Global exponential stability of a class of impulsive neural networks with unstable continuous and discrete dynamics

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Global exponential stability of a class of impulsive neural networks with unstable continuous and discrete dynamics Linna Wei, Wu-Hua Chen

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S0925-2312(14)00865-0 http://dx.doi.org/10.1016/j.neucom.2014.06.072 NEUCOM14406

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Neurocomputing

Received date: 21 March 2014 Revised date: 5 June 2014 Accepted date: 24 June 2014 Cite this article as: Linna Wei, Wu-Hua Chen, Global exponential stability of a class of impulsive neural networks with unstable continuous and discrete dynamics, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2014.06.072 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Global exponential stability of a class of impulsive neural networks with unstable continuous and discrete dynamics ∗

a

Linna Weia,b , Wu-Hua Chenb∗

School of Mathematics, South China University of Technology Guangzhou 510641, Guangdong, P. R. China b College of Mathematics and Information Science, Guangxi University Nanning 530004, Guangxi, P. R. China

Abstract This paper deals with global exponential stability of a class of impulsive neural networks whose continuous and discrete dynamics are unstable. Assuming that the impulsive neural network under consideration can be decomposed into two lower order impulsive systems, a time-varying weighted Lyapunov function associated with the impulse time sequence is introduced for stability analysis. A novel global exponential stability criterion is derived in terms of linear matrix inequalities (LMIs). By employing the newly-obtained stability criterion, a sufficient condition on the existence of a reduced-order impulsive controller is derived. Unlike the previous results concerning impulsive control, the proposed reduced-order impulsive controller only exerts the impulses on a partial set of the state vector. Moreover, the controller gain matrices can be achieved by solving a set of LMIs. Finally, four illustrative examples are given to show the effectiveness of the developed techniques and results. Keywords: Impulsive neural networks; Unstable dynamics; Time-varying Lyapunov function; Reduced-order impulsive control; Linear matrix inequality

1.

Introduction

Since the artificial neural networks model was first proposed in 1982 by J.J. Hopfield [1], various types of models of artificial neural networks have been put forward and widely applied to many areas of science and engineering, such as signal and image processing, pattern recognition, parallel computation and so on. These neural network models include Hopfield neural networks [1], Cohen-Grossberg neural networks [2], cellular neural networks [3], bidirectional associative memory (BAM) neural networks [4], competitive neural networks [5], and delayed neural networks [6], etc. In many applications, stability property of the designed neural networks plays a crucial role. Due to this reason, there has been an increasing interest in studying stability of equilibrium points for neural networks, and many important results have been reported in the literature, see, for example, [7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and the references therein. On the other hand, in the real world, the states of biological systems and electronic systems usually suffer abrupt changes at certain instants during the evolutionary process, which may be ∗

Corresponding author: Wu-Hua Chen. Tel.: +86 771 3273741; fax: +86 771 3232084. E-mail address: wuhua [email protected] (W.-H. Chen).

1

caused by components’ faults, an incremental change in the amplitude or frequency of the input to the circuit or other sudden noise. The neural networks subject to instantaneous perturbation effects are referred to as impulsive neural networks (INNs) [17]. Impulsive effects can affect dynamical behaviors of the neuron systems. A neural network could be stabilized or destabilized by certain impulsive effects. Therefore, in last decade, a lot of efforts have been made to study the stability property of INNs with or without delays. Among these references, to mention a few representative works, the global exponential stability of delayed INNs has been studied in [18] by using Razumikhin type Lyapunov function method, where delay-independent criteria in terms of linear matrix inequalities (LMIs) have been obtained. In [19], the impulsive control problems for the Cohen-Grossberg-type BAM neural networks with time-varying delay have been considered, and some criteria for impulsive stabilization have been proposed by the usage of Lyapunov functionals. Robust exponential stability problem of uncertain INNs with timevarying delays has been studied in [20]. Exponential stability of delayed INNs with partially Lipschitz continuous activation functions has been investigated in [21]. The stability problem of impulsive BAMs with destabilizing impulses and with stabilizing impulses has been addressed in [22] and [23], respectively. Several exponential stability criteria for stochastic neural networks with impulses have been established in [24, 25]. In [26], the stability results for continuous-time INNs proposed in [18] have been extended to the discrete-time case. In [27], some sufficient conditions ensuring the existence and global exponential stability of periodic solution for INNs with distributed delays have been presented. From the viewpoint of control theory, an INN can be viewed as a hybrid system in which the state flows inside the impulsive intervals (described by a differential equation) and jumps at impulse instants (described by a difference equation). Therefore, the stability property of INNs relies on the interaction between the continuous and discrete dynamics. It should be pointed out that most of the research results on the stability of INNs are based on the assumption that either the continuous or discrete dynamics is asymptotically stable, (see, for instance, [18, 22, 23]). In [28], a new concept of time-varying impulses has been proposed to describe a complex impulsive − phenomenon. The time-varying impulses take the form: x(t+ k ) = αk x(tk ), k ∈ N, where |αk | is allowed to be greater than 1 for some k and to be less than 1 for other k. Although such time-varying impulses can consider the stabilizing and destabilizing impulses simultaneously, the obtained stability criterion still requires either the continuous or discrete dynamics of the INNs is asymptotically stable. To the best of the authors’ knowledge, so far no attempt has been made to study the global exponential stability problem of INNs with unstable continuous and discrete dynamics. The case of INNs with unstable continuous and discrete dynamics frequently occurs in at least two different situations. The first situation involves the impulsive stabilization problem via partial state. In practice, some state variables are often not available, not measurable or too expensive to measure. In order to overcome this actual control problem, partial state feedback design can be utilized. In theory, if a neural network can be decomposed into a stable and an unstable subsystem, then it is possible to exert the impulsive control on the state of the unstable subsystem to stabilize the entire neural network. Consequently, the continuous dynamics of the resulting closed-loop system is unstable, while the corresponding discrete dynamics is only marginally stable. Considering the fact that the marginal stability is not robust with respect to small parameter perturbation, it is important and necessary to study the stability of INNs for which both continuous and discrete dynamics are unstable. We note that the impulsive stabilization problem of cellular neural networks via partial state has been investigated in [29, 30]. However, their methods rely heavily on the special structure of cellular neural networks. Moreover, their results only guarantee the cellular neural networks under consideration to be locally exponentially stable. So it would be difficult to apply the techniques of [29, 30] to deal with the global impulsive stabilization of general INNs via partial state. The second situation involves the competitive neural networks [5] and the BAM neural 2

networks [31]. The two kinds of neural networks can be modeled as interconnection of two lower order subsystems. For example, in competitive neural networks, the network consists of a short-term memory (STM) subsystem and a long-term memory subsystem (LTM). Although the stability problem of impulsive BAM neural networks has been studied by several researchers, their results require the impulses either to be stabilizing or to be destabilizing (see, for instance, [22, 23]). However, the impulsive phenomena displayed in real world are very complex. In real artificial neural networks, the impulsive phenomena are usually used to model the abrupt changes in the voltages produced by faulty circuit elements. It is possible that the effects of some part of the impulses are stabilizing, while the effects of other part are destabilizing. Thus, it is of practical significance to quantitatively analyze the effect of complex impulsive phenomena to stability of neural networks. The above discussion shows that the INNs with unstable continuous and discrete dynamics can characterize more complex impulsive phenomena. Moreover, due to the complex hybrid structure characteristics, the stability analysis on INNs with unstable continuous and discrete dynamics is not trivial and remains challenging, which motivates the present research. This paper is devoted to stability analysis of a class of INNs with interconnected hybrid structures. This class of INNs includes the case where both continuous and discrete dynamics are unstable. The basic idea for stability analysis is to introduce a novel time-varying weighted Lyapunov function to explore the interconnected hybrid structure characteristic. By employing the time-varying Lyapunov function, some new criteria for stability and stabilization of INNs are obtained. The main contribution of this paper lies in the following two aspects. First, a new criterion for global exponential stability of INNs is obtained in terms of LMIs. An important feature of the new criterion is that the stability condition relies both on the lower bound and the upper bound of impulsive intervals. Unlike the existing stability criteria for INNs, the derived stability criterion allows both the continuous and discrete dynamics of the INNs to be unstable. It is worth mentioning that the proposed stability criterion can also deal with the stability problem of neural networks with stabilizing/destabilizing impulses. Moreover, when the lower bound (the upper bound) of impulsive intervals is known, the upper bound (the lower bound) of impulsive intervals provided by our result is significantly larger (smaller) than the previously existing results. Second, based on the newly-established stability criterion, a sufficient condition on the existence of reduced-order impulsive controllers is derived. Moreover, the reduced-order impulsive controllers design problem is transformed into an LMI feasibility problem. By solving the established LMIs, the gain matrices of the reduced-order impulsive controllers can be achieved. It should be noted that the synthesis problem of reduced-order impulsive controllers is not considered in [29, 30]. The remainder of the paper is organized as follows. In Section 2, the class of INNs with interconnected hybrid structures is presented, and some definitions and lemmas are stated. In Section 3, we introduce a time-varying weighted Lyapunov function for exponential stability analysis of the INNs under consideration and provide a sufficient condition for the existence of reduced-order impulsive controllers. Four numerical examples are provided to illustrate the effectiveness of the results in Section 4. Finally, some concluding remarks end the paper in Section 5.

2.

Preliminaries

In this paper, if there are no special statements, matrices are assumed to have compatible dimensions. The notation M > ( ≥, <, ≤)0 means that the matrix M is positive-definite(positivesemidefinite, negative-definite, negative-semidefinite).  ·  denotes the Euclidean vector norm or the spectral norm of matrices. N is the set of positive integers. Consider a class of impulsive neural networks (INNs) described by the following nonlinear 3

impulsive differential equation ⎧ ˙ = −Au(t) + T g(u(t)) + J, t > 0, t = tk , ⎨ u(t) Δu(t) = Γ(t, u(t− )), t = tk , k ∈ N, ⎩ u(0) = u0 ,

(1)

where u(t) = (u1 (t), u2 (t), . . . , un (t))T ∈ Rn is the state vector associated with n neurons, A = diag (a1 , a2 , . . . , an ) represents the self-feedback term with ai > 0, i = 1, 2, . . . , n, g(u) = (g1 (u1 ), g2 (u2 ), . . . , gn (un ))T is the neuron activation function, T is the connection weight matrix, J is the constant input vector. The impulse time sequence {tk } is strictly in− creasing and satisfies lim tk = ∞. The notation Δu(tk ) = u(t+ k ) − u(tk ) denotes the jump of k→∞

− the state vector at impulse time instants, where u(t+ k ) = lim u(tk + h), u(tk ) = lim u(tk + h), h→0+

h→0−

k ∈ N. Here, we assume that u(tk ) = u(t+ k ), which means that the state vector of INN (1) is right continuous. The impulsive operator Γ : R+ × Rn → Rn is continuous. u0 ∈ Rn is the initial value. As usual, a vector u∗ ∈ Rn is said to be an equilibrium point of INN (1) if it satisfies  0 = − Au∗ + T g(u∗ ) + J, 0 = Γ(tk , u∗ ), k ∈ N. Through this paper, we assume that some conditions are satisfied such that the equilibrium point u∗ of INN (1) exists. Furthermore, we assume that the impulsive operator Γ(t, u) takes the following form. Γ(t, u) = (C − I + HF (t)E)(u − u∗ ), where C, H, and E are known matrices with appropriate dimensions, and F (t) is an unknown matrix function satisfying  F (t) ≤ 1. For notational convenience, we will shift the equilibrium point u∗ of system (1) to the origin. Set x(t) = u(t) − u∗ , then the INN (1) can be rewritten as ⎧ ˙ = −Ax(t) + T f (x(t)), t > 0, t = tk , ⎨ x(t) x(t) = (C + HF (t)E)x(t− ), t = tk , k ∈ N, (2) ⎩ ∗ x(0) = x0  u0 − u , where f (x) = g(x + u∗ ) − g(u∗ ). In INN (2), the continuous dynamics is described by the ordinary differential equation x(t) ˙ = −Ax(t) + T f (x(t)),

(3)

while the discrete dynamics is described by the difference equation x(k + 1) = (C + HF (k)E)x(k).

(4)

The two different dynamics are interdependent through the impulse time sequence {tk }. Hence, the stability property of INN (2) comes from the interaction between dynamics (3) and dynamics (4). However, most of the stability results for INN (2) require dynamics (3) or dynamics (4) to be asymptotically stable. In this paper, we are interested in a more complex case where both the continuous dynamics (3) and the discrete dynamics (4) may be unstable. For this purpose, we assume that the INN (2) can be decomposed into the following form:        ⎧  A1 0 T11 T12 f1 (x1 (t)) x1 (t) x˙ 1 (t) ⎪ ⎪ =− + , t = tk , ⎪ ⎪ T21 T22 f2 (x2 (t)) ⎪ ⎨ x˙ 2 (t)  0 A2  x2 (t)      x1 (t− )

(5) C11 C12 H1 x1 (t) = t = tk , k ∈ N, + F (t) E1 E2 ⎪ −) , ⎪ (t) C C H (t x x ⎪ 2 21 22 2 2 ⎪ ⎪ ⎩ x1 (0) = x10 , x2 (0) = x20 , 4

where xi = (xi1 , xi2 , . . . , xini )T ∈ Rni , Ai = diag (ai1 , ai2 , . . . , aini ) with aij > 0, j = 1, . . . , ni , fi (xi ) = (fi1 (xi1 ), fi2 (xi2 ), . . . , fini (xini ))T , i = 1, 2, and n1 + n2 = n. Remark 1 INN (5) can be viewed as an interconnection of two impulsive subsystems. Considering the fact that each impulsive subsystem is a hybrid system consisting of a continuous dynamics and a discrete dynamics, this interconnected hybrid structure allows to characterize more general INNs. For example, for the case where A1 is Hurwitz but A2 is not, and C22 is Schur stable but C11 is not, then both the continuous and discrete dynamics of the entire neural network will be unstable if the norms of other parameter matrices are sufficiently small. On the other hand, since each impulsive subsystem contains a stable continuous/discrete dynamics, the entire neural network may maintain stability for appropriate impulse time sequences. We make the following assumption on the activation functions fij , j = 1, 2, . . . , ni , i = 1, 2. (H1) Each fij is continuous with fij (0) = 0, and satisfies κ− ij ≤

fij (α) − fij (β) ≤ κ+ ij , α−β

j = 1, 2, · · · , ni , i = 1, 2,

+ for any α, β ∈ R, α = β. In the above, κ− ij and κij are scalars. The following two notations will be used in next section.

 − + − + − Li1 = − diag κ+ i1 κi1 , κi2 κi2 , . . . , κini κini ,   + − − + − κ + κ κ κ+ + κ + κ ini ini i1 i1 i2 , i2 ,..., , i = 1, 2. Li2 = diag 2 2 2

When the continuous dynamics or discrete dynamics of INN (5) is unstable, the stability of INN will rely on the choice of impulse time sequence {tk }. For given positive scalars τi , i = 1, 2, we use the notation S(τ1 , τ2 ) to denote the set of impulse time sequences {tk } satisfying τ1 ≤ tk − tk−1 ≤ τ2 , k ∈ N. Definition 1 For a given set S(τ1 , τ2 ) of impulse time sequences, INN (5) is said to be uniformly globally exponentially stable (UGES) over S(τ1 , τ2 ) with exponential convergence rate γ, if there exist positive scalars M and γ such that x(t) ≤ M x(t0 )e−γ(t−t0 ) ,

t ≥ t0 ,

for any impulse time sequence {tk } ∈ S(τ1 , τ2 ). The following lemmas are used to prove the theorems. Lemma 1 For any vectors x, y ∈ Rn , matrices D ∈ Rn×nf , N ∈ Rnf ×n , F ∈ Rnf ×nf ,  F ≤ 1, and constant ε > 0, the following inequality holds: DF N + N T F T D T ≤ ε−1 DD T + εN T N. Lemma 2 ([35]) For any n × n matrices X > 0, U , scalar ε > 0, the following matrix inequality holds: U X −1 U T ≥ ε(U + U T ) − ε2 X.

3.

Main results

For any given impulse time sequence {tk } ∈ S(τ1 , τ2 ), we define two piecewise linear functions ρ11 , ρ¯ : R+ → R+ : ρ11 (t) =

t − tk , tk+1 − tk

ρ¯(t) =

1 , tk+1 − tk 5

t ∈ [tk , tk+1 ),

k ∈ N.

(6)

It is easy to see that there exists a function ρ21 (t) : R+ → [0, 1] such that ρ¯(t) =

ρ21 (t) 1 − ρ21 (t) + . τ1 τ2

(7)

Set ρh2 (t) = 1 − ρh1 (t), h = 1, 2. Considering that system (2) can be seen as the interconnection of two lower order impulsive systems, we introduce a time-varying weighted Lyapunov function as follows: V (t, x1 , x2 ) = φ1 (t)x1 T P1 (t)x1 + φ2 (t)x2 T P2 (t)x2 ,

(8)

ρ (t)

where φh (t) = μh11 , Ph (t) = ρ11 (t)Ph1 + ρ12 (t)Ph2 , scalars μh > 0, h = 1, 2, and positivedefinite matrices Phi ∈ Rnh ×nh , h, i = 1, 2. Theorem 1 Consider the INN (5) satisfying (H1). Given a scalar γ > 0, and a set S(τ1 , τ2 ) of impulse time sequences with 0 < τ1 ≤ τ2 , the INN (5) is UGES over S(τ1 , τ2 ) with exponential convergence rate γ, if for given positive scalars μh , h = 1, 2, there exist nh × nh matrices Phi > 0, Qhij > 0, Rhij > 0, nl × nl diagonal matrices Dhlij > 0, and positive scalar ε > 0, h, l, i, j = 1, 2, such that the following matrix inequalities hold: ⎡ ⎤ Ω1ij P1i T11 + L12 D11ij 0 P1i T12 ⎢ ∗ ⎥ μ ¯1 R1ij − D11ij 0 0 ⎥ < 0, i, j = 1, 2, (9) Ξ1ij  ⎢ ⎣ ∗ ⎦ L22 D12ij ∗ −Q2ij + L21 D12ij ∗ ∗ ∗ −R2ij − D12ij ⎡

Ξ2ij

−Q1ij + L11 D21ij ⎢ ∗ ⎢ ⎣ ∗ ∗ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

L12 D21ij −R1ij − D21ij ∗ ∗

TP 0 C11 −μ1 P11 12 TP ∗ −μ2 P21 C12 12 ∗ ∗ −P12 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

⎤ 0 0 TP ⎥ T21 0 2i ⎥ < 0, i, j = 1, 2, Ω2ij P2i T22 + L22 D22ij ⎦ ∗ μ ¯2 R2ij − D22ij (10) TP T ⎤ C21 0 εE 22 1 TP C22 0 εE2T ⎥ 22 ⎥ 0 P12 H1 0 ⎥ ⎥ ≤ 0, (11) 0 ⎥ −P22 P22 H2 ⎥ ∗ −εI 0 ⎦ ∗

∗

−εI

¯2 = max{1, μ1 /μ2 }, and where μ ¯1 = max{1, μ2 /μ1 }, μ ln μh 1 ¯h Qhij + Lh1 Dhhij , h = 1, 2. Phi + (Ph1 − Ph2 ) − Phi Ah − Ah Phi + μ Ωhij = 2γ + τj τj Proof For notational simplicity, we set Qh (t) =

2  i,j=1

ρ1i (t)ρ2j (t)Qhij , Rh (t) =

2 

ρ1i (t)ρ2j (t)Rhij , Dhl (t) =

i,j=1

2  i,j=1

Ωh (t) = (2γ + ρ(t) ln μh )Ph (t) + ρ(t)(Ph1 − Ph2 ) − Ph (t)Ah − Ah Ph (t) +μ ¯h Qh (t) + Lh1 Dhh (t), h, l = 1, 2.

6

ρ1i (t)ρ2j (t)Dhlij ,

Let Ξh (t) =

2  i,j=1

ρ1i (t)ρ2j (t)Ξhij , h = 1, 2. Then

⎤ 0 P1 (t)T12 Ω1 (t) P1 (t)T11 + L12 D11 (t) ⎥ ⎢ ∗ μ ¯1 R1 (t) − D11 (t) 0 0 ⎥, Ξ1 (t) = ⎢ ⎦ ⎣ ∗ L22 D12 (t) ∗ −Q2 (t) + L21 D12 (t) ∗ ∗ ∗ −R2 (t) − D12 (t) ⎤ ⎡ −Q1 (t) + L11 D21 (t) L12 D21 (t) 0 0 T P (t) ⎥ ⎢ ∗ −R1 (t) − D21 (t) T21 0 2 ⎥. Ξ2 (t) = ⎢ ⎣ P2 (t)T22 + L22 D22 (t) ⎦ ∗ ∗ Ω2 (t) ∗ ∗ ∗ μ ¯2 R2 (t) − D22 (t) ⎡

Moreover, the conditions (9)–(10) imply Ξh (t) < 0, ∀ t ≥ 0, h = 1, 2.

(12)

Next, we prove that the condition (11) implies C T (tk )diag (P12 , P22 ) C(tk ) ≤ diag (μ1 P11 , μ2 P21 ) , k ∈ N,     

C11 C12 H1 ,H= , and E = E1 E2 . where C(tk ) = C + HF (t)E with C = C21 C22 H2 In fact, using Schur complement, the matrix inequality (13) is equivalent to   −diag (μ1 P11 , μ2 P21 ) C T (tk )diag (P12 , P22 ) ≤ 0, k ∈ N. Υ(tk )  ∗ −diag (P12 , P22 )

(13)

(14)

We note that Υ(tk ) = Υ0 + Υ1 (tk ), where

 −diag (μ1 P11 , μ2 P21 ) C T diag (P12 , P22 ) , Υ0 = ∗ −diag (P12 , P22 )   T  T   E 0 0 T F (t)[E 0] + F (t) . Υ1 (tk ) = 0 diag (P12 , P22 ) H diag (P12 , P22 ) H 

Applying Lemma 1 to Υ1 (tk ) gives  T  T   0 E 0 −1 [E +ε Υ(tk ) ≤ Υ0 + ε diag (P12 , P22 ) H 0 diag (P12 , P22 ) H On the other hand, applying Schur complement to (11) yields  T  T   0 E 0 [E +ε Υ0 + ε−1 diag (P12 , P22 ) H 0 diag (P12 , P22 ) H

0].

(15)

0] ≤ 0.

(16)

Thus, (14) follows from (15) and (16). That is, (13) holds. Choose a time-varying Lyapunov function candidate given in (8). Set V (t) = V (t, x1 (t), x2 (t)), and W (t) = e2γt V (t). For t ∈ (tk , tk+1 ),  ˙ (t) = e2γt φ1 (t) xT (t) [(2γ + ρ¯(t)(ln μ1 )) P1 (t) + ρ¯(t)(P11 − P12 )] x1 (t) W 1  T +2x1 (t)P1 (t) [−A1 x1 (t) + T11 f1 (x1 (t)) + T12 f2 (x2 (t))]  ¯(t)(ln μ2 )) P2 (t) + ρ¯(t)(P21 − P22 )] x2 (t) + e2γt φ2 (t) xT 2 (t) [(2γ + ρ  T (17) +2x2 (t)P2 (t) [−A2 x2 (t) + T21 f1 (x1 (t)) + T22 f2 (x2 (t))] . 7

From the definitions of φi (t), i = 1, 2, we have φφ21 (t) ¯1 , and φφ12 (t) ¯2 , for all t ≥ 0. It (t) ≤ μ (t) ≤ μ follows that   φ2 (t) T T x (t)Q1 (t)x1 (t) 0 = φ1 (t) −x2 (t)Q2 (t)x2 (t) + φ1 (t) 1   φ1 (t) T x (t)Q (t)x (t) + (t)Q (t)x (t) + φ2 (t) −xT 1 1 2 2 1 φ2 (t) 2

T  ¯ 1 xT ≤ φ1 (t) −x2 (t)Q2 (t)x2 (t) + μ 1 (t)Q1 (t)x1 (t)

T  ¯ 2 xT (18) + φ2 (t) −x1 (t)Q1 (t)x1 (t) + μ 2 (t)Q2 (t)x2 (t) , and

  φ2 (t) T f1 (x1 (t))R1 (t)f1 (x1 (t)) 0 = φ1 (t) −f2T (x2 (t))R2 (t)f2 (x2 (t)) + φ1 (t)   φ1 (t) T T f (x2 (t))R2 (t)f2 (x2 (t)) + φ2 (t) −f1 (x1 (t))R1 (t)f1 (x1 (t)) + φ2 (t) 2

 ¯1 f1T (x1 (t))R1 (t)f1 (x1 (t)) ≤ φ1 (t) −f2T (x2 (t))R2 (t)f2 (x2 (t)) + μ

 ¯2 f2T (x2 (t))R2 (t)f2 (x2 (t)) . + φ2 (t) −f1T (x1 (t))R1 (t)f1 (x1 (t)) + μ

On the other hand, by (H1), we have    − x (t) − f (x (t)) f (x (t)) − κ x (t) , 0 ≤ κ+ ij ij ij ij ij ij ij ij

j = 1, 2, . . . , ni ,

(19)

i = 1, 2.

It follows that for i ∈ {1, 2}, 0≤

ni 2   h=1 j=1

=

2 

   − φh (t)dhij (t) κ+ x (t) − f (x (t)) f (x (t)) − κ x (t) ij ij ij ij ij ij ij ij

 T T φh (t) xT i (t)Li1 Dhi (t)xi (t) + 2xi (t)Li2 Dhi (t)fi (xi (t)) − fi (xi (t))Dhi (t)fi (xi (t)) .

h=1

(20)

Substituting (18)–(20) into (17) yields ˙ (t) ≤ e2γt η T (t) (φ1 (t)Ξ1 (t) + φ2 (t)Ξ2 (t)) η(t), t ∈ (tk , tk+1 ), W where η(t) = [x1 T (t), f1 T (x1 (t)), x2 T (t), f2 T (x2 (t))]T . Then using (12), we obtain ˙ (t) ≤ 0, t ∈ (tk , tk+1 ), W which implies

V (t) ≤ V (tk )e−2γ(t−tk ) , t ∈ [tk , tk+1 ), k ∈ N.

(21)

In the following, we apply (13) to give an estimate of V (t) at impulse instants. Note that ρ11 (t− k ) = 1, ρ11 (tk ) = 0, k ∈ N. It follows from (8) that T − T T V (tk , x1 , x2 ) = xT 1 P12 x1 + x2 P22 x2 , V (tk , x1 , x2 ) = μ1 x1 P11 x1 + μ2 x2 P21 x2 .

Then, by (13), we obtain T V (tk ) = xT 1 (tk )P12 x1 (tk ) + x2 (tk )P22 x2 (tk )

= xT (tk )diag (P12 , P22 ) x(tk ) − T = xT (t− k )C (tk )diag (P12 , P22 ) C(tk )x(tk )

− ≤ xT (t− k )diag (μ1 P11 , μ2 P21 ) x(tk )

− − − T − = μ 1 xT 1 (tk )P11 x1 (tk ) + μ2 x2 (tk )P21 x2 (tk )

= V (t− k ).

(22) 8

Combining (21) and (22) together yields V (t) ≤ V (0)e−2γt , t ≥ 0. It follows that

x(t) ≤



(μ1 λ2 )/(μ2 λ1 )x(0)e−γt , t ≥ 0,

where λ1 = min{λmin (Pij ); i, j = 1, 2}, λ2 = max{λmax (Pij ); i, j = 1, 2}. Thus, we can conclude  that the INN (5) is UGES over S(τ1 , τ2 ) with exponential convergence rate γ. Remark 2 In the Lyapunov function (8), the terms xi Pi (t)xi , i = 1, 2, are used to capture the dynamical characteristics of the i-th impulsive subsystem, the terms φi (t), i = 1, 2, denote the time-varying weights, and the parameters μi , i = 1, 2, characterize the effects of the impulses exerted on the state variable xi (t), i = 1, 2. When μi > 1, i = 1, 2, the whole impulsive effects are destabilizing; when μi ∈ (0, 1], i = 1, 2, the whole impulsive effects are stabilizing; and when μ1 > 1 and μ2 ∈ (0, 1], the impulses exerted on the state variable x1 (t) are destabilizing, while the impulses exerted on the state variable x2 (t) have stabilizing effects. Remark 3 The stability condition in Theorem 1 is expressed in terms of linear matrix inequalities (LMIs), which can be solved numerically in a very efficient manner by using the developed interior point algorithm or the Matlab LMI Toolbox. The number of the variables in the stability condition needed to be determined is 5(n21 + n22 ) + 13(n1 + n2 ) + 1, where n  n1 + n2 is the number of neurons. It can be seen that with the increasing of n, the computational complexity will significantly increase. That is, when applying our results to a neural network possessing a large number of neurons, our LMI-based stability condition may lead to a significant increase in the computational burden. However, with the improvement of the modern computer performance, the computation problem will be resolved with the aid of the high-performance computers. When C12 = 0, C21 = 0, C22 = I, H2 = 0, and E2 = 0, INN (5) reduces to ⎧         ⎪ T11 T12 f1 (x1 (t)) x1 (t) ⎨ x˙ 1 (t) = − A1 0 + , t = tk , T21 T22 f2 (x2 (t)) x˙ 2 (t) 0 A2 x2 (t) ⎪ ⎩ x (t) = (C + H F (t)E ) x (t− ), t = t , k ∈ N. 1

11

1

1

1

(23)

k

We note that in INN (23), only the state vector x1 (t) is subject to impulsive perturbation. Applying Theorem 1 to INN (23), we obtain the following corollary. Corollary 1 Consider the INN (23) satisfying (H1). Given a scalar γ > 0, and a set S(τ1 , τ2 ) of impulse time sequences with 0 < τ1 ≤ τ2 , the INN (23) is UGES over S(τ1 , τ2 ) with exponential convergence rate γ, if for given positive scalars μh , h = 1, 2, there exist nh × nh matrices Phi > 0, Qhij > 0, Rhij > 0, nl × nl diagonal matrices Dhlij > 0, and positive scalar ε > 0, h, l, i, j = 1, 2, such that (9)–(10) and the following matrix inequalities hold: P22 ≤ μ2 P21 ,

⎤ TP 0 −μ1 P11 + εE1T E1 C11 12 ⎣ ∗ −P12 P12 H1 ⎦ ≤ 0. ∗ ∗ −εI ⎡

(24) (25)

Proof Corollary 1 follows from Theorem 1 and the fact that for the case of C12 = 0, C21 = 0,  C22 = I, H2 = 0, and E2 = 0, LMI (11) is equivalent to LMIs (24)–(25). In the sequel, we will consider the problem of designing reduced-order impulsive controllers for continuous neural networks. We assume that in the continuous dynamics of INN (5), only 9

the state variables x2 (t) are accessible. Then, we are interested in the reduced-order impulsive control laws of the form:  x2 (t) = x2 (t− ) + Buk , t = tk , (26) uk = Kx2 (t− k ), k ∈ N, where B ∈ Rn2 ×m is a known impulsive input matrix and K ∈ Rm×n2 is the impulsive gain matrix to be designed. The feedback structure based on reduced-order impulsive control law (26) is shown in Fig. 1. In the framework of reduced-order impulsive control, only samples of state variable x2 (t) at the discrete instants tk are transmitted to the controller. The closed-loop system of the neural network under the impulsive control law (26) takes the following form: ⎧         ⎪ x1 (t) T11 T12 f1 (x1 (t)) ⎨ x˙ 1 (t) = − A1 0 + , t = tk , x˙ 2 (t) 0 A2 x2 (t) T21 T22 f2 (x2 (t)) (27) ⎪ ⎩ x (t) = (I + BK) x (t− ), t = t , k ∈ N. 2 2 k Applying Theorem 1 to impulsively controlled neural network (27), we can obtain a stability criterion for INN (27). Corollary 2 Consider the INN (27) satisfying (H1). Given a matrix K ∈ Rm×n2 , a scalar γ > 0, and a set S(τ1 , τ2 ) of impulse time sequences with 0 < τ1 ≤ τ2 , the INN (27) is UGES over S(τ1 , τ2 ) with exponential convergence rate γ, if for given positive scalars μh , h = 1, 2, there exist nh × nh matrices Phi > 0, Qhij > 0, Rhij > 0, nl × nl diagonal matrices Dhlij > 0, h, l, i, j = 1, 2, such that (9)–(10) and the following matrix inequalities hold: P12 ≤ μ1 P11 , 

−μ2 P21 (I + BK)T P22 ∗ −P22

(28)  ≤ 0.

(29)

Based on Corollary 2, we can present a sufficient condition for the existence of the reducedorder impulsive control laws (26). Theorem 2 Consider the impulsively controlled INN (27) satisfying (H1). Assume that Li1 ≥ 0 1 ¯ i1 = L 2 , i = 1, 2. Given a scalar γ > 0, and a set S(τ1 , τ2 ) of impulse time sequences and set L i1 with 0 < τ1 ≤ τ2 , there exists a matrix K ∈ Rm×n2 such that the INN (27) is UGES over S(τ1 , τ2 ) with exponential convergence rate γ, if for given positive scalars μi , βi , and θhij , h, i, j = 1, 2, ¯ hij > 0, R ¯ hij > 0, and nl ×nl diagonal matrices D ¯ hlij > 0, there exist nh ×nh matrices Xhi > 0, Q h, l, i, j = 1, 2, such that the following matrix inequalities hold: ⎤ ⎡ ¯ ¯ 121j X11 L ¯ 11 Φ11j 0 T12 D 0 0 Ω11j √ ¯ 111j ¯ 111j ⎥ ⎢ ∗ 0 0 0 μ ¯1 D 0 −D ⎥ ⎢ ¯ ¯ ⎢ ∗ 0 0 X21 L21 ⎥ ∗ −Q21j X21 L22 ⎥ ⎢ ⎥ < 0, j = 1, 2, ⎢ ∗ ∗ ∗ Ψ 0 0 0 (30) 11j ⎥ ⎢ ⎥ ⎢ ∗ ¯ 0 0 ∗ ∗ 0 − D 111j ⎥ ⎢ ⎦ ⎣ ∗ ¯ 11j 0 ∗ ∗ 0 ∗ −R ¯ ∗ ∗ ∗ 0 ∗ 0 −D121j

10

⎡ ¯ Φ12j Ω12j ¯ 112j ⎢ ∗ − D ⎢ ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

¯ 122j 0 T12 D 0 0 ¯ 22j X22 L22 −Q ∗ Ψ12j ∗ 0 ∗ 0 ∗ 0 ∗ 0

¯ 11 X12 L 0 √ ¯ 0 μ ¯1 D112j 0 0 0 0 ¯ 112j 0 −D ¯ 12j ∗ −R ∗ ∗ ∗ ∗

¯ 11j X11 L12 0 0 −Q ¯ 211j T T 0 D ∗ Ψ21j 21 ¯ 21j Φ21j ∗ ∗ Ω ¯ 221j ∗ ∗ ∗ −D ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

¯ 12j X12 L12 0 0 −Q T ¯ 0 D212j T21 ∗ Ψ22j ¯ 22j Φ22j ∗ ∗ Ω ¯ 222j ∗ ∗ ∗ −D ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 

where

Φhij Ψ1ij

¯ 11 X11 L 0 0 0 0 0 ¯ 0 X21 L21 0 √ ¯ 0 0 μ ¯2 D221j ¯ 211j −D 0 0 ¯ 0 ∗ −D221j ¯ 21j ∗ 0 −R

¯ 11 X12 L 0 0 0 0 0 ¯ 21 0 X22 L 0 √ ¯ 0 0 μ ¯2 D222j ¯ 0 0 −D212j ¯ 222j 0 ∗ −D ¯ 22j ∗ 0 −R ∗ 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, j = 1, 2, ⎥ ⎥ ⎥ ⎥ ⎦

⎤

(31)

⎥ ⎥ ⎥ ⎥ ⎥ < 0, j = 1, 2, ⎥ ⎥ ⎥ ⎦

(32)

0 0 X22 0 0 0 0 −τj X21

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, j = 1, 2, ⎥ ⎥ ⎥ ⎥ ⎦

X11 ≤ μ1 X12 ,  ¯ TB T −μ2 X21 X21 + K ≤ 0, ∗ −X22

(33) (34) (35)

β2 1 ¯ h1j , (ln μh + 1 − 2βh ) Xh1 + h Xh2 − Ah Xh1 − Xh1 Ah + μ ¯h Q τj τj 1 ¯ h2j , = 2γ + (ln μh − 1) Xh2 − Ah Xh2 − Xh2 Ah + μ ¯h Q τj ¯ hhij + Xhi Lh2 , = Thh D 2 ¯ 2 ¯ ¯ 12ij + θ1ij ¯ 21ij + θ2ij = − (1 + 2θ1ij )D R2ij , Ψ2ij = −(1 + 2θ2ij )D R1ij .

¯ h1j = Ω ¯ h2j Ω

0 X12 0 0 ¯ 21 X22 L 0 0 0 0 0 0 0 ¯ 0 −D122j ∗ −τj X11



2γ +

¯ −1 . Moreover, the impulsive gain matrix K is given by K = KX 21 −1 ¯ −1 , Dhlij = D ¯ −1 , h, l, i, j = 1, 2, and ¯ hij Phi , Rhij = R , Qhij = Phi Q Proof Define Phi = Xhi hij hlij ¯ −1 . Set K = KX 21

Πhij = diag (P1i , Dh1ij , P2i , Dh2ij ) , h, i, j = 1, 2. Note that by Lemma 2, the following matrix inequalities hold: −1 Ph1 , h = 1, 2, −Ph2 ≤ −2βh Ph1 + βh2 Ph1 Ph2

−R2ij ≤ −2θ1ij D12ij + −R1ij ≤ −2θ2ij D21ij +

2 −1 θ1ij D12ij R2ij D12ij , −1 2 θ2ij D21ij R1ij D21ij ,

11

(36)

i, j = 1, 2,

(37)

i, j = 1, 2.

(38)

Pre- and Post-multiplying both sides of the LMIs in (30) with diag(Π11j , I, I, I), and using Schur complement and the inequalities (36) with h = 1 and (37) with i = 1, we obtain (9) with i = 1. Pre- and Post-multiplying both sides of the LMIs in (31) with diag(Π12j , I, I, I, I), and using Schur complement and the inequalities (37) with i = 2, (31) is changed to (9) with i = 2. Preand Post-multiplying both sides of the LMIs in (32) with diag(Π21j , I, I, I), and using Schur complement and the inequalities (36) with h = 2 and (38) with i = 1, we obtain (10) with i = 1. Pre- and Post-multiplying both sides of the LMIs in (33) with diag(Π22j , I, I, I, I), and using Schur complement and the inequalities (38) with i = 2, (33) is changed to (10) with i = 2. Pre- and Post-multiplying both sides of the LMI in (35) with diag{P21 , P22 }, we obtain (29). It is easy to see that the LMI in (34) is equivalent to the LMI in (28). Since all conditions of ¯ −1 is UGES over S(τ1 , τ2 ) with exponential Corollary 2 are satisfied, the INN (27) with K = KX 21 convergence rate γ.  Remark 4 Theorem 2 presents a sufficient condition for the existence of a reduced-order impulsive controller (26). When the LMIs (30)–(35) are feasible, the gain matrix of the reduced-order impulsive controllers can be achieved. It is worth mentioning that the impulsive synchronization problem of Lur’e systems via partial states has been studied in [32]. The synchronization condition derived in [32] is established by employing time-invariant Lyapunov functions. According to the method in [32], the impulse jump matrix I + BK in (27) must satisfy the condition ρ(I + BK) < 1, where ρ(·) denotes the spectral radius of a square matrix. We note that when m < n2 , ρ(I + BK) ≥ 1. This means that the technique of [32] cannot be used to synthesis a reduced-order impulsive controller. Remark 5 We note that in recent years, the control problem of vehicle suspension systems has received considerable attention and many results have been reported in the literature. For instance, the reliable fuzzy H∞ controller for active suspension systems with actuator delay and fault has been designed in [33], while the out-feedback H∞ control for a class of active quarter-car suspension systems with control delay has been investigated in [34]. How to design impulsive controllers for suspension systems via partial state presents an interesting topic for future research.

4.

Illustrated Examples

In this section, we consider four examples to illustrate the effectiveness of the results proposed in this paper. Example 1 Consider the INN (2) with ⎡ ⎤ ⎡ ⎤ −0.1 0.2 0 0.2 1.65 0.3 0.1 0 ⎢ 0 ⎢ −0.1 0.1 0.1⎥ 1.58 0.1 0.01⎥ ⎥, C = ⎢ 0 ⎥. A = diag(2.8, 2.8, 1, 1), T = ⎢ ⎣ 0 ⎦ ⎣ −0.1 4 0.5 0 0 0.18 0 ⎦ −0.5 0 0 4 0 0 0 0.2 It is assumed that f (x) = (f (x1 ), f (x2 ), f (x3 ), f (x4 ))T with f (s) = tanh(s). In the absence of impulse, with the initial values x(0) = (0.01, −0.01, −0.01, 0.01)T , the time responses of the state variables are depicted in Fig. 2. From the graph, we can see that the zero equilibrium of the continuous dynamics is unstable. On the other hand, noting that ρ(C) = 1.65, where ρ(·) denotes the spectral radius of a square matrix, the discrete dynamics of the INN is also unstable. Because both the continuous and discrete dynamics are unstable, the existing stability results for INNs are inapplicable. 12

Now, we rewrite the INN as the form of (5), in which     −0.1 0.2 0 0.2 , T12 = , A1 = 2.8I2 , A2 = I2 , T11 = 0 −0.1 0.1 0.1       0 −0.1 4 0.5 1.65 0.3 , T22 = , C11 = , T21 = −0.5 0 0 4 0 1.58     0.1 0 0.18 0 , C21 = 0, C22 = . C12 = 0.1 0.01 0 0.2 It is easy to see that Li1 = 0, and Li2 = 0.5I2 , i = 1, 2. We assume that the impulse time sequence {tk } ∈ S(0.3, 0.36). Applying Theorem 1 with the choice of μ1 = 2.75 and μ2 = 0.06, it has been found that the LMIs (9)–(11) are feasible. This means that the INN is UGES over S(0.3, 0.36). Moreover, the obtained maximum exponential convergence rate is γ = 0.24. With the initial condition of x(0) = (2, −2, −1.5, 0.9)T , and {tk } ∈ S(0.33, 0.33), the time responses of the state variables of the INN are shown in Fig. 3. Fig. 3 clearly shows that all the state variables converge to zero. Next, we continue to consider the above INN, except that ⎡ ⎤ 0.9 0.3 0.1 0 ⎢ 0 0.3 0.1 0.01⎥ ⎥. C=⎢ ⎣0 0 0.18 0 ⎦ 0 0 0 0.2 It can be seen that ρ(C) = 0.9 < 1. This means that the impulses are stabilizing. Thus, it is possible to apply the results in [18] to ascertain the stability of the INN. It can be shown that using the Theorem 5 in [18], the INN is UGES over S(τ1 , 0.032) for any τ1 ∈ (0, 0.032]. For the case of {tk } ∈ S(τ1 , τ2 ) with τ1 = 0.001, applying our Theorem 1 with the choice of μ1 = 1.439 and μ2 = 0.048, it has been found that the largest value of τ2 such that the INN is UGES over S(τ1 , τ2 ) is 0.44. Taking {tk } ∈ S(0.43, 0.43), and choosing x0 = (2, −2, −1.5, 0.9)T , the time response of the corresponding trajectory is plotted in Fig. 4. The simulation results show that all the state variables asymptotically converge to zero under the impulsive actions. The comparison indicates that when the lower bound of impulsive intervals is known, the upper bound of impulsive intervals provided by our result is significantly larger than that obtained in [18]. Example 2 Consider the following continuous-time neural ⎡ ⎤ ⎡ 3 0 0 −1 0.2 ⎣ ⎦ ⎣ x(t) ˙ = − 0 4 0 x(t) + 0.2 −1 0 0 3.5 0.3 0.2

network ⎤ 0.1 0.1⎦ f (x(t)), −1

(39)

where f (x) = (f (x1 ), f (x2 ), f (x3 ))T with f (s) = 12 (|s + 1| − |s − 1|). By Theorem 3 in [18], system (39) is globally exponentially stable with exponent convergence rate γ = 2.99. With the initial state x(0) = (2, −2, 1.8)T , the time evolution of the state variables is depicted in Fig. 5, which verifies the exponential stability of system (39). Now, we assume that the state variables x1 (t) and x2 (t) are subject to impulsive perturbation: x ˜(t) = (C + HF (t)E)˜ x(t− ), t = tk , {tk } ∈ S(τ1 , τ2 ), where x ˜(t) = (x1 (t), x2 (t))T , and       4 0.1 1 0 − tanh(t/3) 0 C= , H= , F (t) = , E = 0.1I2 . 0 5 0 2 0 tanh(t/2) 13

(40)

It is obvious that the impulsive effects are destabilizing. For the case of τ2 = 2000, applying Corollary 1 to the INN (39)–(40) with the choice of μ1 = 16.92 and μ2 = 0.55, it has been found that the minimum value of τ1 such that the INN (39)–(40) is UGES over S(τ1 , τ2 ) with exponential convergence rate γ = 0.05 is τ1 = 0.479. For simulation studies, take the initial condition as x(0) = (2, −2, 1.8)T , and let {tk } ∈ S(0.5, 0.5). The simulation result, illustrated in Fig. 6, indicates that the stability of system (39) can be retained in spite of the impulsive perturbation on the state variables x1 (t) and x2 (t). We note that the Theorem 1 in [18] guarantees the UGES over S(0.556, τ2 ) for any τ2 ≥ 0.556. The comparison shows that when the upper bound of impulsive intervals is known, the lower bound of impulsive intervals provided by our result is smaller than that obtained in [18]. Example 3 Consider the INN (2) with ⎡ 0 0 ⎢ 0 0 A = diag(6, 5, 0.25, 0.1), T = ⎢ ⎣−0.1 2 1 −0.7

1 3 0 0

⎤ ⎡ ⎤ 1 1.5 0 0.1 0 ⎢ 2⎥ 1.1 0.2 0.01⎥ ⎥, C = ⎢ 0 ⎥. ⎦ ⎣ 0 0.02 0 0.15 0 ⎦ 0 0.1 0.01 0 0.2

It is assumed that f (x) = (f (x1 ), f (x2 ), f (x3 ), f (x4 ))T with f (s) = tanh(s). With the initial value x(0) = (0.01, −0.01, −0.01, 0.01)T , the time responses of the state variables of the INN without impulses are pictured in Fig. 7, which obviously shows the zero equilibrium of the continuous dynamics is unstable. On the other hand, noticing that ρ(C) = 1.5015, the discrete dynamics of the INN is also unstable. We note that the INN can be viewed as an impulsive BAM neural network due to the special structure of the matrix T . However, the stability criteria in [22] are only suitable to the impulsive BAM neural networks with stable continuous dynamics, while the stability criteria in [23] require the discrete dynamics of the impulsive BAM neural networks to be stable. Therefore, they cannot provide any stability results. Now, we assume that the impulse time sequence {tk } ∈ S(0.17, 0.48). Applying our Theorem 1 with the choice of μ1 = 0.76 and μ2 = 0.51, we obtain that the INN is UGES over S(0.17, 0.48) with exponential convergence rate γ = 0.006. With the initial condition of x(0) = (2, −2, −1.5, 0.9)T , and {tk } ∈ S(0.4, 0.4), the time evolution of the state variables of the impulsive BAM neural network is depicted in Fig. 8, which demonstrates the asymptotical stability of the zero equilibrium. Example 4 Consider the following continuous-time neural network: ⎡ ⎤ ⎡ ⎤ 2.8 0 0 0 −0.2 0.5 0 1 ⎢ 0 2.8 0 0⎥ ⎢ −0.3 1 0.2⎥ ⎥ x(t) + ⎢ 0 ⎥ f (x(t)), x(t) ˙ = −⎢ ⎣0 ⎦ ⎣ 0 1 0 0 −0.1 0.82 2 ⎦ 0 0 0 1 0.5 0.2 0 1.5

t = tk

(41)

where f (x) = (f (x1 ), f (x2 ), f (x3 ), f (x4 ))T with f (s) = tanh(s). Fig. 9 shows the time evolution of the state variables with the initial state x(0) = (0.01, −0.01, −0.01, 0.01)T . The simulation result indicates the original of the continuous dynamics is unstable. Now, we assume that the state variables x3 (t) and x4 (t) are available. In the following, we will apply Theorem 2 to design a reduced-order impulsive control with the form of     x3 (t− ) x3 (t) = (I + BK) , t = tk , k ∈ N, (42) x4 (t) x4 (t− ) where B = [1 1]T , and K ∈ R1×2 is the impulsive gain matrix to be designed. Assume that the impulse time sequence {tk } ∈ S(0.35, 0.45). Applying Theorem 2, with the choice of μ1 = μ2 = 0.67, β1 = β2 = 1, θ1ij = θ21j = θ221 = 0.004, i, j = 1, 2, and θ222 = 0.064, 14

it has been found that the maximum value of γ such that the LMIs (30)-(35) are feasible is γ = 0.01. The corresponding impulsive gain matrix K = [−0.1663 − 0.8337]. That is, for any {tk } ∈ S(0.35, 0.45), the neural network (41) can be globally stabilized by the reduced-order impulsive controller (42), and the achieved maximum exponent convergence rate is γ = 0.01. With the initial condition of x(0) = (3, −2.5, −2.8, 3)T , the time responses of the state variables of the neural network (41) under the above reduced-order impulsive controller are depicted in Fig. 10. The graph shows that all the state variables asymptotically converge to the origin via the impulses exerted on the state variables x3 (t) and x4 (t).

5.

Conclusions

The problem of global exponential stability of a class of impulsive neural networks with unstable continuous and discrete dynamics has been addressed. By taking the systems characteristic into account, a time-varying weighted Lyapunov function has been constructed for stability analysis. A new global exponential stability criterion has been derived in terms of LMIs. Based on the newly-established stability criterion, a sufficient condition on the existence of a reduced-order impulsive controller has been derived. The effectiveness of the stability results and the design method has been illustrated through four examples. We note that all the illustration examples are academic examples. How to apply the proposed results to real world systems deserves being investigated in the near future. In addition, in the future work extending the proposed results to delayed neural networks with impulsive effects will be considered. Furthermore, studying the input-to-state stability of impulsive neural networks with unstable continuous and discrete dynamics will be another interesting topic for future research.

Acknowledgment This work was supported in part by the National Natural Science Foundation of China under Grants 61164016, the Key Project of Guangxi Natural Science Foundation (2013GXNSFDA019003), the Guangxi Natural Science Foundation (2011GXNSFA018141).

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[27] X. Li, S. Song, Impulsive control for existence, uniqueness, and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays, IEEE Transactions on Neural Networks and Learning Systems 24 (2013) 868–877. [28] W. Zhang, Y. Tang, J. Fang, X. Wu, Stability of delayed neural networks with time-varying impulses, Neural Networks 36 (2012) 59–63. [29] Q. Wang, X. Liu, Exponential stability of impulsive cellular neural networks with time delay via Lyapunov functionals, Applied Mathematics and Computation 194 (2007) 186–198. [30] X. Lu, W.-H. Chen, R. Du, Stability analysis of delayed cellular neural networks with impulsive effects, IMA Journal of Mathematical Control and Information 28 (2011) 487–506. [31] K. Gopalsamy, X.-Z. He, Delay-independent stability in bidirectional associative memory networks, IEEE Transactions on Neural Networks 5 (1994) 998–1002. [32] W.-H. Chen, X. Lu, F. Chen, Impulsive synchronization of chaotic Lur’e systems via partial states, Physics Letters A 372 (2008) 4210–4216. [33] H.Y. Li, H.H. Liu, H.J. Gao, P. Shi, Reliable fuzzy control for active suspension systems with actuator delay and fault, IEEE Transactions on Fuzzy Systems 20 (2012) 342-357. [34] H.Y. Li, X.J. Jing, H.R. Karimi, Output-feedback-based H∞ control for vehicle suspension systems with control delay, IEEE Transactions on Industrial Electronics 61 (2014) 436-446. [35] W.-H. Chen, W.X. Zheng, Robust stabilization of delayed Markovian jump systems subject to parametric uncertainties, in: Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, Dec. 2007, pp. 3054–3059.

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Captions for figures in the paper Caption for Fig. 1: Feedback structure based on reduced-order impulsive control laws. Caption for Fig. 2: The time response of the continuous dynamics of the INN in Example 1 with the initial condition of x0 = (0.01, −0.01, −0.01, 0.01)T . Caption for Fig. 3: The time response of the INN in Example 1 with the initial condition of x0 = (2, −2, −1.5, 0.9)T and {tk } ∈ S(0.33, 0.33). Caption for Fig. 4: The time response of the second INN in Example 1 with the initial condition of x0 = (2, −2, −1.5, 0.9)T and {tk } ∈ S(0.43, 0.43). Caption for Fig. 5: The state trajectories of neural network (39) without impulses. Caption for Fig. 6: The state trajectories of neural network (39) with partial state subject to impulsive perturbation. Caption for Fig. 7: The state trajectories of the continuous dynamics of the impulsive BAM in Example 3 with the initial condition of x0 = (0.01, −0.01, −0.01, 0.01)T . Caption for Fig. 8: The state trajectories of the impulsive BAM in Example 3 with the initial condition of x0 = (2, −2, −1.5, 0.9)T . Caption for Fig. 9: The state trajectories of neural network (41). Caption for Fig. 10: The state trajectories of neural network (41) under reduced-order impulsive control.

18

uk

­ x Ax  Tf ( x), ®  ¯ x2 (tk ) x2 (tk )  Buk

x2 (t  )

tk

x2 (tk )

Controller

Fig. 1. Feedback structure based on reduced-order impulsive control laws.

1

0.5 x

0 −0.5

0

2

4

6

8

10

0

2

4

6

8

10

0

2

4

6

8

10

0

2

4

6

8

10

2

0.5 x

0 −0.5

x3

5 0 −5

x4

5 0 −5

t

Fig. 2. The time response of the continuous dynamics of the INN in Example 1 with the initial condition of x0 = (0.01, −0.01, −0.01, 0.01)T .

19

x1

2 1 0 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5 t

2

2.5

3

x2

0 −1 −2

x3

0 −1 −2

x4

2 1 0

Fig. 3.

The time response of the INN in Example 1 with the initial condition of x0 = (2, −2, −1.5, 0.9)T and {tk } ∈ S(0.33, 0.33).

x

1

2 1 0 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5 t

2

2.5

3

x2

0 −1 −2

x3

0 −1 −2

x

4

2 1 0

Fig. 4. The time response of the second INN in Example 1 with the initial condition of x0 = (2, −2, −1.5, 0.9)T and {tk } ∈ S(0.43, 0.43).

20

x1

2 0 −2

0

1

2

3

4

5

0

1

2

3

4

5

0

1

2

3

4

5

x2

2 0 −2

x3

2 0 −2

t

Fig. 5. The state trajectories of neural network (39) without impulses.

x1

2 1 0 0

1

2

3

4

5

0

1

2

3

4

5

0

1

2

3

4

5

x2

0 −1 −2

x3

2 1 0 t

Fig. 6. The state trajectories of neural network (39) with partial state subject to impulsive perturbation.

21

x1

0.5 0 −0.5

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10 t

12

14

16

18

20

x2

0.5 0 −0.5

3

5

x

0 −5

4

5

x

0 −5

Fig. 7. The state trajectories of the continuous dynamics of the impulsive BAM neural networks in Example 3 with the initial condition of x0 = (0.01, −0.01, −0.01, 0.01)T .

x

1

2 1 0 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5 t

2

2.5

3

x2

0 −1 −2

x3

0 −1 −2

x

4

2 1 0

Fig. 8. The state trajectories of the impulsive BAM neural networks in Example 3 with the initial condition of x0 = (2, −2, −1.5, 0.9)T .

22

1

x

2

x

0.6 0.4 0.2 0 −0.2 0

5

10

15

20

0

5

10

15

20

0

5

10

15

20

0

5

10 t

15

20

0.6 0.4 0.2 0 −0.2

x3

4 2 0

x4

2 1 0

x2

x1

Fig. 9. The state trajectories of neural network (41).

3 2 1 0

0 −1 −2 −3

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

0

1

2

3

4 t

5

6

7

8

x3

0 −2

x3

−4 3 2 1 0

Fig. 10. The state trajectories of neural network (41) under reduced-order impulsive control.

23