Complexity
Volume 2019 (2019), Article ID 2916364, 8 pages
https://doi.org/10.1155/2019/2916364

Research Article

Synchronization in pth Moment for Stochastic Chaotic Neural Networks with Finite-Time Control

Yuhua Xu,¹ Jinmeng Wang,¹ Wuneng Zhou,² and Xin Wang³

¹School of Finance, Nanjing Audit University, Jiangsu 211815, China
²College of Information Science and Technology, Donghua University, Shanghai 201620, China
³Zhejiang Business Technology Institute, Zhejiang 315012, China

Correspondence should be addressed to Jinmeng Wang; jinmengwang@163.com

Received 21 June 2018; Accepted 5 November 2018; Published 3 January 2019

Academic Editor: Yan-Ling Wei

Copyright © 2019 Yuhua Xu et al. 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.

Abstract.

Finite-time synchronization in pth moment is considered for time varying stochastic chaotic neural networks. Compared with some existing results about finite-time mean square stability of stochastic neural network, we obtain some useful criteria of finite-time synchronization in pth moment for chaotic neural networks based on finite-time nonlinear feedback control and finite-time adaptive feedback control, which are efficient and easy to implement in practical applications. Finally, a numerical example is given to illustrate the validity of the derived synchronization conditions.

1. Introduction

In the past few decades, most of the studies on control of dynamical network were the infinite time control [1–9]. However, in practical engineering, the control of dynamical network is often required in a finite time. From the point of view of the time optimization of the controlled system, finite time convergent of dynamical system is the time optimal control method. Besides the advantages of the optimal convergence performance, the finite-time control of dynamical system has better robust performance and antidisturbance performance compared with the nonfinite-time control of complex system, because of the fractional power in the finite time controller. It makes the finite-time control theory more and more important, and many researchers have studied the finite-time control of dynamical network in detail [10–13].

Moreover, there is inevitable delay in neural networks. The generation of time delays may affect the stability of neural network. Therefore, it is of great practical value and theoretical significance to study synchronization control of dynamic neural networks with time delays. In addition, neural networks with delays are easily affected by stochastic disturbances [14]. At present, some authors studied the finite-time stability of stochastic neural networks [15, 16], and some authors discussed the pth moment stability of stochastic neural networks [11, 17–19]. In these studies, LMI and a matrix equality constraint conditions often were applied to establish sufficient conditions about finite-time mean square asymptotic stability of the stochastic neural network [20–23]. As the LMI software cannot handle large-sized problems and it is not numerically stable, adaptive finite-time control method can be more useful for applying to finite-time mean square asymptotic stability of the stochastic neural network, and the finite-time stability of stochastic chaotic neural networks in pth moment is less studied by the existing works. As the quality of the output signal of the neural network can be measured by the speed of the output signal converging to 0 in pth moment. Therefore, it is of great value to study the finite-time stability in pth moment for stochastic neural networks.

Motivated by the existing works, we considered finite-time synchronization in pth moment for time varying delayed stochastic neural networks in this paper, and we will establish synchronization criterion in pth moment for stochastic delayed neural networks in finite time.

2. Problem Formulation

Consider the following chaotic neural networks:

where , , , , , and . , , for and . satisfies , , where are constants.

Let and ; then

The corresponding response chaotic neural network is

where is the state vector, is an -dimensional Brown moment, is the noise intensity matrix, and is the controller.

Let the errors , , and ; then

Definition 1. The trivial solution of the error system (4) is said to be stability in th moment if

where .

Obviously, Definition 1 is stability in mean square for .

Definition 2 : (finite-time stable). For the error system (4), if there exists a constant such that

and , if , then error system (4) is called stable in finite time.

Assumption 3. Suppose satisfies the following conditions:

Assumption 4. Let ; then

Lemma 5 (see [24]). Letting and , then

Lemma 6 (see [25]). If and are bounded on with probability 1, then

where .

Lemma 7 (see [26]). For some constants ,

then

Lemma 8 (see [27]). For , . Assume that has the unique global solution, if there is a positive definite, twice continuously differentiable, radially unbounded Lyapunov function , and real numbers , such that

then the origin of system is globally stochastically finite-time stable.

3. Main Results

Theorem 9. Under Assumptions 3 and 4, and meet the following conditions:

where , , , , , and .
Then error system (4) is finite-time stability in pth moment by nonlinear feedback controller

where is a positive constant which is to be determined, , and .

Proof. We prove Theorem 9 in two steps.
The First Step. Error system (4) is finite-time stability under nonlinear feedback controller (16).
Let

By computing ,

By using Assumptions 3 and 4 and Lemma 5,

Meanwhile,

Therefore,

where , , , , , and .
From ,

By using Lemma 8, the error system (4) is stability in finite-time.
The Second Step. The stability in pth moment of error system (4) is achieved under nonlinear feedback controller (16).
From (21), we have

so

From the literature [12]

so

From Lemma 8,

Therefore

The stability in pth moment of error system (4) is achieved by using Definition 1.
Combining the first step and the second step, one can get that the fixed-time stability in pth moment of error system (4) is finally realized by nonlinear feedback controller (16).

In fact, may be very large to ensure the stability of error systems (4), and it can be meaningless in some practical applications. Therefore, adaptive feedback control is selected by the following Theorem 10.

Theorem 10. Under Assumptions 3 and 4, and meet the following conditions:

where , , , , , , and is positive constant.
Then, the stability in pth moment of error system (4) is completed by the following adaptive feedback controller:

where is a positive constant which is to be determined, and .

Proof. Similar to the proofs of Theorem 9.
The First Step. The finite-time stability of error system (4) is achieved under adaptive feedback controller (31).
Let

Computing ,

From , have

where .
By using Lemma 8, error system (4) is stability in finite-time.
The Second Step. The stability in pth moment of error system (4) is achieved under adaptive feedback controller (31).
Form (34), we have

The remaining reasoning is similar to the proof process of Theorem 9; we have

By using Definition 1, error system (4) is stability in pth moment.
Combining the first step and the second step, error system (4) is finite-time stability in pth moment by adaptive feedback controller (31).

Remark 11. It can be seen from Theorems 9 and 10 that the greater the noise intensity, the greater the control intensity, so the noisy environment may affect robustness.

Remark 12. In [20–23], research efforts have concentrated on studying finite-time mean stability of stochastic networks based on LMI. In this paper, finite-time nonlinear feedback control and finite-time adaptive feedback control are considered, which can be simpler than LMI.

4. Illustrative Example

In the following, we present an example to illustrate the usefulness of Theorem 10.

The drive systems are

where

The response systems are

The adaptive controllers are

where , , , , and .

We let , , and . Figure 1 shows synchronization errors. Figure 2 shows dynamic curve of the feedback gain. Let . Obviously, when , finite-time synchronization of neural networks is realized (See Figure 3). All numerical simulations illustrate the effectiveness of Theorem 10.

Figure 1: Synchronization errors.

Figure 2: Dynamic curve of the feedback gain.

Figure 3: Dynamic curve of .

5. Conclusion

In this paper, we have discussed finite-time synchronization in pth moment for time-varying stochastic chaotic neural networks via finite-time nonlinear feedback control and finite-time adaptive feedback control, which are efficient and easy to implement in practical applications. Numerical simulations demonstrate the effectiveness of the main results obtained in this paper. Our future work is to study adaptive finite-time pining control in pth moment for time varying stochastic chaotic neural networks, which may help to save control cost and has more practical application in practical engineering.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (61673221); the Natural Science Foundation of Jiangsu Province (BK20181418); “Qing-Lan Engineering” Foundation of Jiangsu Higher Education Institutions; the Fifteenth Batch of Six Talent Peaks Project in Jiangsu Province (DZXX-019); A Project Funded by the Applied Economics of Nanjing Audit University of the Priority Academic Program Development of Jiangsu Higher Education Institutions (Office of Jiangsu Provincial People's Government, No. 201887).

References

Y. Li, J. Lou, Z. Wang, and F. E. Alsaadi, “Synchronization of dynamical networks with nonlinearly coupling function under hybrid pinning impulsive controllers,” Journal of The Franklin Institute, vol. 355, no. 14, pp. 6520–6530, 2018.

J. Lu and D. W. C. Ho, “Globally exponential synchronization and synchronizability for general dynamical networks,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 40, no. 2, pp. 350–361, 2010.

Y. Li, J. Zhong, J. Lu, Z. Wang, and F. E. Alssadi, “On robust synchronization of drive-response Boolean control networks with disturbances,” Mathematical Problems in Engineering, Art. ID 1737685, 9 pages, 2018.

J. Q. Lu and D. W. C. Ho, “Stabilization of complex dynamical networks with noise disturbance under performance constraint,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 1974–1984, 2011.

Y. Tang, H. Gao, W. Zou, and J. Kurths, “Distributed synchronization in networks of agent systems with nonlinearities and random switchings,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 43, no. 1, pp. 358–370, 2013.

Y. Tang, Z. Wang, H. Gao, S. Swift, and J. Kurths, “A constrained evolutionary computation method for detecting controlling regions of cortical networks,” IEEE Transactions on Computational Biology and Bioinformatics, vol. 9, no. 6, pp. 1569–1581, 2012.

Z.-G. Wu, P. Shi, H. Su, and R. Lu, “Dissipativity-Based Sampled-Data Fuzzy Control Design and its Application to Truck-Trailer System,” IEEE Transactions on Fuzzy Systems, vol. 23, no. 5, pp. 1669–1679, 2015.

Y. Li, X. Wu, J.-A. Lu, and J. Lu, “Synchronizability of Duplex Networks,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 63, no. 2, pp. 206–210, 2016.

Y. Wei, J. Qiu, and H.-K. Lam, “A novel approach to reliable output feedback control of fuzzy-affine systems with time delays and sensor faults,” IEEE Transactions on Fuzzy Systems, vol. 25, no. 6, pp. 1808–1823, 2017.

Y. Xu, D. Meng, C. Xie, G. You, and W. Zhou, “A class of fast fixed-time synchronization control for the delayed neural network,” Journal of The Franklin Institute, vol. 355, no. 1, pp. 164–176, 2018.

G. Mei, X. Wu, D. Ning, and J.-A. Lu, “Finite-time stabilization of complex dynamical networks via optimal control,” Complexity, vol. 21, pp. 417–425, 2016.

Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Local synchronization of chaotic neural networks with sampled-data and saturating actuators,” IEEE Transactions on Cybernetics, vol. 44, no. 12, pp. 2635–2645, 2014.

A. Michalak and A. Nowakowski, “Finite-time stability and finite-time synchronization of neural network dual approach,” Journal of The Franklin Institute, vol. 354, no. 18, pp. 8513–8528, 2017.

Y. Wei, J. H. Park, H. R. Karimi, Y.-C. Tian, and H. Jung, “Improved Stability and Stabilization Results for Stochastic Synchronization of Continuous-Time Semi-Markovian Jump Neural Networks With Time-Varying Delay,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 6, pp. 2488–2501, 2018.

Y. Xu, W. Zhou, J. Fang, C. Xie, and D. Tong, “Finite-time synchronization of the complex dynamical network with non-derivative and derivative coupling,” Neurocomputing, vol. 173, pp. 1356–1361, 2016.

T. Jing, F. Chen, and X. Zhang, “Finite-time lag synchronization of time-varying delayed complex networks via periodically intermittent control and sliding mode control,” Neurocomputing, vol. 199, pp. 178–184, 2016.

Y. Xu, H. Yang, D. Tong, and Y. Wang, “Adaptive exponential synchronization in pth moment for stochastic time varying multi-delayed complex networks,” Nonlinear Dynamics, vol. 73, no. 3, pp. 1423–1431, 2013.

X. Yang, J. Cao, and J. Qiu, “Pth moment exponential stochastic synchronization of coupled memristor-based neural networks with mixed delays via delayed impulsive control,” Neural Networks, vol. 65, pp. 80–91, 2015.

X. Liu, X. Yu, and H. Xi, “Finite-time synchronization of neutral complex networks with Markovian switching based on pinning controller,” Neurocomputing, vol. 153, pp. 148–158, 2015.

Y. Zhang, Y. Shi, and P. Shi, “Resilient and robust finite-time H∞ control for uncertain discrete-time jump nonlinear systems,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 49, pp. 612–629, 2017.

R. Sakthivel, R. Sakthivel, B. Kaviarasan, C. Wang, and Y.-K. Ma, “Finite-Time Nonfragile Synchronization of Stochastic Complex Dynamical Networks with Semi-Markov Switching Outer Coupling,” Complexity, vol. 2018, Article ID 8546304, 13 pages, 2018.

Z. Wang, Y. Liu, L. Yu, and X. Liu, “Exponential stability of delayed recurrent neural networks with Markovian jumping parameters,” Physics Letters A, vol. 356, no. 4-5, pp. 346–352, 2006.

P. Shi, Y. Zhang, and R. Agarwal, “Stochastic finite-time state estimation for discrete time-delay neural networks with Markovian jumps,” Neurocomputing, vol. 151, part 1, pp. 168–174, 2014.

X. Liu, D. W. Ho, Q. Song, and J. Cao, “Finite-/fixed-time robust stabilization of switched discontinuous systems with disturbances,” Nonlinear Dynamics, vol. 90, no. 3, pp. 2057–2068, 2017.

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.

B. Ksendal, Stochastic Differential Equations an Introduction with Applications, Springer, New York, 2005.

W. Chen and L. C. Jiao, “Finite-time stability theorem of stochastic nonlinear systems,” Automatica, vol. 46, no. 12, pp. 2105–2108, 2010.