Abstract and Applied Analysis

Volume 2012 (2012), Article ID 231349, 15 pages

doi:10.1155/2012/231349

Research Article

Finite-Time Robust Stabilization for Stochastic Neural Networks

Weixiong Jin,¹ Xiaoyang Liu,² Xiangjun Zhao,² Nan Jiang,³ and Zhengxin Wang⁴

¹Department of Mathematics and Applied Mathematics, Lianyungang Teacher's College, Lianyungang 222006, China

²School of Computer Science and Technology, Jiangsu Normal University, Xuzhou 221116, China

³School of Economics, Jiangsu Normal University, Xuzhou 221116, China

⁴College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210046, China

Received 6 September 2012; Accepted 26 September 2012

Academic Editor: Ju H. Park

Copyright © 2012 Weixiong Jin 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.

This paper is concerned with the finite-time stabilization for a class of stochastic neural networks (SNNs) with noise perturbations. The purpose of the addressed problem is to design a nonlinear stabilizator which can stabilize the states of neural networks in finite time. Compared with the previous references, a continuous stabilizator is designed to realize such stabilization objective. Based on the recent finite-time stability theorem of stochastic nonlinear systems, sufficient conditions are established for ensuring the finite-time stability of the dynamics of SNNs in probability. Then, the gain parameters of the finite-time controller could be obtained by solving a linear matrix inequality and the robust finite-time stabilization could also be guaranteed for SNNs with uncertain parameters. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design method.

1. Introduction

Since the first paper of Ott et al. [1], a large number of monographs and papers studying the stabilization of the nonlinear systems without or with delays have been published [2–5]. These publications have developed many control techniques including continuous feedback and discontinuous feedback. Take [4] for example, the authors studied the pinning stabilization problem of linearly coupled stochastic neural networks, where a minimum number of controllers are used to force the NNs to the desired equilibrium point by fully utilizing the structure of the network.

On the other hand, the well-known Hopfield neural networks, Cohen-Grossberg neural networks and cellular neural networks [6–18], and so forth have been extensively studied in the past decades and successfully applied in many areas such as signal processing, combinatorial optimization, and pattern recognition. Specially, the stability of Hopfield neural networks has received much research attention since, when applied, the neural network is sometimes assumed to have only one globally stable equilibrium [7–9, 19, 20].

Until now, the stability analysis issues for many kinds of neural networks in the presence of stochastic perturbations and/or parameter uncertainties have attracted a lot of research attention. The reasons include twofold: (a) in real nervous systems, because of random fluctuations from the release of neurotransmitters, and other probabilistic causes, the synaptic transmission is indeed a noisy process; (b) the connection weights of the neurons depend on certain resistance and capacitance values that always exist uncertainties. Therefore, the robust stability has been studied for neural networks with parameter uncertainties [21–24] or external stochastic perturbations [7, 19, 25, 26]. However, to the best of the authors’ knowledge, most literature regarding the stability of neural networks is based on the convergence time being large enough, even though we eagerly want the argued network states to become stable as quickly as possible in practical applications. In order to achieve faster stabilization speed and hope to complete stabilization in finite time rather than merely asymptotically [27], an effective method is using finite-time stabilization techniques, which have also demonstrated better robustness and disturbance rejection properties [28].

In this paper, we will focus on the finite-time robust stabilization for neural networks with both stochastic perturbations and parameter uncertainties. The difference of this paper lies in three aspects. First, based on the finite-time stability theorem of stochastic nonlinear systems [29], a new continuous finite-time stabilizator is proposed for a stochastic neural network (SNN). Moreover, in contrast to [30–33], we prove finite-time stabilization by constructing a suitable Lyapunov function and obtain some criteria which are easy to be satisfied. Second, the gain parameters in finite-time stabilizator are designed by solving a linear matrix inequality. Finally, a robust finite-time stabilizator for SNNs with parameter uncertainties is designed as well. Moreover, two illustrative examples are provided to show the effectiveness of the proposed designing.

The notations in this paper are quite standard. and denote, respectively, the -dimensional Euclidean space and the set of all real matrices. The superscript “” denotes the transpose and the notation (resp., ), where and are symmetric matrices, meaning that is positive semidefinite (resp., positive definite). and denote the maximal and minimal eigenvalues of real matrix . Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). stands for the expectation of the stochastic variable with respect to the given probability measure . and represent the identity matrix and a zero matrix, respectively; stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Model Formulation and Preliminaries

Some preliminary knowledge is presented in this section for the derivation of our main results. The deterministic NN can be described by the following differential equation:

or

where is the vector of neuron states; represents the number of neurons in the network; is an constant diagonal matrix with , ; is an interconnection matrix; is a diagonal mapping, where , represents the neuron input-output activation and is a constant external input vector.

To establish our main results, it is necessary to give the following assumption for system (2.1) or (2.2).

Assumption 2.1. The neuron activation function of the NN (2.1) satisfies the following Lipschitz condition:

where is a positive constant for . For convenience, let .

Because of the existence of environmental noise in real neural networks, the stochastic disturbances should be taken into account in the recurrent NN. For this purpose, we modify the system (2.1) as the following SNN:

where is an -dimensional Brownian motion defined on the probability space satisfying the usual conditions (i.e., the filtration contains all -null sets and is right continuous). The white noise is independent of for . The intensity function is the noise intensity function matrix satisfying the following condition:

where is a known constant matrix with compatible dimensions.

In this paper, we want to control the SNN (2.4) to the desired state , which is an equilibrium point of NN (2.1). Based on the discussions in many other papers, the stochastic perturbation will vanish at this equilibrium point , that is, . Without loss of generality, one can shift the equilibrium point to the origin by using the translation , which derives the following stochastic dynamical system:

where .

Consider the SNN (2.6) with parameter uncertainties: the parameter matrices and are unknown but bounded, which are assumed to satisfy

where , , and , .

We denote that = , = , = , = , = , = , = , =   := , = , = , and = , where is the column vector with the th element and elsewhere.

For , let

Then, through simple manipulations, one has

In order to stabilize the SNN (2.4) to the equilibrium point , equivalently, one can stabilize the SNN (2.6) to the origin due to the transformation. Hence, in the remainder of this paper, a controller will be designed for the stabilization of SNN (2.6) in mean square. The controlled SNN can be described by the following stochastic differential equation (SDE):

Similar to [30–33], the controller is designed as follows:

where = , = , constants , are gain coefficients to be determined, and the real number satisfies . In fact, here the continuous function in the SNN (2.10) is the key point for ensuring the finite-time stabilization.

Obviously, when , the controller is a continuous function with respect to , which leads to the continuity of controlled system (2.10) with respect to the state [30–33]. If , turns to be a discontinuous one, which has been considered in [34–36]. If in the controller (2.11), then it becomes the typical stabilization issues which only can realize an asymptotical stabilization in infinite time [3–5].

Similar to the definition of finite-time stability in probability [29], the finite-time stabilization in probability is given through the following definition.

Definition 2.2. The system (2.6) is said to be finite-time stabilized at the original point by the controller (2.11) in probability, that is, the controlled SNN (2.10) is finite-time stable in probability [37] if, for any initial state , there exists a finite-time function such that

where is called the stochastic setting time function satisfying .

The following lemmas are needed for the derivation of our main results in this paper.

Lemma 2.3 (see [38]. (Itô’s formula)). Let ba an n-dimensional Itô’s process on with the stochastic differential

Let . Then, is a real-valued Itô’s process with its stochastic differential given by

where denotes the family of all real-valued functions such that they are continuously twice differentiable in and .

Lemma 2.4 (see [29]). Consider the stochastic differential equation (2.13) with and and assume system (2.13) has a unique global solution. If there exist real numbers and , such that for the function in Lemma 2.3,

then the origin of system (2.13) is globally stochastically finite-time stable, and .

Lemma 2.5 (see [39]). If are positive number and , then

Lemma 2.6 (Boyd et al. [40]). If , , and are real matrices of appropriate dimension with satisfying , then

for all , if and only if there exists a positive constant , such that

3. Main Results

In this section, we first give some theorems in detail to guarantee that the original point of SNN (2.6) is stabilized in finite time, that is, the controlled system (2.10) with (2.11) is finite-time stable in probability. Then, for SNN (2.6) with parameter uncertainties, we provide a sufficient condition under which the controlled system (2.10) is robust finite-time stable in probability. Finally, the control gains and are designed by solving some linear matrix inequalities.

Theorem 3.1. The controlled system (2.10) with (2.11) is finite-time stable in probability, if there exist a constant and a positive-definite matrix such that

Moreover, the upper bound of the stochastic settling time for stabilization can be in terms of the initial errors as .

Proof. Consider the controlled system (2.10) with the controller (2.11), we have

Next, we will prove system (3.2) is finite-time stable in probability based on Definition 2.2. To this end, choose the candidate Lyapunov function and calculate the time derivative of along the trajectories of the augmented system (3.2). By the Itô’s formula, we obtain the stochastic differential as

where

From condition (2.3), using the inequality , where is an arbitrary constant, we have

Combining (2.5), (3.4)-(3.5) results in

From and Lemma 2.5, we get

then,

Thus, based on condition (3.1), taking the expectations on both sides of (3.3), we have

By Lemma 2.4, stochastically converges to zero in a finite time, that is, the controlled system (3.2) is finite-time stable in probability, and the settle time is upper bounded by

This completes the proof.

Remark 3.2. The two gain parameters and in the controller play different roles in ensuring the finite-time stability of the controlled system (3.2). We can see from Theorem 3.1 that, whether or not the controlled system (3.2) could realize the finite-time stability mainly depends on the value of and satisfies condition (3.1) but nothing on . However, the size of the settle time depends on the value of but unrelated to , the only requirement for the gain is satisfying condition (3.1).

Remark 3.3. In [31, 32, 35, 41], the candidate Lyapunov function was chosen as a simple form of and then the upper bound of settle time turns to be . In this paper, in order ro reduce some conservation of conditions in Theorem 3.1, a positive definite matrix parameter is introduced such that condition (3.1) is easier to be satisfied. And the previous conclusions could be included by our results if the matrix is taken, where is a arbitrary constant, just as shown in the next corollary.

Corollary 3.4. The controlled system (3.2) is finite-time stable in probability, if there exist two constants and such that

Moreover, the upper bound of the settle time is

Our next goal is to deal with the design problem, that is, giving a practical design procedure for the controller gains: and , such that the inequalities in Theorem 3.1 or Corollary 3.4 are satisfied. Obviously, those inequalities are difficult to solve, since they are nonlinear and coupled. A meaningful approach to tackling such a problem is to convert the nonlinearly coupled matrix inequalities into linear matrix inequalities (LMIs), while the controller gains are designed simultaneously.

Based on the discussion in Remark 3.2, the parameter gain is one of the primary factors that affect the size of the settle time, which is unrelated to condition (3.11). Hence, in the following discussion, we will fix the gain parameter and mainly focus on the design of control gain . We claim that the desired controller gain can be designed if a linear matrix inequality is feasible.

Theorem 3.5. For a fixed control gain , the finite-time stabilization problem is solvable for the SNN (2.6), if there exist three positive scalars , , and such that

Moreover, the control gain coefficient .

Proof. The result can be proved by pre- and post-multiplying the inequality (3.13) by the block-diagonal matrix and then following from the famous Schur complement lemma and Corollary 3.4 and we omit it here.

Just as mentioned in Introduction, when modelling a dynamic system, one can hardly obtain an exact model. Specially, in practical implementation of neural networks, the firing rates and the weight coefficients of the neurons depend on certain resistance and capacitance values, which are subject to uncertainties. It is thus necessary to take parameter uncertainties into account in the considered neural network. In the following, we consider the robust finite-time stabilization issue for SNN (2.6) under the parametric uncertainties (2.7).

Theorem 3.6. The interval SNN (3.2) with uncertain parameters (2.7) is robust finite-time stable in probability, if there exist three constants , , and a positive-definite matrix such that

where and .

Proof. From Theorems 3.1 and 3.5, we know that the SNN (3.2) is finite-time stable in probability, if there exist a constant and a positive-definite matrix such that the following LMI holds:

Thus, for the uncertain parameters satisfying (2.7), we have

For the second term in the above equality, it is easy to have

Then, based on Lemma 2.6, (3.16) and (3.17), there exist two constants and such that

Then the result can be proved by the famous Schur complement lemma and condition (3.14).

Corollary 3.7. For a fixed control gain , the finite-time robust stabilization problem is solvable for the SNN (2.6) with (2.7), if there exist five positive scalars , , , , and such that

where . Moreover, the control gain coefficient .

Proof. Let and we can prove the result based on Theorem 3.6.

4. Two Numerical Examples

Example 4.1. Consider the following stochastic neural network:

where

, and the activation function is taken as . Then, it is obvious that , where is a identity matrix. The SNN (4.1) with the above-given parameters is depicted in Figure 1 with initial values .

Figure 1: Trajectories of SNN (4.1) without any controller in Example 4.1.

The stabilization controller is designed as

where the parameter is chosen as 0.5 and the initial value . Then, .

According to Theorem 3.5 and using Matlab LMI toolbox, we solve the LMI (3.13), and obtain , , and . Then by Theorem 3.5, the desired controller parameter can be designed as .

By choosing an arbitrary fixed gain , SNN (4.1) can be stabilized in finite time in probability. Taking , for example, we can obtain the upper bound of the settle time .

Simulation result is depicted in Figure 2, which shows the states , , and of the controlled SNN (4.1). The simulation result has confirmed the effectiveness of our main results.

Figure 2: Trajectories of SNN (4.1) under the controller (4.3) with in Example 4.1.

Example 4.2. Still consider the SNN (4.1) with second-order parameter uncertainties:

The parameter in the controller (4.3) is chosen as 0.5 and the initial value . Then, . According to Corollary 3.7 and using Matlab LMI toolbox, we solve the LMI (3.19) and obtain , , , , and . Then by Corollary 3.7, the desired controller parameter can be designed as .

By choosing an arbitrary fixed gain , SNN (4.1) can be robustly stabilized in finite time in probability. Taking , for example, we can obtain the upper bound of the settle time .

Simulation result is depicted in Figure 3, which shows the states and of the second-order controlled SNN (4.1). The simulation result has confirmed the effectiveness of our main results.

Figure 3: Trajectories of SNN (4.1) under the controller (4.3) with in Example 4.2.

5. Conclusions

In this paper, we have investigated the issue of finite-time stabilization for SNNs with noise perturbations by constructing a continuous nonlinear stabilizator. Meanwhile, Based on the Lyapunov-Krasovskii functional method combining with the LMI techniques, a sufficient criterion is derived for the states of the augmented system to be global finite-time stable in probability. Subsequently, for SNNs with parameter uncertainties, the robust finite-time stabilizator could be designed well. Finally, two illustrative examples have been used to demonstrate the usefulness of the main results. It is expected that the theory established in this paper can be widely applied in delayed systems, particularly in those discontinuous cases. It will be an interesting topic in our future research.

Acknowledgments

This research is partially supported by the National Natural Science Foundation of China under Grant no. 61272297, the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant no. 12KJB120002 and by the scientific research support project for teachers with doctor’s degree, Jiangsu Normal University under Grant no. 11XLR19.

References

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990.

E. D. Sontag, “A “universal” construction of Artstein's theorem on nonlinear stabilization,” Systems & Control Letters, vol. 13, no. 2, pp. 117–123, 1989.

J. Lu, D. W. C. Ho, and L. Wu, “Exponential stabilization of switched stochastic dynamical networks,” Nonlinearity, vol. 22, no. 4, pp. 889–911, 2009.

J. Lu, D. W. C. Ho, and Z. Wang, “Pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers,” IEEE Transactions on Neural Networks, vol. 20, no. 10, pp. 1617–1629, 2009.

C. Li, X. Liao, and T. Huang, “Exponential stabilization of chaotic systems with delay by periodically intermittent control,” Chaos, vol. 17, no. 1, Article ID 013103, 7 pages, 2007.

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proceedings of the National Academy of Sciences of the United States of America, vol. 79, no. 8, pp. 2554–2558, 1982.

Z. Wang, Y. Liu, M. Li, and X. Liu, “Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays,” IEEE Transactions on Neural Networks, vol. 17, no. 3, pp. 814–820, 2006.

X. Liu and J. Cao, “On periodic solutions of neural networks via differential inclusions,” Neural Networks, vol. 22, no. 4, pp. 329–334, 2009.

X. Liu, T. Chen, J. Cao, and W. Lu, “Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches,” Neural Networks, vol. 24, no. 10, pp. 1013–1021, 2011.

O. M. Kwon, S. M. Lee, J. H. Park, and E. J. Cha, “New approaches on stability criteria for neural networks with interval time-varying delays,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9953–9964, 2012.

Z. G. Wu, J. H. Park, H. Y. Su, and J. Chu, “New results on exponential passivity of neural networks with time-varying delays,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1593–1599, 2012.

M. J. Park, O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay,” Journal of the Franklin Institute, vol. 349, no. 5, pp. 1699–1720, 2012.

X. Y. Liu and J. D. Cao, “Local synchronization of one-to-one coupled neural networks with discontinuous activations,” Cognitive Neurodynamics, vol. 5, no. 1, pp. 13–20, 2011.

O. M. Kwon, J. H. Park, and S. M. Lee, “On improved passivity criteria of uncertain neural networks with time-varying delays,” Nonlinear Dynamics, vol. 67, no. 2, pp. 1261–1271, 2012.

O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “A new augmented Lyapunov-Krasovskii functional approach to exponential passivity for neural networks with time-varying delays,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 10231–10238, 2011.

O. M. Kwon, S. M. Lee, and J. H. Park, “Improved delay-dependent exponential stability for uncertain stochastic neural networks with time-varying delays,” Physics Letters A, vol. 374, no. 10, pp. 1232–1241, 2010.

X. Y. Liu, J. D. Cao, and W. W. Yu, “Filippov systems and quasi-synchronization control for switched networks,” Chaos, vol. 22, no. 3, Article ID 033110, 12 pages, 2012.

X. Y. Liu, J. D. Cao, and G. Huang, “Complete periodic synchronization of delayed neural networks with discontinuous activations,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 7, pp. 2151–2164, 2010.

H. Huang, D. W. C. Ho, and J. Lam, “Stochastic stability analysis of fuzzy Hopfield neural networks with time-varying delays,” IEEE Transactions on Circuits and Systems II, vol. 52, no. 5, pp. 251–255, 2005.

Z. Wang, Y. Liu, and X. Liu, “On global asymptotic stability of neural networks with discrete and distributed delays,” Physics Letters A, vol. 345, no. 4-6, pp. 299–308, 2005.

S. Xu, J. Lam, and D. W. C. Ho, “Novel global robust stability criteria for interval neural networks with multiple time-varying delays,” Physics Letters A, vol. 342, no. 4, pp. 322–330, 2005.

J. Liang, Z. Wang, Y. Liu, and X. Liu, “Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks,” IEEE Transactions on Neural Networks, vol. 19, no. 11, pp. 1910–1921, 2008.

P. Li, J. Lam, and Z. Shu, “On the transient and steady-state estimates of interval genetic regulatory networks,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 2, pp. 336–349, 2010.

X. Y. Liu and J. D. Cao, “Robust state estimation for neural networks with discontinuous activations.,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 6, pp. 1425–1437, 2010.

Z. Wang, D. W. C. Ho, and X. Liu, “Variance-constrained filtering for uncertain stochastic systems with missing measurements,” IEEE Transactions on Automatic Control, vol. 48, no. 7, pp. 1254–1258, 2003.

S. Blythe, X. Mao, and X. Liao, “Stability of stochastic delay neural networks,” Journal of the Franklin Institute, vol. 338, no. 4, pp. 481–495, 2001.

V. T. Haimo, “Finite time controllers,” SIAM Journal on Control and Optimization, vol. 24, no. 4, pp. 760–770, 1986.

S. Bhat and D. Bernstein, “Finite-time stability of homogeneous systems,” in Proceedings of the American Control Conference, vol. 4, pp. 2513–2514, 1997.

W. S. Chen and L. C. Jiao, “Comments on ‘Finite-time stability theorem of stochastic nonlinear systems [Automatica 46(2010) 2105-2108]’,” Automatica, vol. 47, no. 7, pp. 1544–1545, 2011.

Y. Hong, J. Wang, and D. Cheng, “Adaptive finite-time control of nonlinear systems with parametric uncertainty,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 858–862, 2006.

S. Y. Khoo, L. H. Xie, and Z. H. Man, “Man, Robust finite-time consensus tracking algorithm for multi robot systems,” IEEE/ASME Transactions on Mechatronics, vol. 14, no. 2, pp. 219–228, 2009.

F. Xiao, L. Wang, J. Chen, and Y. P. Gao, “Finite-time formation control for multi-agent systems,” Automatica, vol. 45, no. 11, pp. 2605–2611, 2009.

L. Wang and F. Xiao, “Finite-time consensus problems for networks of dynamic agents,” IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 950–955, 2010.

J. Cortés, “Finite-time convergent gradient flows with applications to network consensus,” Automatica, vol. 42, no. 11, pp. 1993–2000, 2006.

G. Chen, F. L. Lewis, and L. H. Xie, “Finite-time distributed consensus via binary control protocols,” Automatica, vol. 47, no. 9, pp. 1962–1968, 2011.

Q. Hui, W. M. Haddad, and S. P. Bhat, “Finite-time semistability and consensus for nonlinear dynamical networks,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1887–1900, 2008.

H. Deng, M. Krstić, and R. J. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” IEEE Transactions on Automatic Control, vol. 46, no. 8, pp. 1237–1253, 2001.

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

G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1988.

S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994.

X. S. Yang and J. D. Cao, “Finite-time stochastic synchronization of complex networks,” Applied Mathematical Modelling, vol. 34, no. 11, pp. 3631–3641, 2010.