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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 10, OCTOBER 2015
Stochastic Stability of Delayed Neural Networks With Local Impulsive Effects Wenbing Zhang, Yang Tang, Member, IEEE, Wai Keung Wong, and Qingying Miao, Member, IEEE
Abstract— In this paper, the stability problem is studied for a class of stochastic neural networks (NNs) with local impulsive effects. The impulsive effects considered can be not only nonidentical in different dimensions of the system state but also various at distinct impulsive instants. Hence, the impulses here can encompass several typical impulses in NNs. The aim of this paper is to derive stability criteria such that stochastic NNs with local impulsive effects are exponentially stable in mean square. By means of the mathematical induction method, several easy-tocheck conditions are obtained to ensure the mean square stability of NNs. Three examples are given to show the effectiveness of the proposed stability criterion. Index Terms— Impulsive systems, local impulsive effects, neural networks (NNs), stability analysis.
I. I NTRODUCTION
I
N THE past two decades, research relating to neural networks (NNs) has received increasing attention due to their wide applications in many areas such as combinatorial optimization, signal processing, and pattern recognition. In a real-world NN model, axonal signal transmission delays often occur in various NNs, due to the finite speed of switching and transmitting signals, which may cause oscillation or instability. Then, a large number of promising results on the stability or the stabilization of delayed NNs have been published [1]–[7]. For example, in [3], the stability problem was investigated for a class of delayed NNs with time-varying impulses. In [4], by means of a linear matrix inequality approach, the problems of stability and dissipativity analysis were investigated for static NNs with time delay. Manuscript received April 7, 2014; revised October 21, 2014; accepted December 7, 2014. Date of publication December 25, 2014; date of current version September 16, 2015. This work was supported in part by the Natural Science Foundation through the Higher Education Institutions of Jiangsu Province, China, under Grant 14KJB120014, in part by the Natural Science Foundation of Shanghai (Grant 13ZR1421300), in part by the Alexander von Humboldt Foundation of Germany, in part by the National Natural Science Foundation of China under Grant 11426196, Grant 61203235, Grant 61304158, Grant 61473189, and Grant 61375012, and in part by Hong Kong Polytechnic University, Hong Kong, under Project G-YM53. W. Zhang is with the Department of Mathematics, Yangzhou University, Yangzhou 225009, Jiangsu, China (e-mail: [email protected]). Y. Tang is with the Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany, and also with the Institute of Physics, Humboldt University of Berlin, Berlin 12489, Germany (e-mail: [email protected]). W. K. Wong is with the Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hong Kong, and also with the Shenzhen Research Institute, Hong Kong Polytechnic University, Hong Kong (e-mail: [email protected]). Q. Miao is with the School of Continuing Education, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2380451
On the other hand, when designing the NNs or in the implementation of neural systems, stochastic disturbances are almost inevitable [6]. Hence, stochastic modeling is a vital issue. Therefore, it is necessary to investigate effects of both time delays and stochastic perturbations on the stability of NNs. In the past decade, stability analysis of various stochastic NNs with time delays has become an attractive topic of research [8]. Moreover, due to instantaneous disturbances and abrupt changes at certain instants, many real-world systems such as neuronal networks, electronic networks, and biological networks are often subjected to impulsive effects [3]. Therefore, impulsive NNs have gained increasing research attention in the past 20 years [9]–[15]. For example, in [10], by means of the comparison principle, the stability problem was considered for a class of impulsive control systems with time-varying delays. In [12], the global exponential stability analysis problem was investigated for a class of impulsive NNs with variable delay, where three types of impulses are considered, respectively. In [14], the pth moment ( p ≥ 2) and the almostsure stability of stochastic Cohen–Grossberg NNs with mixed time delays and nonlinear impulsive effects were investigated through the Razumikhin type technique. In addition, very recently, we have investigated the stability problem for a class of delayed NNs with time-varying impulses in [3]. However, in most existing results concerning the stability of impulsive NNs, it is implicitly assumed that all the dimensions of the state undergo identical impulsive effects, i.e., the impulsive effects in the dimensions of the system state are equal to each other. Actually, in [16], it was pointed out that, hybrid dynamical systems involve an interacting countable collection of dynamical systems wherein dynamic states are not independent of one another and yet not all the system states are of equal precedence, which means that the impulsive effects in dimension may be different from each other. In addition, it is known that, in theory and practice, impulsive control has been widely used to stabilize and synchronize systems [17]. For instance, in [17], the stability and stabilization were studied for a class of stochastic systems with impulsive effects. However, it should be noted that, in the existing results, it is implicitly assumed that the impulsive effects are added to all the dimensions of the state. Obviously, this mechanism is not affordable in real-world applications, especially when the network size is large. Recently, the pinning control strategy has been widely used to control dynamical networks and agent systems, since only a fraction of nodes are added with controllers [18]. In this sense, this approach is economical and desirable from the perspective of control.
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ZHANG et al.: STOCHASTIC STABILITY OF DELAYED NNs
In addition, some states of dynamical systems are not available, measurable, or too expensive to measure [19]. For example, for a typical network with a number of vehicles, it is desirable to measure only positions or velocities to achieve coordination, since it is expensive to measure positions and velocities at the same time. In practice, there are many control strategies based on the assumption that the states of dynamical systems are not fully measurable, e.g., static output feedback [20], output-based event-triggered control [21], and, partial state impulsive control [19]. Hence, it is more realistic to use the local impulsive controller to control the dynamical systems. Therefore, from this point, the local impulses are promising to model sampled-data control systems and serve as an efficient tool to control dynamical systems. It becomes desirable from the viewpoints of theory and practice to study the stability of stochastic NNs with local impulsive effects, i.e., only a part of substates are subjected to impulsive effects. The integration of the presented local impulses, stochastic perturbations, and time delays poses several difficulties in mathematical derivations of the main results. Actually, how to analyze the stability of delayed NNs with local impulses has not been addressed in the existing results. The results here can also shed light on analysis of networked control systems. Therefore, this paper aims to shorten such a gap using the stochastic analysis techniques and the mathematical induction method. Motivated by the above considerations, in this paper, first, we present local impulses in which only a fraction of dimensions of the state experience different impulsive strengths. Besides, the local impulsive effects can also be distinct at different impulsive instants, and therefore, the presented impulses can encompass recently time-varying impulses in [3] and several typical impulses in [9]–[13]. Then, using the mathematical induction method, stability criteria are established to guarantee that stochastic NNs are exponentially stable in mean square. The main contributions of this paper are listed as follows. 1) Compared with the existing results concerning the stability of impulsive NNs [9]–[13], both stabilizing and destabilizing impulses are considered in this paper. In addition, impulsive effects in different dimensions of NNs here can be distinct from each other, which renders more practical applications of our model. 2) Compared with the time-varying impulses in [3], in this paper, only a fraction of substates are subjected to impulsive effects, which make the implementation of controllers more convenient. 3) Unlike the methods used in the existing results concerning stability of NNs with normal impulses [9]–[13], a time-dependent Lyapunov function will be used to derive the stability conditions of NNs with local impulses. Notations: N+ denotes the set of positive integers. For x ∈ Rn, x T denotes its transpose. The vector norm is defined as |x| = (x T x)1/2 . In denotes an n-dimensional identity matrix. For matrix A ∈ Rn×n, |A| = (λmax (A T A))1/2 , where λmax (·) represents the largest eigenvalue. Moreover, X ≥ Y (respectively, X > Y where X and Y are symmetric matrices)
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means that X − Y is a positive semidefinite (respectively, positive definite) matrix. PC([−r, 0]; Rn ) denotes the family of piecewise continuous functions from [−r, 0] to Rn with the norm |φ|r = sup−r≤θ≤0 φ(θ ). Moreover, let (, F , {Ft }, P) be a complete probability space with filtration {Ft }t ≥0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). Denote by L2F0 ([−τ, 0]; Rn ) the family of all F0 -measurable PC([−τ, 0]; Rn )-valued random variables ξ = {ξ(s) : −τ ≤ s ≤ 0} such that sup−τ ≤s≤0 E|ξ(s)|2 < ∞, where E{·} stands for the mathematical expectation operator with respect to the given probability measure P. II. M ODEL F ORMULATION AND P RELIMINARIES In this section, some preliminaries including model formulation, lemmas, and definitions are given. As in [6], the stochastic NN model can be described as follows: d x(t) = [−C x(t) + B f (x(t)) + D f (x(t − τ (t)))]dt + σ (t, x(t), x(t − τ (t)))dω(t)
(1)
where x(t) = [x 1(t), x 2 (t), . . . , x n (t)]T ∈ Rn is the state vector of the transformed system; C = diag(c1 , c2 , . . . , cn ) > 0 is the self-feedback matrix; B = (bi j )n×n and D = (di j )n×n are the connection weight matrices; f (x(t)) = ( f 1 (x(t)), f2 (x(t)), . . . , fn (x(t)))T denotes the neuron activation function with f i (0) = 0; τ (t) is a time-varying delay satisfying 0 ≤ τ (t) ≤ τ ; and ω(t) is a 1-D Brownian motion. Assume that σ (t, x(t), x(t − τ (t))) satisfies locally Lipschitz continuous and linear growth conditions. Moreover, σ (t, x(t), x(t − τ (t))) satisfies trace[σ T (t, x(t), x(t − τ (t)))σ (t, x(t), x(t − τ (t)))] ≤ |K 1 x(t)|2 + |K 2 x(t − τ (t))|2
(2)
where K 1 and K 2 are constant matrices with appropriate dimensions. The initial value of (1) is given by x(t) = φ(t) ∈ L2F0 ([−τ, 0], Rn ). In this paper, we assume that the nonlinear activation function satisfies the following condition. Assumption 1: There exist scalars li− and li+, such that for any x, y ∈ R, x = y li− ≤ Denote
f i (x) − f i (y) ≤ li+ . x−y
(3)
− l + l1+ l2− + l2+ l − + ln+ L 1 = diag 1 , ,..., n 2 2 2 +− +− + − L 2 = diag l1 l1 , l2 l2 , . . . , ln ln .
It is known from [22] and [23] that under the linear growth condition and Assumption 1, (1) has a unique continuous solution. As the local impulsive effects in (1) are included, we have the following model: ⎧ ⎨d x(t) = [−C x(t) + B f (x(t)) + D f (x(t − τ (t)))]dt + σ (t, x(t), x(t − τ (t)))dω(t), t = tk , k ∈ N+ (4) ⎩ + x(tk ) = νk x(tk− )
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ν1k 0 ∈ Rn×n , ν1k = diag{μ1k , 0 I(n−z) μ2k , . . . , μzk } is an invertible matrix of order z(1 ≤ z ≤ n); the impulsive instant sequence {tk }∞ satisfies k=1 0 < t1 < t2 < · · · < tk < · · · , limk→∞ tk = ∞; and x(tk− ) and x(tk+ ) denote the limit from the left and the right at time tk , respectively. Without loss of generality, in this paper, we assume that x(tk+ ) = x(tk ). It should be noted that the local impulses in (4) do not lose any generality. For example, if the state’s dimensions x i1 , x i2 , . . . , x iz (i j ∈ N+ ) are subjected to impulsive effects, then we can rearrange the order of x iz such that the model with the local impulses can be modeled by (4). Remark 1: From the second equation of (4), we know that only a fraction of states x 1 (t), x 2 (t), . . . , x z (t) undergo impulsive effects. Here, we call such kinds of impulsive effects as local impulsive effects. Unlike the impulses in [14], where the impulsive function is nonlinear, in this paper, only a fraction of dimensions of the state go through impulsive effects. In addition, in [14], only destabilizing impulses were considered. In this paper, both destabilizing and stabilizing impulses are considered. To summarize, the local impulses have the following three characteristics. 1) Only a fraction of dimensions of the state go through impulsive effects. 2) The impulsive effects in the dimensions of the state can be different from each other. 3) The impulsive effects can be distinct at different impulsive instants. It should be noted that the local impulses can take time-varying impulses [3] as a special case. To be more specific, if we assume that z = n and μik = μk , i = 1, 2, . . . , n, then the local impulses turn into the time-varying impulses considered in [3]. From [3], time-varying impulses are general enough to include most of the existing results as special cases [9]–[11], [13], [24]. In a word, the local impulses in this paper encompass the impulses in [9]–[11], [13], and [24]. It should be noted that if we let z = n, then the results can be easily extended to the case that νk = diag{μ1k , μ2k , . . . , μnk } with μik = 0. Remark 2: As in [3], if |μik | < 1, i = 1, 2, . . . , z, then we call such kinds of impulses stabilizing local impulses; if |μik | > 1, i = 1, 2, . . . , z, we call such kinds of impulses destabilizing local impulses. It can be seen from (4) that both destabilizing and stabilizing local impulses are taken account simultaneously. Recently, the impulsive control strategy has received increasing attention [10], [25], [26]. It should be noted that the effects of stabilizing local impulses indicate that we only need to add the impulsive controllers to a fraction of substates of x(t), and therefore, the stabilizing local impulses make control more convenient to implement like pinning impulsive control [27]. In [27], the permissible range of the impulsive strength completely depends on the left eigenvector of the graph Laplacian corresponding to the zero eigenvalue and the pinning node. Compared with the pinning impulsive strategy [27], the local impulses also have the following characteristics. 1) The pinning impulsive strategy is used to drive complex networks to the reference state [27], in which only a where νk =
fraction of nodes are subjected to stabilizing impulsive effects. However, in [27], all the dimensions of each state are subjected to the same strength of impulsive effects, while for the local impulses, only a fraction of dimensions are subjected to impulsive effects, which may be different from each other. From this point, the approaches used in the pinning impulsive strategy cannot be applied to the case of local impulses directly. 2) The pinning impulsive effects only consider synchronizing impulsive effects. However, in the local impulses, both stabilizing and destabilizing impulsive effects are considered. From these two aspects, the local impulses are more general than the pinning impulses in [27]. It is worth pointing out that in this paper, the local impulses include both destabilizing and stabilizing impulses, and the local impulsive effects can also be various at distinct impulsive instants. Hence, the results in [27] cannot be applied to the local impulses in this paper. Remark 3: Recently, in [28], the synchronization problem was investigated for a class of nonlinear dynamical networks with heterogeneous impulses, where in each node, all the dimensions of the state are subjected to impulsive effects. Compared with [28], the differences are as follows. 1) Model Difference: First, stochastic perturbations, destabilizing, and stabilizing impulses are considered here, which means that the method of deterministic systems in [28] cannot be directly applied to the stochastic system here. Second, our results can well fit into the framework of networked control systems. In [28], the results consider the heterogeneous impulses, which focus on describing the heterogeneity in real-world instead of describing the phenomena in networked control systems. Unlike [28], inspired by [29], the local impulses are based on the concept of networked control systems, which can well model eventtriggered control systems. Therefore, the motivation of these two kinds of impulses is distinct. Note that our proposed impulsive effects are also different from [29], since we consider the situation of multiple channel transmissions as well as impulsive strengths. 2) Method Difference: In [28], by means of the comparison principle, the synchronization problem for a class of nonlinear dynamical networks with heterogeneous impulses was investigated. However, in this paper, the mathematical induction method is used to obtain the main results. In addition, instead of the deterministic system approach in [28], stochastic analysis techniques are adopted for analyzing the stability of our model. To derive our main results, we need the following basic definition, lemmas, and assumptions. Definition 1: The NN in (1) is said to be exponentially stable in mean square, if there exist positive scalars 0 and λ, such that for every > 0, there is a positive scalar 1 such
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that E|φ| ≤ 1 implying E|x(t)| ≤ e−λ(t −t0)
(5)
holds for all t ≥ t0 . Lemma 1 [22]: Let x, y ∈ Rn and Q ∈ Rn×n be a positive semidefinite matrix, then the following inequality holds: 2x T Qy ≤ x T Qx + y T Qy. Lemma 2 [30]: The linear matrix inequality is as follows: Q(x) S(x) > 0, S T (x) R(x) where Q(x) = Q T (x) and R(x) = R T (x), is equivalent to either of the following conditions. 1) Q(x) > 0, R(x) − S T (x)Q(x)−1 S(x) > 0. 2) R(x) > 0, Q(x) − S(x)R(x)−1 S T (x) > 0. Lemma 3 [28]: Assume that , X 1 , and X 2 are constant matrices with appropriate dimensions, 0 ≤ ρ(t) ≤ 1, then
+ X1 < 0 (6)
+ X2 < 0
where D1 = diag{d11, d12 , . . . , d1n } > 0, D2 = diag{d21, d22 , . . . , d2n } > 0, and ∗ is used to denote the term that is induced by symmetry. Then the stochastic NN in (1) is exponentially stable in mean square. Proof: From (9), there exist small enough ε0 and ε1 ∈ (0, 1 − μ), such that ⎤ ⎡ ˜ Pm B + D1 L 1 Pm D 0 ⎢∗ −D1 0 0 ⎥ ⎥ ˜ j ml = ⎢ ⎥ 0 P(t) = (1 − ρ(t))P1 + ρ(t)P2 W (t) = eε0 (t −t0 ) V (t). For any given scalar > 0, choose 1 > 0 such that λ1 12 < μλ0 2 , where λ0 = min{λmin (P1 ), λmin (P2 )}. Suppose that the initial value of NN in (4) satisfies E|φ| ≤ 1 . In the following, we will prove that: EW (t) < λ0 2, t ∈ [t0 − τ, +∞).
(13)
To do so, we use the mathematical induction method. First, we prove that EW (t) < λ0 2, t ∈ [t0 − τ, t1 ).
(14)
For t ∈ [t0 − τ, t0 ], from (8) and E|φ| ≤ 1 , t ∈ [t0 − τ, t0 ] EW (t) = Eeε0 (t −t0 ) x T (t)P(t)x(t) ≤ λ1 E|x(t)|2 ≤ λ1 12 < μλ0 2 .
(15)
To prove (14), in the following, we only need to prove that EW (t) < λ0 2, t ∈ (t0 , t1 ). Suppose that it is not true, then there exists some t ∈ (t0 , t1 ) such that
where α ln μ P1 − P2 = −2Pm C + Pm +λ1 |K 1 |2 + Pm + − D1 L 2 μ σ2 σl
j = −α P j − D2 L 2 + λ1 K 2T K 2
(12)
EW (t) ≥ λ0 2 . Set t ∗ = inf{t ∈ [t0 , t1 ]; EW (t) ≥ λ0 2 }.
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From (15), we know that t ∗ ∈ (t0 , t1 ) and
Then, for any diagonal matrices D1 = diag{d11, d12 , . . . , d1n } > 0 and D2 = diag{d21 , d22 , . . . , d2n } > 0
EW (t ∗ ) = λ0 2 .
0 ≤ eε0(t −t0 ) n × d1i li+ x i (t) − f i (x i (t)) × fi (x i (t)) − li− x i (t)
Set t∗ = sup t ∈ (t0 , t ∗ ), EW (t) ≤ μλ0 2 .
i=1 n
Then, from (15), we know that t∗ ∈ (t0 , t ∗ ) and EW (t∗ ) = μλ0 2 . For θ ∈ [−τ, 0], when t ∈ [t∗ , t ∗ ], we have t + θ ∈ (t0 − τ, t ∗ ]. Then we have EW (t + θ ) ≤ λ0 2 .
+
i=1
× f i (x i (t − τ (t))) − li− x i (t − τ (t))
− x T (t)D1 L 2 x(t) + 2x T (t − τ (t))D2 L 1 f (x(t −τ (t))) − f T (x(t − τ (t)))D2 f (x(t − τ (t))) − x T (t − τ (t))D2 L 2 x(t − τ (t)) . (19)
EW (t) ≥ μλ0 2
Set
(16)
1 W˜ (t) = D + W (t) + α W (t) − W (t − τ (t)) − μ1 W (t) μ (17)
where the upper-right Dini derivative D + W (t) is defined as D + W (t) = limh→0+ (W (t + h) − W (t))/ h. From [31], we know that when = t ∈ [tk , tk+1 ), D + EW (t) = ELW (t), where L is the Itô operator. For t ∈ [t∗ , t ∗ ], one has
From (2), we have trace[σ T (t, x(t), x(t − τ (t)))P(t)σ (t, x(t), x(t − τ (t)))] ≤ λ1 x T (t)K 1T K 1 x(t) + x T (t − τ (t))K 2T K 2 x(t − τ (t)) . (20) From the definition of P(t), one has ˙ P(t) =
=e
T
Let
⎡
˘ ⎢∗ (t) = ⎢ ⎣∗ 0
T
− 2x (t)P(t)C x(t) + 2x (t)P(t)B f (x(t)) + 2x T (t)P(t)D × f (x(t − τ (t))) + trace[σ T (t, x(t), x(t − τ (t)))
P(t)B + D1 L 1 −D1 0 0
P(t)D 0 −D2 ∗
⎤ 0 0 ⎥ ⎥ D2 L 1 ⎦ ˘
where
×P(t)σ (t, x(t), x(t − τ (t)))] α + x T (t)P(t)x(t) − αe−ε0 τ x T (t − τ (t)) μ × P(t − τ (t))x(t − τ (t))
˙ . − μ1 x T (t)P(t)x(t) + x T (t) P(t)x(t)
(22)
Then, there exists a function β(t) : [0, +∞) → [0, 1] such that P1 − P2 P1 − P2 ˙ + β(t) . (23) P(t) = (1 − β(t)) σ1 σ2
α + EW (t) − αEW (t − τ (t)) − μ1 EW (t) μ ε0 (t −t0 ) E ε0 x T (t)P(t)x(t) =e
(18)
In view of Assumption 1, it is easy to see that 0 ≤ li+ x i (t) − fi (x i (t)) f i (x i (t)) − li− x i (t) 0 ≤ li+ x i (t − τ (t)) − f i (x i (t − τ (t))) × ( f i (x i (t − τ (t)))
(21)
1 1 1 ≤ ≤ . σ2 tk − tk−1 σ1
(ε0 EV (t) + ELV (t))
− li− x i (t
1 (P1 − P2 ). tk − tk−1
By means of Assumption 2, we have
EW˜ (t) ε0 (t −t0 )
= eε0 (t −t0 ) × 2x T (t)D1 L 1 f (x(t)) − f T (x(t))D1 f (x(t))
Hence
≥ μEW (t + θ ), t ∈ [t∗ , t ∗ ], θ ∈ [−τ, 0].
d2i li+ x i (t − τ (t)) − f i (x i (t − τ (t)))
− τ (t))), i = 1, 2, . . . , n.
˘ = ε0 P(t) − 2P(t)C + α P(t) + λ1 |K 1 |2 μ ln(μ + ε1 ) P1 − P2 P1 − P2 + P(t) + + (1 − β(t)) σ2 σl σ1 P1 − P2 + β(t) − D1 L 2 σ2 ˘ = −αe− 0 τ P(t − τ (t)) − D2 L 2 + λ1 K 2T K 2 .
From (18)–(23), for t ∈ [t∗ , t ∗ ], we have EW˜ (t) ≤ eε0 (t −t0 ) E[η T (t)(t)η(t)]
(24)
where η(t) = [x T (t), f T (x(t)), f T (x(t − τ (t))), x T (t − τ (t))]T.
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Noting that the definition of P(t) = (1 − ρ(t))P1 + ρ(t)P2 , we have (t) = (1 − ρ(t))1 (t) + ρ(t)2 (t). In view of Lemma 3, (t) < 0 is equivalent to ⎤ ⎡ 0 1 P1 B + D1 L 1 P1 D ⎢∗ −D1 0 0 ⎥ ⎥ < 0 (25) 1 (t) = ⎢ ⎣∗ 0 −D2 D2 L 1 ⎦ ˘ 0 0 ∗
and
⎡
2
⎢∗ ⎢ 2 (t) = ⎢ ⎣∗
P2 B + D1 L 1
P2 D
−D1 0
0 −D2
0
∗
0
0
⎤
0 ⎥ ⎥ ⎥ 0 and P22 > 0, there does not exist any C ∈ (0, 1) such that V (tk+ ) ≤ C V (tk− ). Hence, the Lyapunov function considered in [3], [8], [9]–[11], [13], and [33] cannot be applied to Theorem 1 directly. Fortunately, the timedependent Lyapunov function can well solve this problem. It can also be seen from the proof of Theorem 1 that the timedependent Lyapunov function plays a key role in deriving our main results. Remark 6: In Theorem 1, the predefined parameter μ plays an important role in our stability conditions. It can be concluded that a larger μ can make the conditions more feasible. Meanwhile, it will lead to a smaller impulsive interval, which means that more stabilizing impulses are required to enforce the trajectories of (1) to the equilibrium point. IV. C OMPARISON W ITH E XISTING R ESULTS In this section, based on Theorem 1, four corollaries will be derived to illustrate that our model is more general than the existing results. If there are no stochastic perturbations in (1), i.e., the NN in (1) reduces to the following model: x(t) ˙ = −C x(t) + B f (x(t)) + D f (x(t − τ (t))).
(35)
For the NN in (35), the following corollary can be obtained. Corollary 1: Consider the NN in (35) with local impulsive effects. Suppose that Assumptions 1 and 2 hold. If for a prescribed positive scalar μ ∈ (0, 1), there exist positive definite matrices P1 , P2 ∈ Rn×n and a positive scalar α such that the following inequalities hold: ⎡ ⎤ ¯ Pm B + D1 L 1 Pm D 0 ⎢ −D1 0 0 ⎥ ⎥ < 0 (36) ¯ 1ml = ⎢ ∗ ⎣∗ 0 −D2 D2 L 1 ⎦ ¯ 0 0 ∗
−μP1 νkT P2 < 0, m, l = 1, 2 (37) P2 νk −P2 where α ln μ P1 − P2 Pm + Pm + − D1 L 2 μ σ2 σl ¯ = −α Pm − D2 L 2 .
¯ = −2Pm C +
Then the NN in (35) is exponentially stable. In (35), if all the system states are subjected to impulsive effects, i.e., z = n, we have the following corollary. Corollary 2: Consider the NN in (35) with impulsive effects. Suppose that Assumptions 1 and 2 hold. If for a prescribed positive scalar μ ∈ (0, 1), there exist positive definite matrices P1 , P2 ∈ Rn×n and a positive scalar α such that (36) and (37) hold. Then the NN in (35) with impulsive effects is exponentially stable. Remark 7: Corollary 2 investigates the following two special cases. 1) The impulsive effects are stabilizing. 2) The impulsive effects are destabilizing. If |μik | = μ < 1, (i = 1, 2, . . . , n), then Corollary 2 becomes the stability problem of delayed NNs with stabilizing impulses, which has widely been investigated in [11] and [13]. If μik = μk , (i = 1, 2, . . . , n), Corollary 2 becomes the
stability of delayed NNs with time-varying impulses, which has been studied in [3]. It should be noted that in Corollary 2, the impulsive effects in substates can be different from each other, and therefore, the local impulses take the time-varying impulses [3] and stabilizing impulses [11], [13] as special cases. Hence, the local impulses presented in this paper are quite general. Remark 8: In Corollary 2, if the impulsive effects are stabilizing, we can let P(t) = P in (12), then, Corollary 2 becomes the NNs with stabilizing impulsive effects. By Corollary 2, we can derive the analogous result on dynamical system with time-varying delay under stabilizing impulsive control [12]. Corollary 3: Consider the NN in (35) with local impulsive effects. Suppose that Assumptions 1 and 2 hold. If for a prescribed positive scalar μ ∈ (0, 1), there exist positive definite matrices P1 and P2 ∈ Rn×n and a positive scalar α such that the following inequalities hold: ⎤ ⎡ ˆ P B + D1 L 1 P D 0 ⎢ 0 0 ⎥ −D1 ⎥
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 10, OCTOBER 2015
Stochastic Stability of Delayed Neural Networks With Local Impulsive Effects Wenbing Zhang, Yang Tang, Member, IEEE, Wai Keung Wong, and Qingying Miao, Member, IEEE
Abstract— In this paper, the stability problem is studied for a class of stochastic neural networks (NNs) with local impulsive effects. The impulsive effects considered can be not only nonidentical in different dimensions of the system state but also various at distinct impulsive instants. Hence, the impulses here can encompass several typical impulses in NNs. The aim of this paper is to derive stability criteria such that stochastic NNs with local impulsive effects are exponentially stable in mean square. By means of the mathematical induction method, several easy-tocheck conditions are obtained to ensure the mean square stability of NNs. Three examples are given to show the effectiveness of the proposed stability criterion. Index Terms— Impulsive systems, local impulsive effects, neural networks (NNs), stability analysis.
I. I NTRODUCTION
I
N THE past two decades, research relating to neural networks (NNs) has received increasing attention due to their wide applications in many areas such as combinatorial optimization, signal processing, and pattern recognition. In a real-world NN model, axonal signal transmission delays often occur in various NNs, due to the finite speed of switching and transmitting signals, which may cause oscillation or instability. Then, a large number of promising results on the stability or the stabilization of delayed NNs have been published [1]–[7]. For example, in [3], the stability problem was investigated for a class of delayed NNs with time-varying impulses. In [4], by means of a linear matrix inequality approach, the problems of stability and dissipativity analysis were investigated for static NNs with time delay. Manuscript received April 7, 2014; revised October 21, 2014; accepted December 7, 2014. Date of publication December 25, 2014; date of current version September 16, 2015. This work was supported in part by the Natural Science Foundation through the Higher Education Institutions of Jiangsu Province, China, under Grant 14KJB120014, in part by the Natural Science Foundation of Shanghai (Grant 13ZR1421300), in part by the Alexander von Humboldt Foundation of Germany, in part by the National Natural Science Foundation of China under Grant 11426196, Grant 61203235, Grant 61304158, Grant 61473189, and Grant 61375012, and in part by Hong Kong Polytechnic University, Hong Kong, under Project G-YM53. W. Zhang is with the Department of Mathematics, Yangzhou University, Yangzhou 225009, Jiangsu, China (e-mail: [email protected]). Y. Tang is with the Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany, and also with the Institute of Physics, Humboldt University of Berlin, Berlin 12489, Germany (e-mail: [email protected]). W. K. Wong is with the Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hong Kong, and also with the Shenzhen Research Institute, Hong Kong Polytechnic University, Hong Kong (e-mail: [email protected]). Q. Miao is with the School of Continuing Education, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2380451
On the other hand, when designing the NNs or in the implementation of neural systems, stochastic disturbances are almost inevitable [6]. Hence, stochastic modeling is a vital issue. Therefore, it is necessary to investigate effects of both time delays and stochastic perturbations on the stability of NNs. In the past decade, stability analysis of various stochastic NNs with time delays has become an attractive topic of research [8]. Moreover, due to instantaneous disturbances and abrupt changes at certain instants, many real-world systems such as neuronal networks, electronic networks, and biological networks are often subjected to impulsive effects [3]. Therefore, impulsive NNs have gained increasing research attention in the past 20 years [9]–[15]. For example, in [10], by means of the comparison principle, the stability problem was considered for a class of impulsive control systems with time-varying delays. In [12], the global exponential stability analysis problem was investigated for a class of impulsive NNs with variable delay, where three types of impulses are considered, respectively. In [14], the pth moment ( p ≥ 2) and the almostsure stability of stochastic Cohen–Grossberg NNs with mixed time delays and nonlinear impulsive effects were investigated through the Razumikhin type technique. In addition, very recently, we have investigated the stability problem for a class of delayed NNs with time-varying impulses in [3]. However, in most existing results concerning the stability of impulsive NNs, it is implicitly assumed that all the dimensions of the state undergo identical impulsive effects, i.e., the impulsive effects in the dimensions of the system state are equal to each other. Actually, in [16], it was pointed out that, hybrid dynamical systems involve an interacting countable collection of dynamical systems wherein dynamic states are not independent of one another and yet not all the system states are of equal precedence, which means that the impulsive effects in dimension may be different from each other. In addition, it is known that, in theory and practice, impulsive control has been widely used to stabilize and synchronize systems [17]. For instance, in [17], the stability and stabilization were studied for a class of stochastic systems with impulsive effects. However, it should be noted that, in the existing results, it is implicitly assumed that the impulsive effects are added to all the dimensions of the state. Obviously, this mechanism is not affordable in real-world applications, especially when the network size is large. Recently, the pinning control strategy has been widely used to control dynamical networks and agent systems, since only a fraction of nodes are added with controllers [18]. In this sense, this approach is economical and desirable from the perspective of control.
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ZHANG et al.: STOCHASTIC STABILITY OF DELAYED NNs
In addition, some states of dynamical systems are not available, measurable, or too expensive to measure [19]. For example, for a typical network with a number of vehicles, it is desirable to measure only positions or velocities to achieve coordination, since it is expensive to measure positions and velocities at the same time. In practice, there are many control strategies based on the assumption that the states of dynamical systems are not fully measurable, e.g., static output feedback [20], output-based event-triggered control [21], and, partial state impulsive control [19]. Hence, it is more realistic to use the local impulsive controller to control the dynamical systems. Therefore, from this point, the local impulses are promising to model sampled-data control systems and serve as an efficient tool to control dynamical systems. It becomes desirable from the viewpoints of theory and practice to study the stability of stochastic NNs with local impulsive effects, i.e., only a part of substates are subjected to impulsive effects. The integration of the presented local impulses, stochastic perturbations, and time delays poses several difficulties in mathematical derivations of the main results. Actually, how to analyze the stability of delayed NNs with local impulses has not been addressed in the existing results. The results here can also shed light on analysis of networked control systems. Therefore, this paper aims to shorten such a gap using the stochastic analysis techniques and the mathematical induction method. Motivated by the above considerations, in this paper, first, we present local impulses in which only a fraction of dimensions of the state experience different impulsive strengths. Besides, the local impulsive effects can also be distinct at different impulsive instants, and therefore, the presented impulses can encompass recently time-varying impulses in [3] and several typical impulses in [9]–[13]. Then, using the mathematical induction method, stability criteria are established to guarantee that stochastic NNs are exponentially stable in mean square. The main contributions of this paper are listed as follows. 1) Compared with the existing results concerning the stability of impulsive NNs [9]–[13], both stabilizing and destabilizing impulses are considered in this paper. In addition, impulsive effects in different dimensions of NNs here can be distinct from each other, which renders more practical applications of our model. 2) Compared with the time-varying impulses in [3], in this paper, only a fraction of substates are subjected to impulsive effects, which make the implementation of controllers more convenient. 3) Unlike the methods used in the existing results concerning stability of NNs with normal impulses [9]–[13], a time-dependent Lyapunov function will be used to derive the stability conditions of NNs with local impulses. Notations: N+ denotes the set of positive integers. For x ∈ Rn, x T denotes its transpose. The vector norm is defined as |x| = (x T x)1/2 . In denotes an n-dimensional identity matrix. For matrix A ∈ Rn×n, |A| = (λmax (A T A))1/2 , where λmax (·) represents the largest eigenvalue. Moreover, X ≥ Y (respectively, X > Y where X and Y are symmetric matrices)
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means that X − Y is a positive semidefinite (respectively, positive definite) matrix. PC([−r, 0]; Rn ) denotes the family of piecewise continuous functions from [−r, 0] to Rn with the norm |φ|r = sup−r≤θ≤0 φ(θ ). Moreover, let (, F , {Ft }, P) be a complete probability space with filtration {Ft }t ≥0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). Denote by L2F0 ([−τ, 0]; Rn ) the family of all F0 -measurable PC([−τ, 0]; Rn )-valued random variables ξ = {ξ(s) : −τ ≤ s ≤ 0} such that sup−τ ≤s≤0 E|ξ(s)|2 < ∞, where E{·} stands for the mathematical expectation operator with respect to the given probability measure P. II. M ODEL F ORMULATION AND P RELIMINARIES In this section, some preliminaries including model formulation, lemmas, and definitions are given. As in [6], the stochastic NN model can be described as follows: d x(t) = [−C x(t) + B f (x(t)) + D f (x(t − τ (t)))]dt + σ (t, x(t), x(t − τ (t)))dω(t)
(1)
where x(t) = [x 1(t), x 2 (t), . . . , x n (t)]T ∈ Rn is the state vector of the transformed system; C = diag(c1 , c2 , . . . , cn ) > 0 is the self-feedback matrix; B = (bi j )n×n and D = (di j )n×n are the connection weight matrices; f (x(t)) = ( f 1 (x(t)), f2 (x(t)), . . . , fn (x(t)))T denotes the neuron activation function with f i (0) = 0; τ (t) is a time-varying delay satisfying 0 ≤ τ (t) ≤ τ ; and ω(t) is a 1-D Brownian motion. Assume that σ (t, x(t), x(t − τ (t))) satisfies locally Lipschitz continuous and linear growth conditions. Moreover, σ (t, x(t), x(t − τ (t))) satisfies trace[σ T (t, x(t), x(t − τ (t)))σ (t, x(t), x(t − τ (t)))] ≤ |K 1 x(t)|2 + |K 2 x(t − τ (t))|2
(2)
where K 1 and K 2 are constant matrices with appropriate dimensions. The initial value of (1) is given by x(t) = φ(t) ∈ L2F0 ([−τ, 0], Rn ). In this paper, we assume that the nonlinear activation function satisfies the following condition. Assumption 1: There exist scalars li− and li+, such that for any x, y ∈ R, x = y li− ≤ Denote
f i (x) − f i (y) ≤ li+ . x−y
(3)
− l + l1+ l2− + l2+ l − + ln+ L 1 = diag 1 , ,..., n 2 2 2 +− +− + − L 2 = diag l1 l1 , l2 l2 , . . . , ln ln .
It is known from [22] and [23] that under the linear growth condition and Assumption 1, (1) has a unique continuous solution. As the local impulsive effects in (1) are included, we have the following model: ⎧ ⎨d x(t) = [−C x(t) + B f (x(t)) + D f (x(t − τ (t)))]dt + σ (t, x(t), x(t − τ (t)))dω(t), t = tk , k ∈ N+ (4) ⎩ + x(tk ) = νk x(tk− )
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 10, OCTOBER 2015
ν1k 0 ∈ Rn×n , ν1k = diag{μ1k , 0 I(n−z) μ2k , . . . , μzk } is an invertible matrix of order z(1 ≤ z ≤ n); the impulsive instant sequence {tk }∞ satisfies k=1 0 < t1 < t2 < · · · < tk < · · · , limk→∞ tk = ∞; and x(tk− ) and x(tk+ ) denote the limit from the left and the right at time tk , respectively. Without loss of generality, in this paper, we assume that x(tk+ ) = x(tk ). It should be noted that the local impulses in (4) do not lose any generality. For example, if the state’s dimensions x i1 , x i2 , . . . , x iz (i j ∈ N+ ) are subjected to impulsive effects, then we can rearrange the order of x iz such that the model with the local impulses can be modeled by (4). Remark 1: From the second equation of (4), we know that only a fraction of states x 1 (t), x 2 (t), . . . , x z (t) undergo impulsive effects. Here, we call such kinds of impulsive effects as local impulsive effects. Unlike the impulses in [14], where the impulsive function is nonlinear, in this paper, only a fraction of dimensions of the state go through impulsive effects. In addition, in [14], only destabilizing impulses were considered. In this paper, both destabilizing and stabilizing impulses are considered. To summarize, the local impulses have the following three characteristics. 1) Only a fraction of dimensions of the state go through impulsive effects. 2) The impulsive effects in the dimensions of the state can be different from each other. 3) The impulsive effects can be distinct at different impulsive instants. It should be noted that the local impulses can take time-varying impulses [3] as a special case. To be more specific, if we assume that z = n and μik = μk , i = 1, 2, . . . , n, then the local impulses turn into the time-varying impulses considered in [3]. From [3], time-varying impulses are general enough to include most of the existing results as special cases [9]–[11], [13], [24]. In a word, the local impulses in this paper encompass the impulses in [9]–[11], [13], and [24]. It should be noted that if we let z = n, then the results can be easily extended to the case that νk = diag{μ1k , μ2k , . . . , μnk } with μik = 0. Remark 2: As in [3], if |μik | < 1, i = 1, 2, . . . , z, then we call such kinds of impulses stabilizing local impulses; if |μik | > 1, i = 1, 2, . . . , z, we call such kinds of impulses destabilizing local impulses. It can be seen from (4) that both destabilizing and stabilizing local impulses are taken account simultaneously. Recently, the impulsive control strategy has received increasing attention [10], [25], [26]. It should be noted that the effects of stabilizing local impulses indicate that we only need to add the impulsive controllers to a fraction of substates of x(t), and therefore, the stabilizing local impulses make control more convenient to implement like pinning impulsive control [27]. In [27], the permissible range of the impulsive strength completely depends on the left eigenvector of the graph Laplacian corresponding to the zero eigenvalue and the pinning node. Compared with the pinning impulsive strategy [27], the local impulses also have the following characteristics. 1) The pinning impulsive strategy is used to drive complex networks to the reference state [27], in which only a where νk =
fraction of nodes are subjected to stabilizing impulsive effects. However, in [27], all the dimensions of each state are subjected to the same strength of impulsive effects, while for the local impulses, only a fraction of dimensions are subjected to impulsive effects, which may be different from each other. From this point, the approaches used in the pinning impulsive strategy cannot be applied to the case of local impulses directly. 2) The pinning impulsive effects only consider synchronizing impulsive effects. However, in the local impulses, both stabilizing and destabilizing impulsive effects are considered. From these two aspects, the local impulses are more general than the pinning impulses in [27]. It is worth pointing out that in this paper, the local impulses include both destabilizing and stabilizing impulses, and the local impulsive effects can also be various at distinct impulsive instants. Hence, the results in [27] cannot be applied to the local impulses in this paper. Remark 3: Recently, in [28], the synchronization problem was investigated for a class of nonlinear dynamical networks with heterogeneous impulses, where in each node, all the dimensions of the state are subjected to impulsive effects. Compared with [28], the differences are as follows. 1) Model Difference: First, stochastic perturbations, destabilizing, and stabilizing impulses are considered here, which means that the method of deterministic systems in [28] cannot be directly applied to the stochastic system here. Second, our results can well fit into the framework of networked control systems. In [28], the results consider the heterogeneous impulses, which focus on describing the heterogeneity in real-world instead of describing the phenomena in networked control systems. Unlike [28], inspired by [29], the local impulses are based on the concept of networked control systems, which can well model eventtriggered control systems. Therefore, the motivation of these two kinds of impulses is distinct. Note that our proposed impulsive effects are also different from [29], since we consider the situation of multiple channel transmissions as well as impulsive strengths. 2) Method Difference: In [28], by means of the comparison principle, the synchronization problem for a class of nonlinear dynamical networks with heterogeneous impulses was investigated. However, in this paper, the mathematical induction method is used to obtain the main results. In addition, instead of the deterministic system approach in [28], stochastic analysis techniques are adopted for analyzing the stability of our model. To derive our main results, we need the following basic definition, lemmas, and assumptions. Definition 1: The NN in (1) is said to be exponentially stable in mean square, if there exist positive scalars 0 and λ, such that for every > 0, there is a positive scalar 1 such
ZHANG et al.: STOCHASTIC STABILITY OF DELAYED NNs
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that E|φ| ≤ 1 implying E|x(t)| ≤ e−λ(t −t0)
(5)
holds for all t ≥ t0 . Lemma 1 [22]: Let x, y ∈ Rn and Q ∈ Rn×n be a positive semidefinite matrix, then the following inequality holds: 2x T Qy ≤ x T Qx + y T Qy. Lemma 2 [30]: The linear matrix inequality is as follows: Q(x) S(x) > 0, S T (x) R(x) where Q(x) = Q T (x) and R(x) = R T (x), is equivalent to either of the following conditions. 1) Q(x) > 0, R(x) − S T (x)Q(x)−1 S(x) > 0. 2) R(x) > 0, Q(x) − S(x)R(x)−1 S T (x) > 0. Lemma 3 [28]: Assume that , X 1 , and X 2 are constant matrices with appropriate dimensions, 0 ≤ ρ(t) ≤ 1, then
+ X1 < 0 (6)
+ X2 < 0
where D1 = diag{d11, d12 , . . . , d1n } > 0, D2 = diag{d21, d22 , . . . , d2n } > 0, and ∗ is used to denote the term that is induced by symmetry. Then the stochastic NN in (1) is exponentially stable in mean square. Proof: From (9), there exist small enough ε0 and ε1 ∈ (0, 1 − μ), such that ⎤ ⎡ ˜ Pm B + D1 L 1 Pm D 0 ⎢∗ −D1 0 0 ⎥ ⎥ ˜ j ml = ⎢ ⎥ 0 P(t) = (1 − ρ(t))P1 + ρ(t)P2 W (t) = eε0 (t −t0 ) V (t). For any given scalar > 0, choose 1 > 0 such that λ1 12 < μλ0 2 , where λ0 = min{λmin (P1 ), λmin (P2 )}. Suppose that the initial value of NN in (4) satisfies E|φ| ≤ 1 . In the following, we will prove that: EW (t) < λ0 2, t ∈ [t0 − τ, +∞).
(13)
To do so, we use the mathematical induction method. First, we prove that EW (t) < λ0 2, t ∈ [t0 − τ, t1 ).
(14)
For t ∈ [t0 − τ, t0 ], from (8) and E|φ| ≤ 1 , t ∈ [t0 − τ, t0 ] EW (t) = Eeε0 (t −t0 ) x T (t)P(t)x(t) ≤ λ1 E|x(t)|2 ≤ λ1 12 < μλ0 2 .
(15)
To prove (14), in the following, we only need to prove that EW (t) < λ0 2, t ∈ (t0 , t1 ). Suppose that it is not true, then there exists some t ∈ (t0 , t1 ) such that
where α ln μ P1 − P2 = −2Pm C + Pm +λ1 |K 1 |2 + Pm + − D1 L 2 μ σ2 σl
j = −α P j − D2 L 2 + λ1 K 2T K 2
(12)
EW (t) ≥ λ0 2 . Set t ∗ = inf{t ∈ [t0 , t1 ]; EW (t) ≥ λ0 2 }.
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From (15), we know that t ∗ ∈ (t0 , t1 ) and
Then, for any diagonal matrices D1 = diag{d11, d12 , . . . , d1n } > 0 and D2 = diag{d21 , d22 , . . . , d2n } > 0
EW (t ∗ ) = λ0 2 .
0 ≤ eε0(t −t0 ) n × d1i li+ x i (t) − f i (x i (t)) × fi (x i (t)) − li− x i (t)
Set t∗ = sup t ∈ (t0 , t ∗ ), EW (t) ≤ μλ0 2 .
i=1 n
Then, from (15), we know that t∗ ∈ (t0 , t ∗ ) and EW (t∗ ) = μλ0 2 . For θ ∈ [−τ, 0], when t ∈ [t∗ , t ∗ ], we have t + θ ∈ (t0 − τ, t ∗ ]. Then we have EW (t + θ ) ≤ λ0 2 .
+
i=1
× f i (x i (t − τ (t))) − li− x i (t − τ (t))
− x T (t)D1 L 2 x(t) + 2x T (t − τ (t))D2 L 1 f (x(t −τ (t))) − f T (x(t − τ (t)))D2 f (x(t − τ (t))) − x T (t − τ (t))D2 L 2 x(t − τ (t)) . (19)
EW (t) ≥ μλ0 2
Set
(16)
1 W˜ (t) = D + W (t) + α W (t) − W (t − τ (t)) − μ1 W (t) μ (17)
where the upper-right Dini derivative D + W (t) is defined as D + W (t) = limh→0+ (W (t + h) − W (t))/ h. From [31], we know that when = t ∈ [tk , tk+1 ), D + EW (t) = ELW (t), where L is the Itô operator. For t ∈ [t∗ , t ∗ ], one has
From (2), we have trace[σ T (t, x(t), x(t − τ (t)))P(t)σ (t, x(t), x(t − τ (t)))] ≤ λ1 x T (t)K 1T K 1 x(t) + x T (t − τ (t))K 2T K 2 x(t − τ (t)) . (20) From the definition of P(t), one has ˙ P(t) =
=e
T
Let
⎡
˘ ⎢∗ (t) = ⎢ ⎣∗ 0
T
− 2x (t)P(t)C x(t) + 2x (t)P(t)B f (x(t)) + 2x T (t)P(t)D × f (x(t − τ (t))) + trace[σ T (t, x(t), x(t − τ (t)))
P(t)B + D1 L 1 −D1 0 0
P(t)D 0 −D2 ∗
⎤ 0 0 ⎥ ⎥ D2 L 1 ⎦ ˘
where
×P(t)σ (t, x(t), x(t − τ (t)))] α + x T (t)P(t)x(t) − αe−ε0 τ x T (t − τ (t)) μ × P(t − τ (t))x(t − τ (t))
˙ . − μ1 x T (t)P(t)x(t) + x T (t) P(t)x(t)
(22)
Then, there exists a function β(t) : [0, +∞) → [0, 1] such that P1 − P2 P1 − P2 ˙ + β(t) . (23) P(t) = (1 − β(t)) σ1 σ2
α + EW (t) − αEW (t − τ (t)) − μ1 EW (t) μ ε0 (t −t0 ) E ε0 x T (t)P(t)x(t) =e
(18)
In view of Assumption 1, it is easy to see that 0 ≤ li+ x i (t) − fi (x i (t)) f i (x i (t)) − li− x i (t) 0 ≤ li+ x i (t − τ (t)) − f i (x i (t − τ (t))) × ( f i (x i (t − τ (t)))
(21)
1 1 1 ≤ ≤ . σ2 tk − tk−1 σ1
(ε0 EV (t) + ELV (t))
− li− x i (t
1 (P1 − P2 ). tk − tk−1
By means of Assumption 2, we have
EW˜ (t) ε0 (t −t0 )
= eε0 (t −t0 ) × 2x T (t)D1 L 1 f (x(t)) − f T (x(t))D1 f (x(t))
Hence
≥ μEW (t + θ ), t ∈ [t∗ , t ∗ ], θ ∈ [−τ, 0].
d2i li+ x i (t − τ (t)) − f i (x i (t − τ (t)))
− τ (t))), i = 1, 2, . . . , n.
˘ = ε0 P(t) − 2P(t)C + α P(t) + λ1 |K 1 |2 μ ln(μ + ε1 ) P1 − P2 P1 − P2 + P(t) + + (1 − β(t)) σ2 σl σ1 P1 − P2 + β(t) − D1 L 2 σ2 ˘ = −αe− 0 τ P(t − τ (t)) − D2 L 2 + λ1 K 2T K 2 .
From (18)–(23), for t ∈ [t∗ , t ∗ ], we have EW˜ (t) ≤ eε0 (t −t0 ) E[η T (t)(t)η(t)]
(24)
where η(t) = [x T (t), f T (x(t)), f T (x(t − τ (t))), x T (t − τ (t))]T.
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Noting that the definition of P(t) = (1 − ρ(t))P1 + ρ(t)P2 , we have (t) = (1 − ρ(t))1 (t) + ρ(t)2 (t). In view of Lemma 3, (t) < 0 is equivalent to ⎤ ⎡ 0 1 P1 B + D1 L 1 P1 D ⎢∗ −D1 0 0 ⎥ ⎥ < 0 (25) 1 (t) = ⎢ ⎣∗ 0 −D2 D2 L 1 ⎦ ˘ 0 0 ∗
and
⎡
2
⎢∗ ⎢ 2 (t) = ⎢ ⎣∗
P2 B + D1 L 1
P2 D
−D1 0
0 −D2
0
∗
0
0
⎤
0 ⎥ ⎥ ⎥ 0 and P22 > 0, there does not exist any C ∈ (0, 1) such that V (tk+ ) ≤ C V (tk− ). Hence, the Lyapunov function considered in [3], [8], [9]–[11], [13], and [33] cannot be applied to Theorem 1 directly. Fortunately, the timedependent Lyapunov function can well solve this problem. It can also be seen from the proof of Theorem 1 that the timedependent Lyapunov function plays a key role in deriving our main results. Remark 6: In Theorem 1, the predefined parameter μ plays an important role in our stability conditions. It can be concluded that a larger μ can make the conditions more feasible. Meanwhile, it will lead to a smaller impulsive interval, which means that more stabilizing impulses are required to enforce the trajectories of (1) to the equilibrium point. IV. C OMPARISON W ITH E XISTING R ESULTS In this section, based on Theorem 1, four corollaries will be derived to illustrate that our model is more general than the existing results. If there are no stochastic perturbations in (1), i.e., the NN in (1) reduces to the following model: x(t) ˙ = −C x(t) + B f (x(t)) + D f (x(t − τ (t))).
(35)
For the NN in (35), the following corollary can be obtained. Corollary 1: Consider the NN in (35) with local impulsive effects. Suppose that Assumptions 1 and 2 hold. If for a prescribed positive scalar μ ∈ (0, 1), there exist positive definite matrices P1 , P2 ∈ Rn×n and a positive scalar α such that the following inequalities hold: ⎡ ⎤ ¯ Pm B + D1 L 1 Pm D 0 ⎢ −D1 0 0 ⎥ ⎥ < 0 (36) ¯ 1ml = ⎢ ∗ ⎣∗ 0 −D2 D2 L 1 ⎦ ¯ 0 0 ∗
−μP1 νkT P2 < 0, m, l = 1, 2 (37) P2 νk −P2 where α ln μ P1 − P2 Pm + Pm + − D1 L 2 μ σ2 σl ¯ = −α Pm − D2 L 2 .
¯ = −2Pm C +
Then the NN in (35) is exponentially stable. In (35), if all the system states are subjected to impulsive effects, i.e., z = n, we have the following corollary. Corollary 2: Consider the NN in (35) with impulsive effects. Suppose that Assumptions 1 and 2 hold. If for a prescribed positive scalar μ ∈ (0, 1), there exist positive definite matrices P1 , P2 ∈ Rn×n and a positive scalar α such that (36) and (37) hold. Then the NN in (35) with impulsive effects is exponentially stable. Remark 7: Corollary 2 investigates the following two special cases. 1) The impulsive effects are stabilizing. 2) The impulsive effects are destabilizing. If |μik | = μ < 1, (i = 1, 2, . . . , n), then Corollary 2 becomes the stability problem of delayed NNs with stabilizing impulses, which has widely been investigated in [11] and [13]. If μik = μk , (i = 1, 2, . . . , n), Corollary 2 becomes the
stability of delayed NNs with time-varying impulses, which has been studied in [3]. It should be noted that in Corollary 2, the impulsive effects in substates can be different from each other, and therefore, the local impulses take the time-varying impulses [3] and stabilizing impulses [11], [13] as special cases. Hence, the local impulses presented in this paper are quite general. Remark 8: In Corollary 2, if the impulsive effects are stabilizing, we can let P(t) = P in (12), then, Corollary 2 becomes the NNs with stabilizing impulsive effects. By Corollary 2, we can derive the analogous result on dynamical system with time-varying delay under stabilizing impulsive control [12]. Corollary 3: Consider the NN in (35) with local impulsive effects. Suppose that Assumptions 1 and 2 hold. If for a prescribed positive scalar μ ∈ (0, 1), there exist positive definite matrices P1 and P2 ∈ Rn×n and a positive scalar α such that the following inequalities hold: ⎤ ⎡ ˆ P B + D1 L 1 P D 0 ⎢ 0 0 ⎥ −D1 ⎥
