: Under Assumptions 1-6, and similar to the process in Theorem 1, we can obtain Submitting ( 14)-( 15) and ( 40)-( 46) into (39), the following inequalities can be obtained: The parameters in inequality (47) satisfy the following conditions: where  i > 0,  i > 0 for i ∈ 1, 2, … , n are arbitrarily chose constants, one gets Therefore, V(t) ≤ V(0), which shows that e i (t), ̇ei (t),  i (t), Take the limit on both sides of (49) By Lemma 1, one has Therefore, according to Definition 2, the drive-response system can achieve projective lay synchronization under adaptive controllers ( 31) and (32).□ Remark 4. In Hua et al. [24], the synchronization of INNs was investigated, in which the driving second-order INNs were transformed into the first-order system under some variable substitutions; the response networks were also provided by the first-order systems, and some controllers were added on them.Note that the research object is the second-order INNs, it may be more reasonable and meaningful to design second-order response inertial networks with control inputs other than the reduced-order systems.In this paper, without using reduced-order method, the second-order INNs (2) and ( 7) are regarded as the driving and response systems; the controller is added on the original response inertial networks.Obviously, our results are more reasonable.

Corollary 1.
When k i = k, the error can be rewritten as e i (t) = m i (t) − kx i (t − ),  i (t) = r i (t) − ks i (t − ), According to Definition 2, the drive-response system can achieve the projective lay synchronization with the controllers ( 31) and ( 32);  * i , ξ * i ,  * i , π * i are the same as those in Theorem 2.   Proof.The proof is similar to the proof of Theorem 2, so we omitted it here.□ Corollary 3. When k i = −1, the error can be rewritten as e i (t) = m i (t) + x i (t − ),  i (t) = r i (t) + s i (t − ).According to Definition 2, the drive-response system can realize the lay anti-synchronization through controllers (31)   x 1 (t), .
x 2 (t), s 1 (t), s 2 (t), Up to now, we can see that all the conditions in Theorem 1 are satisfied.Thus, according to Theorem 1, we can obtain that system (8) is globally asymptotically stable.This fact is shown by Figure 1.

FIGURE 7 10 FIGURE 8
FIGURE 7Time responses of  i (t), e i (t)(i = 1, 2) without adaptive controllers(31) and(32).[Colour figure can be viewed at wileyonlinelibrary.com] < 1; and Δa i , Δb i , Δc i , ΔD i , Δ Di , Δ Di , Δg i , Δh i , Δp i are uncertain parameters.Firstly, we shall rewrite system (1) as follows: setting s i (31) responses of e The proof is similar to the proof of Theorem 2, so we omitted it here.□Corollary2. When k i = 1, the error can be rewritten as e i (t) = m i (t) − x i (t − ),  i (t) = r i (t) − s i (t − ), According to Definition 2, the drive-response system can realize the lay synchronization through controller controllers(31)and (32);  * i , ξ * i ,  * i , π * i are the same as those in Theorem 2.