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Finite time stabilization of delayed neural networks
Accepted Manuscript Finite time stabilization of delayed neural networks Leimin Wang, Yi Shen, Zhixia Ding PII: DOI: Reference:
S0893-6080(15)00145-8 http://dx.doi.org/10.1016/j.neunet.2015.07.008 NN 3503
To appear in:
Neural Networks
Received date: 26 December 2014 Revised date: 8 May 2015 Accepted date: 16 July 2015 Please cite this article as: Wang, L., Shen, Y., & Ding, Z. Finite time stabilization of delayed neural networks. Neural Networks (2015), http://dx.doi.org/10.1016/j.neunet.2015.07.008 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 proof before it is published in its final 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.
1. Time delays are taken into account in studying the finite time stability problem of neural networks. 2. Provide general conditions on the feedback control law for the finite time stabilization of delayed neural networks (DNNs). 3. Derive some specific conditions by designing two different controllers which include the delay-dependent and delay-independent ones. 4. Discuss the extremum of the settling time functional and realize the optimal stabilization time under fixed control strength. 5. Design a switched controller to optimize the settling time.
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Finite time stabilization of delayed neural networks Leimin Wanga,b , Yi Shena,b,∗, Zhixia Dinga,b b Key
a School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China
Abstract In this paper, the problem of finite time stabilization for a class of delayed neural networks (DNNs) is investigated. The general conditions on the feedback control law are provided to ensure the finite time stabilization of DNNs. Then some specific conditions are derived by designing two different controllers which include the delay-dependent and delay-independent ones. In addition, the upper bound of the settling time for stabilization is estimated. Under fixed control strength, discussions of the extremum of settling time functional are made and a switched controller is designed to optimize the settling time. Finally, numerical simulations are carried out to demonstrate the effectiveness of the obtained results. Keywords: Finite time stabilization, delayed neural networks, delay-dependent controller, delay-independent controller, switched controller, settling time. 1. Introduction Stability is a prerequisite of neural networks in the successful applications like signal processing, pattern recognition and associative memory design (Cohen & Grossberg, 1983; Chua & Yang, 1988; Forti & Tesi, 1995). In the hardware implementation of neural networks, some signal transmission delays are unavoidable, which may cause undesirable dynamical behaviors such as instability and oscillation. Thus, it is necessary to take time delays into consideration in studying stability of neural networks. During the past decades, stability analysis of delayed neural networks (DNNs) has been extensively investigated and many useful stability criteria have been established (Cao & Wang, 2005; Faydasicok & Arik, 2012; Forti et al., 2005; He & Wu, 2006; Huang et al., 2012; Liu & Wang, 2006; Shen & Wang, 2009, 2012; Wang & Shen, 2014; Wang et al., 2014; Wang & Liu, 2005; Wang et al., 2006; Lu, 2002; Zeng & Zheng, 2012). On the other hand, the stabilization of DNNs has been attracting considerable attention and several feedback stabilizing control methods have been proposed, such as linear control (Guo & Wang, 2013; Huang et al., 2013; Phat & Trinh, 2010; Wen et al., 2015; Wu & Zeng, 2012), impulsive control (Guan & Zhang, 2008), and intermittent control (Chen et al., 2014; Huang & Li, 2009; Hu et al., 2010; Zhang & Shen, 2014). It should be pointed out that, in most of the above works, the stability and stabilization of DNNs are asymptotic or exponential, which means the time to approach equilibrium points is infinite. In practical applications, we always hope to obtain fast or even finite time convergent speed, so the concept ∗ Corresponding
author. Tel.: +86 27 87543630; fax: +86 27 87543130. Email addresses: [email protected] (Leimin Wang), [email protected] (Yi Shen) Preprint submitted to Neural Networks
of finite time stability arises naturally. Finite time stability requires the system be Lyapunov stable and its trajectories tend to equilibrium points in finite time. Since Bhat & Bernstein (2000) extended finite time stability to multi-dimensional continuous autonomous systems and proved that there is necessary and sufficient condition for finite time stability, the problem of finite time stability and stabilization have been widely studied (Efimov et al., 2014; Hong & Jiang, 2006; Karafyllis, 2006; Liu et al., 2013, 2014a,b; Moulay & Perruquetti, 2005; Moulay et al., 2008; Sun et al., 2014; Wang & Xiao, 2010; Wu et al., 2015; Yang & Wang, 2012). However, so far there are few results concerning the finite time stability of time-delay systems (Moulay et al., 2008; Yang & Wang, 2012). The reason is that timedelay systems have more complicated dynamic behaviors and are more difficult to deal with than system without delays. As is stated in Moulay et al. (2008), it is difficult to find a Lyapunov functional to satisfy the derivative condition for finite time stability of time-delay systems. Also, it is reported in Efimov et al. (2014) that some key results in Yang & Wang (2012) are incorrect. So finite time stability of time-delay systems is still an open problem that needs further investigation. In recent years, finite time stability or synchronization problem of neural networks has been investigated (Shen & Cao, 2011; Huang & Li, 2014; Hu et al., 2014; Liu et al., 2013, 2014a,b; Shen & Park, 2014; Wu et al., 2015). However, most of the results are restrictive because of the absence of the time delays in the system. For instance, time delays are not taken into account in the finite time stabilization in Liu et al. (2013, 2014a,b) and finite time synchronization in Shen & Cao (2011); Shen & Park (2014). In addition, although different controllers are used in stabilization or finite time stabilization of neural networks, there do not exist one general controller that can finite time stabilize the DNNs. It still needs to answer the question that how to design a proper controller to realize the finite time May 8, 2015
stabilization. Motivated by the above discussions, we investigate the finite time stabilization for a class of DNNs in this paper. The contributions of this paper are fourfold. 1) Time delays are taken into account in the finite time stability problem of neural networks. 2) Based on the finite time Lyapunov stability theory of functional differential equation (Moulay et al., 2008), the general conditions on the feedback control law for finite time stabilization of DMNs are derived, which provide new method to study the finite time stabilization of time-delay systems. 3) Some specific conditions are derived by designing two different types of feedback control algorithms which include the delay-dependent and delay-independent ones. 4) Under fixed control strength, discussions of the extremum of settling time functional are made so as to realize the optimal stabilization time and a switched controller is designed to optimize the settling time. The organization of this paper is as follows. Some preliminaries are introduced in Section 2. In Section 3, general conditions on the feedback control law are provided to ensure the finite time stabilization of DNNs, and some specific conditions are obtained by designing two different types of controllers. In addition, discussions of stabilization time are made and a switched controller is designed to decrease the settling time. Then, two examples are provided to demonstrate the effectiveness of the obtained results in Section 4. Finally, conclusions are drawn in Section 5.
g(x(t − τ(t))) = (g1 (x1 (t − τ1 (t))), . . . , gn (xn (t − τn (t))))T ∈ Rn are the neuron activation functions. τ(t) = (τ1 (t), . . . , τn (t))T is the time-varying delay satisfy that 0 ≤ τ j (t) ≤ τ, j = 1, 2, . . . , n. u(t) = (u1 (t), . . . , un (t))T is the control input. The initial condition of system (1) is: x(s) = φ(s) = (φ1 (s), φ2 (s), . . . , φn (s))T ∈ C([−τ, 0], Rn). (A)The neuron activation functions f j , g j ( j = 1, 2, . . . , n) are bounded and satisfy f j (0) = g j (0) = 0, and there exist constants h j > 0, k j > 0, F j > 0, G j > 0 such that
2. System description and preliminaries
Lemma 2 (Moulay et al., 2008). Consider the system (1) with uniqueness of solutions in forward time. If there exist a continuous functional V : Ω → R+ and two functions ν, r of class K for system (1) such that (i) ν(kφ(0)k) ≤ V(φ), R ε dz < +∞, (ii) D+ V(φ) ≤ −r(V(φ)) with 0 r(z) for all ε > 0, φ ∈ Ω. Then system (1) is finite time stabilizable R V(φ) dz . with a settling time satisfying the inequality T 0 (φ) ≤ 0 r(z) ρ In particularly, if r(V) = λV where λ > 0, ρ ∈ (0, 1), then the settling time satisfies the inequality
| f j (s1 ) − f j (s2 )| ≤ h j |s1 − s2 |, | f j (s1 )| ≤ F j ,
|g j (s1 ) − g j (s2 )| ≤ l j |s1 − s2 |, |g j (s1 )| ≤ G j , for all s1 , s2 ∈ R. Before studying the finite time stabilization of system (1), we give the following definitions and lemmas. Let Ω be an open subset of C([−τ, 0], Rn) containing 0.
Definition 2 (Moulay et al., 2008). The origin of system (1) is finite time stable where ui (t) = 0, if (i) The origin of system (1) is stable, (ii) The origin of system (1) is finite time convergent, i.e. for any initial state φ(s) ∈ Ω, there exists 0 ≤ T (φ) < +∞ such that every solution of system (1) satisfies x(t, φ) = 0 for all t ≥ T (φ). T 0 (φ) = inf{T (φ) ≥ 0 : x(t, φ) = 0 ∀t ≥ T (φ)} is a functional called the settling time of the system (1). Definition 3. The system (1) is finite time stabilizable if there exists a feedback controller ui (t) such that ui (0) = 0 and system (1) is finite time stable.
The √ following notations will be used throughout this paper. kxk = xT x is the Euclidean norm of x ∈ Rn and h., .i denotes the inner product of Euclidean space. C([a, b], Rn) denotes the space of continuous functions φ : [a, b] → Rn with the uniform norm kφk = sup kφ(s)k. AT and A−1 denote the transpose a≤s≤b
and the inverse of the matrix A, respectively. A > 0(A ≥ 0) means that the matrix A is symmetric and positive definite (semi-positive definite). λmax (A) and λmin (A) denote the maximum and minimum eigenvalue of matrix A, respectively. ∗ represents the elements below the main diagonal of a symmetric matrix. I is the identity matrix with compatible dimension. diag{· · · } denotes a block-diagonal matrix. A continuous function s : R+ → R+ belongs to the class K if it is strictly increasing and s(0) = 0. sign(·) denotes the signum function. e refers to the base of natural logarithms. In this paper, we consider a class of neural networks with time-varying delay as follows: x˙(t) = −Dx(t) + A f (x(t)) + Bg(x(t − τ(t))) + u(t),
T 0 (φ) ≤
0
V(φ)
dz V 1−ρ (φ) = . r(z) λ(1 − ρ)
(2)
Lemma 3 (Berman & Plemmons, 1979). For any vectors x, y ∈ Rn , ε > 0 and positive definite matrix Q ∈ Rn×n , the following matrix inequality holds 2xT y ≤ ε−1 xT Q−1 x + εyT Qy.
(1)
where x(t) = (x1 (t), x2 (t), . . . , xn (t))T ∈ Rn is the state vector. D = diag(d1 , d2 , . . . , dn ) is an n × n diagonal matrix with di > 0, i = 1, 2, . . . , n. A = (ai j )n×n , B = (bi j )n×n ∈ Rn×n are the connection weight matrix and delayed connection weight matrix, respectively. f (x(t)) = ( f1 (x1 (t)), . . . , fn (xn (t)))T ∈ Rn and
Z
Lemma 4 (Hardy et al., 1988). If a1 , a2 , . . . , an , r1 , r2 are real numbers and 0 < r1 < r2 , then the following inequality holds [
n n X X (|ai |)r2 ]1/r2 ≤ [ (|ai |)r1 ]1/r1 . i=1
2
i=1
3. Main results
Combining with (4)-(6) and (8)-(10), we derive that T ˙ ≤xT (t)[−PD − DT P + ε−1 PAQ−1 V(t) 1 A P]x(t)
In this section, the finite time stabilization of DNNs is investigated. The feedback control law u(t) = (u1 (t), u2 (t), . . . , un (t))T is given as follows: u(t) = uˆ (t) + uˇ (t).
+ ε f T (x(t))Q1 f (x(t)) + 2xT (t)P(uˆ (t) + uˇ (t)) + 2hPB|g(x(t − τ(t)))|, |x(t)|i
T ≤xT (t)[−PD − DT P + ε−1 PAQ−1 1 A P
+ εH T Q1 H]x(t) − xT (t)Q2 x(t) + 2xT (t)Pˇu(t)
(3)
≤2xT (t)Pˇu(t) n X ≤−δ |xi (t)|µ+1 .
Then we will give general conditions of uˆ (t) and uˇ (t) in order to ensure the finite time stable of system (1). Theorem 1. Under the assumption (A), if there exist three symmetric matrices P > 0, Q1 > 0, Q2 > 0, constants ε > 0, 0 ≤ µ < 1, δ > 0 such that T T − PD − DT P + ε−1 PAQ−1 1 A P − Q2 + εH Q1 H < 0,
hPB|g(x(t − τ(t)))|, |x(t)|i + xT (t)Pˆu(t) 1 ≤ − xT (t)Q2 x(t), 2 n X 2xT (t)Pˇu(t) ≤ −δ |xi (t)|µ+1 ,
Based on Lemma 4, we get the following inequality −[
(4)
n X i=1
On the other hand, we have Z ε dz 0
(7)
T 0 (φ) ≤
˙ =2xT (t)P x˙(t) V(t)
≤
T
+ 2xT (t)P(uˆ (t) + uˇ (t)),
|xi (t)|2 ]1/2 ,
i=1
(12)
=
(13)
2ε(1−µ)/2 −(µ+1)/2 δλmax (P)(1 −
µ)
for all ε > 0. From Lemma 2, one can conclude that system (1) is finite time stabilizable, and the settling time satisfies
Calculating the time derivative of the Lyapunov function along the trajectories of system (1), we have
≤ − x (t)(PD + D P)x(t) + 2hPA| f (x(t))|, |x(t)|i + 2hPB|g(x(t − τ(t)))|, |x(t)|i
−(µ+1)/2 δλmax (P)z(µ+1)/2
< +∞,
Proof. Consider the following Lyapunov function
T
n X
µ+1 −(µ+1)/2 ˙ V(t) ≤ −δλmax (P)V 2 (t).
(6)
where H = diag(h1 , h2 , . . . , hn ). Then system (1) is finite time stabilizable via the controller (3), and the settling time for sta2λmax (P)kφk1−µ . bilization satisfies T 0 (φ) ≤ δ(1 − µ)
V(t) = x (t)Px(t).
|xi (t)|)µ+1 ]1/(µ+1) ≤ −[
for 1 < µ + 1 < 2. Then we have
(5)
i=1
T
(11)
i=1
2V (1−µ)/2 (0) −(µ+1)/2 δλmax (P)(1 − 2λmax (P)kφk1−µ
δ(1 − µ)
µ)
.
(14)
The proof is completed. (8)
3.1. Finite time stabilization via delay-dependent control
where |x(t)| = (|x1 (t)|, |x2 (t)|, . . . , |xn (t)|)T , | f (x(t))| = (| f1 (x1 (t))|, | f2 (x2 (t))|, . . . , | fn (xn (t))|)T , |g(x(t − τ(t)))| = (|g1 (x1 (t − τ1 (t)))|, |g2(x2 (t − τ2 (t)))|, . . . , |gn (xn (t − τn (t)))|)T . According to the assumption (A), there exist h j > 0 such that
In Theorem 1, the general conditions on finite time stabilization of system (1) are obtained. In order to give some specific conditions which are easy to be verified, we design a delayed feedback controller as follows:
| f j (x j (t))| ≤ h j |x j (t)|,
u(t) = − k1 x(t) − BLsign(x(t))|x(t − τ(t))| − k2 sign(x(t))|x(t)|µ ,
(9)
where uˆ (t) = −k1 x(t) − BLsign(x(t))|x(t − τ(t))|, uˇ (t) = −k2 sign(x(t))|x(t)|µ , L = diag(l1 , l2 , . . . , ln ), k1 > 0, k2 > 0, 0 ≤ µ < 1. Noticing that the assumption (A) there exist l j > 0 such that
Now, by using Lemma 3, it is easy to obtain the following inequality T 2hPA| f (x(t))|, |x(t)|i ≤ ε−1 xT (t)PAQ−1 1 A Px(t)
+ ε f T (x(t))Q1 f (x(t)).
(15)
(10) 3
|g j (x j (t − τ j (t)))| ≤ l j |x j (t − τ j (t))|,
(16)
then we have
We can see that (5) and (6) hold obviously, then we get the following result.
hPB|g(x(t − τ(t)))|, |x(t)|i + xT (t)Pˆu(t)
Theorem 3. Under the assumption (A), if there exist three symmetric matrices P > 0, Q1 > 0, constants ε > 0, k3 > 0, k4 > 0, 0 ≤ µ < 1 such that
≤hPBL|x(t − τ(t))|, |x(t)|i + xT (t)Pˆu(t) = − k1 xT (t)Px(t),
2xT (t)Pˇu(t) = −2k2 xT (t)Psign(x(t))|x(t)|µ n X ≤ − 2k2 λmin (P) |xi (t)|µ+1 .
T T −PD − DT P + ε−1 PAQ−1 1 A P − 2k3 P + εH Q1 H < 0, (21)
Then system (1) is finite time stabilizable via the controller (20), and the settling time for stabilization satisfies T 0 (φ) ≤ λmax (P)kφk1−µ . k4 λmin (P)(1 − µ)
i=1
We can see that (5) and (6) hold obviously. Thus, we can get the following results. Theorem 2. Under the assumption (A), if there exist three symmetric matrices P > 0, Q1 > 0, constants ε > 0, k1 > 0, k2 > 0, 0 ≤ µ < 1 such that
Corollary 3. Under the assumption (A), if there exist constants ε > 0, 0 ≤ µ < 1, p > 0, k3 > 0, k4 > 0 such that −p(D + DT ) + ε−1 p2 AAT − 2k3 pI + εH T H < 0.
T T −PD − DT P + ε−1 PAQ−1 1 A P − 2k1 P + εH Q1 H < 0, (17)
Then system (1) is finite time stabilizable via the controller (20), and the settling time for stabilization satisfies T 0 (φ) ≤ kφk1−µ . k4 (1 − µ)
Then system (1) is finite time stabilizable via the controller (15), and the settling time for stabilization satisfies T 0 (φ) ≤ λmax (P)kφk1−µ . k2 λmin (P)(1 − µ) Corollary 1. Under the assumption (A), if there exist symmetric matrix P > 0, constants ε > 0, 0 ≤ µ < 1, k1 > 0, k2 > 0 such that −(PD + DT P) + ε−1 PAAT P − 2k1 P + εH T H < 0.
Remark 1. Although stabilization of DNNs has been investigated in the past few years (Chen et al., 2014; Guo & Wang, 2013; Huang et al., 2013; Huang & Li, 2009; Hu et al., 2010; Phat & Trinh, 2010; Wen et al., 2015; Wu & Zeng, 2012; Zhang & Shen, 2014), so far, there is little work concerning the finite time stabilization of DNNs. In this paper, two different kinds of feedback control law are designed and some sufficient conditions in Theorem 2 and 3 are given to ensure the finite time stabilization of DNNs. Our results complement and extend the previous results which only asymptotic stabilization of DNNs are obtained.
(18)
Then system (1) is finite time stabilizable via the controller (15), and the settling time for stabilization satisfies T 0 (φ) ≤ λmax (P)kφk1−µ . k2 λmin (P)(1 − µ) Corollary 2. Under the assumption (A), if there exist constants ε > 0, 0 ≤ µ < 1, p > 0, k1 > 0, k2 > 0 such that −p(D + DT ) + ε−1 p2 AAT − 2k1 pI + εH T H < 0.
Remark 2. Time-delay systems have more complicated dynamic behaviors and are more difficult to deal with than system without delays. In Liu et al. (2013, 2014a,b), finite time stability of neural networks was studied, but the systems in these papers are without delays. Since delays are unavoidable in the hardware implementation of neural networks, it is necessary to take the delays into consideration in the finite time stability problem. In this paper, the neural networks with delays are considered, which make our results general compared with the existing results (Liu et al., 2013, 2014a,b).
(19)
Then system (1) is finite time stabilizable via the controller (16), and the settling time for stabilization satisfies T 0 (φ) ≤ kφk1−µ . k2 (1 − µ) 3.2. Finite time stabilization via delay-independent control In this section, we design a feedback controller as follows: u(t) = −k3 x(t) − BGsign(x(t)) − k4 sign(x(t))|x(t)|µ ,
(22)
(20) 3.3. Reduce the settling time via a switched controller
where uˆ (t) = −k3 x(t) − BGsign(x(t)), uˇ (t) = −k4 sign(x(t))|x(t)|µ , G = max{G j }( j = 1, 2, . . . , n) is a constant, k3 > 0, k4 > 0, 0 ≤ µ < 1. Then we have
From Corollary 2, we can see that the settling time for stabilization is decided by three parameters: the initial value φ, the control strength k2 of uˇ (t) and µ. Now, we will discuss how to reduce the settling time via a control strategy. Let the control strength k2 of uˇ (t) be given and fixed, and set kφk1−µ , 0 ≤ µ < 1, k2 > 0, kφk > 0. Then T 0 (µ) = k2 (1 − µ)
hPB|g(x(t − τ(t)))|, |x(t)|i + xT (t)Pˆu(t)
≤ − k3 xT (t)Px(t),
2xT (t)Pˇu(t) = −2k4 xT (t)Psign(x(t))|x(t)|µ n X ≤ − 2k4 λmin (P) |xi (t)|µ+1 . i=1
4
dT 0 [(µ − 1) ln(kφk) + 1]kφk1−µ = . dµ k2 (1 − µ)2
(23)
i) If kφk > e (ln kφk > 1), T 0 (µ) has only one critical point 1 where T 0 (µ) reach the minimum value, and at µ = 1 − ln(kφk) ln(kφk) 1/(ln(kφk)) e ln(kφk) kφk = . min T 0 (µ) = k2 k2 µkφk1−µ dT 0 ≥ > 0, ii) If 0 < kφk ≤ e (ln kφk ≤ 1), then dµ k2 (1 − µ)2 namely T 0 (µ) is increasing on (0, 1). So T 0 (µ) reach the minikφk . mum value at µ = 0, and min T 0 (µ) = k2
From Lemma 2, one can conclude that system (1) is finite time stabilizable, and the settling time satisfies T 0 (φ) ≤ ≤
have the following result.
−1 2
T
−p(D + D ) + ε p AA − 2k1 pI + εH H < 0.
≤ −2k2 p where µ¯ = Since Z
0
ε
(
µ∗ , 0,
dz 2k2
¯ z(¯µ+1)/2 p(1−µ)/2
i=1
4
2
2
x (t)
0
−2
−4
−6
(25)
−8 −2
µ¯ + 1 V 2 (t),
k2
−0.5 x (t)
0
0.5
1
Figure 1: Phase plot of system (29) with u(t) = (0, 0)T .
4. Numerical examples
Example 1. Consider the two-dimensional delayed neural networks (Lu, 2002)
(26)
x˙(t) = −Dx(t) + A f (x(t)) + Bg(x(t − τ(t))) + u(t),
(27)
ε(1−µ)/2 ¯ (1 p(1−µ)/2
−1
In this section, two examples are provided to verify the effectiveness of the results obtained in the previous section.
kφk > e, 0 < kφk ≤ e.
=
−1.5
1
¯ |xi (t)|µ+1
(1−µ)/2 ¯
(28)
6
Proof. Let P = pI, then similar to the proof of Theorem 1, we have ˙ ≤ −2k2 p V(t)
0
Remark 5. Theorem 4 gives the sufficient conditions on finite time stabilization of DMNNs via a switched controller. The switch control method can be used to study finite time stabilization of other nonlinear time-delay systems.
Then system (1) is finite time stabilizable via the switched controller (24), and the settling time for stabilization satisfies e + ln(kφk)[e − e1/(ln(kφk)) ] T 0 (φ) ≤ . k2
n X
pe2
Remark 4. It should be pointed out that if the nonlinear system (1) is considered without delays, then the main Theorem 2, 3 and 4 in Liu et al. (2014b) concerning finite time stability of nonlinear systems can be obtained directly from Corollary 1, 2 and Theorem 4, respectively.
1 . Then we ln(kφk)
T
0
Z
p(¯µ−1)/2 dz 2k2 z(¯µ+1)/2 Z pkφk2 (µ∗ −1)/2 p−1/2 dz p dz + ∗ +1)/2 1/2 (µ 2k2 z 2k2 z pe2
The proof is completed.
Theorem 4. Under the assumption (A), if there exist constants ε > 0, p > 0, k1 > 0, k2 > 0 such that T
V(0)
e + ln(kφk)[e − e1/(ln(kφk)) ] . = k2
Remark 3. In order to obtain faster convergence speed and reduce the stabilization time, we make discussion on the extremum of the settling time functional. Under the fixed control strength k2 , the minimum value of T 0 (µ) depends on the value of initial value kφk, which means that we can choose a proper parameter µ to realize the optimal stabilization. If the initial value is big enough (kφk > e), then uˇ (t) is continuous and T 0 (µ) 1 reach the minimum value at µ = 1 − ; If the initial value ln(kφk) is small such that (0 < kφk ≤ e), then uˇ (t) is discontinuous and T 0 (µ) reach the minimum value at µ = 0. Based on the above analysis, we design a switched controller as follows: −k1 x(t) − BLsign(x(t))|x(t − τ(t))| ∗ −k2 sign(x(t))|x(t)|µ , kx(t)k > e, u(t) = (24) −k1 x(t) − BLsign(x(t))|x(t − τ(t))| −k2 sign(x(t)), 0 < kx(t)k ≤ e,
where k1 , k2 , µ are constants and µ∗ = 1 −
Z
− µ) ¯
(29)
where x(t) = (x1 (t), x2 (t))T , D = diag(1, 1), " # " # 2 −0.1 −1.5 −0.1 A= ,B= . −5 4.5 −0.2 −4
< +∞,
τ j (t) = 1, j = 1, 2, and take the activation function as f j (x) = g j (x) = tanh(x), j = 1, 2. Then we have H = L = D =
for all ε > 0. 5
1.5 x1(t)
1.5
x2(t)
1
x (t) 1
x (t) 2
1
0.5
2
0
1
−0.5
x (t) and x (t)
2 1
x (t) and x (t)
0.5
0
−0.5
−1 −1
−1.5 −1.5
−2
0
0.5
1
1.5
2
2.5 time−t
3
3.5
4
4.5
5 −2
Figure 2: State trajectories of variables x1 (t) and x2 (t) of system (29) under the controller (30).
0
0.5
1
1.5
2
2.5 time−t
3
3.5
4
4.5
5
Figure 3: State trajectories of variables x1 (t) and x2 (t) of system (29) under the switch controller (24).
diag(1, 1). The phase plot of system (29) with the initial condition x1 (s) = −1.6, x2 (s) = 1.2, ∀s ∈ [−1, 0) is shown in Fig.1. The controller is designed as 4 3.5
(30)
3
Set k1 = p−1 k, then the condition (19) equals to the following LMI −p(D + DT ) − 2kI pA εH T ∗ −εI 0 < 0. (31) ∗ ∗ −εI
2.5
x2(t)
2 1.5 1 0.5
By using the Matlab LMI Control Toolbox, we can find a solution to the LMI (31): p = 0.0258, ε = 0.6905, k = 0.8269. Then k1 = p−1 k = 32.0825, and by picking an arbitrary fixed k2 > 0, system (29) can be finite time stabilized via the controller (30) according to Corollary 2. Set k2 = 1, the settling kφk1−µ = 2.828. The time for stabilization satisfies T 0 (φ) ≤ k2 (1 − µ) kφk = 2. Under the minimum value of the stabilization time is k2 controller (30), we get the state trajectories of variables x1 (t) and x2 (t) which are shown in Fig.2. System (30) also can be finite time stabilized via the switched controller (24) according to Theorem 4. Set k2 = 1, the settling time for stabilization satisfies T 0 (φ) ≤ 1.6690. Then under the switched controller (24), we get the state trajectories of variables x1 (t) and x2 (t) which are shown in Fig.3.
0 −0.5 −1 −3
−2
−1
0 x1(t)
1
2
3
Figure 4: Phase plot of system (32) with u(t) = (0, 0)T .
4 x1(t) x2(t)
3
x (t) and x (t)
2
Example 2. Consider the two-dimensional delayed neural networks (Hu et al., 2010)
2
− k2 sign(x(t))|x(t)|0.5 .
1
1
u(t) = − k1 x(t) − BLsign(x(t))|x(t − τ(t))|
0
−1
x˙(t) = −Dx(t) + A f (x(t)) + Bg(x(t − τ(t))) + u(t),
(32) −2
where D = diag(1, 1), " " # # 1.8 1.5 −1.4 0.1 A= ,B= . 0.1 1.8 0.1 −1.4
−3
0
0.5
1
1.5
2
2.5 time−t
3
3.5
4
4.5
5
Figure 5: State trajectories of variables x1 (t) and x2 (t) of system (32) with the controller (33).
τ j (t) = et /(1 + et ), j = 1, 2, and take the activation function as 1 f j (x) = g j (x) = (|x + 1| − |x − 1|), j = 1, 2. Then we have 2 6
H = L = D = diag(1, 1), G = 1. The phase plot of system (33) with the initial condition x1 (s) = −3, x2 (s) = 4, ∀s ∈ [−1, 0) is shown in Fig.4. The controller is designed as u(t) = −k3 x(t) − BGsign(x(t)) − k4 sign(x(t))|x(t)|0.5 .
Bhat, S. P., & Bernstein, D. S. (2000). Finite-time stability of continuous autonomous systems,” SIAM Journal on Control and Optimization, 38, 751766. Cao, J., & Wang, J. (2005). Global exponential stability and periodicity of recurrent neural networks with time delays, IEEE Transactions on Circuits and systems I: Regular Papers, 52, 920-931. Chen, W. H., & Zhong, J., & Jiang, Z., & Lu, X. (2014). Periodically intermittent stabilization of delayed neural networks based on piecewise lyapunov functions/functionals, Circuits, Systems, and Signal Processing, 33, 37573782. Chen, X., & Huang, L., & Guo, Z. (2013). Finite time stability of periodic solution for Hopfield neural networks with discontinuous activations, Neurocomputing, 103, 43-49. Chua, L. O., & Yang, L. (1988). Celluar neural networks: Applications, IEEE Transactions on Circuits and Systems, 35, 1273-1290. Cohen, M. A., & Grossberg, S. (1983). Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Transactions on Systems, Man and Cybernetics, 13, 815-826. Efimov, D., & Polyakov, A., & Fridman, E., & Perruquetti, E., & Richard, J. P. (2014). Comments on finite-time stability of time-delay systems, Automatica, 50, 1944-1947. Faydasicok, O., & Arik, S. (2012). Robust stability analysis of a class of neural networks with discrete time delays, Neural Networks, 29-30, 52-59. Forti, M., & Tesi, A. (1995). New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42, 354-366. Forti, M., & Nistri, P., & Papini, D. (2005). Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Transactions on Neural Networks and Learning Systems, 16, 1449-1463. Guan, Z. H., & Zhang, H. (2008). Stabilization of complex network with hybrid impulsive and switching control, Chaos, Solitons and Fractals, 37, 13721382. Guo, Z. Y., & Wang, J., & Yan, Z. (2013). Global exponential dissipativity and stabilization of memristor-based recurrent neural networks with timevarying delays, Neural Networks, 48, 158-172. Hardy, G., & Littlewood, J., & Polya, G. (1988). Inequalities, Cambridge University Press: Cambridge. He, Y., & Wu, M., & She, J. H. (2006). Delay-dependent exponential stability of delayed neural networks with time-varying delay, IEEE Transactions on Circuits Systems II: Express Briefs, 53, 553-557. Hong, Y., & Jiang, Z. P. (2006). Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties, IEEE Transactions on Automatic Control, 51, 1950-1956. Huang, H., & Huang, T., & Chen, X., & Qian, C. (2013). Exponential stabilization of delayed recurrent neural networks: A state estimation based approach, Nueral Networks, 48, 153-157. Huang, J., & Li, C., & Han, Q. (2009). Stabilization of delayed chaotic neural networks by periodically intermittent control, Circuits, Systems and Signal Processing, 28, 567-579. Huang, J., & Li, C., & Huang, T., & He, X. (2014). Finite-time lag synchronization of delayed neural networks, Neurocomputing, 139, 145-149. Huang, T., & Li, C., & Duan, S. & Starzyk, J. A. (2012). Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects, IEEE Transactions on Neural Networks and Learning Systems, 23, 866-875. Hu, C., & Yu, J., & Jiang, H., & Teng, Z. (2010). Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control, Nonlinearity, 23, 2369-2391. Hu, C., & Yu, J., & Jiang, H. (2014). Finite-time synchronization of delayed neural networks with Cohen-Grossberg type based on delayed feedback control, Neurocomputing, 143, 90-96. Karafyllis, I. (2006). Finite-time global stabilization by means of time-varying distributed delay feedback, SIAM Journal on Control and Optimization, 45, 320-342. Liu, Y., & Wang, Z., & Liu, X. (2006). Global exponential stability of generalized recurrent neural networks with discrete and distributed delays, Neural Networks, 19, 667-675. Liu, X., & Park, J. H., & Jiang, N., & Cao, J. (2014a). Nonsmooth finite-time stabilization of neural networks with discontinuous activations, Nueral Net-
(33)
Set k3 = p−1 k, then the condition (23) equals to the same LMI (31). By using the Matlab LMI Control Toolbox, we can find a solution to the LMI (32): p = 0.0297, ε = 0.5754, k = 0.5490. Then k3 = p−1 k = 18.4800, and by picking an arbitrary fixed k4 > 0, system (32) can be finite time stabilized via the controller (33) according to Corollary 3. Set k4 = 1, the settling kφk1−µ time for stabilization satisfies T 0 (φ) ≤ = 3.464. The k4 (1 − µ) e ln(kφk) minimum value of the stabilization time is = 2.9863. k4 Under the controller (33), we get the state trajectories of variables x1 (t) and x2 (t) which are shown in Fig.5. 5. Conclusions Finite time stabilization for a class of delayed neural networks (DNNs) has been investigated in this paper. Firstly, the general conditions on the feedback control law have been provided to ensure the finite time stabilization of DNNs. Then some specific conditions have been obtained by designing two different types of feedback control algorithms which include the delay-dependent and delay-independent ones. It is noted that discussions of the extremum of settling time functional have been made to realize the optimal stabilization time under fixed control strength and a switched controller has been designed to optimize the settling time. Our control method can be used to study the finite time stabilization of other nonlinear time-delay system. On one hand, our results complement and extend the previous results in Liu et al. (2013, 2014a,b) where delays are not taken into consideration in studying finite time stability problem. On the other hand, the new proposed results here achieve a valuable improvement compared with the present works (Chen et al., 2014; Guo & Wang, 2013; Huang et al., 2013; Huang & Li, 2009; Hu et al., 2010; Phat & Trinh, 2010; Wen et al., 2015; Wu & Zeng, 2012; Zhang & Shen, 2014) where only exponential or asymptotic stabilization of DNNs is obtained in these papers. Acknowledgements This work is supported by the Key Program of National Natural Science Foundation of China (Grant no. 61134012), the National Science Foundation of China (Grant nos. 11271146 and 61374150), and the prior developing Field for the Doctoral Program of Higher Education of China (Grant no. 20130142130012). References Berman, A., & Plemmons, R. J. (1979). Nonnegative Matrices in the Mathematical Science, Academic, New York.
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works, 52, 25-32. Liu, X., & Ho, D. W. C., & Yu, W., & Cao, J. (2014b). A new switching design to finite-time stabilization of nonlinear systems with applications to neural networks, Nueral Networks, 57, 94-102. Liu, X., & Jiang, N., & Cao, J., & Wang, S., & Wang, Z. (2013). Finite-time stochastic stabilization for BAM neural networks with uncertainties, Journal of the Franklin Institute, 350, 2109-2123. Liu, Y. (2014). Global finite-time stabilization via time-varying feedback for uncertain nonlinear systems, SIAM Journal on Control and Optimization, 52, 1886-1913. Lu, H. (2002). Chaotic attractors in delayed neural networks, Physics Letters A, 298, 109-116. Moulay, E., & Perruquetti, W. (2005). Finite time stability of differential inclusions, IMA Journal of Mathematical Control and Information, 22, 465-475. Moulay, E., & Dambrine, M., & Yeganefar, N., & Perruquetti, W. (2008). Finite-time stability and stabilization of time-delay systems, Systems and Control Letters, 57, 561-566. Phat, V. N., & Trinh, H. (2010). Exponential stabilization of neural networks with various activation functions and mixed time-varying delays, IEEE Transactions on Neural Networks, 21, 1180-1184. Shen, H., & Park, J. H., & Wu, Z. G. (2014). Finite-time synchronization control for uncertain Markov jump neural networks with input constraints, Nonlinear Dynamics, 77, 1709-1720. Shen, Y., & Wang, J. (2009). Almost sure exponential stability of recurrent neural networks with Markovian switching, IEEE Transactions on Neural Networks, 20, 840-855. Shen, Y., & Wang, J. (2012). Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances, IEEE Transactions on Neural Networks and Learning Systems, 23, 87-96. Shen, J., & Cao, J. (2011). Finite-time synchronization of coupled neural networks via discontinuous controllers, Cognitive neurodynamics, 5, 373-385. Sun, L., & Feng, G., & Wang, Y. (2014). Finite-time stabilization and H∞ control for a class of nonlinear Hamiltonian descriptor systems with application to affine nonlinear descriptor systems, Automatica, 50, 2090-2097. Wang L., & Xiao, F. (2010). Finite-time consensus problems for networks of dynamic agents, IEEE Transactions on Automatic Control, 55, 950-955. Wang L., & Shen, Y. (2014). New results on passivity analysis of memristorbased neural networks with time-varying delays, Neurocomputing, 144, 208214. Wang L., & Shen, Y., & Yin, Q., & Zhang, G. (2014). Adaptive synchronization of memristor-based neural networks with time-varying delays, IEEE Transactions on Neural Networks and Learning Systems, 2014, doi: 10.1109/TNNLS.2014.2361776. Wang, Z., & Liu, Y., & Liu, X. (2005). On global asymptotic stability of neural networks with discrete and distributed delays, Physics Letters A, 345, 299308. Wang, Z., & Liu, Y., & Li, M., & Liu, X. (2006). Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays, IEEE Transactions on Neural Networks and Learning Systems, 17, 814-820. Wen, S., & Huang, T., & Zeng, Z., & Chen, Y., & Li, P. (2015). Circuit design and exponential stabilization of memristive neural networks, Neural Networks, 63, 48-56. Wu, A. L., & Zeng, Z. G.(2012). Exponential stabilization of memristive neural networks with time delays, IEEE Transactions on Neural Networks and Learning Systems, 23, 1919-1929. Wu, R., & Lu, Y., & Chen, L. (2015). Finite-time stability of fractional delayed neural networks, Neurocomputing, 149, 700-707. Yang, R., & Wang, Y. (2012). Finite-time stability and stabilization of a class of nonlinear time-delay systems, SIAM Journal on Control and Optimization, 50, 3113-3131. Zeng, Z., & Zheng, W. X. (2012). Multistability of neural networks with timevarying delays and concave-convex characteristics, IEEE Transactions on Neural Networks and Learning Systems, 23, 293-305. Zhang, G. D., & Shen, Y. (2014). Exponential stabilization of memristorbased chaotic neural networks with time-varying delays via periodically intermittent control, IEEE Transactions on Neural Networks and Learning Systems, doi: 10.1109/TNNLS.2014.2345125.
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S0893-6080(15)00145-8 http://dx.doi.org/10.1016/j.neunet.2015.07.008 NN 3503
To appear in:
Neural Networks
Received date: 26 December 2014 Revised date: 8 May 2015 Accepted date: 16 July 2015 Please cite this article as: Wang, L., Shen, Y., & Ding, Z. Finite time stabilization of delayed neural networks. Neural Networks (2015), http://dx.doi.org/10.1016/j.neunet.2015.07.008 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 proof before it is published in its final 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.
1. Time delays are taken into account in studying the finite time stability problem of neural networks. 2. Provide general conditions on the feedback control law for the finite time stabilization of delayed neural networks (DNNs). 3. Derive some specific conditions by designing two different controllers which include the delay-dependent and delay-independent ones. 4. Discuss the extremum of the settling time functional and realize the optimal stabilization time under fixed control strength. 5. Design a switched controller to optimize the settling time.
*Manuscript Click here to view linked References
Finite time stabilization of delayed neural networks Leimin Wanga,b , Yi Shena,b,∗, Zhixia Dinga,b b Key
a School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China
Abstract In this paper, the problem of finite time stabilization for a class of delayed neural networks (DNNs) is investigated. The general conditions on the feedback control law are provided to ensure the finite time stabilization of DNNs. Then some specific conditions are derived by designing two different controllers which include the delay-dependent and delay-independent ones. In addition, the upper bound of the settling time for stabilization is estimated. Under fixed control strength, discussions of the extremum of settling time functional are made and a switched controller is designed to optimize the settling time. Finally, numerical simulations are carried out to demonstrate the effectiveness of the obtained results. Keywords: Finite time stabilization, delayed neural networks, delay-dependent controller, delay-independent controller, switched controller, settling time. 1. Introduction Stability is a prerequisite of neural networks in the successful applications like signal processing, pattern recognition and associative memory design (Cohen & Grossberg, 1983; Chua & Yang, 1988; Forti & Tesi, 1995). In the hardware implementation of neural networks, some signal transmission delays are unavoidable, which may cause undesirable dynamical behaviors such as instability and oscillation. Thus, it is necessary to take time delays into consideration in studying stability of neural networks. During the past decades, stability analysis of delayed neural networks (DNNs) has been extensively investigated and many useful stability criteria have been established (Cao & Wang, 2005; Faydasicok & Arik, 2012; Forti et al., 2005; He & Wu, 2006; Huang et al., 2012; Liu & Wang, 2006; Shen & Wang, 2009, 2012; Wang & Shen, 2014; Wang et al., 2014; Wang & Liu, 2005; Wang et al., 2006; Lu, 2002; Zeng & Zheng, 2012). On the other hand, the stabilization of DNNs has been attracting considerable attention and several feedback stabilizing control methods have been proposed, such as linear control (Guo & Wang, 2013; Huang et al., 2013; Phat & Trinh, 2010; Wen et al., 2015; Wu & Zeng, 2012), impulsive control (Guan & Zhang, 2008), and intermittent control (Chen et al., 2014; Huang & Li, 2009; Hu et al., 2010; Zhang & Shen, 2014). It should be pointed out that, in most of the above works, the stability and stabilization of DNNs are asymptotic or exponential, which means the time to approach equilibrium points is infinite. In practical applications, we always hope to obtain fast or even finite time convergent speed, so the concept ∗ Corresponding
author. Tel.: +86 27 87543630; fax: +86 27 87543130. Email addresses: [email protected] (Leimin Wang), [email protected] (Yi Shen) Preprint submitted to Neural Networks
of finite time stability arises naturally. Finite time stability requires the system be Lyapunov stable and its trajectories tend to equilibrium points in finite time. Since Bhat & Bernstein (2000) extended finite time stability to multi-dimensional continuous autonomous systems and proved that there is necessary and sufficient condition for finite time stability, the problem of finite time stability and stabilization have been widely studied (Efimov et al., 2014; Hong & Jiang, 2006; Karafyllis, 2006; Liu et al., 2013, 2014a,b; Moulay & Perruquetti, 2005; Moulay et al., 2008; Sun et al., 2014; Wang & Xiao, 2010; Wu et al., 2015; Yang & Wang, 2012). However, so far there are few results concerning the finite time stability of time-delay systems (Moulay et al., 2008; Yang & Wang, 2012). The reason is that timedelay systems have more complicated dynamic behaviors and are more difficult to deal with than system without delays. As is stated in Moulay et al. (2008), it is difficult to find a Lyapunov functional to satisfy the derivative condition for finite time stability of time-delay systems. Also, it is reported in Efimov et al. (2014) that some key results in Yang & Wang (2012) are incorrect. So finite time stability of time-delay systems is still an open problem that needs further investigation. In recent years, finite time stability or synchronization problem of neural networks has been investigated (Shen & Cao, 2011; Huang & Li, 2014; Hu et al., 2014; Liu et al., 2013, 2014a,b; Shen & Park, 2014; Wu et al., 2015). However, most of the results are restrictive because of the absence of the time delays in the system. For instance, time delays are not taken into account in the finite time stabilization in Liu et al. (2013, 2014a,b) and finite time synchronization in Shen & Cao (2011); Shen & Park (2014). In addition, although different controllers are used in stabilization or finite time stabilization of neural networks, there do not exist one general controller that can finite time stabilize the DNNs. It still needs to answer the question that how to design a proper controller to realize the finite time May 8, 2015
stabilization. Motivated by the above discussions, we investigate the finite time stabilization for a class of DNNs in this paper. The contributions of this paper are fourfold. 1) Time delays are taken into account in the finite time stability problem of neural networks. 2) Based on the finite time Lyapunov stability theory of functional differential equation (Moulay et al., 2008), the general conditions on the feedback control law for finite time stabilization of DMNs are derived, which provide new method to study the finite time stabilization of time-delay systems. 3) Some specific conditions are derived by designing two different types of feedback control algorithms which include the delay-dependent and delay-independent ones. 4) Under fixed control strength, discussions of the extremum of settling time functional are made so as to realize the optimal stabilization time and a switched controller is designed to optimize the settling time. The organization of this paper is as follows. Some preliminaries are introduced in Section 2. In Section 3, general conditions on the feedback control law are provided to ensure the finite time stabilization of DNNs, and some specific conditions are obtained by designing two different types of controllers. In addition, discussions of stabilization time are made and a switched controller is designed to decrease the settling time. Then, two examples are provided to demonstrate the effectiveness of the obtained results in Section 4. Finally, conclusions are drawn in Section 5.
g(x(t − τ(t))) = (g1 (x1 (t − τ1 (t))), . . . , gn (xn (t − τn (t))))T ∈ Rn are the neuron activation functions. τ(t) = (τ1 (t), . . . , τn (t))T is the time-varying delay satisfy that 0 ≤ τ j (t) ≤ τ, j = 1, 2, . . . , n. u(t) = (u1 (t), . . . , un (t))T is the control input. The initial condition of system (1) is: x(s) = φ(s) = (φ1 (s), φ2 (s), . . . , φn (s))T ∈ C([−τ, 0], Rn). (A)The neuron activation functions f j , g j ( j = 1, 2, . . . , n) are bounded and satisfy f j (0) = g j (0) = 0, and there exist constants h j > 0, k j > 0, F j > 0, G j > 0 such that
2. System description and preliminaries
Lemma 2 (Moulay et al., 2008). Consider the system (1) with uniqueness of solutions in forward time. If there exist a continuous functional V : Ω → R+ and two functions ν, r of class K for system (1) such that (i) ν(kφ(0)k) ≤ V(φ), R ε dz < +∞, (ii) D+ V(φ) ≤ −r(V(φ)) with 0 r(z) for all ε > 0, φ ∈ Ω. Then system (1) is finite time stabilizable R V(φ) dz . with a settling time satisfying the inequality T 0 (φ) ≤ 0 r(z) ρ In particularly, if r(V) = λV where λ > 0, ρ ∈ (0, 1), then the settling time satisfies the inequality
| f j (s1 ) − f j (s2 )| ≤ h j |s1 − s2 |, | f j (s1 )| ≤ F j ,
|g j (s1 ) − g j (s2 )| ≤ l j |s1 − s2 |, |g j (s1 )| ≤ G j , for all s1 , s2 ∈ R. Before studying the finite time stabilization of system (1), we give the following definitions and lemmas. Let Ω be an open subset of C([−τ, 0], Rn) containing 0.
Definition 2 (Moulay et al., 2008). The origin of system (1) is finite time stable where ui (t) = 0, if (i) The origin of system (1) is stable, (ii) The origin of system (1) is finite time convergent, i.e. for any initial state φ(s) ∈ Ω, there exists 0 ≤ T (φ) < +∞ such that every solution of system (1) satisfies x(t, φ) = 0 for all t ≥ T (φ). T 0 (φ) = inf{T (φ) ≥ 0 : x(t, φ) = 0 ∀t ≥ T (φ)} is a functional called the settling time of the system (1). Definition 3. The system (1) is finite time stabilizable if there exists a feedback controller ui (t) such that ui (0) = 0 and system (1) is finite time stable.
The √ following notations will be used throughout this paper. kxk = xT x is the Euclidean norm of x ∈ Rn and h., .i denotes the inner product of Euclidean space. C([a, b], Rn) denotes the space of continuous functions φ : [a, b] → Rn with the uniform norm kφk = sup kφ(s)k. AT and A−1 denote the transpose a≤s≤b
and the inverse of the matrix A, respectively. A > 0(A ≥ 0) means that the matrix A is symmetric and positive definite (semi-positive definite). λmax (A) and λmin (A) denote the maximum and minimum eigenvalue of matrix A, respectively. ∗ represents the elements below the main diagonal of a symmetric matrix. I is the identity matrix with compatible dimension. diag{· · · } denotes a block-diagonal matrix. A continuous function s : R+ → R+ belongs to the class K if it is strictly increasing and s(0) = 0. sign(·) denotes the signum function. e refers to the base of natural logarithms. In this paper, we consider a class of neural networks with time-varying delay as follows: x˙(t) = −Dx(t) + A f (x(t)) + Bg(x(t − τ(t))) + u(t),
T 0 (φ) ≤
0
V(φ)
dz V 1−ρ (φ) = . r(z) λ(1 − ρ)
(2)
Lemma 3 (Berman & Plemmons, 1979). For any vectors x, y ∈ Rn , ε > 0 and positive definite matrix Q ∈ Rn×n , the following matrix inequality holds 2xT y ≤ ε−1 xT Q−1 x + εyT Qy.
(1)
where x(t) = (x1 (t), x2 (t), . . . , xn (t))T ∈ Rn is the state vector. D = diag(d1 , d2 , . . . , dn ) is an n × n diagonal matrix with di > 0, i = 1, 2, . . . , n. A = (ai j )n×n , B = (bi j )n×n ∈ Rn×n are the connection weight matrix and delayed connection weight matrix, respectively. f (x(t)) = ( f1 (x1 (t)), . . . , fn (xn (t)))T ∈ Rn and
Z
Lemma 4 (Hardy et al., 1988). If a1 , a2 , . . . , an , r1 , r2 are real numbers and 0 < r1 < r2 , then the following inequality holds [
n n X X (|ai |)r2 ]1/r2 ≤ [ (|ai |)r1 ]1/r1 . i=1
2
i=1
3. Main results
Combining with (4)-(6) and (8)-(10), we derive that T ˙ ≤xT (t)[−PD − DT P + ε−1 PAQ−1 V(t) 1 A P]x(t)
In this section, the finite time stabilization of DNNs is investigated. The feedback control law u(t) = (u1 (t), u2 (t), . . . , un (t))T is given as follows: u(t) = uˆ (t) + uˇ (t).
+ ε f T (x(t))Q1 f (x(t)) + 2xT (t)P(uˆ (t) + uˇ (t)) + 2hPB|g(x(t − τ(t)))|, |x(t)|i
T ≤xT (t)[−PD − DT P + ε−1 PAQ−1 1 A P
+ εH T Q1 H]x(t) − xT (t)Q2 x(t) + 2xT (t)Pˇu(t)
(3)
≤2xT (t)Pˇu(t) n X ≤−δ |xi (t)|µ+1 .
Then we will give general conditions of uˆ (t) and uˇ (t) in order to ensure the finite time stable of system (1). Theorem 1. Under the assumption (A), if there exist three symmetric matrices P > 0, Q1 > 0, Q2 > 0, constants ε > 0, 0 ≤ µ < 1, δ > 0 such that T T − PD − DT P + ε−1 PAQ−1 1 A P − Q2 + εH Q1 H < 0,
hPB|g(x(t − τ(t)))|, |x(t)|i + xT (t)Pˆu(t) 1 ≤ − xT (t)Q2 x(t), 2 n X 2xT (t)Pˇu(t) ≤ −δ |xi (t)|µ+1 ,
Based on Lemma 4, we get the following inequality −[
(4)
n X i=1
On the other hand, we have Z ε dz 0
(7)
T 0 (φ) ≤
˙ =2xT (t)P x˙(t) V(t)
≤
T
+ 2xT (t)P(uˆ (t) + uˇ (t)),
|xi (t)|2 ]1/2 ,
i=1
(12)
=
(13)
2ε(1−µ)/2 −(µ+1)/2 δλmax (P)(1 −
µ)
for all ε > 0. From Lemma 2, one can conclude that system (1) is finite time stabilizable, and the settling time satisfies
Calculating the time derivative of the Lyapunov function along the trajectories of system (1), we have
≤ − x (t)(PD + D P)x(t) + 2hPA| f (x(t))|, |x(t)|i + 2hPB|g(x(t − τ(t)))|, |x(t)|i
−(µ+1)/2 δλmax (P)z(µ+1)/2
< +∞,
Proof. Consider the following Lyapunov function
T
n X
µ+1 −(µ+1)/2 ˙ V(t) ≤ −δλmax (P)V 2 (t).
(6)
where H = diag(h1 , h2 , . . . , hn ). Then system (1) is finite time stabilizable via the controller (3), and the settling time for sta2λmax (P)kφk1−µ . bilization satisfies T 0 (φ) ≤ δ(1 − µ)
V(t) = x (t)Px(t).
|xi (t)|)µ+1 ]1/(µ+1) ≤ −[
for 1 < µ + 1 < 2. Then we have
(5)
i=1
T
(11)
i=1
2V (1−µ)/2 (0) −(µ+1)/2 δλmax (P)(1 − 2λmax (P)kφk1−µ
δ(1 − µ)
µ)
.
(14)
The proof is completed. (8)
3.1. Finite time stabilization via delay-dependent control
where |x(t)| = (|x1 (t)|, |x2 (t)|, . . . , |xn (t)|)T , | f (x(t))| = (| f1 (x1 (t))|, | f2 (x2 (t))|, . . . , | fn (xn (t))|)T , |g(x(t − τ(t)))| = (|g1 (x1 (t − τ1 (t)))|, |g2(x2 (t − τ2 (t)))|, . . . , |gn (xn (t − τn (t)))|)T . According to the assumption (A), there exist h j > 0 such that
In Theorem 1, the general conditions on finite time stabilization of system (1) are obtained. In order to give some specific conditions which are easy to be verified, we design a delayed feedback controller as follows:
| f j (x j (t))| ≤ h j |x j (t)|,
u(t) = − k1 x(t) − BLsign(x(t))|x(t − τ(t))| − k2 sign(x(t))|x(t)|µ ,
(9)
where uˆ (t) = −k1 x(t) − BLsign(x(t))|x(t − τ(t))|, uˇ (t) = −k2 sign(x(t))|x(t)|µ , L = diag(l1 , l2 , . . . , ln ), k1 > 0, k2 > 0, 0 ≤ µ < 1. Noticing that the assumption (A) there exist l j > 0 such that
Now, by using Lemma 3, it is easy to obtain the following inequality T 2hPA| f (x(t))|, |x(t)|i ≤ ε−1 xT (t)PAQ−1 1 A Px(t)
+ ε f T (x(t))Q1 f (x(t)).
(15)
(10) 3
|g j (x j (t − τ j (t)))| ≤ l j |x j (t − τ j (t))|,
(16)
then we have
We can see that (5) and (6) hold obviously, then we get the following result.
hPB|g(x(t − τ(t)))|, |x(t)|i + xT (t)Pˆu(t)
Theorem 3. Under the assumption (A), if there exist three symmetric matrices P > 0, Q1 > 0, constants ε > 0, k3 > 0, k4 > 0, 0 ≤ µ < 1 such that
≤hPBL|x(t − τ(t))|, |x(t)|i + xT (t)Pˆu(t) = − k1 xT (t)Px(t),
2xT (t)Pˇu(t) = −2k2 xT (t)Psign(x(t))|x(t)|µ n X ≤ − 2k2 λmin (P) |xi (t)|µ+1 .
T T −PD − DT P + ε−1 PAQ−1 1 A P − 2k3 P + εH Q1 H < 0, (21)
Then system (1) is finite time stabilizable via the controller (20), and the settling time for stabilization satisfies T 0 (φ) ≤ λmax (P)kφk1−µ . k4 λmin (P)(1 − µ)
i=1
We can see that (5) and (6) hold obviously. Thus, we can get the following results. Theorem 2. Under the assumption (A), if there exist three symmetric matrices P > 0, Q1 > 0, constants ε > 0, k1 > 0, k2 > 0, 0 ≤ µ < 1 such that
Corollary 3. Under the assumption (A), if there exist constants ε > 0, 0 ≤ µ < 1, p > 0, k3 > 0, k4 > 0 such that −p(D + DT ) + ε−1 p2 AAT − 2k3 pI + εH T H < 0.
T T −PD − DT P + ε−1 PAQ−1 1 A P − 2k1 P + εH Q1 H < 0, (17)
Then system (1) is finite time stabilizable via the controller (20), and the settling time for stabilization satisfies T 0 (φ) ≤ kφk1−µ . k4 (1 − µ)
Then system (1) is finite time stabilizable via the controller (15), and the settling time for stabilization satisfies T 0 (φ) ≤ λmax (P)kφk1−µ . k2 λmin (P)(1 − µ) Corollary 1. Under the assumption (A), if there exist symmetric matrix P > 0, constants ε > 0, 0 ≤ µ < 1, k1 > 0, k2 > 0 such that −(PD + DT P) + ε−1 PAAT P − 2k1 P + εH T H < 0.
Remark 1. Although stabilization of DNNs has been investigated in the past few years (Chen et al., 2014; Guo & Wang, 2013; Huang et al., 2013; Huang & Li, 2009; Hu et al., 2010; Phat & Trinh, 2010; Wen et al., 2015; Wu & Zeng, 2012; Zhang & Shen, 2014), so far, there is little work concerning the finite time stabilization of DNNs. In this paper, two different kinds of feedback control law are designed and some sufficient conditions in Theorem 2 and 3 are given to ensure the finite time stabilization of DNNs. Our results complement and extend the previous results which only asymptotic stabilization of DNNs are obtained.
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Then system (1) is finite time stabilizable via the controller (15), and the settling time for stabilization satisfies T 0 (φ) ≤ λmax (P)kφk1−µ . k2 λmin (P)(1 − µ) Corollary 2. Under the assumption (A), if there exist constants ε > 0, 0 ≤ µ < 1, p > 0, k1 > 0, k2 > 0 such that −p(D + DT ) + ε−1 p2 AAT − 2k1 pI + εH T H < 0.
Remark 2. Time-delay systems have more complicated dynamic behaviors and are more difficult to deal with than system without delays. In Liu et al. (2013, 2014a,b), finite time stability of neural networks was studied, but the systems in these papers are without delays. Since delays are unavoidable in the hardware implementation of neural networks, it is necessary to take the delays into consideration in the finite time stability problem. In this paper, the neural networks with delays are considered, which make our results general compared with the existing results (Liu et al., 2013, 2014a,b).
(19)
Then system (1) is finite time stabilizable via the controller (16), and the settling time for stabilization satisfies T 0 (φ) ≤ kφk1−µ . k2 (1 − µ) 3.2. Finite time stabilization via delay-independent control In this section, we design a feedback controller as follows: u(t) = −k3 x(t) − BGsign(x(t)) − k4 sign(x(t))|x(t)|µ ,
(22)
(20) 3.3. Reduce the settling time via a switched controller
where uˆ (t) = −k3 x(t) − BGsign(x(t)), uˇ (t) = −k4 sign(x(t))|x(t)|µ , G = max{G j }( j = 1, 2, . . . , n) is a constant, k3 > 0, k4 > 0, 0 ≤ µ < 1. Then we have
From Corollary 2, we can see that the settling time for stabilization is decided by three parameters: the initial value φ, the control strength k2 of uˇ (t) and µ. Now, we will discuss how to reduce the settling time via a control strategy. Let the control strength k2 of uˇ (t) be given and fixed, and set kφk1−µ , 0 ≤ µ < 1, k2 > 0, kφk > 0. Then T 0 (µ) = k2 (1 − µ)
hPB|g(x(t − τ(t)))|, |x(t)|i + xT (t)Pˆu(t)
≤ − k3 xT (t)Px(t),
2xT (t)Pˇu(t) = −2k4 xT (t)Psign(x(t))|x(t)|µ n X ≤ − 2k4 λmin (P) |xi (t)|µ+1 . i=1
4
dT 0 [(µ − 1) ln(kφk) + 1]kφk1−µ = . dµ k2 (1 − µ)2
(23)
i) If kφk > e (ln kφk > 1), T 0 (µ) has only one critical point 1 where T 0 (µ) reach the minimum value, and at µ = 1 − ln(kφk) ln(kφk) 1/(ln(kφk)) e ln(kφk) kφk = . min T 0 (µ) = k2 k2 µkφk1−µ dT 0 ≥ > 0, ii) If 0 < kφk ≤ e (ln kφk ≤ 1), then dµ k2 (1 − µ)2 namely T 0 (µ) is increasing on (0, 1). So T 0 (µ) reach the minikφk . mum value at µ = 0, and min T 0 (µ) = k2
From Lemma 2, one can conclude that system (1) is finite time stabilizable, and the settling time satisfies T 0 (φ) ≤ ≤
have the following result.
−1 2
T
−p(D + D ) + ε p AA − 2k1 pI + εH H < 0.
≤ −2k2 p where µ¯ = Since Z
0
ε
(
µ∗ , 0,
dz 2k2
¯ z(¯µ+1)/2 p(1−µ)/2
i=1
4
2
2
x (t)
0
−2
−4
−6
(25)
−8 −2
µ¯ + 1 V 2 (t),
k2
−0.5 x (t)
0
0.5
1
Figure 1: Phase plot of system (29) with u(t) = (0, 0)T .
4. Numerical examples
Example 1. Consider the two-dimensional delayed neural networks (Lu, 2002)
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x˙(t) = −Dx(t) + A f (x(t)) + Bg(x(t − τ(t))) + u(t),
(27)
ε(1−µ)/2 ¯ (1 p(1−µ)/2
−1
In this section, two examples are provided to verify the effectiveness of the results obtained in the previous section.
kφk > e, 0 < kφk ≤ e.
=
−1.5
1
¯ |xi (t)|µ+1
(1−µ)/2 ¯
(28)
6
Proof. Let P = pI, then similar to the proof of Theorem 1, we have ˙ ≤ −2k2 p V(t)
0
Remark 5. Theorem 4 gives the sufficient conditions on finite time stabilization of DMNNs via a switched controller. The switch control method can be used to study finite time stabilization of other nonlinear time-delay systems.
Then system (1) is finite time stabilizable via the switched controller (24), and the settling time for stabilization satisfies e + ln(kφk)[e − e1/(ln(kφk)) ] T 0 (φ) ≤ . k2
n X
pe2
Remark 4. It should be pointed out that if the nonlinear system (1) is considered without delays, then the main Theorem 2, 3 and 4 in Liu et al. (2014b) concerning finite time stability of nonlinear systems can be obtained directly from Corollary 1, 2 and Theorem 4, respectively.
1 . Then we ln(kφk)
T
0
Z
p(¯µ−1)/2 dz 2k2 z(¯µ+1)/2 Z pkφk2 (µ∗ −1)/2 p−1/2 dz p dz + ∗ +1)/2 1/2 (µ 2k2 z 2k2 z pe2
The proof is completed.
Theorem 4. Under the assumption (A), if there exist constants ε > 0, p > 0, k1 > 0, k2 > 0 such that T
V(0)
e + ln(kφk)[e − e1/(ln(kφk)) ] . = k2
Remark 3. In order to obtain faster convergence speed and reduce the stabilization time, we make discussion on the extremum of the settling time functional. Under the fixed control strength k2 , the minimum value of T 0 (µ) depends on the value of initial value kφk, which means that we can choose a proper parameter µ to realize the optimal stabilization. If the initial value is big enough (kφk > e), then uˇ (t) is continuous and T 0 (µ) 1 reach the minimum value at µ = 1 − ; If the initial value ln(kφk) is small such that (0 < kφk ≤ e), then uˇ (t) is discontinuous and T 0 (µ) reach the minimum value at µ = 0. Based on the above analysis, we design a switched controller as follows: −k1 x(t) − BLsign(x(t))|x(t − τ(t))| ∗ −k2 sign(x(t))|x(t)|µ , kx(t)k > e, u(t) = (24) −k1 x(t) − BLsign(x(t))|x(t − τ(t))| −k2 sign(x(t)), 0 < kx(t)k ≤ e,
where k1 , k2 , µ are constants and µ∗ = 1 −
Z
− µ) ¯
(29)
where x(t) = (x1 (t), x2 (t))T , D = diag(1, 1), " # " # 2 −0.1 −1.5 −0.1 A= ,B= . −5 4.5 −0.2 −4
< +∞,
τ j (t) = 1, j = 1, 2, and take the activation function as f j (x) = g j (x) = tanh(x), j = 1, 2. Then we have H = L = D =
for all ε > 0. 5
1.5 x1(t)
1.5
x2(t)
1
x (t) 1
x (t) 2
1
0.5
2
0
1
−0.5
x (t) and x (t)
2 1
x (t) and x (t)
0.5
0
−0.5
−1 −1
−1.5 −1.5
−2
0
0.5
1
1.5
2
2.5 time−t
3
3.5
4
4.5
5 −2
Figure 2: State trajectories of variables x1 (t) and x2 (t) of system (29) under the controller (30).
0
0.5
1
1.5
2
2.5 time−t
3
3.5
4
4.5
5
Figure 3: State trajectories of variables x1 (t) and x2 (t) of system (29) under the switch controller (24).
diag(1, 1). The phase plot of system (29) with the initial condition x1 (s) = −1.6, x2 (s) = 1.2, ∀s ∈ [−1, 0) is shown in Fig.1. The controller is designed as 4 3.5
(30)
3
Set k1 = p−1 k, then the condition (19) equals to the following LMI −p(D + DT ) − 2kI pA εH T ∗ −εI 0 < 0. (31) ∗ ∗ −εI
2.5
x2(t)
2 1.5 1 0.5
By using the Matlab LMI Control Toolbox, we can find a solution to the LMI (31): p = 0.0258, ε = 0.6905, k = 0.8269. Then k1 = p−1 k = 32.0825, and by picking an arbitrary fixed k2 > 0, system (29) can be finite time stabilized via the controller (30) according to Corollary 2. Set k2 = 1, the settling kφk1−µ = 2.828. The time for stabilization satisfies T 0 (φ) ≤ k2 (1 − µ) kφk = 2. Under the minimum value of the stabilization time is k2 controller (30), we get the state trajectories of variables x1 (t) and x2 (t) which are shown in Fig.2. System (30) also can be finite time stabilized via the switched controller (24) according to Theorem 4. Set k2 = 1, the settling time for stabilization satisfies T 0 (φ) ≤ 1.6690. Then under the switched controller (24), we get the state trajectories of variables x1 (t) and x2 (t) which are shown in Fig.3.
0 −0.5 −1 −3
−2
−1
0 x1(t)
1
2
3
Figure 4: Phase plot of system (32) with u(t) = (0, 0)T .
4 x1(t) x2(t)
3
x (t) and x (t)
2
Example 2. Consider the two-dimensional delayed neural networks (Hu et al., 2010)
2
− k2 sign(x(t))|x(t)|0.5 .
1
1
u(t) = − k1 x(t) − BLsign(x(t))|x(t − τ(t))|
0
−1
x˙(t) = −Dx(t) + A f (x(t)) + Bg(x(t − τ(t))) + u(t),
(32) −2
where D = diag(1, 1), " " # # 1.8 1.5 −1.4 0.1 A= ,B= . 0.1 1.8 0.1 −1.4
−3
0
0.5
1
1.5
2
2.5 time−t
3
3.5
4
4.5
5
Figure 5: State trajectories of variables x1 (t) and x2 (t) of system (32) with the controller (33).
τ j (t) = et /(1 + et ), j = 1, 2, and take the activation function as 1 f j (x) = g j (x) = (|x + 1| − |x − 1|), j = 1, 2. Then we have 2 6
H = L = D = diag(1, 1), G = 1. The phase plot of system (33) with the initial condition x1 (s) = −3, x2 (s) = 4, ∀s ∈ [−1, 0) is shown in Fig.4. The controller is designed as u(t) = −k3 x(t) − BGsign(x(t)) − k4 sign(x(t))|x(t)|0.5 .
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(33)
Set k3 = p−1 k, then the condition (23) equals to the same LMI (31). By using the Matlab LMI Control Toolbox, we can find a solution to the LMI (32): p = 0.0297, ε = 0.5754, k = 0.5490. Then k3 = p−1 k = 18.4800, and by picking an arbitrary fixed k4 > 0, system (32) can be finite time stabilized via the controller (33) according to Corollary 3. Set k4 = 1, the settling kφk1−µ time for stabilization satisfies T 0 (φ) ≤ = 3.464. The k4 (1 − µ) e ln(kφk) minimum value of the stabilization time is = 2.9863. k4 Under the controller (33), we get the state trajectories of variables x1 (t) and x2 (t) which are shown in Fig.5. 5. Conclusions Finite time stabilization for a class of delayed neural networks (DNNs) has been investigated in this paper. Firstly, the general conditions on the feedback control law have been provided to ensure the finite time stabilization of DNNs. Then some specific conditions have been obtained by designing two different types of feedback control algorithms which include the delay-dependent and delay-independent ones. It is noted that discussions of the extremum of settling time functional have been made to realize the optimal stabilization time under fixed control strength and a switched controller has been designed to optimize the settling time. Our control method can be used to study the finite time stabilization of other nonlinear time-delay system. On one hand, our results complement and extend the previous results in Liu et al. (2013, 2014a,b) where delays are not taken into consideration in studying finite time stability problem. On the other hand, the new proposed results here achieve a valuable improvement compared with the present works (Chen et al., 2014; Guo & Wang, 2013; Huang et al., 2013; Huang & Li, 2009; Hu et al., 2010; Phat & Trinh, 2010; Wen et al., 2015; Wu & Zeng, 2012; Zhang & Shen, 2014) where only exponential or asymptotic stabilization of DNNs is obtained in these papers. Acknowledgements This work is supported by the Key Program of National Natural Science Foundation of China (Grant no. 61134012), the National Science Foundation of China (Grant nos. 11271146 and 61374150), and the prior developing Field for the Doctoral Program of Higher Education of China (Grant no. 20130142130012). References Berman, A., & Plemmons, R. J. (1979). Nonnegative Matrices in the Mathematical Science, Academic, New York.
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