Adaptive almost sure asymptotically stability for neutral-type neural networks with stochastic perturbation and Markovian switching

doi:10.1016/j.neucom.2014.12.069

Neurocomputing

Volume 156, 25 May 2015, Pages 151-156

https://doi.org/10.1016/j.neucom.2014.12.069 Get rights and content

Abstract

The problem of stability via adaptive controller is considered for time-delay neutral-type neural networks with stochastic noise and Markovian switching in this paper. A new criterion of almost sure (a.s.) asymptotic stability for a general neutral-type stochastic differential equation is proposed. Based on this criterion, and by using of the generalized Itô׳s formula and the M-matrix method, a delay dependent sufficient condition is established to ensure the almost sure asymptotic stability for neutral-type neural networks with stochastic perturbation and Markovian switching. Meanwhile, the update law of the feedback control is determined. A numerical example is provided to verify the usefulness of the criterion proposed in this paper.

Introduction

Recently, the stability of neutral-type neural networks, which depend on the derivative of the state and the delay state have attracted a lot of attention (see e.g. [1], [2], [3], [4], [5], [6], [7], [8] and the references therein) since the fact that some physical systems in the real world can be described by neutral-type models (see [9]).

As we know, the synaptic transmission in real nervous systems can be viewed as a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes [10]. In general, Gaussian noise has been regarded as the disturbance arising in neural networks (see e.g. [11], [12], [13], [14], [15], [16], [17], [18], [19], and the references therein).

Also, it has been shown that many neural networks may experience abrupt changes in their structure and parameters due to the phenomena such as component failures or repairs, changing subsystem interconnections and abrupt environmental disturbances. In this situation, neural networks may be treated as systems which have finite modes, and the modes may switch from one to another at different times, and can be described by finite-state Markov chains. The stability analysis problem for neural networks with Markovian switching has therefore received much research attention (see e.g. [13], [14], [17], [20], [21], [22], and the references therein).

Although the importance of adaptive stabilization has been widely recognized, no related results have been established for time-delay neutral-type neural networks with Markovian switching and stochastic perturbation. Motivated by the studies mentioned above, we aim to tackle the problem of almost sure asymptotic stability for time-delayed neural networks with stochastic noise and Markovian switching via adaptive control. A new criterion of almost sure (a. s.) asymptotic stability for a general neutral-type stochastic differential equation is proposed. Based on this criterion, and by using of the generalized Itô׳s formula and the M-matrix method, a delay dependent sufficient condition is established to ensure the almost sure asymptotic stability for neutral-type neural networks with stochastic perturbation and Markovian switching. Meanwhile, the update law of the feedback control is determined. A numerical example is provided to verify the usefulness of the criterion proposed in this paper.

The attributions of this work lie in two aspects. Firstly, a new criterion of almost sure asymptotic stability for a general neutral-type stochastic differential equation is proposed which extends the existing results. The second one is that we concern with the M-matrix method to obtain the delay dependent sufficient condition of the almost sure asymptotic stability for neutral-type neural networks with stochastic perturbation and Markovian switching.

Section snippets

System and problem description and preliminaries

Consider an n-dimensional time-delay neutral-type neural network with Markovian switching and stochastic noise of the form $d [x (t) - D (r (t)) x (t - τ)] = [- A (r (t)) x (t) + W (r (t)) φ (x (t)) + W_{d} (r (t)) φ (x (t - τ)) + U (r (t))] dt + g (t, r (t), x (t), x (t - τ)) d ω (t)$ where $x (t) = {[x_{1} (t) \dots x_{n} (t)]}^{T} \in R^{n}$ is the state vector associated with the n neurons. $φ (x (t)) = {[φ_{1} (x_{1} (t)) \dots φ_{n} (x_{n} (t))]}^{T}$ denotes the neuron activation function, τ denotes the constant time-delay.

${r (t), t \geq 0}$ is a right-continuous Markov chain on the complete probability space $(Ω, F, {F$

Main results

We are now in a position to derive the condition under which the neutral-type time-delay neural networks (1) with stochastic disturbance and Markovian switching is almost surely asymptotically stable. We will divide the discussion into two parts: (1) $p \geq 3$ and (2) p=2.

Theorem 1

Let Assumption 1, Assumption 2, Assumption 3 hold, and $p \geq 3$ . Assume that $M := - diag {{\underset{︸}{η, η, \dots, η}}_{S}} - Γ$ is a nonsingular M-matrix, where $η = - p ς + (1 / 2) (p - 2) (κ^{2} + {\bar{α}}^{2} + 2 Φ^{2} + (p - 1) (G_{1} + G_{2})) + (1 / 2) p (β^{2} + β_{d}^{2}),$ with $\bar{α} = \max_{i \in S} \max_{1 \leq j \leq n} a_{j}^{i}$ , $β = \max_{i \in S} (ρ (W_{i}))$ , $β_{d} = \max$

Numerical simulation

One example is presented here in order to show the usefulness of our results. Our aim is to examine the adaptive a. s. stability for the given neural networks with stochastic noise and Markovian switching.

Let the state space of Markov chain $r {(t)}_{t \geq 0}$ be $S = {1, 2}$ with generator $Γ = [\begin{matrix} - 1.2 & 1.2 \\ 0.5 & - 0.5 \end{matrix}] .$ Consider a time-delay neural network (1) with Markovian switching and the following network parameters: $A_{1} = [\begin{matrix} 1 & 0 \\ 0 & 0.9 \end{matrix}], A_{2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], W_{1} = [\begin{matrix} 2.7 & 8 \\ 0.4 & 2.7 \end{matrix}], W_{2} = [\begin{matrix} 3 & 8 \\ 0.3 & 2.5 \end{matrix}], W_{d 1} = [\begin{matrix} - 4.3 & 1 \\ 0.7 & - 4.3 \end{matrix}], W_{d 2} = [\begin{matrix} - 5 & 0.3 \\ 0.3 & - 5 \end{matrix}], D_{1} = [\begin{matrix} 0.1 & 0 \\ 0 & 0.2 \end{matrix}], D$

Conclusion

In this paper, we have dealt with the problem of almost sure asymptotically stability via adaptive controller for neutral-type time-delay neural networks with stochastic noise and Markovian switching. By using Lyapunov functional and the M-matrix technique, a sufficient condition depended on the switching mode and time-delay is presented to guarantee the almost sure asymptotically stability of the neutral-type time-delay neural networks, and the desired controller can be also obtained. An

Liuwei Zhou received his B.S. degree in mathematics from Zhejiang Normal University, Zhejiang, China, in 2007. He obtained his M.Sc. degree in mathematical finance from University of York, UK, in 2010. Now he is under a Ph.D. candidate in control science and engineering from Donghua University, Shanghai, China. His current research interests include stochastic control with Levy process, control of neural networks, and their applies on financial risk management, derivative pricing and portfolio

References (25)

X. Li
Global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type
Appl. Math. Comput.
(2010)
J.H. Park
Synchronization of cellular neural networks of neutral type via dynamic feedback controller
Chaos Solitons Fractals
(2009)
Q. Zhu et al.
Adaptive synchronization for stochastic neural networks of neutral-type with mixed time-delays
Neurocomputing
(2013)
R. Lu et al.
H-infinity filtering for singular systems with communication delays
Signal Process.
(2010)
R. Lu et al.
New delay-dependent robust stability criteria for uncertain neutral systems with mixed delays
J. Frankl. Inst.
(2014)
Z. Wang et al.
Robust stability for stochastic hopfield neural networks with time delays
Nonlinear Anal.Real World Appl.
(December 2006)
W. Yu et al.
Synchronization control of stochastic delayed neural networks
Physica AStat. Mech. Appl.
(January 2007)
D. Tong et al.
Adaptive synchronization for stochastic ts fuzzy neural networks with time-delay and Markovian jumping parameters
Neurocomputing
(October 2013)
Z. Wu et al.
Improved result on stability analysis of discrete stochastic neural networks with time delay
Phys. Lett. A
(April 2009)
Y. Liu et al.
An LMI approach to stability analysis of stochastic high-order Markovian jumping neural networks with mixed time delays
Nonlinear Anal.: Hybrid Syst.
(March 2008)

Y. Xu et al.

Output feedback stabilization for markov-based nonuniformly sampled-data networked control systems

Syst. Control Lett.

(2013)

Y. Xu et al.

Stability analysis of networked control systems with round-Robin scheduling and packet dropouts

J. Frankl. Inst.

(2013)

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Zhijie Wang was born in ZheJiang, China, in 1969. He received his Bachelor, Master, and Doctor degrees in Electrical Engineering from Donghua University, Shanghai, China, in 1991, 1994 and 1997, respectively. From 2000 to 2002, he was a visiting researcher in Aihara Laboratory, Department of Mathematical Engineering, The University of Tokyo. He is currently a professor at College of Information Science and Technology, Donghua University. His research interests include neural networks, fuzzy logic, and intelligent systems.

Xiantao Hu was born in 1976. He received his B.S. degree and Master degree in Huazhong University of science and technology in 1999 and 2004, respectively. His current research interests include database management, mass data processing, network analysis.

Bo Chu received the Bachelor degree of Engineering from Donghua University, Shanghai, China, in 2013. Now, she is a Master degree candidate in Department of Software Engineering at Donghua University, Shanghai, China. Her current research interests include database management, big data and web development.

Wuneng Zhou received the B.S. degree in mathematics from Huazhong Normal University, China, in 1982 and the Ph.D. degree in control science and engineering from Zhejiang University, China, in 2005. Now he is a professor in Donghua University, Shanghai, China. His current research interests include the stability, the synchronization and control for neural networks, wireless sensor networks and complex networks.

^☆: This work is partially supported by the National Natural Science Foundation of China under Grant no. 61075105, and the Natural Science Foundation of Shanghai under Grant no. 15ZR1401800, no. 12ZR1440200 and the China Scholarship Council.

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Adaptive almost sure asymptotically stability for neutral-type neural networks with stochastic perturbation and Markovian switching☆

Abstract

Introduction

Section snippets

System and problem description and preliminaries

Main results

Numerical simulation

Conclusion

Appl. Math. Comput.

Chaos Solitons Fractals

Neurocomputing

Signal Process.

J. Frankl. Inst.

Nonlinear Anal.Real World Appl.

Physica AStat. Mech. Appl.

Neurocomputing

Phys. Lett. A

Nonlinear Anal.: Hybrid Syst.

Syst. Control Lett.

J. Frankl. Inst.