Global Robust Exponential Synchronization of Multiple Uncertain Neural Networks Subject to Event-Triggered Strategy

is paper proposes the event-triggered strategy (ETS) for multiple neural networks (NNs) with parameter uncertainty and time delay. By establishing event-triggeredmechanism and usingmatrix inequality techniques, several sucient criteria are obtained to ensure global robust exponential synchronization of coupling NNs. In particular, the coupling matrix need not be the Laplace matrix in this paper. In addition, the lower bounds of sampling time intervals are also found by the established event-triggered mechanism. Eventually, three numerical examples are oered to illustrate the obtained results.


Introduction
Multisystem network is a complex dynamical network, which has received great attraction due to its many applications in various elds, such as secure communications and biological systems.e dynamical characteristics of multisystem networks have been paid more and more attention in the elds of science and engineering (see [1][2][3][4]).As we know, synchronization of a coupling system means that its all subsystems produce common behavior under di erent initial values.
e coupling scheme describing the interaction rules between subsystems plays an important role in ensuring the synchronization.In particular, the synchronization of multisystem networks has become a hotpot of nonlinear scienti c research because the synchronization of multisystem networks can better describe many observed natural phenomena and can create ordered multisystem networks.As a kind of control, coupling is the key to ensure the synchronization of complex dynamical networks; hence, several coupling schemes are proposed to realize the synchronization.However, most of the existing works require that the coupling matrix is always the Laplace matrix; that is, the row sums of the coupling matrix all are zero and the nondiagonal elements are nonnegative, which greatly restrict us to design a good performance coupling controller.
In fact, we notice that there are some non-Laplacian coupling in complex dynamical networks, so we can try to achieve the synchronization of multisystem networks by constructing non-Laplacian coupling matrix.In addition, in recent decades, NNs have become a hot research topic because of its rich content and wide application; therefore, there are many results in the research of NNs, such as in [1,[5][6][7][8][9][10][11][12].e authors in [1] proposed a new ETS to achieve the synchronization of multiple NNs (MNNs).
e dynamical characteristics of nonautonomous fractional-order delay NNs were studied in [5].Wu and Zeng [9] derived two anti-synchronization algorithms to realize exponential antisynchronization of memristive recurrent NNs.In [10], some results were established to ensure the Lagrange stability of NNs with memristive synapses.Wu and Zeng [11] discussed a class of memristive NNs, and several su cient criteria were established for exponential stabilization by using matrix inequality techniques.Wu et al. [12] proposed a novel and e cacious method to study the periodic NNs, and some new results for the periodic NNs were obtained.erefore, it is an interesting and far-reaching research topic to study the synchronization of MNNs by constructing a proper non-Laplacian coupling matrix.
e uncertainty of parameters must be considered in actual system because the parameters of the model in the process of industrial control are often uncertain.It is fortunate that uncertain parameters have been considered in many models in order to describe practical problems more accurately.In [13], the dynamical characteristics of stochastic nonlinear systems with parametric uncertainty were concerned.Maharajan et al. [14] investigated the problem of enhanced results on robust finite-time passivity for uncertain discretetime Markovian jumping BAM delayed NNs. e authors in [17] provided several novel delay-dependent stability criteria to ensure robust stability of uncertain stochastic systems.Huang et al. [20] investigated robust state estimation of uncertain neural networks by designing robust state estimator and using new bounding technique.Zhu et al. [22] discussed the stability of uncertain neutral systems, and some new criteria were provided to guarantee the stability of the model.Moreover, fractional-order system is a charming research field, which describes the real world more accurately than integer-order system.Recently, Zhang [21] studied uncertain fractional-order system and its application.
e uncertain parameters of the above models are required to be normbounded; it is a common way to deal with these uncertain parameters by using inequality techniques.With the mature of the technology, some problems in MNNs have been well solved; however, it is still a great challenge to design a feedback controller with good performance when we analyze multisystem networks.ere are many papers about the topic of parametric uncertainty, but the issue about trusted control in the MNNs with uncertain parameters still has a long way to go and much further work is worth studying.In addition, time delay is often inevitable in practical systems, which may lead to instability of the system; in other words, time delay is the important reason for instability of the model.Although parameter uncertainty and time delay bring difficulties to the theoretical analysis of dynamical characteristics of system, in practical application, the analysis of model is often unreasonable without considering uncertain parameter and time delay.Hence, theoretical knowledge and practical experience urge us to study the dynamics problems of MNNs with parameter uncertainty and time delay.
Event-triggered mechanism is a very important sampling mechanism, that is, the emergence of certain events rely on the state of the system and this is also the difference between event-triggered feedback control and the traditional sampling mechanism.Because continuous-time control requires the continuous information, it is expensive and unrealistic in the real world.Unlike continuous-time control, event-triggered mechanism is a discontinuous-time control, which only requires the local communication data.Considering the sampling period, the sampling data control scheme usually adopts zero-order hold to keep the last sampling system state and control signal which is sent to the next event.In eventtriggered mechanism, the sampling will not start and the controller will not be updated, unless its size reaches the specified threshold; therefore, the greatest advantage of eventtriggered mechanism is to reduce communication data.Furthermore, event-triggered mechanism has been widely studied due to its effectiveness in practical systems, for example, [23][24][25][26] and references therein.Dolk and Heemels [23] investigated a networked control system subject to eventtriggered control and its application to packet losses.In [24], a new decentralized event-triggered item for distributed networked systems was proposed to reduce the waste of network resources.e event-triggered rule was proposed to solve the problem of excessive use of communication resources in [25].
e authors in [26] studied neural networks by the proposed event-triggered rule, and the rule effectively solved a large number of computational problems.It is not difficult to find that event-triggered mechanism is very useful and meaningful sampling mechanism because many practical problems can be solved via ETS, but there are few research results on the combination of event-triggered mechanism and MNNs.
Based on the above discussion, the synchronization of MNNs with parameter uncertainty and time delay is studied and emphasized.By establishing the event-triggered mechanism, this paper derives several sufficient criteria to guarantee the synchronization of the systems; it shows that these results are different from the previous ones.Furthermore, the lower bounds of sampling time intervals are also obtained; that is, if the triggering time t k is known, we can predict the next triggering time t k+1 .Roughly speaking, this paper has three highlights: (1) Most existing works on multisystem networks require that the coupling matrix is always the Laplace matrix, in which row sums of coupling matrix all are zero and nondiagonal elements are nonnegative.Because of the particularity of Laplacian matrix, the design of coupling controller is greatly limited.In fact, we have found some non-Laplacian coupling in coupled dynamic networks.In this paper, the coupling matrix need not be the Laplace matrix; that is to say, row sums of coupling matrix can be nonzero constant and nondiagonal elements are arbitrary.(2) Some criteria are proposed to ensure the synchronization of MNNs with parameter uncertainty and time delay by establishing event-triggered mechanism and using matrix inequality techniques.Compared with the most of the existing papers, the results obtained are more simple and convenient in this paper.(3) We can find the lower bounds of the sampling time intervals by the established event-triggered mechanism, and then, we can know that the Zeno behavior will not happen.
e remaining part of this paper is designed as follows.Section 2 introduces the model of MNNs with parameter uncertainty and time delay and some preliminary knowledge.Section 3 gives several sufficient criteria to achieve the global robust exponential synchronization, and the lower bounds of the sampling time intervals are also found.Section 4 provides three numerical examples to demonstrate the obtained results.In the end, Section 5 summarizes the paper's relevant conclusion. ; furthermore, A T stands for its transpositive, and A > 0, A < 0, A ≥ 0, and A ≤ 0 mean that A are positive definite, negative definite, positive semidefinite, and negative semidefinite, respectively.diag • { } is representative of diagonal matrix.ϕ refers to empty set.If no otherwise specified, matrices are supposed to have compatible dimensions.

Preliminaries and Problem Formulation
Consider a group of r NNs with parameter uncertainty and time delay, and the model of kth NNs with parameter uncertainty and time delay is given as follows: dz where is the real diagonal positive-definite matrix standing for the neuron self-inhibitions; J(t) ∈ R n is the input or bias; τ > 0 represents the transmission delay; for k ∈ I and s ≥ 1, where t ks   refers to the sampling time sequence, λ > 0 represents coupling gain, and K � (y km ) r×r stands for the coupling matrix such that  r m�1 y km � 9 ∈ R. ( Next, we define the following distributed event-triggered function: and the sampling time sequence t ks   satisfies ETS: for k ∈ I and s ≥ 1, where α k > 0 refers to the control parameter and e initial conditions of model ( 6) are assumed to be where Let z(t, t 0 , φ) stands for the solution of model ( 6); in order to avoid this paper being too long, we assume that there exists a unique solution of system (6).
System ( 6) is called to be globally robustly exponentially synchronized, namely, each subsystem in ( 6) can achieve global robust exponential synchronization if there are constants M ≥ 1 and h > 0 satisfying for any t ≥ 0 and k, m ∈ I, where h stands for convergence rate.
For the sake of discussion, we give two basic assumptions.
Assumption 1 where M i and N i (i � 1, 2, 3) are constant matrices with proper dimensionality and R i (t) are the unknown matrix with Assumption 2. For any u, v ∈ R and k ∈ I, the activation functions f k (u) and g k (u) are all continuous in R and satisfy where p k > 0 and q k > 0 represent Lipschitz constants.Let Complexity 3 Remark 2. Assumption 1 guarantees the boundedness of ΔB, ΔC, and Δ D. Assumption 2 ensures the speciality of the activation functions f k (u) and g k (u).e Lipschitz constants p k > 0 and q k > 0 are dependent on the activation functions f k (u) and g k (u) rather than fixed.Lipschitz continuity is a smoother condition than uniform continuity; intuitively, Lipschitz continuity restricts the speed of function change.Furthermore, the slope of the function satisfying Lipschitz condition must be less than a real number called Lipschitz constant.In differential equation theory, Lipschitz condition is a core condition in the existence and uniqueness theorem of solutions under initial conditions.Next, two important lemmas are introduced as follows.
Lemma 1 (see [19]).Let F, G, and S(t) be real matrices and η > 0 is a constant; if S T (t)S(t) ≤ I, then Lemma 2 (see [18]).Let F, G, P, and Q(t) be real matrices; if For scalar ξ > 0 and vectors z 1 and z 2 of proper dimensionality (2) For matrix P > 0 of suitable dimensions:

Main Results
Let ε km (t) � z k (t) − z m (t) be the synchronization error and en, by establishing event-triggered mechanism and using matrix inequality techniques, we obtain several novel conditions to achieve global robust exponential synchronization of system (6), and the lower bounds of sampling time interval are also found.

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Remark 3. By eorem 1, we can know α k > 0 and I k ≠ ϕ implying that the Zeno behavior does not happen for sampling time sequence t ks  .In addition, if α k > 0 and I k ≠ ϕ, there is no Zeno behavior for model (6) under ETS (5).
where 8 Complexity en, one gets By eorem 1, we obtain  60) and (61), we have From (56), we can know that there is a h > 0 such that Similar to eorem 2, by (63), we get then us, model ( 6) is globally robustly exponentially synchronized with convergence rate h under ETS ( 5). e proof is finished.
Remark 5. From eorems 2-5, it is easy to find that they are closely related.In eorems 2 and 3, the coupling matrix can be unstable.However, in eorems 4 and 5, the coupling matrix must be stable.eorems 4 and 5 can be derived from eorems 2 and 3, respectively, when the coupling matrix is stable, but eorems 3 and 4 are more simpler and convenient than eorems 2 and 3.
10 Complexity Remark 6.Most existing works on multisystem networks (see [2][3][4]) require that the coupling matrix is always the Laplace matrix, in which row sums of coupling matrix all are zero and nondiagonal elements are nonnegative.In this paper, the coupling matrix need not be the Laplace matrix; that is to say, row sums of coupling matrix can be nonzero constant and nondiagonal elements are arbitrary.Compared with the previous works, the advantage of this paper is that the coupling matrix need not be the Laplace matrix, and the criteria obtained are more simple and convenient.
Remark 7. Actually, if we consider MNNs with non-Laplace coupling matrix and norm-unbounded uncertainty terms in this paper, we can get eorem 1 by changing ( 17)-( 22) and get eorems 2 and 3 by changing (36) and (37).Similarly, we also get eorems 4 and 5 by changing (72) and (74).In addition, if we consider time-varying delay instead of constant delay into system (6), when the time-varying delay τ ks (t) is bounded, namely, there exists a constant σ satisfying 0 < τ ks (t) < σ, and several similar conclusions can be obtained.
Remark 8.It can be seen that the row sums of coupling matrices in Examples 1, 2, and 3 all are 0.01, − 1, and − 2, respectively, and there are nondiagonal elements in Examples 2 and 3. us, our coupling matrices all are non-Laplacian.

Concluding Remarks
is paper has introduced a novel ETS for MNNs with parameter uncertainty and time delay.By using matrix inequality techniques and establishing the event-triggered mechanism, some su cient criteria are derived for judging global robust exponential synchronization of the system; in addition, the lower bounds of the sampling time intervals are also obtained.In particular, the most existing papers on multisystem networks require that the coupling matrix is Laplace matrix, in which row sums of coupling matrix all are zero and nondiagonal elements are nonnegative.However, the coupling matrix need not be the Laplace matrix in this paper; that is to say, row sums of coupling matrix can be nonzero constant and nondiagonal elements are arbitrary.In

Figure 5 :Figure 6 :
Figure 5: Time evolutions of the event-triggered controller and the synchronization errors in Example 3.
Complexity 15 roughout the paper, R n represents the n-dimensional Euclidean space.I stands for an identity matrix with proper dimensionality.LetC τ ≔ C([− τ, 0], R n ) be aBanach space that composed of all the continuous functions φ : [− τ, 0] to 2 Complexity R n .‖•‖ 2 and ‖•‖ inf stand for 2-norm and inf-norm, respectively, and let us write ‖•‖ 2 in terms of ‖•‖ for the sake of simplicity.For a matrix A � (a ij ) n×n , a max � max |a ij |  , and a min � min |a ij | the neuron activation functions; B � (b km ) n×n is the weight matrix and C � (c km ) n×n is the delay weight matrix; ΔB, ΔC, and Δ D are the norm-bounded uncertainty terms; and u k (t) ∈ R n denotes the control input.
ks )‖ be the sth threshold on the kth neuron.It is not difficult to find that Φ km determines sampling time of the kth neuron, and Φ km is closely related to the control parameter α k and subset I k .In particular, if α k � 0 or I k � ϕ, the kth neuron will have sampling time at any t ≥ t 0 ; that is to say, the Zeno behavior does happen when α k � 0 or I k � ϕ.It should be noted that the transmission event is triggered and the controller is updated when the measurement error ‖z k (t) − z k (t ks )‖ exceeds the threshold Φ km in the event-triggered controller; unlike the event-triggered controller, the impulse controller samples on the determined impulse time sequence t ks  .us,we can know that the sampling time sequence t ks   satisfies t k(s+1) � inf t > t ks , H ks > 0   in the event-triggered controller, and the sampling time sequence t ks   is known in the impulse controller.en,we can rewritte model (1) as follows: dz r k stands for the number of elements in I k , and r max � max 1≤k≤r r k  .I\ l { }, t ks ⟶ ∞ as s ⟶ ∞, then, there exists a s ≥ 1 satisfying t ∈ [t ks , t k(s+1) ): Figure 1: Time evolutions of the event-triggered controller and the synchronization errors in Example 1.
Figure 3: Time evolutions of the event-triggered controller and the synchronization errors in Example 2.