Finite-Time Passivity of Stochastic Coupled Complex Networks

The finite-time passivity problem is, respectively, investigated for stochastic coupled complex networks (SCCNs) with and without time-varying delay. Firstly, we present several new concepts about finite-time passivity in the sense of expectation on the basis of existing passivity definition. By designing appropriate controllers, the finite-time passivity of SCCNs with and without time-varying delay is obtained. In addition, the definition of finite-time synchronization in the sense of expectation is proposed. Under some sufficient conditions and designed controllers, finite-time passivity derives finite-time synchronization. Finally, two examples are given to demonstrate the effectiveness of finite-time passive and synchronization criteria.


Introduction
In the real world, complex networks can be seen everywhere such as food webs, communication networks, World Wide Web, and many others [1][2][3]. Due to various uncertainties in the actual system, complex network systems may be affected by noise. In recent years, the stability of stochastic systems has been extensively studied. At the same time, the synchronization and stability of stochastic complex networks have gradually become a topic of widespread concern for scholars in various fields [4][5][6][7][8].
Passivity is one part of dissipativeness. e main property of passivity is keeping the systems internally stable. e passivity theory has been extensively applied in many fields such as stability, complexity, signal processing, chaos control, synchronization fuzzy control, and so on [9][10][11][12].
ese are the main reasons why the passivity theory has been one of the most active research areas. In [13], the problem of passivity analysis was studied for discrete-time stochastic Markovian jump neural networks with both discrete and distributed delays. In [14], the problem of passivity analysis is investigated for a class of discrete-time stochastic neural networks with time-varying delays.
It is well known that passivity theory can provide a powerful tool to analyze synchronization of complex networks. However, in many existing works, synchronization is defined over the infinite time interval. Most of the theoretical methods on the synchronization of complex networks can only realize the network [15] or exponential asymptotical synchronization [16] which guarantees that error tends to 0 when t tends to infinity.
at is to say, achieving asymptotically stable convergence will be in infinite time. No further consideration has been given to the time and speed of synchronization. However, in practical engineering, people usually expect faster convergence rate and predict the required convergence time. Consequently, in order to achieve better control, the idea of finite-time synchronization has been proposed, and more and more attention has been paid by researchers. is kind of method can predict the synchronization time in advance and has better robustness, anti-interference, and better control effect. It has important research significance in theory and practice. erefore, it is more meaningful to study finite-time synchronization [17][18][19][20][21]. In [22], the authors study finite-time passivity of multi-weighted coupled neural networks with and without coupling delays. As far as we know, very few scholars have discussed finite-time passivity of stochastic complex networks in recent years. Motivated by the above discussions, we will investigate finite-time passivity of stochastic coupled complex networks (SCCNs). e main novelty and contributions of this paper can be summarized as follows. Firstly, we give three concepts of finite-time passivity in the sense of expectation. Secondly, we develop several finite-time passivity criteria. Lastly, we establish the relationship between finite-time passivity and finite-time synchronization in the sense of expectation.

Lemmas and Definitions
In this section, we will give some lemmas and definitions.

Lemmas
Lemma 1 (see [23]). Assume that a continuous, positivedefinite function W(t) satisfies the following differential inequality: where ϱ > 0 and 0 < μ < 1 are constants. en, for any given t 0 , W(t) satisfies the following inequality: with t 1 given by Lemma 2 (see [24]). For any b i ∈ R, i � 1, . . . , n, 0 < p ≤ 1, the following inequality holds: Lemma 3 (see [25]). For any vectors x, y ∈ R n and matrix 0 < P ∈ R n×n , the following inequality holds: 2.2. Definitions. Next, we will give three definitions about finite-time passivity in the sense of expectation. E · { } in these definitions stands for the mathematical expectation operator with respect to the given probability. Definition 1. A stochastic system with input u(t) ∈ R n and output y(t) ∈ R n is said to be finite-time passive in the sense of expectation if there exists a nonnegative function V such that for some α ∈ (0, 1) and β > 0.
Definition 2. A stochastic system with input u(t) ∈ R n and output y(t) ∈ R n is finite-time input strictly passive in the sense of expectation if there exists a nonnegative function V such that for some α ∈ (0, 1), β > 0, and c 1 > 0.
Definition 3. A stochastic system with input u(t) ∈ R n and output y(t) ∈ R n is finite-time output strictly passive in the sense of expectation if there exists a nonnegative function V such that for some α ∈ (0, 1), β > 0, and c 2 > 0.
Definition 4 (see [25]). Let A � (a ij ) m×m ∈ R m×n and B � (b ij ) p×q ∈ R p×q . en, the Kronecker product of A and B is defined as the matrix roughout this paper, we make the following assumptions.
(H1) (see [26]) e function f(·) is in the QUAD class, that is, there exist diagonal matrices 0 < P � diag (p 1 , p 2 , . . . , p n ) ∈ R n×n and Δ � diag(δ 1 , δ 2 , . . . , δ n ) ∈ R n×n , such that for all x, y ∈ R n and some λ > 0. (H2) For arbitrary u, v ∈ R n , there exists a positive constant L such that the following inequality holds: Remark 1 (see [25]). It can be verified that many of the benchmark chaotic systems belong to "function class QUAD," such as the Lorenz system, the Chen system, and the L € u system.

Network Model.
In this paper, we will consider the following stochastic coupled complex networks model: 2 Discrete Dynamics in Nature and Society where z i (t) � (z i1 (t), z i2 (t), . . . , z in (t)) T ∈ R n is the state vector of the ith node; N corresponds to the number of neurons; . . , f n (z in (t))) T ∈ R n denotes the neuron activation function and satisfies assumption (H1); , ω 2 (t), . . . , ω n (t)) T ∈ R n is a n− dimensional Brownian motion defined on a complete probability space (Ω, P); a is a positive real number which represents the overall coupling strength; Γ denotes the inner coupling matrix; and G � (G ij ) N×N represents the topological structure of the network, where G ij is defined as follows: if there exists a connection between node i and node j, then G ij � G ji > 0; otherwise, G ij � G ji � 0, (i ≠ j), and the diagonal elements of matrix G are defined by

Finite-Time Passivity. Set synchronization function z(t) satisfies
where where i � 1, 2, . . . , N. y i (t) ∈ R n refers to the output vector of (15) and is defined as follows: where A 1 , A 2 ∈ R n×n , e controller for network (12) is defined as follows: where

Theorem 1. Under assumptions (H1) and (H2), network model (15) is finite-time passive in the sense of expectation
where Proof. For network (15), the Lyapunov functional is chosen as follows: According to Ito's lemma, we acquire from (15) and (17) Here , According to (H1), we can obtain Discrete Dynamics in Nature and Society 3 We can get the following from (H2): Here λ M (P) represents maximum eigenvalue of matrix P, λ 0 � Lλ M (P). us, where Set (28) From (19) and (26)-(28), where en, we can obtain erefore, network (15) (17) if there exist matrix Q � diag(Q 1 , Q 2 , . . . , Q N ) ∈ R nN×nN and a positive real number c 1 such that Proof. We will choose the same V 1 (t) as (21) for network (15). By (26)-(28), one can get Taking the mathematical expectation on both sides above, one can derive that erefore, network (15) where 1 have the same meanings as in eorem 1.
Proof. Firstly we calculate the following equality: For the last step, we utilize the important properties of the Kronecker product: Select the same V 1 (t) as (21) for network (15). We can obtain Discrete Dynamics in Nature and Society 5 Taking the mathematical expectation on both sides above, one can derive that (40) erefore, network (15) is finite-time output strictly passive in the sense of expectation under controller (17). □

Finite-Time Synchronization.
In this section, we will verify finite-time synchronization in the sense of expectation for SCCNs (12). Firstly, the definition of finite-time synchronization is given as follows.  Proof. e network model (15) is finite-time passive in the sense of expectation with respect to V(t) under controller (17), that is to say, there exist α ∈ (0, 1) and β > 0 such that

Theorem 4. Assume that a continuous, positive-definite function V(t) satisfies the following inequality:
Considering u(t) � 0, one obtains According to the property of mathematical expectation, Choosing t 0 � 0 in Lemma 1, we can obtain V(t) ≡ 0 for t ⩾ t 1 , where t 1 � (V 1− α (0)/β (1 − α)). On the one hand, since one has for t ≥ t 1 . Since φ 1 (s) � 0 if and only if s � 0. en, we can conclude that On the other hand, V(t) is continuous, so Taking the limit t ⟶ t − 1 on both sides of (46), we will get lim Namely, SCCN (12) is finite-time synchronized in the sense of expectation under controller (17).
Similarly, it is easy to prove that SCCN (12) is also finitetime synchronized in the sense of expectation under controller (17) if network model (15)

Theorem 5. Under assumptions (H1) and (H2), network model (53) is finite-time passive in the sense of expectation under controller (55) if there exist matrices
where , Proof. Choose the following Lyapunov functional for network (51): where e(t) � (e T 1 (t), e T 2 (t), . . . , e T N (t)) T . According to Ito's lemma, we acquire from (53) and (55) Here According to Lemma 3, we can take It is not difficult to obtain From the above, one has Discrete Dynamics in Nature and Society us, where u(t) � (u T 1 (t), u T 2 (t), . . . , u T N (t)) T , y(t) � (y T 1 (t), y T 2 (t), . . . , y T N (t)) T , ζ(t) � (e T (t), u T (t)) T . Taking the mathematical expectation on (59), we can obtain Consequently, network model (53) is finite-time passive in the sense of expectation under controller (55). nN×nN and a positive real number c 3 such that

Theorem 6. Under assumptions (H1) and (H2), network model (53) is finite-time input strictly passive in the sense of expectation under controller (55) if there exist matrices
where W 1 , Ω 1 have the same meanings as in eorem 5.
Proof. We also select the same V 2 (t) as (58) for network (53). By (64), we get Taking the mathematical expectation on (59), we can obtain Discrete Dynamics in Nature and Society where W 1 , Ω 1 have the same meanings as in eorem 5.

Finite-Time Synchronization
Theorem 8. Assume that a continuous, positive-definite function V(t) satisfies the following inequality: Here we omit the proof of the theorem. e readers can refer to the proof of eorem 4.
By eorem 4, network (74) under finite-time output strictly passive can achieve finite-time synchronization. Figure 3 shows the simulation results.
Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.