Finite-Time Passivity of Stochastic Coupled Complex Networks

Yin, Xunwu; Cao, Min

doi:https://doi.org/10.1155/2021/9905125

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Analysis, Control and Applications of Passivity in Complex Networks 2021

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 9905125 | https://doi.org/10.1155/2021/9905125

Finite-Time Passivity of Stochastic Coupled Complex Networks

Xunwu Yin¹and Min Cao²

Academic Editor: A. E. Matouk

Received30 Apr 2021

Accepted13 Jun 2021

Published09 Jul 2021

Abstract

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.

1. Introduction

In the real world, complex networks can be seen everywhere such as food webs, communication networks, World Wide Web, and many others [1–3]. 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 [4–8].

Passivity is one part of dissipativeness. The main property of passivity is keeping the systems internally stable. The passivity theory has been extensively applied in many fields such as stability, complexity, signal processing, chaos control, synchronization fuzzy control, and so on [9–12]. These 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. That 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. This 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. Therefore, it is more meaningful to study finite-time synchronization [17–21]. 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). The 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.

2. Lemmas and Definitions

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

2.1. Lemmas

Lemma 1. (see [23]). Assume that a continuous, positive-definite function satisfies the following differential inequality:where and are constants. Then, for any given , satisfies the following inequality:with given by

Lemma 2. (see [24]). For any , the following inequality holds:

Lemma 3. (see [25]). For any vectors and matrix , the following inequality holds:

2.2. Definitions

Next, we will give three definitions about finite-time passivity in the sense of expectation. in these definitions stands for the mathematical expectation operator with respect to the given probability.

Definition 1. A stochastic system with input and output is said to be finite-time passive in the sense of expectation if there exists a nonnegative function such thatfor some and .

Definition 2. A stochastic system with input and output is finite-time input strictly passive in the sense of expectation if there exists a nonnegative function such thatfor some , and .

Definition 3. A stochastic system with input and output is finite-time output strictly passive in the sense of expectation if there exists a nonnegative function such thatfor some , and .

Definition 4. (see [25]). Let and . Then, the Kronecker product of and is defined as the matrixThroughout this paper, we make the following assumptions. (H1) (see [26]) The function is in the QUAD class, that is, there exist diagonal matrices and , such that for all and some . (H2) For arbitrary , there exists a positive constant 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 system.

3. Finite-Time Passivity of SCCNs

3.1. Network Model

In this paper, we will consider the following stochastic coupled complex networks model:where is the state vector of the th node; corresponds to the number of neurons; denotes the neuron activation function and satisfies assumption (H1); is a varying external input vector; denotes the control input; satisfies assumption (H2); is a dimensional Brownian motion defined on a complete probability space ; is a positive real number which represents the overall coupling strength; denotes the inner coupling matrix; and represents the topological structure of the network, where is defined as follows: if there exists a connection between node and node , then ; otherwise, , and the diagonal elements of matrix are defined by

3.2. Finite-Time Passivity

Set synchronization function satisfieswhere .

Define . Then, we havewhere .

refers to the output vector of (15) and is defined as follows:where ,

The controller for network (12) is defined as follows:where , is defined in (H1), and

Theorem 1. Under assumptions (H1) and (H2), network model (15) is finite-time passive in the sense of expectation under controller (17) if there exists matrix such thatwhere

Proof. For network (15), the Lyapunov functional is chosen as follows:According to Ito’s lemma, we acquire from (15) and (17)HereAccording to (H1), we can obtainWe can get the following from (H2):Here represents maximum eigenvalue of matrix , .
Thus,where , .
Considering sign and Lemma 2, we can easily conclude thatSet ; consequently,From (19) and (26)–(28),where .
Considering , consequentlyThen, we can obtainTherefore, network (15) is finite-time passive in the sense of expectation under controller (17).

Theorem 2. Under assumptions (H1) and (H2), network model (15) is finite-time input strictly passive in the sense of expectation under controller (17) if there exist matrix and a positive real number such thatwhere have the same meanings as in Theorem 1.

Proof. We will choose the same as (21) for network (15).
By (26)–(28), one can getTaking the mathematical expectation on both sides above, one can derive thatTherefore, network (15) is finite-time input strictly passive in the sense of expectation under controller (17).

Theorem 3. Under assumptions (H1) and (H2), network model (15) is finite-time output strictly passive in the sense of expectation under controller (17) if there exist matrix and a positive real number such thatwhereK₁, E₁ have the same meanings as in Theorem 1.

Proof. Firstly we calculate the following equality:For the last step, we utilize the important properties of the Kronecker product:Select the same as (21) for network (15). We can obtainTaking the mathematical expectation on both sides above, one can derive thatTherefore, network (15) is finite-time output strictly passive in the sense of expectation under controller (17).

3.3. 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.

Definition 5. The SCCN (12) is finite-time synchronized in the sense of expectation if there exists a constant such thatfor , where .

Theorem 4. Assume that a continuous, positive-definite function satisfies the following inequality:where is continuous and strictly monotonically increasing function and is positive for with . If network (15) is finite-time passive (finite-time input strictly passive, finite-time output strictly passive) in the sense of expectation with respect to , then SCCN (12) is finite-time synchronized in the sense of expectation under controller (17).

Proof. The network model (15) is finite-time passive in the sense of expectation with respect to under controller (17), that is to say, there exist and such thatConsidering , one obtainsAccording to the property of mathematical expectation,Choosing in Lemma 1, we can obtain for , where . On the one hand, sinceone hasfor . Since if and only if . Then, we can conclude thatOn the other hand, is continuous, soTaking the limit on both sides of (46), we will getNamely, 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 finite-time synchronized in the sense of expectation under controller (17) if network model (15) is finite-time input strictly passive or finite-time output strictly passive in the sense of expectation.

4. Finite-Time Passivity of SCCNs with Time-Varying Delay

4.1. Network Model

In this section, the network model is described bywhere is the time delay and satisfies .

4.2. Finite-Time Passivity

Let also satisfy

Define . Then, we have

The output vector of network (53) is defined as follows:

Design the following controller for network (51):where have the same meanings as in (17).

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 such thatwhere

Proof. Choose the following Lyapunov functional for network (51):where .
According to Ito’s lemma, we acquire from (53) and (55)HereAccording to Lemma 3, we can takeIt is not difficult to obtainFrom the above, one hasThus,where .
Taking the mathematical expectation on (59), we can obtainConsequently, network model (53) is finite-time passive in the sense of expectation under controller (55).

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 and a positive real number such thatwhere have the same meanings as in Theorem 5.

Proof. We also select the same as (58) for network (53). By (64), we getTaking the mathematical expectation on (59), we can obtainTherefore, network (53) is finite-time input strictly passive in the sense of expectation under controller (55).

Theorem 7. Under assumptions (H1) and (H2), network model (53) is finite-time output strictly passive in the sense of expectation under controller (55) if there exist matrices and a positive real number such thatwhereW₁, Ω₁ have the same meanings as in Theorem 5.

Proof. Select the same as (58) for network (53). By (37) and (64), we getTaking the mathematical expectation on (59), we can obtainTherefore, network (53) is finite-time output strictly passive in the sense of expectation under controller (55).

4.3. Finite-Time Synchronization

Theorem 8. Assume that a continuous, positive-definite function satisfies the following inequality:where is continuous and strictly monotonically increasing function and is positive for with . If network (51) is finite-time passive (finite-time input strictly passive, finite-time output strictly passive) in the sense of expectation with respect to , then SCCN (53) is finite-time synchronized in the sense of expectation under controller (55).

Here we omit the proof of the theorem. The readers can refer to the proof of Theorem 4.

5. Numerical Examples

Example 1. The following SCCNs are discussed:where ,Obviously, holds under the condition that , and . Choose andTake . According to Theorem 1, the SCCNs (74) can realize finite-time passivity in the sense of expectation under controller (17). Then, we can easily find the parameters and satisfying the condition of Theorems 2 and 3. The simulation results are shown in Figures 1 and 2.
By Theorem 4, network (74) under finite-time output strictly passive can achieve finite-time synchronization. Figure 3 shows the simulation results.

Example 2. The following SCCNs with time-varying delay are discussed:where . Taking , we can get . The matrix is chosen asObviously, holds under the condition that , and . Choose andTake . According to Theorem 5, the SCCNs (77) can realize finite-time passivity under controller (55). Then, we can easily find the parameters and satisfying the condition of Theorems 6 and 7. The simulation results are shown in Figures 4 and 5.
By Theorem 8, network (77) under finite-time output strictly passive can achieve finite-time synchronization. Figure 6 shows the simulation results.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

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

Acknowledgments

This study was supported by the National Natural Science Foundation of China under grant nos. 61773285 and 11871371.

References

J. Ruths and D. Ruths, “Control profiles of complex networks,” Science, vol. 343, no. 6177, pp. 1373–1376, 2014.
View at: Publisher Site | Google Scholar
A. R. Benson, D. F. Gleich, and J. Leskovec, “Higher-order organization of complex net- works,” Science, vol. 353, no. 6295, pp. 163–165, 2016.
View at: Publisher Site | Google Scholar
J. Gao, B. Barzel, and A. Barabsi, “Universal resilience patterns in complex networks,” Nature, vol. 530, no. 7590, pp. 307–312, 2016.
View at: Publisher Site | Google Scholar
J. Liang, Z. Wang, and X. Liu, “Exponential synchronization of stochastic delayed discrete time complex networks,” Nonlinear Dynamics, vol. 53, pp. 153–165, 2008.
View at: Publisher Site | Google Scholar
C. Li, L. Chen, and K. Aihara, “Stochastic synchronization of genetic oscillator networks,” BMC Systems Biology, vol. 1, pp. 1–6, 2007.
View at: Publisher Site | Google Scholar
Y. Wang, Z. Wang, and J. Liang, “A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances,” Physics Letters A, vol. 372, no. 9, pp. 6066–6073, 2008.
View at: Publisher Site | Google Scholar
Z. Wang, Y. Wang, and Y. Liu, “Global synchronization for discrete time stochastic complex networks with randomly occurred nonlinearities and mixed time delays,” IEEE Transactions on Neural Networks, vol. 21, no. 1, pp. 11–26, 2010.
View at: Publisher Site | Google Scholar
H. Li and D. Yue, “Synchronization of Markovian jumping stochastic complex networks with distributed time delays and probabilistic interval discrete time varying delays,” Journal of Physics A: Mathematical and Theoretical, vol. 43, no. 10, 2010.
View at: Publisher Site | Google Scholar
X. Lou and B. Cui, “Passivity analysis of integro-differential neural networks with time-varying delays,” Neurocomputing, vol. 70, pp. 1071–1078, 2007.
View at: Publisher Site | Google Scholar
J. Yao, H. O. Wang, Z.-H. Guan, and W. Xu, “Passive stability and synchronization of complex spatio-temporal switching networks with time delays,” Automatica, vol. 45, pp. 1721–1728, 2009.
View at: Publisher Site | Google Scholar
J. L. Wang, H. N. Wu, and L. Guo, “Passivity and stability analysis of reaction-diffusion neural networks with dirichlet boundary conditions,” IEEE Transactions on Neural Networks, vol. 22, pp. 2105–2116, 2011.
View at: Publisher Site | Google Scholar
X. W. Yin, X. N. Song, and M. Wang, “Passive fuzzy control design for a class of nonlinear distributed parameter systems with time-varying Delay,” International Journal of Control, Automation and Systems, vol. 18, no. 4, pp. 911–921, 2020.
View at: Publisher Site | Google Scholar
Z. G. Wu, P. Shi, and J. Chu, “Passivity analysis for discrete-time stochastic Markovian jump neural networks with mixed time delays,” IEEE Transactions on Neural Networks, vol. 22, no. 10, pp. 1566–1575, 2011.
View at: Publisher Site | Google Scholar
Q. Song, J. Liang, and Z. Wang, “Passivity analysis of discrete-time stochastic neural networks with time-varying delays,” Neurocomputing, vol. 72, pp. 1782–1788, 2009.
View at: Publisher Site | Google Scholar
H. Y. Du, W. S. Sun, G. Hu, and L. H. Qi, “Function projective synchronization of two fractional-order complex dynamical networks,” Acta Automatica Sinica, vol. 42, no. 2, pp. 226–234, 2016.
View at: Google Scholar
V. Andrieu, B. Jayawardhana, and S. Tarbouriech, “Some results on exponential synchronization of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 63, no. 4, pp. 1213–1219, 2018.
View at: Publisher Site | Google Scholar
D. Zhang, Y. Shen, and J. Mei, “Finite-time synchronization of multi-layer nonlinear coupled complex networks via intermittent feedback control,” Neurocomputing, vol. 225, pp. 129–138, 2017.
View at: Publisher Site | Google Scholar
X. Lu, X. Zhang, and Q. Liu, “Finite-time synchronization of nonlinear complex dynamical networks on time scales via pinning impulsive control,” Neurocomputing, vol. 275, pp. 2104–2110, 2018.
View at: Publisher Site | Google Scholar
M. Tan, X. Li, and Y. Liu, “Finite-time stability and synchronization of the coupled switched neural networks with nodes of different dimensions,” Neural Processing Letters, vol. 49, no. 1, pp. 285–303, 2019.
View at: Publisher Site | Google Scholar
X. Yang and J. Lu, “Finite-time synchronization of coupled networks with Markovian topology and impulsive effects,” IEEE Transactions on Automatic Control, vol. 61, no. 8, pp. 2256–2261, 2016.
View at: Publisher Site | Google Scholar
E. Moulay and M. Dambrine, “Finite-time stability and stabilization of time-delay systems,” System & Control Letter, vol. 57, no. 7, pp. 561–566, 2008.
View at: Publisher Site | Google Scholar
J. L. Wang, M. Xu, H. N. Wu, and T. W. Huang, “Finite-time passivity of coupled neural networks with multiple weights,” IEEE Transcations on Network Science and Engineering, vol. 5, no. 3, pp. 184–197, 2018.
View at: Publisher Site | Google Scholar
Y. Tang, “Terminal sliding mode control for rigid robots,” Automatica, vol. 34, pp. 51–56, 1998.
View at: Publisher Site | Google Scholar
X. Huang, W. Lin, and B. Yang, “Global finite-time stabilization of a class of uncertain nonlinear systems,” Automatica, vol. 41, pp. 881–888, 2005.
View at: Publisher Site | Google Scholar
J. L. Wang, H. N. Wu, and S. Y. Ren, Passivity of Complex Dynamical Networks, Springer, Berlin, Germany, 2021.
W. L. Lu and T. P. Chen, “New approach to synchronization analysis of linearly coupled ordinary differential systems,” Physica D, vol. 213, no. 2, pp. 214–230, 2006.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Xunwu Yin and Min Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

172

Downloads

449

Citations