Passivity and Synchronization of Coupled Different Dimensional Delayed Reaction-Diffusion Neural Networks with Dirichlet Boundary Conditions

Two types of coupled dierent dimensional delayed reaction-diusion neural network (CDDDRDNN) models without and with parametric uncertainties are analyzed in this paper. On the one hand, passivity and synchronization of the raised network model with certain parameters are studied through exploiting some inequality techniques and Lyapunov stability theory, and some adequate conditions are established. On the other hand, the problems of robust passivity and robust synchronization of CDDDRDNNs with parameter uncertainties are solved. Finally, two numerical examples are given to testify the eectiveness of the derived passivity and synchronization conditions.


Introduction
In recent years, complex networks (CNs) have attracted much attention since they are ubiquitous under the circumstance of our daily life, for instance, communication networks, food webs, and social networks. Coupled neural networks (CNNs), as a particular type of CNs, have been put into use successfully in various elds, e.g., pattern recognition, chaos generators design, and brain science [1][2][3]. Strictly speaking, these applications in a large extent are depending on some properties of dynamics in CNNs (e.g., synchronization). us, the problem of synchronization for CNNs has attracted comprehensive attention and been developed into a hot research topic. So far, many important and interesting results have been derived on this topic recently [4][5][6][7][8][9][10]. rough exploiting the properties of random variables, Yang et al. [4] acquired some synchronization conditions for randomly delayed CNNs. In [7], several synchronization criteria were gained for delayed CNNs in accordance with the Lyapunov functional strategy.
Nevertheless, most of the synchronization results in aforesaid literatures [4][5][6][7][8][9][10] neglected reaction-di usion phenomena. In fact, the reaction-di usion phenomena are unavoidable for CNs such as neural networks and cellular networks when they are implemented by means of electric circuits in practical situations [11]. Hence, it is necessary to research coupled reactiondi usion neural networks (CRDNNs). Recently, some signicant synchronization results for CRDNNs have been acquired [12][13][14][15][16][17]. In [12], the authors researched synchronization of CRDNNs and presented some synchronization criteria. By designing appropriate pinning controllers, several synchronization criteria were established for CRDNNs in [17]. Furthermore, owing to the existence of external interferences, the noises of environment, and equipment restrictions, it is very di cult to ensure network models containing the certain parameter values in some practice situations. Consequently, some authors studied parametric uncertainties of neural networks [18][19][20][21][22], and a few interesting results have been derived regarding robust synchronization of CNNs with uncertain parameters [23][24][25]. In [24], the scholars presented the memristive CNNs model with parametric uncertainties and obtained some adequate conditions for guaranteeing the robust synchronization for the considered network. By utilizing impulsive functional strategy combined with the stability theory, Li et al. [25] presented some robust synchronization criteria for the CNNs with uncertainties.
Actually, passivity is also one of the most important behaviors of dynamics for CNs which can guarantee a system's internal stabilization in system theory. Due to the potential applications in plenty of fields, e.g., fuzzy control and sliding mode control, the passivity problem of CNNs has been investigated extensively in the past years [26][27][28][29][30][31][32][33]. In [27], the authors proposed several passivity criteria for delayed CNNs. Ren et al. [28] analyzed the model of CRDNNs and established some passivity and pinning passivity conditions for the considered network. Unfortunately, the passivity results in aforementioned studies [26][27][28][29][30][31][32][33] are based on the case that the dimension of input is identical with output. As far as we know, only few scholars have addressed the problem of passivity for the network with nonidentical dimensional output and input [34][35][36]. e authors in [34] established some conditions for ensuring that the CRDNNs with the input and output in different dimensions achieve passivity. Ren et al. [36] discussed the (pinning) passivity problems of CNNs with nonidentical dimensional output and input and acquired some corresponding passivity criteria.
Note that the networks are composed of identical nodes in the aforementioned works . Unfortunately, this case is very rare in the real-world networks. Consequently, the CNs consisting of nonidentical nodes in the same dimension [37][38][39][40][41] have been discussed firstly by researchers. Zhao et al. [39] dealt with the synchronization problem of CNs with nonidentical nodes. As a matter of fact, the networks constructed by nonidentical nodes of different dimensions can reflect more real networks in many circumstances. Note that the networks constructed by nonidentical nodes in different dimensions can exhibit different and even more sophisticated dynamical behaviors, which makes the passivity and synchronization methods for the networks with the same dimensional nonidentical nodes or identical nodes in the abovementioned works  invalid. Hence, it is necessary and meaningful to develop some new stabilization and synchronization strategies for the network with nonidentical nodes of different dimensions [42][43][44][45]. In [43], by designing appropriate decentralized controllers, the authors devoted to establishing stabilization and synchronization criteria for CNs consisting of nonidentical nodes. Up to now, only a few researchers considered CNNs constructed by the nonidentical nodes of different dimensions [46][47][48]. In [47], the authors investigated generalized synchronization of delayed CNNs with different dimensional nodes by making use of the Lyapunov functional method. To the best of knowledge, the problems of synchronization and passivity for CDDDRDNNs have not yet been investigated. Consequently, it is essential to put forth some efforts to study passivity and synchronization of CDDDRDNNs.
In terms of the above introduction, the main aim in this paper is to address passivity and synchronization of CDDDRDNNs without and with parametric uncertainties. On the one hand, we discuss the CDDDRDNNs without parametric uncertainties, and several conditions are derived to guarantee the considered network to achieve passivity and synchronization. On the other hand, the problems of robust synchronization and robust passivity for CDDDRDNNs with parametric uncertainties are also studied.

Preliminaries
Let the matrix G ∈ R n×n , the notation G < 0 (G > 0, G ≤ 0, G ≥ 0) signifies G is symmetric and negative (positive, seminegative, and semipositive) definite. λ m (·)(λ M (·)) represents the minimum (maximum) eigenvalue of the corresponding matrix. An open-bounded domain in R 9 with smooth boundary zΩ is defined by Definition 1 (see [49]). Let u(m, t) ∈ R p and y(m, t) ∈ R q denote the input and output of a system. Assume that there exists a storage function S : for any t 0 , t l ∈ [0, +∞) and t 0 ≤ t l , then the system with supply rate Π(u, y) is dissipative. Moreover, a system is passive if the system is dissipative with where the matrix Q ∈ R q×p . Furthermore, assume that a system is dissipative with > 0, and Q ∈ R q×p , then the system is strictly passive. Especially, if M 1 > 0, then the system is called to be output-strictly passive; if M 2 > 0, then the system is called to be input-strictly passive.

Passivity and Synchronization of CDDDRDNNs
e CDDDRDNNs considered in this section is stated as follows: , and φ (i) l (·)(l � 1, 2, · · · , ξ i ) denote the activation functions for the lth neuron in neural network i; J i � (J (i) 1 , J (i) 2 , · · · , J (i) ξ i ) T ; u i (m, t) ∈ R p i denotes the control input; the inner coupling matrix is defined by H ij ∈ R ξ i ×ξ j ; G i ∈ R ξ i ×p i is a known matrix; C � (c ij ) N×N is the coupling configuration matrix denoting coupling weight, which satisfies c ij ≠ 0(i ≠ j) if node i and node j are connected, or else c ij � 0. In addition, the time-varying delay τ i (t) satisfies 0 ⩽ τ i (t) ⩽ τ i and _ τ i (t) ⩽ δ i < 1.

Remark 1.
As a special type of CNs, CNNs have attracted much attention due to their extensive applications on plenty of fields. So far, a large number of scholars have acquired some interesting research results about synchronization and passivity of CNNs . Nevertheless, the considered CNs in the aforementioned literature are made up of identical nodes. As a matter of fact, it is utterly impractical that the networks have totally identical nodes in many practical situations. For instance, due to the differences of the parameters, it is impossible that the neurons in the nervous system of neural networks are entirely the same as each other. Consequently, it is significant to study CNNs composed of nonidentical nodes. To our knowledge, a few investigators have discussed CNs consisting of the same dimensional nonidentical nodes in recent years [37][38][39][40][41]. However, the networks with nonidentical nodes of different dimensions can describe more practical networks. In addition, the considered networks in these existing works did not take the reaction-diffusion terms into account. erefore, we pay our attention on the CRDNNs with different dimensional nodes. As far as we know, this is our first step toward addressing the passivity and synchronization problems of CDDDRDNNs. For network (6), the initial value condition and boundary value condition are described by where i � 1, 2, · · · , N, ϑ i (m, t) is bounded and continuous on Ω × [− ρ, 0] and ρ � max i�1,2,···,N τ i .
Let the constant vector O � (o T 1 , o T 2 , · · · , o T N ) T be an equilibrium point of an isolated node of network (6). en, (11) is given as follows: where K (i) 1 ∈ R q i ×ξ i and K (i) 2 ∈ R q i ×p i are the known matrices. In the whole paper, we denote 3.2. Passivity Analysis. It follows from (11) and (12) that the following system with output y i (m, t) ∈ R q i and input u i (m, t) ∈ R p i can be obtained as follows: where H ij � c ij H ij and i � 1, 2, · · · , N.

Remark 2.
To the best of our knowledge, the concept of passivity is proposed for the first time in circuit analysis and has discovered comprehensive potential applications in lots of areas after that. Over the past few decades, some scholars have investigated the passivity of CNNs and CRDNNs and derived many meaningful results [26][27][28][29][30][31][32][33]. Note that the passivity problem is solved in the abovementioned works based on the case that the input has identical dimension as Complexity 7 the output. As far as we know, only a few passivity results have been obtained for the CRDNNs with different dimensional input and output until now [34][35][36]. However, the networks considered in these works are coupled by identical nodes. erefore, we devote ourselves to studying the passivity of CRDNNs consisting of different dimensional nodes (i.e., CDDDRDNNs), in which the input has different dimensions as output. Hence, the acquired passivity results are more general and less conservative in this paper. (6) achieves synchronization if for all i � 1, 2, · · · , N,
Proof. Choose the same Lyapunov functional as (17) for system (11), then, where From (17) and (40), one knows that all terms of V(t) are bounded and V(t) is nonincreasing. Hence, V(t) converges to a finite nonnegative real number. According to (41), we can infer that lim t⟶+∞ t 0 ‖z(·, t)‖ 2 2 dt exists and is finite.

Network Model.
Taking the parameter uncertainties into account, a CDDDRDNN with uncertain parameters is stated as follows: ), H ij , and c ij represent the same senses as in model (6). e parameters D i , B i , A i , and Z i can be changed within a certain parameter range of precisions as follows: In addition, for convenience, we denote where g � 1, 2, · · · , ξ i and h � 1, 2, · · · , ξ i , i � 1, 2, · · · , N. According to (10), we can get the error system z i (m, t) of network (49) as follows: where the ranges of D i , B i , A i , and Z i are described by using (50).

Robust Passivity Analysis.
e output vector y i (m, t) of system (52) is defined similarly as (12). In terms of (12) and (52), the error system with output y i (m, t) ∈ R q i and input u i (m, t) ∈ R p i can be described by Complexity where H ij � c ij H ij and i � 1, 2, · · · , N.

Theorem 3. If there are matrices
where

then, system (53) under the given ranges of parameters (50) is output-robustly passive.
Proof. Choose the same Lyapunov functional as (17) for system (52). en, Obviously, Similar to the deduction of (25), one can get In addition, 10 Complexity According to (56)-(59), one gets Hence, From (54), one can get for any t 0 , t l ∈ [0, +∞) and t l ≥ t 0 . In other words, in which S(t) � V(t)/2. e following results can be deduced by using the similar method.

Corollary 3. System (53) under the given ranges of parameters (50) is robustly passive if there are matrices
where (49) under the given ranges of parameter (50) realizes robust synchronization if for all D i ∈ D (P) , B i ∈ B (P) , A i ∈ A (P) , and Z i ∈ Z (P) , i � 1, 2, · · · , N,

Robust Synchronization Analysis. Definition 3. Network
under the condition u i (m, t) � 0. where Proof. Select the same Lyapunov functional as (17) for system (52); then, where en, lim t⟶+∞ ‖z(·, t)‖ 2 � 0 can be proved similarly as in eorem 2. Consequently, network (49) under the given ranges of parameters (50) realizes robust synchronization. □ Remark 3. In Section 3, we deal with the passivity and synchronization problems of CDDDRDNNs and establish some adequate conditions to achieve the passivity and synchronization of network (6). Note that the parameters in matrices D i , B i , A i , and Z i of network model (6) are fixed. However, it is utterly unrealistic for the networks with some certain parameters due to the noises of environment and equipment restrictions in some practical situations [18][19][20][21][22][23][24][25].
us, it is necessary to consider the case that the parameters in matrices D i , B i , A i , and Z i of network model (6) belong to some given ranges and investigate robust dynamical properties of the considered network. As far as we know, the robust synchronization and robust passivity of CDDDRDNNs with parametric uncertainties have never been studied. In this section, we present several robust synchronization and robust passivity criteria of CDDDRDNNs with uncertain parameters in eorems 3 and 4 and Corollaries 3 and 4, respectively.
Case 2. ψ il , ψ il , and O are the same as those in Case 1. By exploiting the MATLAB Toolbox, the matrices P satisfying (68) can be computed as follows: According to eorem 4, network (76) under the given ranges of parameters defined in (78) achieves robust synchronization. Figures 6-8 show the simulation results. Remark 4. Section 3 is devoting to investigating the synchronization and passivity of CDDDRDNNs. First, the network model of CDDDRDNNs is presented in Section 3.1. In Section 3.2, the passivity of CDDDRDNNs with certain parameters is studied. Moreover, we establish some adequate conditions to ensure the network being outputstrictly passive in eorem 1, passive in Corollary 1, and input-strictly passive in Corollary 2, respectively. en, in eorem 2 of Section 3.3, a synchronization criterion is obtained for the considered network. Because the precise values of parameters are difficult to acquire because of noises of environment and equipment limitations, we address the problems of robust synchronization and passivity for CDDDRDNNs with parametric uncertainties in Section 4. e main distinction on two kinds of CDDDRDNNs in Section 3 and Section 4 is whether D i , B i , A i , and Z i in the network model of CDDDRDNNs is uncertain or not. More precisely, in Section 4.1, we give the CDDDRDNN model with parametric uncertainties firstly. After that, a robust output-strict passivity criterion is proposed for CDDDRDNNs with parameter uncertainties in eorem 3 of Section 4.2. Meanwhile, we establish the related robust passivity condition in Corollary 3 and robust input-strict passivity condition in Corollary 4, respectively. en, a robust synchronization criterion is obtained for the considered network in eorem 4 of Section 4.3. In Section 5, we select the most intricate output-strict passivity condition for verifying the validity of the obtained passivity results. Actually, the related passivity or input-strict passivity condition can be illustrated similarly. In order to avoid repetition, we omit the simulation results for showing the effectiveness of the obtained passivity results in corollaries. erefore, in Case 1 and Case 2 of Example 1, the correctness of the output-strict passivity and synchronization conditions in eorems 1 and 2 for

Conclusion
e synchronization and passivity of CDDDRDNNs with and without parameter uncertainties have been investigated in this paper. First, several new criteria for CDDDRDNNs with parametric certainties have been derived to guarantee the passivity and synchronization by taking advantage of the Lyapunov functional method. Second, we have also studied the problems of robust synchronization and robust passivity for CDDDRDNNs with uncertain parameters. ird, two numerical examples have been provided to display the effectiveness of the obtained passivity and synchronization results. In our future work, it would be very interesting to study the pinning adaptive passivity and synchronization of CDDDRDNNs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.