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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 615947, 15 pages
Sliding Intermittent Control for BAM Neural Networks with Delays
1Department of Mathematics, Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 210096, China
2Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway
3Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Received 9 May 2013; Accepted 29 May 2013
Academic Editor: Zidong Wang
Copyright © 2013 Jianqiang Hu 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.
This paper addresses the exponential stability problem for a class of delayed bidirectional associative memory (BAM) neural networks with delays. A sliding intermittent controller which takes the advantages of the periodically intermittent control idea and the impulsive control scheme is proposed and employed to the delayed BAM system. With the adjustable parameter taking different particular values, such a sliding intermittent control method can comprise several kinds of control schemes as special cases, such as the continuous feedback control, the impulsive control, the periodically intermittent control, and the semi-impulsive control. By using analysis techniques and the Lyapunov function methods, some sufficient criteria are derived for the closed-loop delayed BAM neural networks to be globally exponentially stable. Finally, two illustrative examples are given to show the effectiveness of the proposed control scheme and the obtained theoretical results.
Since the bidirectional associative memory (BAM) neural networks were first proposed by Kosko  which are well known as the extension of the unidirectional autoassociators such as the Hopfield neural networks, they have been widely studied due to their extensive applications such as pattern recognition, signal or image processing, solving optimization problems, and automatic control [2####^~^~^~^~^~^####x2013;7]. Later, constant delays are introduced in  to the BAM neural networks, and it is proved that the delayed versions of the neural networks are significant for handling certain motion-related optimization problems . For more results concerning the dynamical behaviors of the BAM neural networks with delays, we refer to [10, 11].
In practice, most of the neural networks are unstable or convergent with a rate far less than the requirement. Under such cases we need to try to stabilize them or speed up the convergence rate of the neural system in order to make the system work more efficiently. Therefore, the designing of appropriate control input becomes extremely urgent. When it comes to the problem of stabilizing a nonlinear system, it is natural to consider the feedback strategies. There are two basic kinds of feedback control: the state feedback control and the output feedback control. When referring to the control methods, different kinds of schemes have been utilized to stabilize the nonlinear system such as static feedback control , delayed feedback control , adaptive control , fuzzy control , sampled control , sliding mode control , and random control [18, 19]. In terms of the control time, the controllers are classified with continuous control and discontinuous control. Compared with the continuous control, the discontinuous control including the impulsive control [20, 21] and the intermittent control [22, 23] has attracted much more attention, and it is very effective, practical, and applicable in many areas, especially in secure communication .
In the literature, the impulsive neural networks have been extensively studied from the following two aspects: either the system is subject to the impulsive state displacements at fixed time instants or the system is imposed by external impulsive control . The main idea of periodically intermittent control  is that when the system signal becomes weak to a low level, the external control will be imposed to supplement the loss of signal; after some period of time the external control is stopped; in the next control period the external control is needed again. Compared with the method of periodically intermittent control, the system with the impulsive control is activated only at some isolated time moments. Both the impulsive control and the intermittent control have their own benefits and disadvantages. The main difference between these two control techniques lies in the length of control period; the former has zero duration, while the later has a nonzero control width. Meanwhile, the cost of the intermittent control is much higher.
The aim of this paper is to design a sliding intermittent controller by combining the advantages of both the impulsive control and the periodically intermittent control. More specifically, in one control period, we will impose the continuous state feedback control at the preceding control width and the impulsive control in the latter control width. The sliding intermittent control method is very flexible and could achieve the expected control performance. The sketch of such a controller is shown in Figure 1. Motivated by the name of the slide rheostat in the physical electronic circuitry, we named such a joint controller as the sliding intermittent controller.
In this paper, we will investigate the exponential stability problem of the delayed BAM neural networks under the proposed sliding intermittent control. The closed-loop neural system becomes a switched network where the switching rules are dependent on the time index. To the best of the authors####^~^~^~^~^~^####x2019; knowledge, this is the first time in the literature to consider such a joint controller. The rest of the paper is organized as follows. In Section 2, the model discussed in this paper and the novel sliding intermittent control idea are introduced, and some preliminaries are also given. In Section 3, several sufficient criteria are established to ensure the delayed BAM neural networks to be exponentially stable under the sliding intermittent control scheme. Meanwhile, several particular cases are discussed. In Section 4, two illustrative examples are given to demonstrate the effectiveness of the proposed results. And finally, the paper is concluded in Section 5.
2. Problem Formulation and Some Preliminaries
The delayed bidirectional associative memory (BAM) neural networks have been investigated in [8, 9] as follows: where the index set, ;, are the activations of theth neuron and theth neuron, respectively;, are positive constants denoting the rates with which the cellsandreset their potential to the resting states when isolated from the other cells and inputs; time delaysandare nonnegative constants corresponding to the finite speeds of axonal signal transmission with;andare the delayed connection weights denoting the strengths of connectivity between the cellsand;anddenote, respectively, theth and theth components of an external input source introduced from the network outside to the cellsand;andare bounded nonlinear activation functions, and throughout this paper they are assumed to satisfy the following conditions: where, and, ; positive scalars, are known.
Remark 1. The above conditions are general in the literature to study the existence and uniqueness of the equilibrium for the delayed BAM neural networks (1) without assuming the activation functions to be monotonic or differentiable [27####^~^~^~^~^~^####x2013;29].
Lettingwith and be an equilibrium of the system (1) and denoting, the equilibrium of the network (1) can be transformed to the origin of the following system: whereand.
It is generally known that, under some specific cases such as the abrupt changes of system parameters or the occurrence of time delays, the network may present unstable dynamics such as bifurcation, oscillation, divergence, or instability. In this paper, we will focus on the unstable delayed BAM neural system (3). To stabilize the network (3), the novel sliding intermittent control scheme is imposed as follows: where; and ; constantandare scalars denoting the control width and the control period, respectively;, and, ,, and . The impulsive sequencesatisfies , are controllers imposed on the system (3) when, and, are the impulsive operators imposed at the impulsive moments. Here we use the linear state feedback and linear impulsive control strategies; that is, where, , , and are gain constants.
Remark 2. The sliding intermittent control idea is illustrated in Figure 1 with the adjustable parameter(i.e., the control widthshown in Figure 1). The closed-loop system (4) can be the continuous controlled neural networks (), the impulsive controlled neural networks (), or the hybrid controlled neural networks ().
The system (4) is supplemented with initial function given by,,, where,, andrepresents the set of all-dimensional continuous functions defined on the interval. Obviously, the solution of (4) is piecewise left-hand continuous with possible discontinuity atfor.
The following definition and lemmas are introduced before we give the main results of the paper.
Definition 3. The system (4) is said to be globally exponentially stable if there exist constants, such that for all, holds for alland.
Lemma 4 (see ). Letbe a continuous function such that is satisfied for. If, then where,is the unique positive real root of the equation.
Lemma 5 (see ). Let,,, andbe constants, and assume thatis a piecewise continuous nonnegative function satisfying If there exists constantsuch that hold for, then where,, andis the unique positive root of the equation.
3. Main Results
In this section, some sufficient criteria will be given based on the sliding intermittent control scheme. First, the global exponential stability of the closed-loop hybrid neural networks (4) is analyzed, and then several criteria are obtained by setting different parameters in the sliding intermittent controller.
Theorem 6. Assume the upper bound delayand the external imposed impulsive strengths satisfy, . Under the sliding intermittent control, the closed-loop control system (4) is globally exponentially stable if there exist constantsand, such that the following conditions hold:
where;;;;;is used to denote the last impulsive moment on the intervalwith;;;with functionsdefined as and, are, respectively, the unique positive real root of the equationsand.
More specifically, we have the following inequality: with.
Proof. For the functionsanddefined in (16), from the condition (13), it is clear that
Sinceandare continuous on and ,, there must exist constantsandsuch that and. By setting, one obtains that, for any,
Now consider the Lyapunov function defined as follows:
whereand. Obviously,is a positive definite function for.
When, one is easy to have Calculating the upper right Dini derivative ofalong the solutions of network (4), from the above inequality, we get where and. By Lemma 4, one has whereis the unique positive real root of the equation.
When and, , , it is easy to have Without loss of generality, suppose, , whereis assumed to be the last impulsive moment on the interval. It follows from the inequality (24) that where;.
Whenand, Considering the condition that, , we have and which infers that From condition (14) and Lemma 5, one has where and is the unique positive real root of the equation.
Now, we are ready to estimatebased on the inequalities (23) and (29) with the method of induction.####^~^~^~^~^~^####x2009;When, one obtains ####^~^~^~^~^~^####x2009;When, one can derive ####^~^~^~^~^~^####x2009;When, we have ####^~^~^~^~^~^####x2009;When, one can derive
By induction, one can derive the following estimation offor any integer:####^~^~^~^~^~^####x2009;when, ####^~^~^~^~^~^####x2009;and when,
By settingand substituting it to the above two inequalities, one has, for, and, for, Therefore, we have which means that, for any, where. And the conclusion that the origin of network (4) is exponentially stable. The proof is complete.
Remark 7. In the above Theorem 6, the control widthwhich does not include the boundary cases. If the parameter, the controlled system approximates to impulsive neural networks. If the parameter, the controlled system approximates to continuous neural networks. And the controlled system can be handled, respectively, by the methods of the impulsive system and the continuous system. In order to avoid such boundary cases, usually we can take; that is, the periodically intermittent controller and the impulsive controller play a significant role in the process of control; this results a switched neural networks.
Remark 8. Most of the literatures [28, 32] concerning the global exponential stability of the delayed BAM neural networks with impulses have focused on the stable system. Namely, without the impulsive disturbance, the original neural networks are stable, and under the impulsive disturbance the system can still be kept stable with particular conditions. While, in this article, the impulses can be viewed as either control input or impulsive disturbance, at the same time, the original system is not required to be stable initially.
If adjustable parameter, the controlled system turns out to be the following model; that is, the original unstable system (3) is controlled with the continuous feedback controller. Under such case, the conditions in the following proposition will guarantee the closed-loop system to be globally exponentially stable:
Proposition 9. Under the continuous feedback control scheme, the origin of the closed-loop control system (40) is globally exponentially stable if there exist constantsandsuch that Furthermore, one has the following inequality: where the parametersandare consistent with the ones in Theorem 6.
Proof. Consider the Lyapunov function defined as follows: The upper right Dini derivative ofwith respect to timealong the solutions of the network (40) can be calculated as follows: which means that. Hence we have , and this completes the proof.
If the impulsive strengths, namely, there are no impulsive controls on the latter control interval in each control period, which means the closed-loop system is only subject to the continuous feedback control in the preceding control width of each control period. Such a case is then reduced to the pure periodically intermittent control, and the neural network system (4) turns into the following controlled neural network (45). The conditions in the following proposition will guarantee the closed-loop system to be globally exponentially stable. In order to obtain the main result, the following lemma is given firstly
Lemma 10 (see ). Letbe a continuous function such that is satisfied for. If,, then
Proposition 11. Assuming the upper bound delay, under the periodically intermittent control, the closed-loop control system (45) is globally exponentially stable if the control gains andsatisfy the following conditions:(i)(ii)Moreover, we have whereandis any positive constant; , and the parameters, , , , and are consistent with those in Theorem 6.
Proof. Considering the same Lyapunov function as that in Theorem 6, by similar analytical technique, one can get the following results on the control period and the control width as follows.(1)When, we have. By Lemma 4, one obtains that, for any, .(2)When, we have. By Lemma 10, it is derived that, for any, .
From the above inequality relationships, by similar estimation procedure, we can get the following conclusion: By the notational expression, one can further obtain Hence it follows that, for any, and this completes the proof.
If the adjustable parameter, that is, the original unstable system (3) is subjected to the impulsive controller, then system (4) becomes the following impulsive neural networks (54). Under such a case, it would be natural to assume that the frequency of the impulses should not be too low; that is, some restrictions on the impulsive periods and the impulsive strengths are needed. The conditions in the following proposition will guarantee the closed-loop system (54) to be globally exponentially stable:
Proposition 12. Assume the external imposed impulsive strengths. The origin of the closed-loop control system (54) is globally exponentially stable if there exists constantsuch that, for, the following conditions hold: More specifically, we have where the parameters, , , , andare consistent with those in Theorem 6.
Proof. Considering the same Lyapunov function as that in Theorem 6, by similar analytical technique, one can get the following results on the impulsive interval and the impulse moments as follows.(1)When, the upper right Dini derivative ofalong the solution of (54) is depicted as.(2)When,.
From Lemma 5 and the conditions in (55), it follows that which means and the proof is completed.
If the continuous feedback control gains, that is, there is no feedback on the preceding control interval in a control period, and only impulsive control is imposed on the latter control interval, this means the system is only under the piecewise impulsive (not the uniformly distributed) control. As for such case, the impulsive system is reduced to the following one: From (59), it is noticed that the occurrence of the impulses is not uniformly distributed since the impulses never occur on the interval, whereas they frequently occur on the interval. By observing the proof of Proposition 12, the result is somewhat more conservative especially when the control periodis very large and the control widthapproaches to one. In order to describe the conservatism for such case, we will utilize the notation of average impulsive interval proposed in  to characterize the occurrence frequency of the impulses. The definition of average impulsive interval and the corresponding impulsive differential inequality are given firstly.
Definition 13 ( average impulsive interval). The average impulsive interval of the impulsive sequenceis equal toif there exist positive integerand positive numbersuch that wheredenotes the number of impulsive times of the impulsive sequenceon the interval.
Lemma 14 (see Lakshmikantham et al., Theorem, [33, 34]). Let() denote the set of piecewise continuous (piecewise continuously differentiable) functions with first kind of discontinuities at, fromto. If the following conditions hold, where, ,andare real constants, then
Lemma 15. Let,,, andbe constants, and assume thatis a piecewise continuous nonnegative function satisfying and the average impulsive interval of the impulsive sequenceis equal to. If the following inequality holds, then one has where,is the unique positive root of the equation####^~^~^~^~^~^####x2009;+####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;+####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;+####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;.
Proof. By Lemma 14 and the definition of average impulsive interval, it follows from (63) that, for,
Denote. Since, andand, one knows thathas a unique positive root.
Next, it is claimed that, for all, When,
Supposing (67) is not always true for, there must exist one time pointsuch that From inequalities (66) and (69), one can obtain that Considering the equality, one has which contradicts with the first inequality of (69). Therefore, the inequality (67) holds, and this completes the proof.
The following proposition will guarantee the system (59) to be globally exponentially stable.
Proposition 16. Assume the external imposed impulsive strengths, , and the average impulsive interval of the impulsive sequenceis equal to. The impulsive system (59) is globally exponentially stable if the following condition holds: Moreover, we have where, and, , and are defined as in Theorem 6 andis the unique positive root of the equation.
Proof. Considering the same Lyapunov function as that in Theorem 6, the following results on the impulsive interval and the impulse moments can be obtained.(1)When, the upper right Dini derivative ofalong the solutions of (59) is depicted as.(2)When, ,.
From the above two inequality relationships, and Lemma 15, one has whereis the unique positive root of the equation. The proof is completed.
Remark 17. It should be pointed out that, in the preceding control width of the control period, other kinds of continuous controllers can also be used to achieve the same performance. For example, in , the adaptive control scheme has been employed in the control width instead of the continuous state feedback, where the adjusting gains can be designed based on different norms. We can borrow such an idea to the sliding intermittent control design. Moreover, the sliding width parameter can berather than the fixed width, and the period can berather than constant. By doing so, we might obtain more general conditions. On the other hand, the phenomena of stochastic nonlinearities are extremely ubiquitous in practical controlled systems [36####^~^~^~^~^~^####x2013;39]; hence it is more reasonable to consider neural networks with random nonlinearities, and this will be our future works.
4. Illustrative Example
In this section, we present some examples to illustrate the applicability and efficiency of the proposed control scheme.
Example 1. Considering the following extensively studied BAM neural system, with,,,,, and. Obviously,. The initial condition is given as, , . With the above system parameters, the phase diagram of system (75) is given in Figure 2. Obviously, the origin of system (75) is not stable. We will design the sliding intermittent controller to stabilize it.
Applying the sliding intermittent controller to the unstable system (75), one can derive the following system:
In the following, we will give the convergence results by simulating the system (76). Firstly, by setting the continuous feedback gains,and the impulsive strengths. With the above parameters setting, calculations show thatand