Boundary Controlling Synchronization and Passivity Analysis for Multi-Variable Discrete Stochastic Inertial Neural Networks

: The current paper considers discrete stochastic inertial neural networks (SINNs) with reaction diffusions. Firstly, we give the difference form of SINNs with reaction diffusions. Secondly, stochastic synchronization and passivity-based control frames of discrete time and space SINNs are newly formulated. Thirdly, by designing a boundary controller and constructing a Lyapunov-Krasovskii functional, we address decision theorems for stochastic synchronization and passivity-based control for the aforementioned discrete SINNs. Finally, to illustrate our main results, a numerical illustration is provided.


Introduction
Neural networks (NNs) can be considered as complicated nonlinear models coupled with numerous internal nodes, and they are capable of offering an effective approach to solving many difficult tasks in the fields of engineering.Due to their huge potential in real-world applications, they have become a significant research topic over the last few decades and have garnered increasing interest in many areas of technology (please refer to refs.[1][2][3][4][5][6][7]).On the other hand, it is necessary to address practical problems by studying the dynamic properties of non-linear neural networks not only in the over-damped case but also under weakly damped conditions [8].Hence, inertial neural networks (INNs), which can act as second-order differential systems, have been extensively studied.Additionally, numerous publications have addressed synchronization problems, including finite-time synchronization [9], nonfragile H ∞ synchronization [10], event-triggered impulsive synchronization [11], fuzzy synchronization [12], Mittag-Leffler synchronization [13], and others.
Passivity, as a specific form of dissipativity, constitutes a fundamental characteristic of physical problems.A system is considered passive when dissipative elements are present in the modeled system, and the accumulated energies remain lower than the external input over a certain time span.Consequently, passivity ensures internal stability of the systems.Due to its widespread applicability in mechanical and electrical systems, the concept of passivity has garnered increasing attention, leading to extensive studies on the passivity of nonlinear systems.In the literature [14], Zhou et al. discussed passivitybased boundary control for stochastic delay reaction-diffusion systems with boundary input-output.Padmaja and Balasubramaniam [15] analyzed passivity-based stability in fractional-order delayed gene regulatory networks.By leveraging Lyapunov-Krasovskii functionals, novel linear matrix inequality conditions were developed to guarantee certain levels of passivity performance in the networks.For further details on this topic, please consult the references [16][17][18].
Widely, NNs were implemented through IC in engineering applications; spatial diffusions invariably occur when electronic motion takes place in an inhomogeneous electromagnetic domain.Therefore, it is important to consider NNs that incorporate the impact of spatial diffusions.In recent years, greater attention has been devoted to NNs with spatial diffusions; please refer to papers [19][20][21][22][23][24].Stochastic neural networks have received substantial attention in our everyday reality.Typically, actions of random networks are heavily time-and space-dependent.As a result, reaction diffusion must be taken into account.Relevant research topics are discussed in references [14,19,20,22,25,26], etc.While there have been reports on space-time discrete models [27][28][29] to date, the problems of synchronization and passivity-based control for discrete-time SINNs involving diffusions have not been explored.
It is well known that discrete systems,(DSs) can be utilized to simulate a wide range of phenomena, including biological dynamics and artificial NNs, among others.In many scenarios, it has been demonstrated that DSs outperform continuous systems.As a result, the theory of DSs holds significant importance; please refer to references [30][31][32][33][34][35][36][37][38].Reports [35][36][37][38] have explored various types of discrete INNs.However, they have not focused on the effects of other variables, such as spatial variables.Addressing this gap, the present paper investigates the issues of stochastic synchronization and passivity-based control for time and space discrete SINNs by designing a novel boundary controller.
Our main contributions include the following: (1) Establishment of a discrete space and time SINNs model, which complements the continuous cases in literature [22][23][24] and the discrete-time cases in literature [35][36][37][38].(2) Unlike prior works in the literature [22][23][24], a controller is formulated at the boundary to achieve synchronization and passivity-based control of discrete space and time SINNs.
In what follows, Section 2 establishes the discrete space and time SINNs based on prior works in the literature [27,29].Section 3 discusses synchronization and passivitybased control of the discrete SINNs.In Section 4, in order to illustrate our main results, a numerical illustration is provided.Finally, the conclusions and perspectives are described in Section 5.
Hereon, INNs Equation ( 1) can be regarded as slaver networks and the isolated node w ∈ R n satisfies the master networks below The initial condition of INNs Equation ( 5) is described as where φ • 0 and φ • 0 are ℱ 0 -adaptive and ℱ 1 -adaptive, respectively.Let u i = z i − w, then the error networks of INNs Equations ( 1) and ( 5) are described by where f (u i With the help of Equations ( 3) and ( 6), the initial condition for INNs in Equation ( 7) can be derived, as depicted by To study INNs Equation (1) effectively, let where ε > 0 is a controlling parameter, which can be adjusted freely, i = 1, 2, . . ., N.Then, the first equation in INNs Equation ( 7) is changed into The vector forms of INNs Equations ( 9) and ( 10) are written as where I N denotes the N-order identity matrix.Hereby, (A) ⊗ := I N ⊗ A and (A) ⊗B := B ⊗ A.
In accordance with Equations ( 8) and ( 9), the initial condition of INNs Equation ( 11) is expressed by where ι ∈ [0, ] Z , i = 1, 2, . . ., N, ψ The current discussion will establish a boundary controller to synchronize and passivitybased control the master INNs Equations ( 5) and slave INNs (1), which will be demonstrated in Section 3.
Hereon, we need the following assumption for activation functions.
(F) L f and L g are n-order matrices ensuring
Using Lemma 3, we get where P is defined as in Lemma 3.

Stochastic Synchronization and Passivity-Based Control
The slave INNs Equation ( 1) is said to be stochastically synchronized with the master INNs Equation (5) if the error vector networks Equation (11) achieves globally asymptotically stability in mean square, i.e.,

Stochastic Synchronization
Define where Θ ∈ R n×n .Set D := I−e where

K.
Proof.Let us define a Lyapunov-Krasovskii function, which is described by In the line with the first segment of the error networks Equation ( 11), we can derive According to the second equation of networks Equation ( 11), we get where k ∈ Z 0 .
According to Lemmas 1-3 and boundary conditions in Equation ( 11), we calculate With the help of (F), we have u,k , (24) and by using ê u,• and Lemma 4, it gets Considering Equation ( 20), we have for all k ∈ Z 0 .
From Lemma 4, the following inequality is valid: where k ∈ Z 0 .Further, So, we have the following: Corollary 1. Assuming that (F) is valid, we pre-give values of ε > 0 and β ∈ [0, 1].Additionally, we assume that D and M ε are nonsingular, and we define Θ as indicated in Theorem 1.Under these conditions, the slave INNs Equation (1) stochastically synchronize with the master INNs Equation (5), meaning that the model Equation ( 11) achieves global mean-squared asymptotic stability.This holds true if the model has positive constants λ f , λ g , and positive definite n-order matrices P, Q, H, and K such that the Õ matrix defined in Equation ( 31) is negative definite.
Remark 1. Reports [22,24] addressed the issues of synchronization for inertial neural networks with reaction-diffusion terms.However, the networks in reports [22,24] were involved in the Dirichlet boundary condition and the controller is embedded in the model of the networks.In this article, the controller does not exist in the model of the networks, but it is designed in the boundary.

Passivity-Based Control
The error vector networks described by Equation (11) with respect to a supply rate can be represented as This system is stochastically passive if there exists a nonnegative mapping θ that satisfies Theorem 2. Let Hypothesis (F) be satisfied, ε > 0 be given, and D, M ε be nonsingular.Additionally, let the controller gain Θ be as provided in Theorem 1.The error networks Equation ( 11) are stochastically passive if there exist positive constants λ f , λ g and n-order positive definite matrices P, Q, H, K, 1 , 2 , 3 such that where and the other unmentioned block matrices O ij in O are equal to O ij in O for i, j = 1, 2, . . ., 6.
Proof.Define the Lyapunov-Krasovskii function V for the error vector networks Equation (11), following the approach described in Section 3.1.Additionally, introduce an output vector Y ∈ R Nn to the error vector networks Equation ( 11) using the expression Y = (I N ⊗ 1 )e u + (I N ⊗ 2 )e v + (I N ⊗ 3 )γ.
Similar to the argument in Equation ( 17), we get Meanwhile, similar to the estimates in inequalities Equations ( 18)-( 23), we obtain from Equation (33) the following: ∀k ∈ Z 0 .By employing Equations ( 16)-( 26) and ( 33) and ( 34), we can compute where η [ι] for k ∈ Z 0 , ι = 1, 2, . . ., .In accordance with Equation (35), we get Accordingly, INNs Equation ( 11) is stochastic passive.This completes the proof.So, we have the following: Corollary 2. Assuming that (F) is satisfied, ε > 0 and β ∈ [0, 1] are pre-given, D and M ε are nonsingular, and the controller gain Θ is provided in Theorem 1, the error network Equation ( 11) is stochastically passive if there exist positive constants λ f , λ g , and n-order positive definite matrices P, Q, H, K, 1 , 2 , and 3 such that Õ < 0. Here, Õ = ( Õij)1 ≤ i, j ≤ 7 is defined as O in Theorem 2 except for the following modifications: According to Theorems 1 and 2, a realizable algorithm for stochastic synchronization or passivity of INNs Equations ( 1) and ( 5) is designed as Algorithm 1, and its O-chart is described in Figure 1.Remark 2. Papers [44,45] investigated the passivity of inertial neural networks without reactiondiffusion terms.This paper considers the effects of the reaction diffusions, which complements the works in the literature [44,45].
(5)uming that (F) is valid, and ε > 0 is given in advance, D and M ε are nonsingular.The slaver INNs Equation (1) stochastically synchronizes with the master INNs Equation(5); in other words, model Equation (11) is globally mean-squared asymptotically stable if it has positive constants λ f , λ g and n-order matrices P −D• h D • .Theorem 1.