Sliding Intermittent Control for BAM Neural Networks with Delays

Abstract

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.

1. Introduction

Since the bidirectional associative memory (BAM) neural networks were first proposed by Kosko [1] 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 [2–7]. Later, constant delays are introduced in [8] 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 [9]. 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 [12], delayed feedback control [13], adaptive control [14], fuzzy control [15], sampled control [16], sliding mode control [17], 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 [24].

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 [25]. The main idea of periodically intermittent control [26] 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.

Figure 1 

Sketch map of the sliding intermittent control.

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’ 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: 𝑤̇_𝑖(𝑡)=−𝑎_𝑖𝑤_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑧_𝑗𝑡−𝜏_𝑖𝑗+𝑐_𝑖,𝑖∈ℐ_𝑛,𝑧̇_𝑗(𝑡)=−𝑏_𝑗𝑧_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑤_𝑖𝑡−𝜎_𝑗𝑖+𝑑_𝑗,𝑗∈ℐ_𝑚, (1) where the index setℐ_𝑛={1,2,…,𝑛}, ℐ_𝑚={1,2,…,𝑚};𝑤_𝑖(𝑡), 𝑧_𝑗(𝑡)∈ℝare the activations of the𝑖th neuron and the𝑗th neuron, respectively;𝑎_𝑖, 𝑏_𝑗are positive constants denoting the rates with which the cells𝑖and𝑗reset their potential to the resting states when isolated from the other cells and inputs; time delays𝜏_𝑖𝑗and𝜎_𝑗𝑖are nonnegative constants corresponding to the finite speeds of axonal signal transmission with𝜏^∗=max_𝑖,𝑗{𝜏_𝑖𝑗,𝜎_𝑗𝑖};𝑝_𝑖𝑗and𝑞_𝑗𝑖are the delayed connection weights denoting the strengths of connectivity between the cells𝑗and𝑖;𝑐_𝑖and𝑑_𝑗denote, respectively, the𝑖th and the𝑗th components of an external input source introduced from the network outside to the cells𝑖and𝑗;𝑓_𝑗and𝑔_𝑖are bounded nonlinear activation functions, and throughout this paper they are assumed to satisfy the following conditions: 𝑓_𝑗𝜈₁−𝑓_𝑗𝜈₂≤𝐿^𝑓_𝑗𝜈₁−𝜈₂,𝑔_𝑖𝜈₁−𝑔_𝑖𝜈₂≤𝐿^𝑔_𝑖𝜈₁−𝜈₂, (2) where𝜈₁, 𝜈₂∈ℝand𝑖∈ℐ_𝑛, 𝑗∈ℐ_𝑚; positive scalars𝐿^𝑓_𝑗, 𝐿^𝑔_𝑖are 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 [27–29].
Letting(𝑤^∗,𝑧^∗)with𝑤^∗=(𝑤^∗₁,𝑤^∗₂,…,𝑤^∗_𝑛)^𝑇∈ℝ^𝑛 and 𝑧^∗=(𝑧^∗₁,𝑧^∗₂,…,𝑧^∗_𝑚)^𝑇∈ℝ^𝑚be an equilibrium of the system (1) and denoting𝑥_𝑖(𝑡)=𝑤_𝑖(𝑡)−𝑤^∗_𝑖,𝑦_𝑗(𝑡)=𝑧_𝑗(𝑡)−𝑧^∗_𝑗, the equilibrium of the network (1) can be transformed to the origin of the following system: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗,𝑖∈ℐ_𝑛,𝑦̇_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖,𝑗∈ℐ_𝑚, (3) where𝑓_𝑗(𝑦_𝑗)=𝑓_𝑗(𝑦_𝑗+𝑧^∗_𝑗)−𝑓_𝑗(𝑧^∗_𝑗)and𝑔_𝑖(𝑥_𝑖)=𝑔_𝑖(𝑥_𝑖+𝑤^∗_𝑖)−𝑔_𝑖(𝑤^∗_𝑖).

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: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗+𝑢_𝑖⁽¹⁾(𝑡),[()𝑇),𝑥̇𝑡∈𝑙𝑇,𝑙+𝜃_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗,𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇),𝑡≠𝑡_𝑘,Δ𝑥_𝑖(𝑡)=𝐼_𝑘⁽¹⁾𝑥_𝑖(𝑡),𝑡=𝑡_𝑘,𝑦̇_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖+𝑢_𝑗⁽²⁾(𝑡),[()𝑇),𝑦̇𝑡∈𝑙𝑇,𝑙+𝜃_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖,𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇),𝑡≠𝑡_𝑘,Δ𝑦_𝑗(𝑡)=𝐼_𝑘⁽²⁾𝑦_𝑗(𝑡),𝑡=𝑡_𝑘, (4) where𝑖∈ℐ_𝑛,𝑗∈ℐ_𝑚;𝑙∈ℕ⁺₀≜{0,1,2,…,} and 𝑘∈ℕ⁺≜{1,2,…}; constant𝜃∈[0,1]and𝑇are scalars denoting the control width and the control period, respectively;Δ𝑥_𝑖(𝑡_𝑘)=𝑥_𝑖(𝑡⁺_𝑘)−𝑥_𝑖(𝑡⁻_𝑘), Δ𝑦_𝑗(𝑡_𝑘)=𝑦_𝑗(𝑡⁺_𝑘)−𝑦_𝑗(𝑡⁻_𝑘)and𝑥_𝑖(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑥_𝑖(𝑡_𝑘+ℎ), 𝑥_𝑖(𝑡_𝑘)=𝑥_𝑖(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑥_𝑖(𝑡_𝑘+ℎ),𝑦_𝑗(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑦_𝑗(𝑡_𝑘+ℎ), and 𝑦_𝑗(𝑡_𝑘)=𝑦_𝑗(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑦_𝑗(𝑡_𝑘+ℎ). The impulsive sequence{𝑡_𝑘}^∞_𝑘=1satisfies 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯,lim_𝑘→∞𝑡_𝑘𝑡=∞,_𝑘∈_𝑙∈ℕ+0((𝑙+𝜃)𝑇,(𝑙+1)𝑇]. (5)𝑢_𝑖⁽¹⁾(𝑡), 𝑢_𝑗⁽²⁾(𝑡)are controllers imposed on the system (3) when𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇), and𝐼_𝑘⁽¹⁾(⋅), 𝐼_𝑘⁽²⁾(⋅)are the impulsive operators imposed at the impulsive moments{𝑡_𝑘}^∞_𝑘=1. Here we use the linear state feedback and linear impulsive control strategies; that is, 𝑢_𝑖⁽¹⁾(𝑡)=𝑘_𝑖⁽¹⁾𝑥_𝑖(𝑡),𝑢_𝑗⁽²⁾(𝑡)=𝑘_𝑗⁽²⁾𝑦_𝑗(𝑡),𝐼_𝑘⁽¹⁾𝑥_𝑖𝑡_𝑘=ℎ⁽¹⁾_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝐼_𝑘⁽²⁾𝑦_𝑗𝑡_𝑘=ℎ⁽²⁾_𝑗𝑘𝑦_𝑗𝑡_𝑘, (6) where𝑘_𝑖⁽¹⁾, 𝑘_𝑗⁽²⁾, ℎ⁽¹⁾_𝑖𝑘, and ℎ⁽²⁾_𝑗𝑘are gain constants.

Remark 2. The sliding intermittent control idea is illustrated in Figure 1 with the adjustable parameter𝜃(i.e., the control width𝜃shown in Figure 1). The closed-loop system (4) can be the continuous controlled neural networks (𝜃=1), the impulsive controlled neural networks (𝜃=0), or the hybrid controlled neural networks (0<𝜃<1).
The system (4) is supplemented with initial function given by𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑦_𝑗(𝑠)=𝜓_𝑗(𝑠),−𝜏^∗≤𝑠≤0, where𝜑(𝑠)=(𝜑₁(𝑠),𝜑₂(𝑠),…,𝜑_𝑛(𝑠))^𝑇∈𝐶([−𝜏^∗,0],ℝ^𝑛),𝜓(𝑠)=(𝜓₁(𝑠),𝜓₂(𝑠),…,𝜓_𝑚(𝑠))^𝑇∈𝐶([−𝜏^∗,0],ℝ^𝑚), and𝐶([−𝜏^∗,0],ℝ^𝑛)represents the set of all𝑛-dimensional continuous functions defined on the interval[−𝜏^∗,0]. Obviously, the solution of (4) is piecewise left-hand continuous with possible discontinuity at𝑡=𝑡_𝑘for𝑘∈ℕ⁺.

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 constants𝛼>0, 𝛽>0such that for all𝑡>0, ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡)≤𝛽𝑒^−𝛼𝑡sup_{−𝜏∗≤𝑠≤0}^𝑛_𝑖=1𝜑_𝑖(𝑠)+^𝑚_𝑗=1𝜓_𝑗(𝑠) (7) holds for all𝜑(⋅)∈𝐶([−𝜏^∗,0],ℝ^𝑛)and𝜓(⋅)∈𝐶([−𝜏^∗,0],ℝ^𝑚).

Lemma 4 (see [30]). Let𝑉(⋅)∶[𝑡₀−𝜏,∞)→[0,∞)be a continuous function such that 𝑉̇(𝑡)≤−𝑎𝑉(𝑡)+𝑏𝑉(𝑡) (8) is satisfied for𝑡≥𝑡₀. If𝑎>𝑏>0, then 𝑉(𝑡)≤𝑉𝑡₀𝑒^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀, (9) where𝑉(𝑡)=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑠),𝜆>0is the unique positive real root of the equation−𝑎+𝜆+𝑏𝑒^𝜆𝜏=0.

Lemma 5 (see [31]). Let𝑞≥0,𝜏>0,𝜇_𝑘>0(𝑘=1,2,…), and𝑝be constants, and assume that𝑉(𝑡)is a piecewise continuous nonnegative function satisfying 𝐷⁺𝑉(𝑡)≤𝑝𝑉(𝑡)+𝑞𝑉(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘𝑉𝑡⁺_𝑘≤𝜇_𝑘𝑉𝑡_𝑘,𝑘=1,2,…. (10) If there exists constant𝛽such that ln𝜇_𝑘𝑡_𝑘−𝑡_𝑘−1≤𝛽,𝑝+𝑑𝑞+𝛽<0, (11) hold for𝑘=1,2,…, then 𝑉(𝑡)≤𝑑𝑉𝑡₀𝑒^{−𝜆(𝑡−𝑡0)}, (12) where𝑉(𝑡)=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑠),𝑑=sup_𝑘{𝑒^{𝛽(𝑡𝑘−𝑡𝑘−1)},𝑒^{−𝛽(𝑡𝑘−𝑡𝑘−1)}}<∞, and𝜆is the unique positive root of the equation𝜆+𝑝+𝑑𝑞𝑒^𝜆𝜏+𝛽=0.

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 delay𝜏^∗<min{𝜃𝑇,(1−𝜃)𝑇}and the external imposed impulsive strengths satisfyℎ⁽¹⁾_𝑖𝑘, ℎ⁽²⁾_𝑗𝑘≠−1. Under the sliding intermittent control, the closed-loop control system (4) is globally exponentially stable if there exist constants𝛽and𝑘_𝑖⁽¹⁾, 𝑘_𝑗⁽²⁾such that the following conditions hold:
(i)𝑎_𝑖−𝑘_𝑖⁽¹⁾>𝐿^𝑔_𝑖^𝑚_𝑗=1𝑞_𝑗𝑖,𝑖∈ℐ_𝑛,𝑏_𝑗−𝑘_𝑗⁽²⁾>𝐿^𝑓_𝑗^𝑛_𝑖=1𝑝_𝑖𝑗,𝑗∈ℐ_𝑚, (13)
(ii)ln𝜇_𝑘+1𝑡_𝑘+1−𝑡_𝑘≤𝛽,𝜌+𝑑𝑏̃₁+𝛽<0,𝑘∈ℕ⁺⧵𝑖_𝑙∣𝑙=0,1,2,…, (14)
(iii)𝑎₁>𝑏̃₁,𝜚≜𝜂+𝜆₁𝜃−𝛾−𝜆₂𝜃+𝛾−ln𝑑𝑇>0, (15)
where𝑎₁=min_𝑖,𝑗{𝑎_𝑖−𝑘_𝑖⁽¹⁾−𝜂,𝑏_𝑗−𝑘_𝑗⁽²⁾−𝜂};𝑏̃₁=max_𝑖{𝐿^𝑔_𝑖∑^𝑚_𝑗=1|𝑞_𝑗𝑖|𝑒^{𝜂𝜎𝑗𝑖}}+max_𝑗{𝐿^𝑓_𝑗∑^𝑛_𝑖=1|𝑝_𝑖𝑗|𝑒^{𝜂𝜏𝑖𝑗}};𝜇_𝑘=max_𝑖,𝑗{|1+ℎ⁽¹⁾_𝑖𝑘|,|1+ℎ⁽²⁾_𝑗𝑘|};𝜌=max_𝑖,𝑗{𝜂−𝑎_𝑖,𝜂−𝑏_𝑗};𝑑=sup_𝑘{𝑒^{𝛽(𝑡𝑘+1−𝑡𝑘)},𝑒^{−𝛽(𝑡𝑘+1−𝑡𝑘)}};𝑡_𝑖𝑙is used to denote the last impulsive moment on the interval[(𝑙+𝜃)𝑇,(𝑙+1)𝑇)with𝑡_𝑖𝑙<(𝑙+1)𝑇−𝜏^∗;𝛾=𝜏^∗/𝑇;0<𝜂≤𝜂^∗;𝜂^∗=min_𝑖,𝑗{𝜃^∗_𝑖,𝜗^∗_𝑗∣𝐹_𝑖(𝜃^∗_𝑖)=0,𝐺_𝑗(𝜗^∗_𝑗)=0}with functions𝐹_𝑖(⋅),𝐺_𝑗(⋅)defined as 𝐹_𝑖𝜃_𝑖=𝑎_𝑖−𝑘_𝑖⁽¹⁾−𝜃_𝑖−^𝑚_𝑗=1𝑞_𝑗𝑖𝐿^𝑔_𝑖𝑒^{𝜃𝑖𝜎𝑗𝑖},𝜃_𝑖∈[),𝐺0,∞_𝑗𝜗_𝑗=𝑏_𝑗−𝑘_𝑗⁽²⁾−𝜗_𝑗−^𝑛_𝑖=1𝑝_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝜗𝑗𝜏𝑖𝑗},𝜗_𝑗∈[0,∞), (16) and𝜆₁, 𝜆₂are, respectively, the unique positive real root of the equations−𝑎₁+𝜆₁+𝑏̃₁𝑒^𝜆1𝜏∗=0and𝜆₂+𝜌+𝑑𝑏̃₁𝑒^𝜆2𝜏∗+𝛽=0.
More specifically, we have the following inequality: ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡)≤𝑀𝑒^−𝜚𝑡sup_{−𝜏∗≤𝑠≤0}^𝑛_𝑖=1𝜑_𝑖(𝑠)+^𝑚_𝑗=1𝜓_𝑗(𝑠),𝑡>0, (17) with𝑀=max{𝑒^{𝜆1(𝜃−𝛾)𝜃𝑇},𝑒^{𝜆2(1−𝜃−𝛾)𝜃𝑇+(1−𝜃)ln𝑑}}.

Proof. For the functions𝐹_𝑖(⋅)and𝐺_𝑗(⋅)defined in (16), from the condition (13), it is clear that 𝐹_𝑖(0)=𝑎_𝑖−𝑘_𝑖⁽¹⁾−^𝑚_𝑗=1𝑞_𝑗𝑖𝐿^𝑔_𝑖𝐺>0,_𝑗(0)=𝑏_𝑗−𝑘_𝑗⁽²⁾−^𝑛_𝑖=1𝑝_𝑖𝑗𝐿^𝑓_𝑗>0. (18) Since𝐹_𝑖(⋅)and𝐺_𝑗(⋅)are continuous on[0,∞) and lim_{𝜃𝑖→∞}𝐹_𝑖(𝜃_𝑖)=−∞,lim_{𝜗𝑗→∞}𝐺_𝑗(𝜗_𝑗)=−∞, there must exist constants𝜃^∗_𝑖>0and𝜗^∗_𝑗>0such that𝐹_𝑖(𝜃^∗_𝑖)=0 and𝐺_𝑗(𝜗^∗_𝑗)=0. By setting𝜂^∗=min{𝜃^∗₁,𝜃^∗₂,…,𝜃^∗_𝑛,𝜗^∗₁,𝜗^∗₂,…,𝜗^∗_𝑚}, one obtains that, for any0<𝜂≤𝜂^∗, 𝐹_𝑖𝜂=𝑎_𝑖−𝑘_𝑖⁽¹⁾−𝜂−^𝑚_𝑗=1𝑞_𝑗𝑖𝐿^𝑔_𝑖𝑒^{𝜂𝜎𝑗𝑖}𝐺≥0,_𝑗𝜂=𝑏_𝑗−𝑘_𝑗⁽²⁾−𝜂−^𝑛_𝑖=1𝑝_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝜂𝜏𝑖𝑗}≥0. (19) Now consider the Lyapunov function defined as follows: 𝑉(𝑡)=^𝑛_𝑖=1𝑢_𝑖(𝑡)+^𝑚_𝑗=1𝑣_𝑗(𝑡),𝑡≥0, (20) where𝑢_𝑖(𝑡)=𝑒^𝜂𝑡|𝑥_𝑖(𝑡)|and𝑣_𝑗(𝑡)=𝑒^𝜂𝑡|𝑦_𝑗(𝑡)|. Obviously,𝑉(𝑡)is a positive definite function for𝑡≥0.
(1) When𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇),𝑙∈ℕ⁺₀, one is easy to have 𝐷⁺𝑢_𝑖(𝑡)≤−𝑎_𝑖−𝑘_𝑖⁽¹⁾−𝜂𝑢_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝜂𝜏𝑖𝑗}𝑣_𝑗𝑡−𝜏_𝑖𝑗,𝐷⁺𝑣_𝑗(𝑡)≤−𝑏_𝑗−𝑘_𝑗⁽²⁾−𝜂𝑣_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝐿^𝑔_𝑖𝑒^{𝜂𝜎𝑗𝑖}𝑢_𝑖𝑡−𝜎_𝑗𝑖. (21) Calculating the upper right Dini derivative of𝑉(𝑡)along the solutions of network (4), from the above inequality, we get 𝐷⁺𝑉(𝑡)≤−^𝑛_𝑖=1𝑎_𝑖−𝑘_𝑖⁽¹⁾−𝜂𝑢_𝑖(𝑡)−^𝑚_𝑗=1𝑏_𝑗−𝑘_𝑗⁽²⁾−𝜂𝑣_𝑗(𝑡)+^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝜂𝜏𝑖𝑗}𝑣_𝑗𝑡−𝜏_𝑖𝑗+^𝑚𝑛_𝑗=1𝑖=1𝑞_𝑗𝑖𝐿^𝑔_𝑖𝑒^{𝜂𝜎𝑗𝑖}𝑢_𝑖𝑡−𝜎_𝑗𝑖≤−𝑎₁𝑉(𝑡)+𝑏̃₁sup_{𝑡−𝜏∗≤𝑠≤𝑡}𝑉(𝑠), (22) where𝑎₁=min_𝑖,𝑗{𝑎_𝑖−𝑘_𝑖⁽¹⁾−𝜂,𝑏_𝑗−𝑘_𝑗⁽²⁾−𝜂} and𝑏̃₁=max_𝑖{𝐿^𝑔_𝑖∑^𝑚_𝑗=1|𝑞_𝑗𝑖|𝑒^{𝜂𝜎𝑗𝑖}}+max_𝑗{𝐿^𝑓_𝑗∑^𝑛_𝑖=1|𝑝_𝑖𝑗|𝑒^{𝜂𝜏𝑖𝑗}}. By Lemma 4, one has 𝑉(𝑡)≤𝑒^{−𝜆1(𝑡−𝑙𝑇)}sup_{𝑙𝑇−𝜏∗≤𝑠≤𝑙𝑇}𝑉(𝑠)=𝑒^{−𝜆1(𝑡−𝑙𝑇)}𝑉(),[()𝑇),𝑙𝑇𝑡∈𝑙𝑇,𝑙+𝜃 (23) where𝜆₁is the unique positive real root of the equation−𝑎₁+𝜆₁+𝑏̃₁𝑒^𝜆1𝜏∗=0.
(2) When𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇) and𝑡≠𝑡_𝑘, 𝑙∈ℕ⁺₀, 𝑘∈ℕ⁺, it is easy to have 𝐷⁺𝑢_𝑖(𝑡)≤−𝑎_𝑖−𝜂𝑢_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝜂𝜏𝑖𝑗}𝑣_𝑗𝑡−𝜏_𝑖𝑗,𝐷⁺𝑣_𝑗(𝑡)≤−𝑏_𝑗−𝜂𝑣_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝐿^𝑔_𝑖𝑒^{𝜂𝜎𝑗𝑖}𝑢_𝑖𝑡−𝜎_𝑗𝑖. (24) Without loss of generality, suppose𝑡∈(𝑡_𝑘−1,𝑡_𝑘], 𝑘≤𝑖_𝑙, where𝑡_𝑖𝑙is assumed to be the last impulsive moment on the interval[(𝑙+𝜃)𝑇,(𝑙+1)𝑇). It follows from the inequality (24) that 𝐷⁺𝑉(𝑡)≤^𝑛_𝑖=1𝜂−𝑎_𝑖𝑢_𝑖(𝑡)+^𝑚_𝑗=1𝜂−𝑏_𝑗𝑣_𝑗(𝑡)+^𝑛_𝑖=1𝐿^𝑔_𝑖^𝑚_𝑗=1𝑒^{𝜂𝜎𝑗𝑖}𝑞_𝑗𝑖𝑢_𝑖𝑡−𝜎_𝑗𝑖+^𝑚_𝑗=1𝐿^𝑓_𝑗^𝑛_𝑖=1𝑒^{𝜂𝜏𝑖𝑗}𝑝_𝑖𝑗𝑣_𝑗𝑡−𝜏_𝑖𝑗≤𝜌𝑉(𝑡)+𝑏̃₁sup_{𝑡−𝜏∗≤𝑠≤𝑡}𝑉(𝑠), (25) where𝜌=max_𝑖,𝑗{𝜂−𝑎_𝑖,𝜂−𝑏_𝑗};𝑏̃₁=max_𝑖{𝐿^𝑔_𝑖∑^𝑚_𝑗=1𝑒^{𝜂𝜎𝑗𝑖}|𝑞_𝑗𝑖|}+max_𝑗{𝐿^𝑓_𝑗∑^𝑛_𝑖=1𝑒^{𝜂𝜏𝑖𝑗}|𝑝_𝑖𝑗|}.
When𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇)and𝑡=𝑡_𝑘, 𝑥_𝑖𝑡⁺_𝑘=𝑥_𝑖𝑡_𝑘+𝐼_𝑘⁽¹⁾𝑥_𝑖𝑡_𝑘=1+ℎ⁽¹⁾_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑦_𝑗𝑡⁺_𝑘=𝑦_𝑗𝑡_𝑘+𝐼_𝑘⁽²⁾𝑦_𝑗𝑡_𝑘=1+ℎ⁽²⁾_𝑗𝑘𝑦_𝑗𝑡_𝑘. (26) Considering the condition thatℎ⁽¹⁾_𝑖𝑘, ℎ⁽²⁾_𝑗𝑘≠−1, we have𝜇_𝑘>0 and 𝑢_𝑖𝑡⁺_𝑘=𝑒^{𝜂𝑡+𝑘}𝑥_𝑖𝑡⁺_𝑘≤1+ℎ⁽¹⁾_𝑖𝑘𝑢_𝑖𝑡_𝑘≤𝜇_𝑘𝑢_𝑖𝑡_𝑘,𝑖∈ℐ_𝑛,𝑣_𝑗𝑡⁺_𝑘=𝑒^{𝜂𝑡+𝑘}𝑦_𝑗𝑡⁺_𝑘≤1+ℎ⁽²⁾_𝑗𝑘𝑣_𝑗𝑡_𝑘≤𝜇_𝑘𝑣_𝑗𝑡_𝑘,𝑗∈ℐ_𝑚, (27) which infers that 𝑉𝑡⁺_𝑘=^𝑛_𝑖=1𝑢_𝑖𝑡⁺_𝑘+^𝑚_𝑗=1𝑣_𝑗𝑡⁺_𝑘≤𝜇_𝑘𝑉𝑡_𝑘. (28) From condition (14) and Lemma 5, one has 𝑉(𝑡)≤𝑑𝑒^{−𝜆2(𝑡−(𝑙+𝜃)𝑇)}𝑉)𝑇),)()𝑇),((𝑙+𝜃𝑡∈[(𝑙+𝜃𝑇,𝑙+1 (29) where𝑑=sup_𝑘{𝑒^{𝛽(𝑡𝑘−𝑡𝑘−1)},𝑒^{−𝛽(𝑡𝑘−𝑡𝑘−1)}} and 𝜆₂is the unique positive real root of the equation𝜆₂+𝜌+𝑑𝑏̃₁𝑒^𝜆2𝜏∗+𝛽=0.
(3) Now, we are ready to estimate𝑉(𝑡)based on the inequalities (23) and (29) with the method of induction. When𝑡∈[0,𝜃𝑇), one obtains 𝑉(𝑡)≤𝑒^−𝜆1𝑡𝑉(0). (30) When𝑡∈[𝜃𝑇,𝑇), one can derive 𝑉(𝑡)≤𝑑𝑒^{−𝜆2(𝑡−𝜃𝑇)}𝑉(𝜃𝑇)≤𝑑𝑒^{−[𝜆2(𝑡−𝜃𝑇)+𝜆1(𝜃𝑇−𝜏∗)]}𝑉(0). (31) When𝑡∈[𝑇,(1+𝜃)𝑇), we have 𝑉(𝑡)≤𝑒^{−𝜆1(𝑡−𝑇)}𝑉(𝑇)≤𝑑𝑒^{−[𝜆2(𝑇−𝜏∗−𝜃𝑇)+𝜆1(𝑡−(1−𝜃)𝑇−𝜏∗)]}𝑉(0). (32) When𝑡∈[(1+𝜃)𝑇,2𝑇), one can derive 𝑉(𝑡)≤𝑑𝑒^{−𝜆2(𝑡−(1+𝜃)𝑇)}𝑉((1+𝜃)𝑇)≤𝑑²𝑒^{−[𝜆2(𝑡−2𝜃𝑇−𝜏∗)+2𝜆1(𝜃𝑇−𝜏∗)]}𝑉(0). (33)
By induction, one can derive the following estimation of𝑉(𝑡)for any integer𝑙∈ℕ⁺₀: when𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇), 𝑉(𝑡)≤𝑑^𝑙𝑒^{−[𝑙𝜆2(𝑇−𝜃𝑇−𝜏∗)+𝜆1(𝑡−𝑙(𝑇−𝜃𝑇+𝜏∗))]}𝑉(0), (34) and when𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇), 𝑉(𝑡)≤𝑑^𝑙+1𝑒^{−[𝜆2(𝑡−(𝑙+1)𝜃𝑇−𝑙𝜏∗)+(𝑙+1)𝜆1(𝜃𝑇−𝜏∗)]}𝑉(0). (35)
By setting𝛾=𝜏^∗/𝑇and substituting it to the above two inequalities, one has, for𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇), 𝑉(𝑡)≤𝑑^𝑙𝑒^{−[𝜆2𝑙𝑇(1−𝜃−𝛾)+𝜆1(𝑡−𝑙𝑇)+𝜆1(𝜃−𝛾)𝑙𝑇]}𝑉(0)≤𝑑^𝑙𝑒^{[𝜆2(𝜃+𝛾)𝑙𝑇−𝜆1(𝜃−𝛾)𝑙𝑇]}𝑉(0)≤𝑑^𝑙𝑒^{[𝜆2(𝜃+𝛾)𝑡+𝜆1(𝜃−𝛾)(−𝑡+𝜃𝑇)]}𝑉(0)=𝑑^𝑙𝑒^{−[𝜆1(𝜃−𝛾)−𝜆2(𝜃+𝛾)]𝑡}𝑒^{𝜆1(𝜃−𝛾)𝜃𝑇}𝑉(0)≤𝑒^{(ln𝑑/𝑇)𝑡}𝑒^{−[𝜆1(𝜃−𝛾)−𝜆2(𝜃+𝛾)]𝑡}𝑒^{𝜆1(𝜃−𝛾)𝜃𝑇}𝑉(0)≤𝑒^{−[𝜆1(𝜃−𝛾)−𝜆2(𝜃+𝛾)−((ln𝑑)/𝑇)]𝑡}𝑀𝑉(0), (36) and, for𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇), 𝑉(𝑡)≤𝑑^𝑙+1𝑒^{−[𝜆2(𝑡−(𝑙+1)𝜃𝑇−𝑙𝛾𝑇)+𝜆1(𝜃−𝛾)(𝑙+1)𝑇]}𝑉(0)≤𝑑^𝑙+1𝑒^{[𝜆2𝜃𝑇+𝜆2𝑙𝑇(𝜃+𝛾)−𝜆1(𝜃−𝛾)𝑡]}𝑉(0)≤𝑑^𝑙+1𝑒^{[𝜆2𝜃𝑇+𝜆2(𝑡−𝜃𝑇)(𝜃+𝛾)−𝜆1(𝜃−𝛾)𝑡]}𝑉(0)=𝑑^𝑙+1𝑒^{[𝜆2(𝜃+𝛾)𝑡−𝜆1(𝜃−𝛾)𝑡]+𝜆2𝜃𝑇(1−𝜃−𝛾)}𝑉(0)≤𝑒^{((𝑡/𝑇)−𝜃+1)ln𝑑}𝑒^{−[𝜆1(𝜃−𝛾)−𝜆2(𝜃+𝛾)]𝑡}𝑒^{𝜆2𝜃𝑇(1−𝜃−𝛾)}𝑉(0)≤𝑒^{−[𝜆1(𝜃−𝛾)−𝜆2(𝜃+𝛾)−((ln𝑑)/𝑇)]𝑡}𝑀𝑉(0). (37) Therefore, we have 𝑉(𝑡)≤𝑒^{−[𝜆1(𝜃−𝛾)−𝜆2(𝜃+𝛾)−((ln𝑑)/𝑇)]𝑡}𝑀𝑉(0),𝑡>0, (38) which means that, for any𝑡>0, ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡)≤𝑒^{−[𝜂+𝜆1(𝜃−𝛾)−𝜆2(𝜃+𝛾)−((ln𝑑)/𝑇)]𝑡}𝑀𝑉(0), (39) where𝑀=max{𝑒^{𝜆1(𝜃−𝛾)𝜃𝑇},𝑒^{𝜆2(1−𝜃−𝛾)𝜃𝑇+(1−𝜃)ln𝑑}}. 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 width𝜃∈(0,1)which does not include the boundary cases. If the parameter𝜃→0, the controlled system approximates to impulsive neural networks. If the parameter𝜃→1, 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 take𝜃=0.5; 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 parameter𝜃=1, 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: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗+𝑢_𝑖⁽¹⁾(𝑡),𝑖∈ℐ_𝑛,𝑦̇_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖+𝑢_𝑗⁽²⁾(𝑡),𝑗∈ℐ_𝑚. (40)

Proposition 9. Under the continuous feedback control scheme, the origin of the closed-loop control system (40) is globally exponentially stable if there exist constants𝑘_𝑖⁽¹⁾and𝑘_𝑗⁽²⁾such that 𝑎_𝑖−𝑘_𝑖⁽¹⁾>𝐿^𝑔_𝑖^𝑚_𝑗=1𝑞_𝑗𝑖,𝑖∈ℐ_𝑛,𝑏_𝑗−𝑘_𝑗⁽²⁾>𝐿^𝑓_𝑗^𝑛_𝑖=1𝑝_𝑖𝑗,𝑗∈ℐ_𝑚. (41) Furthermore, one has the following inequality: ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡≤𝑏̃₁)𝜏^∗+1𝑒^−𝜂𝑡sup_{−𝜏∗≤𝑠≤0}^𝑛_𝑖=1𝜑_𝑖(𝑠+_𝑚𝑗=1)𝜓_𝑗(𝑠),𝑡>0, (42) where the parameters𝜂and𝑏̃_𝑖are consistent with the ones in Theorem 6.

Proof. Consider the Lyapunov function defined as follows: 𝑊(𝑡)(𝑡)+=𝑉_{𝑛𝑚𝑖=1𝑗=1}𝑝_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝜂𝜏𝑖𝑗𝑡}_{𝑡−𝜏𝑖𝑗}𝑣_𝑗(𝑠)+𝑞d𝑠_𝑗𝑖𝐿^𝑔_𝑖𝑒^{𝜂𝜎𝑗𝑖𝑡}_{𝑡−𝜎𝑗𝑖}𝑢_𝑖(𝑠)d𝑠,𝑡≥0. (43) The upper right Dini derivative of𝑊(𝑡)with respect to time𝑡along the solutions of the network (40) can be calculated as follows: 𝐷⁺𝑊(𝑡)=^𝑛_𝑖=1𝐷⁺𝑢_𝑖(𝑡)+^𝑚_𝑗=1𝐷⁺𝑣_𝑗(𝑡)+^𝑛𝑚_𝑖=1𝑗=1𝑝_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝜂𝜏𝑖𝑗}𝑣_𝑗(𝑡)−𝑣_𝑗𝑡−𝜏_𝑖𝑗+^𝑚𝑛_𝑗=1𝑖=1𝑞_𝑗𝑖𝐿^𝑔_𝑖𝑒^{𝜂𝜎𝑗𝑖}𝑢_𝑖(𝑡)−𝑢_𝑖𝑡−𝜎_𝑗𝑖≤−^𝑛_𝑖=1𝑎_𝑖−𝑘_𝑖⁽¹⁾−𝜂−^𝑚_𝑗=1𝑞_𝑗𝑖𝐿^𝑔_𝑖𝑒^{𝜂𝜎𝑗𝑖}𝑢_𝑖(𝑡)−^𝑚_𝑗=1𝑏_𝑗−𝑘_𝑗⁽²⁾−𝜂−^𝑛_𝑖=1𝑝_𝑖𝑗𝐿^𝑓_𝑗𝑒^{𝜂𝜏𝑖𝑗}𝑣_𝑗(𝑡)=−^𝑛_𝑖=1𝐹_𝑖𝜂𝑢_𝑖(𝑡)−^𝑚_𝑗=1𝐺_𝑗𝜂𝑣_𝑗(𝑡)≤0 (44) which means that𝑊(𝑡)≤𝑊(0). Hence we have ∑^𝑛_𝑖=1|𝑥_𝑖(𝑡)|+∑^𝑚_𝑗=1|𝑦_𝑗(𝑡)|≤(𝑏̃₁𝜏^∗+1)𝑒^−𝜂𝑡𝑉(0), and this completes the proof.

If the impulsive strengthsℎ⁽¹⁾_𝑖𝑘=ℎ⁽²⁾_𝑗𝑘≡0, 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 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗+𝑢_𝑖⁽¹⁾(𝑡),()𝑥̇𝑙𝑇≤𝑡<𝑙+𝜃𝑇,_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗,()()𝑦̇𝑙+𝜃𝑇≤𝑡<𝑙+1𝑇,_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖+𝑢_𝑗⁽²⁾(𝑡),()𝑦̇𝑙𝑇≤𝑡<𝑙+𝜃𝑇,_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖,(𝑙+𝜃)𝑇≤𝑡<(𝑙+1)𝑇. (45)

Lemma 10 (see [23]). Let𝑉(⋅):[𝑡₀−𝜏,∞)→[0,∞)be a continuous function such that 𝑉̇(𝑡)≤𝑎𝑉(𝑡)+𝑏sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑠) (46) is satisfied for𝑡≥𝑡₀. If𝑎>0,𝑏>0, then 𝑉(𝑡)≤sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑠)≤sup_{𝑡0−𝜏≤𝑠≤𝑡0}𝑉(𝑠)𝑒^{(𝑎+𝑏)(𝑡−𝑡0)},𝑡≥𝑡₀. (47)

Proposition 11. Assuming the upper bound delay𝜏^∗<min{𝜃𝑇,(1−𝜃)𝑇}, under the periodically intermittent control, the closed-loop control system (45) is globally exponentially stable if the control gains𝑘_𝑖⁽¹⁾ and𝑘_𝑗⁽²⁾satisfy the following conditions:(i)𝑎_𝑖−𝑘_𝑖⁽¹⁾>𝐿^𝑔_𝑖^𝑚_𝑗=1𝑞_𝑗𝑖,𝑖∈ℐ_𝑛,𝑏_𝑗−𝑘_𝑗⁽²⁾>𝐿^𝑓_𝑗^𝑛_𝑖=1𝑝_𝑖𝑗,𝑗∈ℐ_𝑚, (48)(ii)𝑎₁>𝑏̃₁,𝜂+𝜆₁𝜃−𝛾−𝜌+𝑏̃₁(1−𝜃)>0. (49)Moreover, we have ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡≤)𝑀𝑒^{−[𝜂+𝜆1(𝜃−𝛾)−(}𝜌₊𝑏̃_{1)(1−𝜃)]𝑡}×sup_{−𝜏∗≤𝑠≤0}^𝑛_𝑖=1𝜑_𝑖(𝑠)+^𝑚_𝑗=1𝜓_𝑗(𝑠),𝑡>0, (50) where𝜌=max_𝑖,𝑗{𝜂−𝑎_𝑖,𝜂−𝑏_𝑗,𝜅},and𝜅is any positive constant; 𝑀=𝑒^{𝜆1𝜃𝑇(𝜃−𝛾)}, and the parameters𝑎₁, 𝑏̃₁, 𝜂, 𝜆₁, and 𝛾are 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)When𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇),𝑙∈ℕ⁺₀, we have𝐷⁺𝑉(𝑡)≤−𝑎₁𝑉(𝑡)+𝑏̃₁𝑉(𝑡). By Lemma 4, one obtains that, for any𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇), 𝑉(𝑡)≤𝑒^{−𝜆1(𝑡−𝑙𝑇)}𝑉(𝑙𝑇).(2)When𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇),𝑙∈ℕ⁺₀, we have𝐷⁺𝑉(𝑡)≤𝜌𝑉(𝑡)+𝑏̃₁𝑉(𝑡). By Lemma 10, it is derived that, for any𝑡∈[(𝑙+𝜃)𝑇, (𝑙+1)𝑇),𝑉(𝑡)≤𝑒⁽𝜌⁺𝑏̃_{1)(𝑡−(𝑙+𝜃)𝑇)}𝑉((𝑙+𝜃)𝑇).
From the above inequality relationships, by similar estimation procedure, we can get the following conclusion: 𝑉(𝑡)≤𝑒^{−𝜆1(𝑡−𝑙𝑇)+𝑙(}𝜌⁺𝑏̃_{1)(1−𝜃)𝑇−𝜆1𝑙(𝜃−𝜏}^∗₎𝑉(0),[()𝑇),𝑉(𝑡)𝑡∈𝑙𝑇,𝑙+𝜃≤𝑒^{−𝜆1(𝑙+1)(𝜃𝑇−𝜏∗)+𝑙(}𝜌₊𝑏̃_{1)(1−𝜃)𝑇+(}𝜌₊𝑏̃_{1)(𝑡−𝑙𝑇−𝜃𝑇)}𝑉(0),𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇). (51) By the notational expression𝜏^∗=𝛾𝑇, one can further obtain 𝑉(𝑡)≤𝑒^{[−𝜆1(𝜃−𝛾)+(}𝜌⁺𝑏̃_{1)(1−𝜃)]𝑡}𝑒^{𝜆1𝜃𝑇(𝜃−𝛾)}𝑉(0),[()𝑇),𝑉(𝑡)𝑡∈𝑙𝑇,𝑙+𝜃≤𝑒^{[−𝜆1(𝜃−𝛾)+(}𝜌₊𝑏̃_{1)(1−𝜃)]𝑡}𝑉(0),𝑡∈[(𝑙+𝜃)𝑇,(𝑙+1)𝑇). (52) Hence it follows that, for any𝑡>0, ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡≤)𝑀𝑒^{−[𝜂+𝜆1(𝜃−𝛾)−(}𝜌₊𝑏̃_{1)(1−𝜃)]𝑡}×sup_{−𝜏∗≤𝑠≤0}^𝑛_𝑖=1𝜑_𝑖(𝑠)+^𝑚_𝑗=1𝜓_𝑗(𝑠), (53) and this completes the proof.

If the adjustable parameter𝜃=0, 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: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_j=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗,𝑡≥0,𝑡≠𝑡_𝑘,𝑘∈ℕ⁺,Δ𝑥_𝑖𝑡_𝑘=𝐼_𝑘⁽¹⁾𝑥_𝑖𝑡_𝑘,𝑘∈ℕ⁺,𝑖∈ℐ_𝑛,𝑦̇_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖,𝑡≥0,𝑡≠𝑡_𝑘,𝑘∈ℕ⁺,Δ𝑦_𝑗𝑡_𝑘=𝐼_𝑘⁽²⁾𝑦_𝑗𝑡_𝑘,𝑘∈ℕ⁺,𝑗∈ℐ_𝑚. (54)

Proposition 12. Assume the external imposed impulsive strengthsℎ⁽¹⁾_𝑖𝑘,ℎ⁽²⁾_𝑗𝑘≠−1. The origin of the closed-loop control system (54) is globally exponentially stable if there exists constant𝛽such that, for𝑘=1,2,…, the following conditions hold: ln𝜇_𝑘𝑡_𝑘−𝑡_𝑘−1≤𝛽,𝜌+𝑑𝑏̃₁+𝛽<0. (55) More specifically, we have ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡)≤𝑑𝑒^{−(𝜂+𝜆2)𝑡}sup_{−𝜏∗≤𝑠≤0}^𝑛_𝑖=1𝜑_𝑖(𝑠+_𝑚𝑗=1)𝜓_𝑗(𝑠),𝑡>0, (56) where the parameters𝜇_𝑘, 𝜌, 𝑑, 𝑏̃₁, and𝜂are 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)When𝑡∈(𝑡_𝑘−1,𝑡_𝑘], the upper right Dini derivative of𝑉(𝑡)along the solution of (54) is depicted as𝐷⁺𝑉(𝑡)≤𝜌𝑉(𝑡)+𝑏̃₁(sup_{𝑡−𝜏∗≤𝑠≤𝑡}𝑉(𝑠)).(2)When𝑡=𝑡_𝑘,𝑘∈ℕ⁺,𝑉(𝑡⁺_𝑘)≤𝜇_𝑘𝑉(𝑡_𝑘).
From Lemma 5 and the conditions in (55), it follows that 𝑉(𝑡)≤𝑑sup_{0−𝜏∗≤𝑠≤0}𝑉(𝑠)𝑒^{−𝜆2(𝑡−0)}, (57) which means ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡)≤𝑒^−𝜂𝑡𝑑𝑒^−𝜆2𝑡𝑉(0)≤𝑑𝑒^{−(𝜂+𝜆2)𝑡}𝑉(0),𝑡>0 (58) and the proof is completed.

If the continuous feedback control gains𝑘_𝑖⁽¹⁾=𝑘_𝑗⁽²⁾≡0, 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: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗,𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇)∪[(𝑙+𝜃)𝑇,(𝑙+1)𝑇)⧵𝑡≠𝑡_𝑘^∞_𝑘=1,Δ𝑥_𝑖(𝑡)=𝐼_𝑘⁽¹⁾𝑥_𝑖(𝑡),𝑡=𝑡_𝑘,𝑦̇_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖,𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇)∪[(𝑙+𝜃)𝑇,(𝑙+1)𝑇)⧵𝑡≠𝑡_𝑘^∞_𝑘=1,Δ𝑦_𝑗(𝑡)=𝐼_𝑘⁽²⁾𝑦_𝑗(𝑡),𝑡=𝑡_𝑘. (59) From (59), it is noticed that the occurrence of the impulses is not uniformly distributed since the impulses never occur on the interval[𝑙𝑇,(𝑙+𝜃)𝑇),𝑙∈ℕ⁺₀, whereas they frequently occur on the interval((𝑙+𝜃)𝑇,(𝑙+1)𝑇),𝑙∈ℕ⁺₀. By observing the proof of Proposition 12, the result is somewhat more conservative especially when the control period𝑇is very large and the control width𝜃approaches to one. In order to describe the conservatism for such case, we will utilize the notation of average impulsive interval proposed in [21] 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 ([21] average impulsive interval). The average impulsive interval of the impulsive sequence{𝑡_𝑘}^∞_𝑘=1is equal to𝑇_𝑎if there exist positive integer𝑁₀and positive number𝑇_𝑎such that 𝑁𝜅₂,𝜅₁≥𝜅₂−𝜅₁𝑇_𝑎−𝑁₀,∀𝜅₂≥𝜅₁≥0, (60) where𝑁(𝜅₂,𝜅₁)denotes the number of impulsive times of the impulsive sequence{𝑡_𝑘}^∞_𝑘=1on the interval(𝜅₁,𝜅₂).

Lemma 14 (see Lakshmikantham et al., Theorem1.4.1, [33, 34]). Let𝑃𝐶(ℝ₊,ℝ)(𝑃𝐶¹(ℝ₊,ℝ)) denote the set of piecewise continuous (piecewise continuously differentiable) functions with first kind of discontinuities at𝑡_𝑘,𝑘=1,2,…, fromℝ₊toℝ. If the following conditions hold, 𝑚(𝑡)∈𝑃𝐶¹ℝ₊,ℝ𝑖𝑠𝑙𝑒𝑓𝑡𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡𝑡_𝑘,𝐷𝑘=1,2,…,₊𝑚(𝑡)≤𝑝(𝑡)𝑚(𝑡)+𝑞(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑚𝑡⁺_𝑘≤𝑑_𝑘𝑚𝑡_𝑘+𝑏_𝑘,𝑘=1,2,…, (61) where𝑝(𝑡), 𝑞(𝑡)∈𝑃𝐶(ℝ₊,ℝ),𝑑_𝑘≥0and𝑏_𝑘are real constants, then 𝑚(𝑡)≤𝑚𝑡_{0𝑡0<𝑡𝑘<𝑡}𝑑_𝑘exp^𝑡_𝑡0𝑝(𝑠)d𝑠+_{𝑡0<𝑡𝑘<𝑡𝑡0<𝑡𝑗<𝑡}𝑑_𝑗exp^𝑡_𝑡𝑘𝑝(𝑠)d𝑠𝑏_𝑘+^𝑡_{𝑡0𝑠<𝑡𝑘<𝑡}𝑑_𝑘exp^𝑡_𝑠𝑝(𝜉)d𝜉𝑞(𝑠)d𝑠,𝑡≥𝑡₀. (62)

Lemma 15. Let𝑞≥0,𝜏>0,0<𝜇<1, and𝑝be constants, and assume that𝑉(𝑡)is a piecewise continuous nonnegative function satisfying 𝐷⁺𝑉(𝑡)≤𝑝𝑉(𝑡)+𝑞𝑉(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑉𝑡⁺_𝑘≤𝜇𝑉𝑡_𝑘,𝑘=1,2,…, (63) and the average impulsive interval of the impulsive sequence{𝑡_𝑘}^∞_𝑘=1is equal to𝑇_𝑎. If the following inequality holds, 𝑝+ln𝜇𝑇_𝑎+𝜇^−𝑁0𝑞<0, (64) then one has 𝑉(𝑡)≤𝜇^−𝑁0𝑉𝑡₀𝑒^{−𝜆(𝑡−𝑡0)}, (65) where𝑉(𝑡)=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑠),𝜆is the unique positive root of the equation𝜆 + 𝑝 + (ln𝜇/𝑇_𝑎) + 𝜇^−𝑁0𝑞𝑒^𝜆𝜏 = 0.

Proof. By Lemma 14 and the definition of average impulsive interval, it follows from (63) that, for𝑡≥𝑡₀, 𝑉(𝑡)≤𝑉𝑡₀𝜇^{𝑁(𝑡,𝑡0)}𝑒^{𝑝(𝑡−𝑡0)}+^𝑡_𝑡0𝑒^{𝑝(𝑡−𝑠)}𝜇^{𝑁(𝑡,𝑠)}𝑞𝑉(𝑠)≤d𝑠𝑉𝑡₀𝜇^{(((𝑡−𝑡0)/𝑇𝑎)−𝑁0)}𝑒^{𝑝(𝑡−𝑡0)}+^𝑡_𝑡0𝑒^{𝑝(𝑡−𝑠)}𝜇^{(((𝑡−𝑠)/𝑇𝑎)−𝑁0)}𝑞𝑉(𝑠)d𝑠≤𝜇^−𝑁0𝑉𝑡₀𝑒^{[𝑝+(ln𝜇/𝑇𝑎)](𝑡−𝑡0)}+^𝑡_𝑡0𝑒^{[𝑝+(ln𝜇/𝑇𝑎)](𝑡−𝑠)}𝑞𝑉(𝑠)d𝑠. (66) Denote𝜙(𝜆)=𝜆+𝑝+(ln𝜇/𝑇_𝑎)+𝜇^−𝑁0𝑞𝑒^𝜆𝜏. Since𝜙(0)=𝑝+(ln𝜇/𝑇_𝑎)+𝜇^−𝑁0𝑞<0, andlim_𝜆→∞𝜙(𝜆)=+∞and𝜙^′(𝜆)=1+𝜇^−𝑁0𝑞𝜏𝑒^𝜆𝜏>0, one knows that𝜙(𝜆)=0has a unique positive root.
Next, it is claimed that, for all𝑡>𝑡₀, 𝑉(𝑡)<𝜇^−𝑁0sup_{𝑡0−𝜏≤𝑠≤𝑡0}𝑉(𝑠)𝑒^{−𝜆(𝑡−𝑡0)}. (67) When𝑡∈[𝑡₀−𝜏,𝑡₀], 𝑉(𝑡)<𝑉𝑡₀<𝜇^−𝑁0𝑉𝑡₀𝑒^{−𝜆(𝑡−𝑡0)}. (68)
Supposing (67) is not always true for𝑡>𝑡₀, there must exist one time point𝑡^∗>𝑡₀such that 𝑉𝑡^∗=𝜇^−𝑁0𝑉𝑡₀𝑒^{−𝜆(𝑡∗−𝑡0)},𝑉(𝑡)<𝜇^−𝑁0𝑉𝑡₀𝑒^{−𝜆(𝑡−𝑡0)},𝑡₀−𝜏≤𝑡<𝑡^∗. (69) From inequalities (66) and (69), one can obtain that 𝑉𝑡^∗≤𝜇^−𝑁0𝑉𝑡₀𝑒^{[𝑝+(ln𝜇/𝑇𝑎)](𝑡∗−𝑡0)}+^𝑡∗_𝑡0𝑒^{[𝑝+(ln𝜇/𝑇𝑎)](𝑡∗−𝑠)}𝑞𝑉(𝑠)d𝑠<𝜇^−𝑁0𝑉𝑡₀𝑒^{[𝑝+(ln𝜇/𝑇𝑎)](𝑡∗−𝑡0)}+^𝑡∗_𝑡0𝑒^{[𝑝+(ln𝜇/𝑇𝑎)](𝑡∗−𝑠)}𝑞𝜇^−𝑁0𝑉𝑡₀𝑒^{−𝜆(𝑠−𝜏−𝑡0)}d𝑠=𝜇^−𝑁0𝑉𝑡₀𝑒^{[𝑝+(ln𝜇/𝑇𝑎)](𝑡∗−𝑡0)}+𝑞𝜇^−2𝑁0𝑉𝑡₀𝑒^{𝜆𝜏𝑡∗}_𝑡0𝑒^{[𝑝+(ln𝜇/𝑇𝑎)](𝑡∗−𝑠)}𝑒^{−𝜆(𝑠−𝑡0)}d𝑠. (70) Considering the equality𝜓(𝜆)=𝜆+𝑝+(ln𝜇/𝑇_𝑎)+𝜇^−𝑁0𝑞𝑒^𝜆𝜏=0, one has 𝑉𝑡^∗<𝜇^−𝑁0𝑉𝑡₀𝑒^{−𝜆(𝑡∗−𝑡0)}×𝑒^{[−𝜇0−𝑁𝑞𝑒𝜆𝜏](𝑡∗−𝑡0)}+𝑞𝜇^−𝑁0𝑒^{𝜆𝜏𝑡∗}_𝑡0𝑒^{[−𝜇0−𝑁𝑞𝑒𝜆𝜏](𝑡∗−𝑠)}d𝑠=𝜇^−𝑁0𝑉𝑡₀𝑒^{−𝜆(𝑡∗−𝑡0)}𝑒^{−[𝜇0−𝑁𝑞𝑒𝜆𝜏]𝑡∗}×𝑒^{[𝜇0−𝑁𝑞𝑒𝜆𝜏]𝑡0}+𝑒^{[𝜇0−𝑁𝑞𝑒𝜆𝜏]𝑡∗}−𝑒^{[𝜇0−𝑁𝑞𝑒𝜆𝜏]𝑡0}=𝜇^−𝑁0𝑉𝑡₀𝑒^{−𝜆(𝑡∗−𝑡0)} (71) 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 strengthsℎ⁽¹⁾_𝑖𝑘, ℎ⁽²⁾_𝑗𝑘∈(−2,−1)∪(−1,0), and the average impulsive interval of the impulsive sequence{𝑡_𝑘}^∞_𝑘=1is equal to𝑇_𝑎. The impulsive system (59) is globally exponentially stable if the following condition holds: 𝜌+ln𝜇𝑇_𝑎+𝜇^−𝑁0𝑏̃₁<0. (72) Moreover, we have ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^m_𝑗=1𝑦_𝑗(𝑡)≤𝜇^−𝑁0𝑒^{−(𝜂+𝜆)𝑡}sup_{−𝜏∗≤𝑠≤0}^𝑛_𝑖=1𝜑_𝑖(𝑠)+^𝑚_𝑗=1𝜓_𝑗(𝑠),𝑡>0, (73) where𝜇=sup_𝑘{𝜇_𝑘}∈(0,1), and𝜇_𝑘, 𝜌, and 𝑏̃₁are defined as in Theorem 6 and𝜆is the unique positive root of the equation𝜆+𝑝+(ln𝜇/𝑇_𝑎)+𝜇^−𝑁0𝑞e^𝜆𝜏=0.

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)When𝑡∈[𝑙𝑇,(𝑙+𝜃)𝑇)⧵{𝑡_𝑘}^∞_𝑘=1,𝑙∈ℕ⁺₀, the upper right Dini derivative of𝑉(𝑡)along the solutions of (59) is depicted as𝐷⁺𝑉(𝑡)≤𝜌𝑉(𝑡)+𝑏̃₁𝑉(𝑡).(2)When𝑡=𝑡_𝑘, 𝑘∈ℕ⁺,𝑉(𝑡⁺_𝑘)≤𝜇𝑉(𝑡_𝑘).
From the above two inequality relationships, and Lemma 15, one has ^𝑛_𝑖=1𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑦_𝑗(𝑡)≤𝜇^−𝑁0𝑒^{−(𝜂+𝜆)𝑡}sup_{𝑡0−𝜏≤𝑠≤𝑡0}𝑉(𝑠),𝑡>0, (74) where𝜆is the unique positive root of the equation𝜆+𝑝+(ln𝜇/𝑇_𝑎)+𝜇^−𝑁0𝑞𝑒^𝜆𝜏=0. 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 [35], 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 be{𝜃_𝑖}rather than the fixed width𝜃, and the period can be{𝑇_𝑖}rather than constant𝑇. 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 [36–39]; 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, 𝑥̇(𝑡)(𝑡)𝑦(),𝑦̇(𝑡)(𝑡)(𝑥(=−𝑎𝑥+𝑝𝑓𝑡−𝜏=−𝑏𝑦+𝑞𝑔𝑡−𝜎)) (75) with𝑎=1.922,𝑝=9.8501,𝑏=1.1631,𝑞=8.2311,𝜏=𝜎=3, and𝑓(𝑥)=𝑔(𝑥)=1/(1+𝑒^−𝑥)−1/2. Obviously,𝐿^𝑓=𝐿^𝑔=0.25. The initial condition is given as𝑥(𝑡)=−0.43, 𝑦(𝑡)=0.42, 𝑡∈[−3,0]. 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.

Figure 2 

The phase diagram of the original system (75).

Applying the sliding intermittent controller to the unstable system (75), one can derive the following system: 𝑥̇(𝑡)=−𝑎𝑥(𝑡)+𝑝𝑓𝑦(𝑡−𝜏)+𝑘₁𝑥(𝑡)[()𝑇)𝑥̇(𝑡)(𝑡)𝑦(,)()𝑇),𝑡∈𝑙𝑇,𝑙+𝜃=−𝑎𝑥+𝑝𝑓𝑡−𝜏)𝑡∈[(𝑙+𝜃𝑇,𝑙+1,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=ℎ₁𝑥𝑡_𝑘,𝑡=𝑡_𝑘,𝑦̇(𝑡)=−𝑏𝑦(𝑡)+𝑞𝑔(𝑥(𝑡−𝜎))+𝑘₂𝑦(𝑡),[()𝑇),𝑦̇(𝑡)(𝑡)(𝑥(,)()𝑇)𝑡∈𝑙𝑇,𝑙+𝜃=−𝑏𝑦+𝑞𝑔𝑡−𝜎))𝑡∈[(𝑙+𝜃𝑇,𝑙+1,𝑡≠𝑡_𝑘,Δ𝑦𝑡_𝑘=ℎ₂𝑦𝑡_𝑘,𝑡=𝑡_𝑘. (76)

In the following, we will give the convergence results by simulating the system (76). Firstly, by setting the continuous feedback gains𝑘₁=−4.8512,𝑘₂=−4.6378and the impulsive strengthsℎ₁=ℎ₂=−0.15. With the above parameters setting, calculations show that𝜂^∗=0.2679and𝜇=0.85. By setting the control period𝑇=10, we have control width0.3<𝜃<0.7and𝛾=0.3. Here we take the parameter𝜃=0.5. If we utilize uniform distributed impulsive sequences (𝑡_𝑘−𝑡_𝑘−1=0.01) in the latter control width of the control period, it is easy to check that when𝜂=0.0221, we have𝜌=−1.1410,𝑎₁=5.7788, and𝑏̃₁=4.8302. By choosing𝛽=−4.8519, one has𝑑=1.1765,ln𝜇/(𝑡_𝑘−𝑡_𝑘−1)−𝛽=−11.4000, 𝜌+𝑑𝑏̃₁+𝛽=−0.3104,𝜆₁=0.0565,𝜆₂=0.0168, and 𝜂+𝜆₁(𝜃−𝛾)−𝜆₂(𝜃+𝛾)−ln𝑑/𝑇=0.0037, which means all the conditions in Theorem 6 hold. The simulation result under the sliding intermittent control is given in Figure 3.

Figure 3 

Simulation of the system (75) under the sliding intermittent control.

Next, the corresponding convergence results under the propositions obtained in this article will be illustrated.

If we utilize the continuous feedback control, by taking the same continuous feedback gains as the above sliding intermittent control, the conditions in Proposition 9 hold, and the simulation result under the continuous feedback control is given in Figure 4.

Figure 4 

Simulation of the system (75) under the continuous feedback control.

If we use pure periodically intermittent control, by setting the continuous feedback gains𝑘₁=−6.8512,𝑘₂=−6.6378, we get𝜂^∗=0.3682,𝜌=𝜅=0.0001. It is easy to know when𝜂=0.0002, we have𝑎₁=7.8007,𝑏̃₁=4.5230, and 𝜆₁=0.1742. In Proposition 11, by taking𝑇=100, the relationships among the upper bound delay, the control width, and the control period infer that0.03<𝜃<0.97, while the last inequality in Proposition 11 means𝜃>0.9640. Here we set the𝜃=0.9650, and the simulation result under the periodically intermittent control is given in Figure 5.

Figure 5 

Simulation of the system (75) under the periodically intermittent control.

If we use full impulsive control with uniform distributed impulsive sequences (𝑡_𝑘−𝑡_𝑘−1=0.04) and the impulsive strengths𝜇_𝑘=0.85, 𝑘=1,2,…, it is easy to check when𝛽=−3.1262,𝜂=0.0002, the parameters𝑑=1.1332,𝜌=−1.1629, and 𝑏̃₁=4.5230, all the conditions in Proposition 12 satisfied. If we use the semi-impulsive control scheme and set the impulsive sequences satisfying the average impulsive interval with 𝑇_𝑎=0.038and𝑁₀=1; it is easy to check𝜌+(ln𝜇/𝑇_𝑎)+𝜇^−𝑁0𝑏̃₁=−0.1185<0, which infers that the condition in Proposition 16 holds. The simulation results for system (75) with the full impulsive control and the semi-impulsive control are given in Figures 6 and 7.

Figure 6 

Simulation of the system (75) under the full impulsive control.

Figure 7 

Simulation of the system (75) under the semi impulsive control.

Remark 18. From the above verifying process, it can be found that the sliding intermittent control is much better than the pure periodically intermittent control. More specifically, in the periodically intermittent control, the control period𝑇=100and the control width𝜃=0.965, while in the sliding intermittent control, the control period𝑇=10and the control width𝜃=0.5. As for the impulsive control, the full impulsive control is better than the semi-impulsive control in that the earlier converges faster. When dealing with the nonuniformly distributed impulsive sequence, the result derived in Proposition 16 is less conservative.

Example 2. Consider the following unstable delayed BAM neural network, and we will show that the sliding intermittent control benefits the stabilization of the unstable system: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+³_𝑗=1𝑝_𝑖𝑗𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗,𝑖∈ℐ₃,𝑦̇_𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗(𝑡)+⁴_𝑖=1𝑞_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑗𝑖,𝑗∈ℐ₄, (77) where𝐴=diag{3.1220,2.3156,2.2683},𝐵=diag{2.9631,2.3456,2.6341,3.0726}, ,.𝑃=7.85012.30703.22804.71913.24636.05895.37512.26092.01591.78032.66015.7647𝑄=4.23314.37412.24594.88302.32591.28572.40221.33773.79302.13512.67593.4719 (78)

The activations functions𝑓(𝑥)=𝑔(𝑥)=1/(1+𝑒^−𝑥)−1/2with𝐿^𝑓=𝐿^𝑔=0.25and the time delays𝜏=𝜎=3. The system has an unstable equilibrium0under the above parameters with the initial functions𝑥(𝑡)=[−0.73,−0.79,−0.82]^𝑇, 𝑦(𝑡)=[0.72,0.87,0.83,0.76]^𝑇, 𝑡∈[−3,0].

In the following, we will check the convergence results for system (77) under the sliding intermittent controller. Setting the continuous feedback gains𝑘⁽¹⁾=diag{−8.0350,−10.4875,−13.1075},𝑘⁽²⁾=diag{−11.1950,−8.4125,−5.7425,−4.1475}, the impulsive strengths are identical withℎ⁽¹⁾_𝑖𝑘=ℎ⁽²⁾_𝑗𝑘=−0.25, 𝑖∈ℐ_𝑛, 𝑗∈ℐ_𝑚, 𝑘∈ℕ⁺, and the control period𝑇=15; some calculations show that𝜂^∗=0.2604,𝜇=0.75. We can get the control width0.2<𝜃<0.8and𝛾=0.2; here we take the parameter𝜃=0.4. If we utilize the uniform distributed impulsive sequences (𝑡_𝑘−𝑡_𝑘−1=0.02) in the latter control width of the control period, it is easy to know when𝜂=0.1250, we have𝜌=−2.1433,𝑎₁=7.0951, and 𝑏̃₁=4.9664. By choosing𝛽=−4.6841, one hasln𝜇/(𝑡_𝑘−𝑡_𝑘−1)−𝛽=−9.7000,𝑑=1.333,𝜌+𝑑𝑏̃₁+𝛽=−0.2055,𝜆₁=0.1135,𝜆₂=0.1924, and 𝜂 + 𝜆₁(𝜃−𝛾) − 𝜆₂(𝜃+𝛾) − ln𝑑/𝑇 = 0.0131, which mean all the conditions in Theorem 6 hold. The simulation result under the sliding intermittent control is given in Figure 8.

Figure 8 

Simulation of the system (77) under the sliding intermittent controller.

5. Conclusions

In this paper, a sliding intermittent controller has been proposed by unifying the periodically intermittent control with the impulsive control together with continuous feedback control. More specifically, the continuous feedback control is employed as the preceding control width, and the impulsive control is resorted in the latter control width. Furthermore, the adjustable parameter𝜃∈[0,1]is very flexible in that the continuous feedback control (𝜃=1), the impulsive control (𝜃=0), the periodically intermittent control, and the semi-impulsive control (0<𝜃<1) are all possible cases. Based on the analysis technique and the Lyapunov function approach, the conditions have been constructed for the exponential stability of the delayed BAM neural networks under the proposed control schemes. Finally, numerical simulations are used to illustrate the effectiveness of the control technique.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61174136, the Natural Science Foundation of Jiangsu Province of China under Grant BK2011598, the Program for New Century Excellent Talents in University under Grant NCET-12-0117, and the Fundamental Research Funds for the Central Universities under Grant 2242012155.