Event-triggered synchronization of uncertain delayed generalized RDNNs

This paper investigates the exponential synchronization analysis of master–slave chaotic uncertain delayed generalized reaction–diffusion neural networks (GRDNNs) with event-triggered control scheme. A delay GRDNNs system model for the analysis is constructed by investigating the effect of the network transmission delay. By constructing a novel Lyapunov–Krasovskii functional and using a delay system approach for designing event-triggered controllers and some inequality techniques like Jensen’s inequality, Wirtinger’s inequality and Halanay’s inequality, the criteria are obtained for the event-triggered synchronization analysis and control synthesis of delayed GRDNNs. The synchronization criteria are formulated in terms of linear matrix inequalities. Finally, we conclude that the slave systems synchronize with the master systems. Two examples show the proposed theoretical results are feasible and effective.


Introduction
Neural networks (NNs) have been rather hot topic and been extensively studied in brain science and engineering fields due to their potential applications such as mammalian brains, solving certain optimization problems and fault diagnosis (Rakkiyappan et al. 2015;Wang et al. 2017;Kwon et al. 2013;Sibel Senana 2017). In general, NNs are categorized as either discrete type (Liu et al. 2009) or continuous type (Kwon et al. 2013;Sibel Senana 2017).
The prominence of continuous-time NNs with time-varying delays has been emphasized in many references. It is necessary to investigate the models of delayed NNs. As pointed in Zeng et al. (2015); Liu et al. 2015;Zhang et al. 2014;Zhang and Han 2011), in order to study the dynamical evolution law of NNs, two basic mathematical models are commonly adopted such as local field NN models (Xu et al. 2004) or static NN models (Zeng et al. 2015) by the application of local field states or neural states of neurons. As said in Liang and Cao (2006), the above network models possibly relocated equally from one to another with some suppositions, but these suppositions do not always hold for a large number of applications. Hence, static network models and local field network models are never identical. Recently, some researchers have studied new unified NNs called generalized NNs (GNNs) to combine these two NN models and provided some criteria with a unified frame suitable for both local field NNs and static NNs (Liu et al. 2015;Zhang et al. 2014;Zhang and Han 2011;Gan et al. 2016).
Time delays are frequently encountered in hardware installations of NNs and many biological systems, which are ubiquitous in nature because of the finite signal communication time, switching speed of amplifiers and memory effects, and they may induce instability, poor performance and oscillation of systems (Rakkiyappan et al. 2015;Wang et al. 2017;Kwon et al. 2013;Sibel Senana 2017). Thus, time delays always appear in domains ranging from population dynamics and network control to biological networks. NNs with time-varying delays are usually found and examined when one analyzes the dynamic behavior of systems (Rakkiyappan et al. 2015;Wang et al. 2017Wang et al. , 2021Kwon et al. 2013Kwon et al. , 2020Vadivel et al. 2020). In the previous studies on NNs, the time delay introduced is mainly time-varying delay signals s t ð Þ (given as s t ð Þ s), which belong to an upper bound interval 0 s t ð Þ s Li et al. 2008;He et al. 2007;Saravanakumar et al. 2017). In general, the lower bound value of s t ð Þ is not needed to be zero, and thus, s t ð Þ has been stretched to the interval with lower and upper bounds, 0 s 1 s t ð Þ s 2 (Zhang et al. 2014;Zhang and Han 2011;Wang et al. 2021;Kwon et al. 2020). In (Wang et al. 2021), an improved Lyapunov-Krasovskii functional is proposed by considering more important information about delay states, integral terms and the activation function. New less conservative delaydependent stability conditions for generalized neural networks with interval time-varying delays are established by constructing a suitable Lyapunov-Krasovskii functional and applying the linear matrix inequality formulation. The authors in Gan et al. (2016) investigated the synchronization for a class of GNNs based on Lyapunov stability theory and obtained both delay-derivative-dependent and delay-range-dependent conditions in terms of linear matrix inequalities. In addition, parameter uncertainties are unavoidable when modeling and implementing real NNs owing to measure errors, the parameter fluctuation and external disturbance. We all know that the stability or synchronization of a well-designed system may often be destroyed due to its unavoidable uncertainty. Parametric uncertainties will also be introduced into the models of NNs (Vadivel et al. 2020;Liao et al. 2001;Li et al. 2004). If the uncertainty of a system is only due to the deviations and perturbations of its parameters, and if these deviations and perturbations are all bounded, then the system is called an interval system (Li et al. 2004).
In 1665, Huygens found a fundamental natural phenomenon in the universe, that is, two weakly coupled pendulum clocks with hanging at the same beam synchronous in phase (Pecora and Carroll 1990). Chaos synchronization of two coupled NNs is an important phenomenon, which have drawn considerable attention because of wide practical applications in some fields with parallel recognition, secure communication and so on. For example, a lot of NNs rely on a synchronous behavior for a proper functioning including pattern recognition and information transmission (Rakkiyappan et al. 2015;Wang et al. 2017). NNs are easy to physically implement and hence have promising applications especially in secure communications based on synchronization (Kwon et al. 2013;Sibel Senana 2017). In the past years, NNs synchronization problems are hot topic and have been successfully used to many areas (Sibel Senana 2017;Zhang et al. 2016;Chen et al. 2019;Wu et al. 2017;Lu et al. 2018;Li et al. 2013;Sheng and Zeng 2018). In fact, some natural systems can synchronize by themselves, but others cannot attain synchronization by themselves. In this case, some controllers need to be designed and are used to compel the systems to synchronize. In general, master NNs and slave NNs start to adjust from different initial state via the proper control strategy. Some control methods have been applied to obtain synchronization of NNs, such as pinning control (Wang et al. 2017), decentralized eventtriggered control (Sibel Senana 2017), intermittent control (Chen et al. 2019), sampled-data control (Lu et al. 2018), adaptive learning control (Li et al. 2013) and impulsive control (Sheng and Zeng 2018;Xie et al. 2019). In (Lu et al. 2018), the exponential synchronization for a class of hybrid coupled delayed RDNNs was considered under intermittent control with spacial sampled data. In (Xie et al. 2019), the authors studied the synchronization analysis of coupled delayed RDNNs via a novel pinning impulsive controller.
In addition, networked control system has many remarkable advantages besides low cost, easy diagnosis, long distance control and so on. In (Yue et al. 2013), the authors have proposed a novel event-triggering scheme, and then, many papers focused on event-triggered control approach of many systems (Sibel Senana 2017;Huang and Liu 2019;He et al. 2019;Fei et al. 2017;Wen et al. 2017;Fan et al. 2019;Li et al. 2019;. In (Wen et al. 2017), eventtriggered synchronization control was studied for switched delayed NNs with communication delays. In , event-triggered synchronization control was investigated for delayed NNs with quantization and actuator saturation. As we all known, maximizing the time intervals is an effective strategy to decrease the communication. In order to reduce the workload of communication network, the event-triggered control approach was applied to the network-based synchronization system to decrease the amount of data packets sent by the sensor and thereby saved the network resources (Yue et al. 2013;Huang and Liu 2019;He et al. 2019;Fei et al. 2017;Wen et al. 2017;Fan et al. 2019;Chen et al. 2491). However, there are few results on eventtriggered control synchronization and analysis of uncertain GRDNNs. This paper will develop this control strategy to uncertain GRDNN, and think about communication time delays between sensor and controller nodes for a class of distributed parameter networks system. At the same time, to decrease the cost, the GRDNNs may allocate a tiny embedded chip, which is often including few energy resources and limited computing capabilities. These limitations encourage the development of event-triggered control approach in the digital form (Fei et al. 2017;Wen et al. 2017;Fan et al. 2019).
Although research on synchronization has attracted so much attention, few of that had been devoted to the synchronization properties of the spatial and temporal GRDNNs. To the best of our knowledge, the synchronization of uncertain GRDNNs has not yet been considered. Therefore, it is important and interesting to study the synchronization of uncertain GRDNNs. In most existing works, it is assumed that the node state is only dependent on the time. Therefore, it is essential to investigate the relationship synchronization for the uncertain GRDNNs. The purpose of this study is to establish synchronization conditions of uncertain RGDNNs by applying the Lyapunov functional theory and event-triggered communication scheme. In event-triggered control, the measured error acts a key role during the event-triggered controller design. An event will be triggered to update the event-triggered controller when its magnitude reaches the prescribed value. This paper will develop this control strategy to uncertain delayed GRDNNs and consider the communication delays between sensor and controller nodes. We extend uncertain delay GRDNNs model with communication delays and establish criteria for the synchronization by utilizing some inequality techniques. Also, we propose an even-triggered control method to synchronize two coupled uncertain GRDNNs systems.
The paper is organized as follows. Section 2 provides some mathematical preliminaries and formulates the model of uncertainty delayed GRDNNs, and some necessary definitions, lemmas and hypotheses are presented. In Sect. 3, some criteria for the synchronization sufficient of such coupled uncertain delayed GRDNNs are derived. In Sect. 4, numerical example is given to show the validity of the obtained results. The conclusions of this paper are presented in Sect. 5.

Notation
The notation A T means the transpose of A. For symmetric matrices A and B, the notation A [ B means that the A À B is positive definite. For a matrix C and symmetric matrices A and B, A C Ã B means symmetric matrix, in which the notation Ã denotes the entries implied by symmetry. Let I N be an N-dimensional identity matrix and R n denote the n-dimensional Euclidean space, and R nÂm is the set of n Â m real matrices. X ¼ 0; l ½ is a compact set with smooth boundary oX and mesX [ 0, where l [ 0 is a constant; L 2 X ð Þ is the space of real functions on X which are L 2 for the Lebesgue measure. It is a Banach space for the norm where u t; x ð Þ ¼ u 1 t; x ð Þ; :::; u n t; x ð Þ ð Þ T and u i t; x ð Þ k . In addition, we define 2 Model description and preliminaries Consider a class of robust delayed GRDNNs master system as follows: where x ¼ x 1 ; x 2 ; :::; x m ð Þ T 2 X, u t; x ð Þ ¼ u 1 t; x ð Þ; u 2 t; x ð Þ; ð . . .; u n t; x ð ÞÞ T 2 R n denotes the state vector, D ¼ diag D 1 ; :::; D n ð Þ ; D i [ 0 represents the transmission diffusion coefficient, A ¼ diag a 1 ; :::; a n ð Þ is a known positive diagonal matrix with a i [ 0, M ¼ diag m 1 ; :::; m n ð Þ is a known diagonal matrix, B; C and W¼ w ij À Á nÂn represent the appropriately dimensioned known connection weight matrices between neurons, gðWuðÁ; xÞÞ ¼ ðg 1 ðW 1 uðÁ; xÞÞ; :::; g n ðW n uðÁ; xÞÞÞ T denotes the activation vector function, W j represents the jth row vector of the matrix W; and J ¼ J 1 ; J 2 ; :::; J n ð Þ T denotes a constant external input vector. s t ð Þ denotes the transmission time-varying delay, which satisfies 0 s t ð Þ s; _ s t ð Þ l; where l and s are constants, i¼1; 2; :::; n: DA; DB; DC represent time-varying parameter uncertainties, which are supposed to be of the form where X; t 1 ; t 2 and t 3 are known real constant matrices, and F t ð Þ is an unknown time-varying matrix that satisfies where I is the identity matrix with appropriate dimension.
Event-triggered synchronization of uncertain delayed generalized RDNNs 13245 In this paper, we assume the following conditions hold.
Lemma 1 (Extended Wirtinger's Inequality, Hardy et al. 1988 Lemma 2 (Gu et al. 2003) Let scalars c 1 and c 2 satisfy c 2 [ c 1 , a vector function x: c 1 ; c 2 ½ !R n such that the integrations concerned are defined, for any matrix M [ 0, then Lemma 3 (Halanay's Inequality, Halanay 1966). Let V : Þbe an absolutely continuous function and 0\d 1 \2d satisfying where d 1 and d are positive constants, and a [ 0 is a unique positive solution of a ¼ d À d 1 e 2ah 2 : Lemma 4 (Syed Ali et al. 2015) Let X; JðtÞ and g be real matrices of appropriate dimensions and J t ð Þ satisfy J t ð Þ T J t ð Þ I. Then, for any constant e [ 0; the following inequality holds: Remark 1 It is worth pointing out that the structure of the parameter uncertainties including (2a) and (2b) has been widely utilized in many references (Sibel Senana 2017;Vadivel et al. 2020;Liao et al. 2001;Li et al. 2004) to deal with the stability or synchronization problems for uncertain NNs and other systems. The motivation for considering system (1) with uncertainties DA; DB and DC stems from the fact that, in practice, it is almost impossible to obtain exact mathematical models of dynamic systems because of the complexity for such systems. In (2a) and (2b), DA; DB and DC represent time-varying parameter uncertainties with appropriate dimensions, and F t ð Þ is the unknown and uncertain matrix with Lebesgue measurable elements bounded satisfying F T t ð ÞF t ð Þ I, which is an important condition in solving the robust stability or synchronization problems for uncertain systems. Lemma 4 plays an important role in the reported criteria, and (2b) is a key and necessary condition to ensure Lemma 4 holds. Furthermore, it is reasonable and unavoidable that the model of the controlled system with uncertainties is owing to the existence of external disturbance, modeling error and parameter fluctuation during the implementation. The parameter uncertainties in the system considered are said to be admissible if (2a) and (2b) hold. Therefore, it is of theoretical and practical importance to investigate robust stability, synchronization, etc., of the delayed NNs (Sibel Senana 2017;Vadivel et al. 2020;Liao et al. 2001;Li et al. 2004).
Consider system (1) with the Neumann boundary conditions. Set the points 0 ¼ The sampling intervals on time and space may be variable and satisfy here the measured output of system (4) is sampled and then transmitted to the controller of the slave system as follows: in which e t k ; x j À Á is utilized to decide if the control system needs to send z t kþj ; x j À Á to the controller by the following inequality: in which K 1 ; K 2 are positive weighting matrices, and j ¼ 1; 2; :::; d 2 0; 1 ½ Þ:z t kþj ; x j À Á will not be sent until it cannot satisfy condition (12).
Suppose the time-varying communication delays s k 2 0;s ð Þ;s [ 0; k ¼ 1; 2; :::; from the sampling termination of the error system to the slave system in practical implementation can be detected by the difference between the instants at sampled states e t k ; x j À Á and the control states e t k þ s k ; x j À Á . Thus, we can get Hence the novel control law is designed as with the gain matrix K ¼ diag k 1 ; k 2 ; :::; k n ð Þ to be determined,x j ¼ x j þx jþ1 2 ; j ¼ 0; :::; N À 1: By (12)-(14), we obtain Next, we consider two cases: where h is a sampling period, considering the following intervals Froms k s;we can find that there exists N 0 such that moreover, x t k ; x j À Á and x t k þ ih; x j À Á with i ¼ 1; 2; :::; N 0 satisfy (15). In ; thus we can get and 0 s k h M ; for all t 2 t k þ s k ; t kþ1 þ s kþ1 ½ Þ ; Therefore, From definition of e t; x j À Á and function h t ð Þ, we can have control condition (12) as According to the sampled error output z t; x ð Þ, one can obtain We

Main results
In this section, our event-triggering scheme can ensure the stability of the closed-loop system (25). The following novel criteria can guarantee synchronous of the master system (1) and slave system (3) with DA ¼ DB ¼ DC ¼ 0 via the event-trigger scheme (12).
Proof Define the Lyapunov-Krasovskii functional as We can get the following equality by calculating the derivative of V t ð Þ, The derivative of V 1 t ð Þ, t 2 t k þ s k ; t kþ1 þ s kþ1 ½ Þ , along the trajectories of (22) can be found The derivatives of V 2 t ð Þ; V 3 t ð Þ and V 4 t ð Þ, t 2 t k þ s k ; t kþ1 þ s kþ1 ½ Þ , are as follows: By using Lemma 2 combined with reciprocally convex approach (Park et al. 2011 where : Event-triggered synchronization of uncertain delayed generalized RDNNs 13249 Using the free-weighting matrix method, we can get that x j e s t; s ð Þds.

Integrating by parts and substitution of the Dirichlet boundary conditions, one can lead to
From Young's inequality for any scalar e 1 [ 0; the following inequality (40) is true: According to Lemma 1, we have the following inequality Choosing next e 1 ¼ D p e 1 , we get from (40) and (41) Thus, from (40)-(42), we have Similarly to (42) and (43), by Young's inequality for any scalar e 2 ; e 3 [ 0; we find inequality (44) and equality (45) as follows: From (24), we can conclude Similar to inequality (39), we know From assumption (5), we have where v t; x ð Þ ¼ e t; x ð Þ T f We t; x ð Þ ð Þ T À Á T , and I i is the unit column vector having 1 element on its i-th row and zeros elsewhere. Thus, for any appropriately dimensioned diagonal matrix C [ 0, the following inequality holds (Wang et al. 2006): where L 1 ¼ diag L À 1 L þ 1 ; :::; L À n L þ n À Á ; L 2 ¼ diag L À 1 þL þ 1 2 ; :::; Similarly, for any appropriately dimensioned diagonal matrix C [ 0, we have the following inequality: Combining (31)-(38), we get Event-triggered synchronization of uncertain delayed generalized RDNNs 13251 Substitute (39) and (47) into (38) and add the right-hand sides of (52), one can use the derived (39)-(47), (24) Event-triggered synchronization of uncertain delayed generalized RDNNs 13253 According to Lemma 3 and the conditions of Theorem 1, we have ð54Þ then the error system (4) with DA ¼ DB ¼ DC ¼ 0 under the event-triggered control (14) is stable. This implies that the two systems (1) and (3) are synchronized. This completes the proof.
Remark 2 In (30), the Lyapunov-Krasovskii functional V t ð Þ is continuous, derivative and positive definite. As we know, the choice of the Lyapunov-Krasovskii functional is crucial for obtaining less conservative criteria. Various kinds of the Lyapunov-Krasovskii functional have been constructed to discuss the NNs with time-varying delay. For the derivative of the Lyapunov-Krasovskii functional, it is necessary to estimate the derivative for deriving the criteria. In this paper, some techniques including the freeweighting matrix, Jensen's inequality, Wirtinger's inequality, Halanay's inequality, combined with reciprocally convex, and so on have been used by constructing a novel Lyapunov-Krasovskii functional and applying a delay system approach for designing event-triggered controllers of delayed GRDNNs. The synchronization criteria are formulated in terms of linear matrix inequalities.
Remark 3 In recent years, a great number of research investigations have analyzed interval-delayed NNs (Zeng et al. 2015;Liu et al. 2015;Zhang et al. 2014;Zhang and Han 2011;Gan et al. 2016;Vadivel et al. 2020;Wang et al. 2021;Saravanakumar et al. 2017;Liao et al. 2001;Li et al. 2004). Development of delay-dependent stability or synchronization conditions has received increasing attention from research communities, which have become important topics of research. Recently, many effective methods and techniques including the delay-partitioning method (Zhang et al. 2010), reciprocally convex combination method (Zhang and Han 2011), free-weighting matrix approach (Gan et al. 2016;Saravanakumar et al. 2017) and augmented Lyapunov-Krasovskii functionals approach (Wang et al. 2021) have been utilized to derive much less conservative results. For interval-delayed NNs, the authors in Zhang and Han (2011);Li et al. 2008;Zhang et al. 2010) have found the admissible maximum upper bound s of time-varying delay s t ð Þ for guaranteeing stability of system considered by giving different values of scalar l. The corresponding results are shown and listed in tables. In this paper, we investigate the exponential synchronization analysis of master-slave chaotic uncertain delayed GRDNNs with event-triggered control scheme. The event-triggered synchronization criteria obtained for delayed GRDNNs are delay-dependent. In fact, Theorem 1 is stabilization criterion of system considered, too. Interval influence stability of such system can be discussed by using approach similar to references (Zhang and Han 2011;Li et al. 2008;Zhang et al. 2010).
Remark 4 To reduce the network workload, event-triggered control scheme of finite-dimensional systems has been investigated extensively over the past decades. There are a few references on event-triggered control of distributed parameter systems (DPSs), which are potentially of great interest in a long distance control of chemical reactors (Smagina and Sheintuch 2006) or air-polluted areas (Court et al. 2012). In (Yao and El-Farra 2013), event-triggered control of DPSs was used via model reduction approach leading to local results concerning practical stability. Furthermore, this method seems to be inapplicable to the systems with spatially dependent diffusion coefficients. In , the authors propose distributed event-triggered control of DPSs under the point measurements and under the averaged measurements. However, there are few theoretical results for reaction diffusion neural networks using the event-triggered control approach. In the present work, we develop eventtriggered controllers for DPSs governed by uncertain semilinear diffusion equations to solve distributed parameter neural networks synchronization control problems.
Remark 5 In Theorem 1, the Lyapunov-based analysis of (25) with the ''compensation'' of the terms KM R x x j e s t À h t ð Þ; s ð Þ ds with _ h t ð Þ ¼ 1 for t 6 ¼ t k and KM R x x j e s t; s ð Þds is difficult. In this paper, the approach that we have developed is based on the combination of the Lyapunov-Krasovskii functional for (25) with Halanay's inequality, which was applied to solve robust sampled-data control problem for parabolic systems governed by uncertain semilinear diffusion equations with distributed control on a finite interval in Fridman (2012).
Remark 6 Theorem 1 provides a sufficient condition on synchronization under the event-triggered control. We assume a sampled-data controller design for a one-dimensional delayed generalized RDNNs, where the sampled data in time measurements of the state are taken in a finite number of fixed sampling spatial points. It is suggested that the sampling intervals in time and in space may be variable, but bounded. The sampling instants (in time) may be uncertain. The approach in this paper is based on the novel combination of Lyapunov-Krasovskii functionals with Wirtinger's, Young's and Halanay's inequalities.
Proof Replace A; B; C in LMI (29) by A; B; C, respectively.

An illustrative example
An illustrative example is shown to examine the eventtriggered synchronization characteristic of delayed GRDNNs.
The change processes of the state u i t; x ð Þ and error e i t; x ð Þ; i ¼ 1; 2; are shown in Fig. 1, 2, 3 and 4. Therefore, one can know from Theorem 1 that the master-slave systems are exponential synchronization in this example (Figs. 5,6).
Example 2 In this example, we consider the following master system (1) and slave system (3) with system parameters: We can see that g j Á ð Þ; j ¼ 1; 2 satisfy Assumption (A1). We can verify that L 1 ¼ 0; L 2 ¼ 0:5I: A straightforward calculation gives _ s t ð Þ 0:5\1; s ¼ 0:5; l ¼ 0:4: Fig. 1 Change of the master system u 1 t; x ð Þ described in Example 1 Fig. 3 Evolution of the error system e 1 t; x ð Þ described in Example 1 Fig. 4 Evolution of the error system e 2 t; x ð Þ described in Example 1 Fig. 2 Change of the master system u 2 t; x ð Þ described in Example 1 Choose D ¼ p 4 ; e 1 ¼ e 2 ¼ e 3 ¼ 1; a ¼ 0:2; h M ¼ 0:01; d ¼ 0:2; a 1 ¼ 2: It can solve the LMIs (26)- (29) and (55) in Theorem 2 by applying the MATLAB LMI control Toolbox. We can have the feasible solutions as follows:  . 7 Change of the master system u 1 t; x ð Þ described in Example 2 Fig. 6 Change of the error system e 2 t; x ð Þ when x ¼ 0:6 in Example 1 Fig. 8 Change of the master system u 2 t; x ð Þ described in Example 2 Fig. 9 Evolution of the error system e 1 t; x ð Þ described in Example 2   Table 1 for various values of d, and we find the corresponding upper bounds for sampling interval h which is essentially smaller. We assume the initial conditions of u i t; x ð Þ and u i t; x ð Þ; i ¼ 1; 2 are u 1 t; x ð Þ ¼ 0:8; u 2 t; x ð Þ ¼ 1:1;ũ 1 t; x ð Þ ¼ 0:5;ũ 2 t; x ð Þ ¼ 1:2; in addition, let the boundary conditions The simulation results are shown in Figs. 7,8,9,10,11,12 and 13,where Figs. 7 and 8 show the change processes of u 1 t; x ð Þ and u 2 t; x ð Þ in system (1), respectively. Figures 9 and 10 exhibit change processes of the synchronization errors e 1 t; x ð Þ and e 2 t; x ð Þ; respectively. Figures 11 and 12 depict change processes of the synchronization errors e 1 t; x ð Þ and e 2 t; x ð Þ when x ¼ 0:6, respectively. In this example, we can take event-triggering schemes to show the release instants and release intervals with known control gain and triggered matrix, which are illustrated in Fig. 13.The diagram of networked master-slave synchronization for the uncertain delayed GRDNNs is shown in Fig. 14.
According to Theorem 2, the slave system (3) and the master system (1) are exponential synchronized as shown in Figs. 9 and 10. The numerical simulations obviously demonstrate the effectiveness of the developed eventtriggered control method to the exponential synchronization of chaotic uncertain delayed GRDNNs with Dirichlet boundary conditions. Remark 7 In (Zhang and Han 2011), a generalized NN model without reaction-diffusion terms was firstly proposed, which includes both the local field NN model and static NN model as its special cases, and model stability criteria were given. In fact, diffusion effects always exist in the NNs and electric circuits when electrons are moving in asymmetric electromagnetic fields. Hence, one desires to study the activation of neurons varying in space as well as in time. At the same time, RDNNs have been investigated to show unpredictable behaviors such as bifurcation, chaotic attractors and periodic oscillations, which induced to study on its chaos synchronization, which is a key step for both understanding brain science and designing RDNNs for practical use. Inspired by the analysis above, in this paper, synchronization problem for a class of generalized NN model with reaction-diffusion terms was studied. As is well known, when D ¼ 0; C ¼ 0 and W ¼ I, model (1) becomes the local field NNs (Liu et al. 2015;Zhang et al. 2014;Zhang and Han 2011); when D ¼ 0; B ¼ 0; C ¼ 0 and W ¼ I, model (1) becomes the static NNs (Zeng et al. 2015;Zhang and Han 2011;Gan et al. 2016); and when D ¼ 0 and W ¼ I, model (1) becomes a classic delayed cellular NNs model (Liang and Cao 2006;Gan et al. 2016;Liao et al. 2001). Thus, in case of D 6 ¼ 0, model (1) becomes corresponding NNs model with reaction-diffusion terms. Therefore, model (1) can be called a generalized RDNNs model, and we make full use of information with the reaction-diffusion terms in the studies for synchronization problem.

Remark 8
The motivation for presenting the experimental results is to demonstrate that the proposed event-triggered communication scheme can be utilized to synchronization strategy for delayed uncertain GRDNNs. The improved synchronization novel results are obtained via given controllers. The simulation is performed applying MATLAB software. In Theorem 2, the size of the matrix inequality deduces bigger because more relationships have been taken consideration of, and this will make it difficult to find feasible solutions for these linear matrix inequalities. In order to solve this problem, some optimization algorithms will be used to take consideration of both the upper bounds of certain parameters and the computation complexity. By taking advantage of the classical implicit format solving the partial differential equations and the method of steps for differential difference equations, we provide two illustrative numerical examples and their simulations to show the feasibility and effectiveness of the theoretical results.

Conclusions
This paper studied the event-triggered synchronization control for uncertain delayed generalized RDNNs with communication delays. An event-triggered algorithm was used to synchronize the uncertain delayed generalized RDNNs. A model is presented for the synchronization error system with state delay by utilizing the delay system approach. The criteria for the event-triggered synchronization problem and synthesis of uncertain delayed generalized RDNNs are established in terms of LMIs. Two