Passivity Analysis of Coupled Stochastic Neural Networks with Multiweights

Tianjin Key Laboratory of Autonomous Intelligences Technology and System, School of Computer Science and Technology, Tiangong University, Tianjin 300387, China School of Mathematical Sciences, Tiangong University, Tianjin 300387, China Academy of Science and Technology, Tiangong University, Tianjin 300387, China School of Information Science and Technology, Linyi University, Linyi 276005, China School of Mechanical Engineering, Tiangong University, Tianjin 300387, China


Introduction
In recent decades, neural networks (NNs) have potential applications in the image encryption, pseudorandom number generators, optimization, and other areas [1][2][3], which depend on the dynamical behaviors of NNs including stability and passivity. erefore, the stability [4][5][6][7][8] and passivity [9][10][11][12][13][14] for various NNs have received special attention in recent years. Mou et al. [4] considered the asymptotic stability problem for Hopfield NNs with time delay via combining the Lyapunov functional and delay fractioning approach. Yang et al. [6] discussed the stability for a kind of NNs with time-varying delays and gave several delay-dependent stability conditions by taking advantage of integral inequality. In [9], a class of NNs with time-varying delays and parameter uncertainties was took into account, and some exponential passivity criteria were established by exploiting weighted integral inequalities. Xiao et al. [11] studied the passivity for a type of memristive NNs with inertial term, obtained some criteria of asymptotic stability by utilizing the passivity, and discussed the case that parameters are uncertain but bounded.
As it is known to all, stochastic perturbations are unavoidable in the implementation of NNs and may cause undesirable dynamical behaviors in NNs [15,16]. erefore, the dynamical behaviors including the stability [17][18][19][20][21] and passivity [22][23][24][25][26][27] have been widely investigated by numerous researchers for NNs with stochastic perturbations in recent years. In [18], several sufficient conditions on mean square stability for stochastic neural networks (SNNs) with local impulses were derived by using the mathematical induction method. Yang and Li [20] coped with the stability problem for switched SNNs with parameter uncertainties, and derived several conditions to guarantee the robust stability by utilizing the state-dependent switching method. In [24], the authors took into account one type of uncertain SNNs with distributed and discrete time-varying delays and gave some passivity criteria with the help of integral inequality technique. Nagamani et al. [25], respectively, discussed the passivity and dissipativity for Markovian jump stochastic NNs with two types of time-varying delays and obtained several delay-dependent passivity and dissipativity criteria by taking a suitable Lyapunov functional.
Coupled neural networks (CNNs) comprised of a number of NNs have tremendous potential applications in many areas of engineering [28][29][30]. Hence, the dynamical behaviors of CNNs have attracted much attention; especially, the passivity [31][32][33][34][35] and synchronization [36][37][38][39][40] for many types of CNNs have been deeply discussed. In [34], the authors not only obtained several passivity criteria for the directed CNNs based on the developed adaptive control strategies but also discussed the case that topologies are undirected. Qi et al. [37] derived several global exponential synchronization criteria for quaternion-valued CNNs with impulses by exploiting Lyapunov functional and matrix inequalities. In [38], the authors considered CNNs with mixed impulses and developed some exponential synchronization conditions for the proposed network model through using delayed impulsive differential inequality. Particularly, a number of authors have discussed the dynamical behavior of coupled SNNs (CSNNs) in recent years [41][42][43][44][45]. Chen et al. [44] utilized the adaptive feedback controller to deal with the exponential synchronization for CSNNs. In [45], the authors respectively employed timetriggered and event-triggered impulsive control methods to investigate the synchronization of discrete time CSNNs with multidelays. Unfortunately, the passivity of CSNNs has not yet been investigated.
In this paper, the passivity for two types of MWCSNNs with incompatible input and output dimensions is investigated. e main contributions have three aspects. First, we present several new definitions of passivity for stochastic systems with incompatible dimensions of output and input. Second, some sufficient conditions to ensure the passivity of MWCSNNs are obtained by taking advantage of the Lyapunov functional method and stochastic analysis techniques, and a synchronization criterion is also developed by utilizing the result of output-strictly passivity.
ird, we further address the passivity and synchronization for CSNNs with multiple delay couplings (CSNNMDCs).

Notations.
(Ω, F, F t t ≥ 0 , P) is a complete probability space with the natural filtration F t t ≥ 0 satisfying the usual conditions. C 1,2 (R + × R n ; R + ) represents the family of all nonnegative functions V(t, κ(t)) on R + × R n , which are once differentiable in t and twice continuously differentiable in κ(t). Tr(·) stands for the trace of a matrix. P ≥ 0 is used to denote a symmetric semipositive definite matrix. λ M (·) and λ m (·), respectively, denote the maximum and minimum eigenvalue of a real symmetric matrix.

Lemmas
Lemma 1 (Ito formula, see [49]). A stochastic system can be described by in which κ(t) ∈ R n represents the state of system, f(·): R + × R n ⟶ R n is continuous nonlinear function, g(·): R + × R n ⟶ R n×n is noise intensity function, and w(t) is an n-dimensional Brownian motion (Wiener process) defined on (Ω, F, F t t ≥ 0 , P).
Lemma 2 (see [50]). For any matrices M ∈ R m×n and 0 ≤ P ∈ R m×m , one obtains

Definitions
Definition 1. A stochastic system with input β(t) ∈ R p and output η(t) ∈ R q is passive if for any ϑ p , ϑ 0 ∈ R + and ϑ p ≥ ϑ 0 , in which F ∈ R q×p and S is a nonnegative function.
2 Discrete Dynamics in Nature and Society Definition 2. A stochastic system with input β(t) ∈ R p and output η(t) ∈ R q is input-strictly passive if for any ϑ p , ϑ 0 ∈ R + and ϑ p ≥ ϑ 0 , in which F ∈ R q×p , 0 < A 1 ∈ R p×p , and S is a nonnegative function.
for any ϑ p , ϑ 0 ∈ R + and ϑ p ≥ ϑ 0 , in which F ∈ R q×p , 0 < A 2 ∈ R q×q , and S is a nonnegative function.

Network Model.
e MWCSNNs in this paper is considered as follows: where n � 1, 2, . . . , s, G � (G zj ) m×m ∈ R m×m , G zj ∈ R, represents the strength of the j th neuron on the z th neuron, H ∈ R m×p is the known matrix, κ z (t) � (κ z1 (t), κ z2 (t), . . . , κ zm (t)) T ∈ R m is the state vector of the z th node, β z (t) ∈ R p represents the external input, 0 < D � diag(d 1 , d 2 , . . . , d m ) ∈ R m×m , 0 < d z ∈ R, represents the rate with which the z th neuron will reset its potential to the resting state when disconnected from the network and external input, . . , f m (κ zm (t))) T ∈ R m , 0 < b n ∈ R, denotes coupling strength, and C n � (C n zh ) M×M ∈ R M×M represents the outer coupling matrix, where C n zh satisfies the following conditions: if there exists a connection between nodes z and h, then In this paper, the following assumptions are made.

Remark 1.
On the one hand, the passivity for various CNNs has been investigated and some meaningful results have been obtained [31][32][33][34][35]. However, the passivity of CSNNs has not yet been discussed. On the other hand, some researchers have dealt with the synchronization problem for CSNNs [41][42][43][44][45]. Given that passivity has been developed as a powerful tool to solve the synchronization problem of CNNs, the investigation on synchronization for CSNNs based on the passivity is apparently very valuable. Regrettably, the result about this topic has not yet been reported.

Passivity Analysis.
Suppose that s(t) � (s 1 (t), s 2 (t), . . . , s m (t)) T ∈ R m is an arbitrary desired solution of the isolated node of system (8), then it satisfies Discrete Dynamics in Nature and Society Letting we can obtain from (8) and (11) that e output vector η z (t) ∈ R q of network (12) is defined as follows: where According to (12), we have where σ(t) � I M ⊗ (σ(κ(t)) − σ(s(t))). where Proof. For convenience, we denote then network (15) can be rewritten as Choose the following Lyapunov functional for network (15): 4 Discrete Dynamics in Nature and Society In light of Lemma 1, we can obtain Obviously, From Lemma 2 and Assumption 2, we have Furthermore, By (20)-(23), one obtains From (24), we have where From (26), we obtain According to (27) and Lemma 1, we can acquire □ Discrete Dynamics in Nature and Society where W 1 and E 1 have the same meanings as these in eorem 1 and Proof. Construct the same V 1 (t) as (19) for network (15), and we can easily obtain From (29), we have Similarly, we can derive Theorem 3. Network (15) is output-strictly passive if there exist matrices F ∈ R Mq×Mp , 0 < A 2 ∈ R Mq×Mq , and 0 < P ∈ R Mm×Mm satisfying where Proof. We select the same V 1 (t) as (19) for network (15), and we can easily obtain From (47), one has Similarly, we can derive 6 Discrete Dynamics in Nature and Society □

Synchronization in Passive MWCSNNs
Theorem 4. If network (15) is output-strictly passive with regard to storage function K(t) � V 1 (t)/2 and Z 1 ∈ R m×m is nonsingular, then MWCSNNs (8) can achieve synchronization.
Proof. If network (15) is output-strictly passive with respect to storage function K(t), then there exist matrices R Mm×Mm ∋ A 2 > 0 and F ∈ R Mm×Mm such that for any t ∈ R + and δ > 0. en, we can easily obtain By taking limit δ ⟶ 0 in (39), one has Letting β z (t) � 0(z � 1, 2, . . . , M), we can get from (40) that From (41) and definition of V 1 (t), we can get that lim t⟶+∞ E K(t) { } exists. erefore, we can conclude that Suppose that en, there obviously exists 0 < t * ∈ R such that By (41) and (44), we have From (45), we derive which results in a contradiction. Hence, lim t⟶+∞ E ‖α(t)‖ { } � 0. at is, network (8) realizes synchronization. e following conclusion can be obtained from eorems 3 and 4.

Corollary 1. Network (8) achieves synchronization if there exist matrices
where Discrete Dynamics in Nature and Society 7

Network Model.
e CSNNMDCs in this paper is considered as follows: where τ n (n � 1, 2, . . . , s) are coupling delays and κ z (t), β z (t), f(κ z (t)), D, G, B, H, b n , C n zh , and Γ n have the same meanings as those in Section 3.
In light of Lemma 1, we can obtain (57) From (57) and (58), we have (59) From (59), we have (60) Discrete Dynamics in Nature and Society By (53), we can acquire By employing similar proof methods in eorem 5, we can get the following conclusions.
where W 3 and E 3 have the same meanings as those in eorem 5 and where
e results can be easily obtained by employing similar proof method in the eorem 4. e following conclusion can be obtained from eorems 7 and 8. □ Corollary 2. Network (48) achieves synchronization if there exist matrices F ∈ R Mm×Mm , 0 < A 2 ∈ R Mm×Mm , 0 < P ∈ R Mm×Mm , and N n � diag(N 1 n , N 2 n , . . . , N M n ) ∈ R Mm×Mm , n � 1, 2, . . . , s, satisfying where Remark 2. In this paper, two types of network models are proposed (see (8) and (48)), some sufficient conditions for ensuring the passivity of networks (8) and (48) are acquired by employing the stochastic analysis techniques and Lyapunov functional method (see eorems 1-3 and eorems 5-7), and several synchronization criteria for networks (8) and (48) are established in view of the output-strictly passivity (see Corollaries 1 and 2).

Conclusion
Two kinds of MWCSNNs models have been proposed, in which the dimension of output is incompatible with input.
On the one hand, we have analyzed the passivity, input-strict passivity, and output-strict passivity for MWCSNNs by employing stochastic analysis techniques. Moreover, two synchronization criteria for MWCSNNs and CSNNMDCs have been derived on the basis of output-strict passivity. Finally, the correctness of the passivity and synchronization criteria has been verified through two numerical examples.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.