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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 597298, 8 pages
Finite-Time Boundedness for a Class of Delayed Markovian Jumping Neural Networks with Partly Unknown Transition Probabilities
College of Information Sciences and Technology, Hainan University, Haikou 570228, China
Received 11 November 2013; Accepted 8 December 2013; Published 6 January 2014
Academic Editor: Zhengguang Wu
Copyright © 2014 Li Liang. 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.
This paper is concerned with the problem of finite-time boundedness for a class of delayed Markovian jumping neural networks with partly unknown transition probabilities. By introducing the appropriate stochastic Lyapunov-Krasovskii functional and the concept of stochastically finite-time stochastic boundedness for Markovian jumping neural networks, a new method is proposed to guarantee that the state trajectory remains in a bounded region of the state space over a prespecified finite-time interval. Finally, numerical examples are given to illustrate the effectiveness and reduced conservativeness of the proposed results.
Over the past decades, delayed neural networks have been successfully applied in the pattern recognition, signal processing, image processing, and pattern recognition problems. However, these successful applications mostly rely on the dynamic behaviors of delayed neural networks and some of these applications are dependent on stability of the equilibria of neural networks. Up to now, there have been a large number of results related to dynamical behaviors of delayed neural networks [1–8].
On the one, in the past few decades, Markovian jump systems have gained special research attention. Such class of systems is a special class of stochastic hybrid systems, which may switch from one to another at the different time. Such as component failures, sudden environmental disturbance and abrupt variations of a nonlinear system [9–11]. Moreover, it is shown that such jumping can be decided by a Markovian chain . For the linear Markovian jumping systems, many important issues have been devoted extensively such as stability, stabilization, control synthesis, and filter design [13–16]. In reality, however, it is worth mentioning that most of the gotten results are based on the implicit assumptions that the complete knowledge of transition probabilities is known. It is known that in most situations, the transition probabilities rate of Markovian jump systems and networks is not known; it is difficult to obtain all the transition probabilities. Therefore, it is of great importance to investigate the partly unknown transition probabilities. Very recently, the systems with partially unknown transition probabilities have been fully investigated and many important results have been obtained; for a recent survey on this topic and related questions, one can refer to [17–23]. However, it has been shown that the existing delay-dependent results are conservative.
On the other hand, the practical problems which described system stay as not exceeding a given threshold over finite-time interval are considered. Compared with classical Lyapunov stability, finite-time stability was studied to tackle the transient behavior of systems in the finite-time interval. Recently, the concept of finite-time stability has been revisited in the terms of linear matrix inequalities (LMIs); some results have been obtained to guarantee that system is finite-time stable and finite-time bounded [24–39]. To the best of our knowledge, the finite-time stability analysis for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates has not been tackled, and such a situation motivates our present study.
The main contribution of this paper lies in proposing a novel method for finite-time boundedness of delayed Markovian jumping neural networks with partly unknown transition probabilities. The considered system is more general than the systems with completely known or completely unknown transition probabilities, which can be regarded as two special cases of the one tackled here. In contrast to study on Markovian jumping neural networks with time delays, the knowledge of the unknown elements is not required in our method. By employing the appropriate Lyapunov-Krasovskii functional, the sufficient conditions are obtained to ensure that the system does not exceed a given threshold in a specified time interval. The finite-time bounded criteria can be tackled in the form of LMIs. Finally, numerical examples are given to demonstrate that the derived results are less conservative and more useful than some existent ones.
Given a probability space where , and , respectively, represents the sample space, the algebra of events and the probability measure which defined on . In this paper, we consider the following -neuron Markovian jumping neural network over the space described by where represents the neural state vector of the system, is the nonlinear activation function with the initial condition , describes the rate with each neuron which would reset its potential to resting state in isolation, and are the connection weight matrices and the delayed connection weight matrices, respectively, and denotes a constant external input vector. are the time-varying delays which satisfy where and are constant scalars and , .
Remark 1. This assumption is often employed to investigate the stability of neural networks. It is worth noting that if this assumption is not true, corresponding time-delays are not a continuous function belonging to a given interval; neither the lower nor upper bounds for time-varying delays are available. Therefore, it may lead to more conservativeness.
Let the random form process be the Markovian stochastic process taking values on the finite set with transition rate matrix , ; namely, for , , one has where , , and , denote switching rate from mode at time to mode at time . For all , . Moreover, the Markovian process transition matrix is defined as follows:
Moreover, the transition rates of jumping process in this paper are considered to be partly accessed; that is, some elements in matrix are unknown. Therefore, the transition rates matrix which is Markovian jump system (1) may be as follows: where represents the inaccessible elements. For notational clarity, for all , we denote and we denote that
Moreover, if , and can be further described, respectively, as where represents the th known element with the index in the th row of matrix . represents the th unknown element with the index in the th row of matrix .
Set contains modes of system (1) and, for , the system matrices of the th mode are denoted by , , and , which are considered to be real known with appropriate dimensions.
Remark 2. The Markovian jump process in the literature is always assumed ether to be completely known () or completely unknown (). Therefore, our transition probabilities matrix considered in this paper is more general than the Markovian jump systems and therefore covers the existing ones.
Assumption 3. The neuron state-based nonlinear function considered in Markovian jump system (1) is bounded and satisfies for all , , with being known real constants with .
It should be noted that by using the Brouwer fixed-point theorem, there should exist at least the one equilibrium point for system (1). Assuming that is the equilibrium point of (1) and using the transformation , system (1) can be converted to the following system: where , , and , . According to Assumption 3, one can obtain that
Definition 5 (see ). Let be a stochastic positive functional and define its weak infinitesimal operator as
3. Finite-Time Performance Analysis
In this section, one method would be employed to analyze the finite-time stability of Markovian jump systems with partial information on transition probabilities.
Theorem 6. Given a time constant , the delayed Markovian jumping neural networks (1) are stochastically finite-time bounded with respect to , if there exist a positive constant , mode-dependent symmetric positive-definite matrices , , , , , a set of symmetric matrices , any appropriately dimensioned matrices , , , and scalars such that the following matrix inequalities hold: where
Proof. We consider the following the stochastic Lyapunov-Krasovskii functional:
with , , , , , and being positive definite matrices and
For notational simplicity, let
Let be the infinitesimal generator of random process ; then for each , , we can obtain that Similar to the process above, it yields
From (18) and (19), we obtain that
Also, it results from (10) that for any appropriately dimensioned matrices , , , one can obtain From (16)–(24), we have where
By the fact that , we can rewrite as
Thus, from (6), we have
Then, for and if , can be guaranteed. On the other hand, for and if , can be further expressed as
Similarly, (18) and (19) can be rewritten, respectively, as
It is well known that ; according to (6), one can also obtain On the other hand, from (32) and the needed constant , it yields that from which we can easily get that Note that ; we can obtain the following inequality: On the other hand, from (16), we can get Then, we can obtain
By condition (14), we can obtain
By Definition 4, we conclude that Markovian jump system (1) is stochastically finite-time bounded with respect to .
Remark 7. In this paper, it is in contrast with existing results for delay-dependent Markovian jump systems with partly unknown transition probabilities, and another different method is presented to tackle the unknown elements in the transition matrix. Compared with , some slack matrix variables are introduced in this paper based on the probability identity , which leads to less conservativeness than .
Remark 8. Theorem 6 develops a finite-time bounded criterion of Markovian jumping neural networks with time-varying delays and partially known transition rates. Theorem 6 makes full use of the information of the subsystems’ upper bounds of the time-varying delays, which also brings us the less conservativeness.
Remark 9. In our paper, and may indicate the different upper bounds during various time-delay intervals which satisfies condition (2), respectively. However, in existing work, for example, , and are always extended to and , respectively, which may inevitably lead to the conservativeness. Therefore, in order to reduce the conservatism, the cases above are taken into account by employing the stochastic Lyapunov-Krasovskii functional (16).
4. Illustrative Example
Example 1. Consider a class of delayed Markovian jumping neural networks (9) with two operation modes in :
The mode switching is governed by a Markov chain that has the following transition rate matrix:
In this paper, let the initial values for , , , and time-varying delay be , which means that and . Through Theorem 6 and optimization over value , it yields that delayed Markovian jumping neural networks (9) are finite-time bounded with respect to with minimal while minimal in  is 5.4296, which shows the less conservative result in this paper.
Example 2. Consider a class of delayed Markovian jumping neural networks (9) with partially known transition rates and operation modes described as follows:
The three cases of the transition rates matrices are considered as
With the same mode switching rates, initial values and time-varying delays, through Theorem 6 and optimization over value , it yields that in Case I, ; in Case II, ; in Case III, . Therefore, the delayed Markovian jumping neural networks (9) are finite-time bounded with respect to .
Remark 10. The accessibility of the jumping process in the existing literature is commonly assumed to be completely accessible or completely unaccessible. Note that the transition probabilities are still viewed as accessible in this paper. Therefore, the transition probabilities matrix considered in this paper is more general assumption than Markovian jump systems.
Unlike most existing research results focusing on Lyapunov stability property of Markovian jump system, our paper investigated finite-time stability which concerns the boundedness of state during the delayed Markovian jump interval. In this paper, we have examined the problems of finite-time boundedness for a class of delayed Markovian jumping neural networks with partly unknown transition probabilities. Based on the analysis result, the static state feedback finite-time boundedness is given. Although the derived result is not in LMIs form, we can turn it into LMIs feasibility problem by fixing some parameters. At last, numerical examples are also given to demonstrate the effectiveness of the proposed approach.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Natural Science Foundation of Hainan province (111002).
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