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A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks
Journal Pre-proof A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks Chuan Chen, Lixiang Li, Haipeng Peng, Yixian Yang, Ling Mi, Hui Zhao
PII: DOI: Reference:
S0893-6080(19)30430-7 https://doi.org/10.1016/j.neunet.2019.12.028 NN 4360
To appear in:
Neural Networks
Received date : 16 April 2019 Revised date : 1 November 2019 Accepted date : 27 December 2019 Please cite this article as: C. Chen, L. Li, H. Peng et al., A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks. Neural Networks (2020), doi: https://doi.org/10.1016/j.neunet.2019.12.028. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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.
© 2020 Elsevier Ltd. All rights reserved.
Journal Pre-proof A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks First author:
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Chuan Chen 1. School of Cyber Security, Qilu University of Technology, Jinan 250353, China 2. Shandong Provincial Key Laboratory of Computer Networks, Jinan, 250353, China [email protected]
Corresponding author:
Chuan Chen 1. School of Cyber Security, Qilu University of Technology, Jinan 250353, China 2. Shandong Provincial Key Laboratory of Computer Networks, Jinan, 250353, China [email protected]
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Lixiang Li Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [email protected]
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Ling Mi School of Mathematics and Statistics, Qilu University of Technology, Jinan 250353, China [email protected]
Other authors:
Haipeng Peng, Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [email protected]
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Yixian Yang Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [email protected] Hui Zhao
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School of Information Science and Engineering, University of Jinan, Jinan 250022, China [email protected]
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Manuscript
Click here to view linked References
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A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks Chuan Chena,b,∗, Lixiang Lic,∗, Haipeng Pengc , Yixian Yangc , Ling Mid,∗, Hui Zhaoe a School
of Cyber Security, Qilu University of Technology, Jinan 250353, China Provincial Key Laboratory of Computer Networks, Jinan, 250353, China c Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China d School of Mathematics and Statistics, Qilu University of Technology, Jinan 250353, China e School of Information Science and Engineering, University of Jinan, Jinan 250022, China b Shandong
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Abstract
In this paper, we derive a new fixed-time stability theorem based on definite integral, variable substitution and some inequality techniques. The fixed-time stability criterion and the upper bound estimate formula for the settling time
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are different from those in the existing fixed-time stability theorems. Based on the new fixed-time stability theorem, the fixed-time synchronization of neural networks is investigated by designing feedback controller, and sufficient conditions are derived to guarantee the fixed-time synchronization of neural networks. To show the usability and superiority of the obtained theoretical results, we pro-
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pose a secure communication scheme based on the fixed-time synchronization of neural networks. Numerical simulations illustrate that the new upper bound estimate formula for the settling time is much tighter than those in the existing fixed-time stability theorems. What is more, the plaintext signals can be recovered according to the new fixed-time stability theorem, while the plaintext signals can not be recovered according to the existing fixed-time stability theorems.
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Keywords: Fixed-time stability, Fixed-time synchronization, Neural networks
∗ Corresponding
author Email addresses: [email protected] (Chuan Chen), li [email protected] (Lixiang Li), [email protected] (Ling Mi)
Preprint submitted to Journal of LATEX Templates
November 1, 2019
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1. Introduction Stability [1, 2, 3] is an important dynamical behavior of nonlinear system,
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and the stability analysis of nonlinear systems has been paid much attention in recent years [4, 5, 6, 7]. Especially, asymptotic stability [8, 9, 10, 11, 12] and exponential stability [13, 14, 15] have been investigated intensively. Asymptotic stability and exponential stability belong to infinite-time stability, which means stability can only be reached when time tends to infinity. Compared with asymptotic stability and exponential stability, finite-time stability [16, 17, 18] has an obvious advantage in practical applications. For finite-time stability, stability can be reached within a finite time (the settling time), which can be
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estimated based on the initial value of the studied system. However, if the accurate initial value of the studied system is unknown beforehand, the settling time can’t be estimated.
In 2012, fixed-time stability was put forward [19]. For fixed-time stability
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[19, 20, 21], stability can also be reached within the settling time. What is more, the settling time has a fixed upper bound for any initial value of the studied system. Although we know the settling time of fixed-time stability has a fixed upper bound, deriving the tightest upper bound estimate formula is impossible due to the difficulty of theoretical analysis. However, referring to the upper
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bound estimate formulas in some existing fixed-time stability theorems, can we derive a tighter upper bound estimate formula for the settling time? Considering the current over-dependence on the existing fixed-time stability theorems in [19, 20, 21], developing a new fixed-time stability theorem with a tighter upper bound estimate formula for the settling time will be very meaningful. On the other hand, since the synchronization problem of nonlinear systems can be converted into the stability problem of error system, some fixed-time
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stability theorems have been used to study the fixed-time synchronization of nonlinear systems [20, 22, 23, 24, 25, 26, 27, 28, 29]. In [24], the fixed-time synchronization of complex-valued neural networks was studied. By designing non-chattering controllers, the fixed-time synchronization of complex networks
2
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with impulsive effects was considered in [29]. As we know, the stability of neural network has important applications in
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associative memory [30, 31, 32] and optimization [33]. Take associative memory for example, memory mode is designed as a stable state (i.e. attractor) of neural network. A neural network can have more than one attractor, and the memory capacity of neural network is the number of attractors. Each attractor has an attraction basin, which is the domain that the attractor can attract. If a given input belongs to the attraction basin of an attractor, neural network can stabilize to this attractor, that is, the stored memory mode is recalled. Meanwhile, the synchronization of neural networks also has important applications in many
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fields, such as secure communication [34], image encryption [35], the cooperative control of intelligent robots, etc. If synchronization time and robustness are taken into account, we must consider the fixed-time synchronization of neural networks.
In this paper, we establish a novel fixed-time stability theorem. The fixed-
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time stability criterion and the upper bound estimate formula for the settling time are different from those in [19, 20, 21]. Theoretical analysis shows that the new upper bound estimate formula for the settling time in this paper is tighter than that in [19]. In the proof of the new fixed-time stability theorem,
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we depend heavily on definite integral, variable substitution and some inequality techniques. The proof of the new fixed-time stability theorem is a strict mathematical proof, and it is very general. By adopting the proof techniques in the new fixed-time stability theorem, the existing finite/fixed-time stability theorems can also be given strict mathematical proofs. To demonstrate the usability of the new fixed-time stability theorem, we investigate the fixed-time synchronization of neural networks by means of the new fixed-time stability theorem. In fact, the new fixed-time stability theorem and the analysis techniques
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in this paper can also be used to study the fixed-time synchronization/stability of other nonlinear systems. To show the application of the obtained theoretical results, we propose a secure communication scheme based on the fixed-time synchronization of neural networks. Numerical simulations illustrate that the new 3
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R ij ,i,j=1,2,...,n
... ...
R1n
R 2n
...
...
...
R12
R 22 R 21 R11
Rn2
...
R n1
Cn
...
x2 t
xn t
R2
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R1
...
f 2
...
Rn
f n
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f1
R nn
C2
C1
x1 t
Jn
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J2
J1
Fig. 1. The circuit implementation of a neural network without time delay. upper bound estimate formula for the settling time is much tighter than those
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in [19, 20, 21]. What is more, the plaintext signals can be recovered according to the new fixed-time stability theorem, while the plaintext signals can not be recovered according to the existing fixed-time stability theorems. The rest of this article is arranged as follows. Some necessary preliminaries are included in Section 2, and theoretical results are obtained in Section 3. In Section 4, a secure communication scheme is proposed, and numerical simulations are provided to verify the theoretical results. Conclusion is given in
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Section 5.
4
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2. Preliminaries A neural network without time delay can be implemented by a very large
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scale integration (VLSI) circuit as illustrated in Fig.1. By Kirchoff’s current law, the equation of the ith subsystem can be written as follows: n
Ci x˙ i (t) +
xi (t) X fj (xj (t))signij − xi (t) = + Ji , Ri Rij j=1
(1)
i = 1, 2, ..., n, where xi (t) denotes the voltage of the ith capacitor; fj (·) represents the activation function; Ci represents the capacitance of the ith capacitor, and Ci is the input capacitance of fi (xi (t)); Ri represents the resistance of the
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ith parallel-resistor, and Ri is the input resistance of fi (xi (t)); Rij expresses the resistance between fj (xj (t)) and xi (t); Ji denotes external input current; signij = −1 if i = j, otherwise signij = 1. From the perspective of a biological neural network, xi (t) denotes the instantaneous input to the ith neuron; fj (·) is a nonlinear amplifier; fj (xj (t)) denotes the instantaneous output of the jth
refer to Ref.[36].
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neuron. For more details about the circuit of neural network, the readers can
Equation (1) can be rewritten as
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x˙ i (t) = −σi xi (t) +
n X
pij fj (xj (t)) + Ii ,
(2)
j=1
i = 1, 2, ..., n, where σi > 0 denotes the rate of neuron self-inhibition; pij expresses connection weight; Ii denotes external input. σi , pij and Ii are given by n X 1 1 1 signij Ji σi = + , pij = , Ii = . Ci Ri j=1 Rij Ci Rij Ci
(3)
System (2) is called the master system, and this is the corresponding slave
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system:
y˙ i (t) = −σi yi (t) +
n X
pij fj (yj (t)) + Ii + ui (t),
j=1
i = 1, 2, ..., n, where ui (t) denotes controller.
5
(4)
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The synchronization errors are defined as ei (t) = yi (t) − xi (t), i = 1, 2, ..., n.
e˙ i (t) = −σi ei (t) +
n X j=1
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Then error system can be described by pij [fj (yj (t)) − fj (xj (t))] + ui (t),
(5)
i = 1, 2, ..., n. In this paper, let e(t) = (e1 (t), e2 (t), ..., en (t))T .
Assumption 1. Activation functions are Lipschitz continuous, i.e., for ∀µ, ν ∈ R, there exist constants lj > 0 such that
(6)
|fj (µ) − fj (ν)| ≤ lj |µ − ν| , j = 1, 2, ..., n.
Lemma 1 [37]. If α1 , α2 , ..., αn ≥ 0, 0 < ξ ≤ 1, η > 1, we have
i=1
αiξ
≥
n X
αi
i=1
!ξ
n X
n X
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n X
,
αiη
i=1
≥n
1−η
αi
i=1
!η
.
(7)
Definition 1. The origin of system (5) can achieve finite-time stability, if there exists a constant T (e(0)) > 0 such that
lim
t→T (e(0))
ke(t)k1 = 0 and ke(t)k1 = 0
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for ∀t > T (e(0)), where T (e(0)) is called the settling time. Definition 2. The origin of system (5) can achieve fixed-time stability, if two conditions can be satisfied: (i) The origin of system (5) can achieve finite-time stability; (ii) For any e(0), there exists a fixed constant Tmax > 0 such that
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T (e(0)) ≤ Tmax .
Lemma 2 [19]. Suppose V (·) : Rn → R+ ∪ {0} is a continuous radially unbounded function, and the following two conditions hold: (i) V (e(t)) = 0 ⇔ e(t) = 0;
(ii) Any solution e(t) of system (5) satisfies D+ V (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)),
(8)
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for some a, b > 0, 0 < ξ < 1 and η > 1, where D+ V (e(t)) denotes the upper right-hand Dini derivative of V (e(t)). Then the origin of system (5) can achieve fixed-time stability, and 1 Tmax =
1 1 + . a(1 − ξ) b(η − 1) 6
(9)
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Lemma 3 [20]. Suppose V (·) : Rn → R+ ∪ {0} is a continuous radially un(i) V (e(t)) = 0 ⇔ e(t) = 0;
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bounded function, and the following two conditions hold:
(ii) Any solution e(t) of system (5) satisfies
V˙ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)), for some a, b > 0, 0 < ξ < 1 and η > 1.
Then the origin of system (5) can achieve fixed-time stability, and 1−ξ 1 1 a η−ξ 1 2 + Tmax = · . a b 1−ξ η−1
(10)
(11)
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Lemma 4 [21]. Suppose V (·) : Rn → R+ ∪ {0} is a continuous radially unbounded function, and the following two conditions hold: (i) V (e(t)) = 0 ⇔ e(t) = 0;
(ii) Any solution e(t) of system (5) satisfies
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D+ V (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)), for some a, b > 0, ξ = 1 −
1 2β
and η = 1 +
1 2β ,
(12)
where β > 1.
Then the origin of system (5) can achieve fixed-time stability, and (13)
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πβ 3 Tmax =√ . ab
3. Main results
First, by using definite integral, variable substitution and some inequality techniques, we derive a novel fixed-time stability theorem. Theorem 1. Suppose V (·) : Rn → R+ ∪{0} is a continuous radially unbounded function, and the following two conditions hold:
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(i) V (e(t)) = 0 ⇔ e(t) = 0;
(ii) Any solution e(t) of system (5) satisfies V˙ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)) − cV (e(t)), 7
(14)
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for some a, b, c > 0, 0 < ξ < 1 and η > 1. Then the origin of system (5) can achieve fixed-time stability, and 1 c 1 c ln(1 + ) + ln(1 + ). c(1 − ξ) a c(η − 1) b
Proof. Let W (s) = V 1−ξ (s), then we have V˙ (s) =
(15)
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4 Tmax =
1 ˙ W (s)V ξ (s). 1−ξ
(16)
It follows from V˙ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)) − cV (e(t)) that
1 ˙ W (e(t))V ξ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)) − cV (e(t)). 1−ξ 1
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Since V (s) = W 1−ξ (s), we can obtain
˙ (e(t)) ≤ (1 − ξ) −a − bV η−ξ (e(t)) − cV 1−ξ (e(t)) ≤ −a(1 − ξ). W
(17)
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˙ (e(t)) ≤ −a(1 − ξ), there always exists a constant T (e(0)) = Because W > 0 such that
lim
t→T (e(0))
W (e(t)) = 0 and W (e(t)) = 0 for ∀t > T (e(0)).
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W (e(0)) a(1−ξ)
In view of W (e(t)) = 0 ⇔ V (e(t)) = 0 ⇔ e(t) = 0, we can say that there exists a constant T (e(0)) > 0 such that
lim
t→T (e(0))
ke(t)k1 = 0 and ke(t)k1 = 0 for
∀t > T (e(0)). According to Definition 1, the origin of system (5) can achieve finite-time stability.
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Since V˙ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)) − cV (e(t)), it follows that dt 1 ≥ , dV (e(t)) −aV ξ (e(t)) − bV η (e(t)) − cV (e(t))
(19)
then we have
T (e(0)) ≤
Z
0
V (e(0))
1 ds = ξ −as − bsη − cs
Z
V (e(0))
asξ
0
1 ds. + bsη + cs
(20)
Next, two cases are considered separately.
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(i) If 0 ≤ V (e(0)) ≤ 1, T (e(0)) ≤
Z
0
1
1 ds ≤ asξ + bsη + cs
8
Z
0
1
1 ds. asξ + cs
(21)
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Therefore, T (e(0)) ≤ (ii) If V (e(0)) > 1, Z 1 T (e(0)) ≤ 0
1
0
1 c ln(1 + ). c(1 − ξ) a
(23)
Z V (e(0)) 1 1 ds + ds ξ η ξ as + bs + cs as + bsη + cs 1 Z +∞ 1 1 ds + ds. ξ η as + cs bs + cs 1
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≤
Z
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Let z = s1−η , we have dz = (1 − η)s−η ds, thus Z +∞ Z 0 Z 1 1 1 sη 1 1 ds = dz = dz bsη + cs 1 − η 1 bsη + cs η − 1 0 b + cs1−η 1 Z 1 1 1 1 c dz = ln(1 + ). = η − 1 0 b + cz c(η − 1) b Therefore,
T (e(0)) ≤
(22)
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Let w = s1−ξ , we have dw = (1 − ξ)s−ξ ds, thus Z 1 Z 1 Z 1 1 sξ 1 1 1 dw ds = dw = ξ ξ 1 − ξ 0 as + cs 1 − ξ 0 a + cs1−ξ 0 as + cs Z 1 1 1 c 1 = dw = ln(1 + ). 1 − ξ 0 a + cw c(1 − ξ) a
1 c 1 c ln(1 + ) + ln(1 + ). c(1 − ξ) a c(η − 1) b
(24)
(25)
(26)
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According to Definition 2, the origin of system (5) can achieve fixed-time stability, and
4 Tmax =
1 c 1 c ln(1 + ) + ln(1 + ). c(1 − ξ) a c(η − 1) b
Remark 1. As we know, if λ > 0, ln(1 + λ) < λ. Since ln(1 +
c a)
<
c a,
ln(1 +
c b)
<
c b,
c a
> 0,
c b
> 0, we have
then
1 c 1 c 1 1 ln(1 + ) + ln(1 + ) < + . c(1 − ξ) a c(η − 1) b a(1 − ξ) b(η − 1)
(27)
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Therefore, compared with Lemma 2, Theorem 1 can give Tmax a more accurate estimate.
Remark 2. In Lemma 2, Lemma 3 and Theorem 1, ξ and η are only required to 1 satisfy 0 < ξ < 1 and η > 1, while in Lemma 4, ξ and η must satisfy ξ = 1 − 2β
9
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1 2β ,
and η = 1 +
where β > 1. Therefore, in terms of applicability, Lemma 2,
Lemma 3 and Theorem 1 have some advantages over Lemma 4.
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To study the fixed-time synchronization of systems (2) and (4), we design the following controller: ξ
η
ui (t) = −ki1 ei (t) − ki2 sign(ei (t)) |ei (t)| − ki3 sign(ei (t)) |ei (t)| , i = 1, 2, ..., n,
(28)
where 0 < ξ < 1, η > 1, positive constants ki1 , ki2 , ki3 are control gains, i = 1, 2, ..., n.
Theorem 2. If Assumption 1 holds and ki1 >
n P
j=1
|pij |lj +|pji |li −σi , 2
i = 1, 2, ..., n,
Furthermore, 5 Tmax =
c 2 c 2 ln(1 + ) + ln(1 + ), c(1 − ξ) a c(η − 1) b
ξ+1 2
i
|pji | li ) + 2σi }.
, b = min{ki3 } · n i
1−η 2
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where a = min{ki2 } · 2
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systems (2) and (4) can achieve fixed-time synchronization under controller (28).
·2
η+1 2
, c = min{2ki1 − i
(29) n P
(|pij | lj +
j=1
Proof. Choose the following Lyapunov function: n
1 T 1X e (t)e(t) = (ei (t))2 . 2 2 i=1
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V (e(t)) =
(30)
We calculate the derivative of V (e(t)) along system (5): V˙ (e(t)) = =
n X
i=1 n X i=1
ei (t)e˙ i (t) n
ei (t) − σi ei (t) +
n X j=1
o (31) pij [fj (yj (t)) − fj (xj (t))] + ui (t) .
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It follows from Assumption 1 that |fj (yj (t)) − fj (xj (t))| ≤ lj |yj (t) − xj (t)| = lj |ej (t)| ,
10
(32)
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then we have
≤
n X
ei (t)
i=1 j=1 n X n X i=1 j=1
|pij | lj
(ej (t))2 (ei (t))2 + = 2 2
i=1
ei (t)ui (t) = −
n X i=1
ki1 (ei (t))2 −
n X i=1
|ei (t)|
i=1 n X n X i=1 j=1
It is obvious that n X
n X
pij [fj (yj (t)) − fj (xj (t))] ≤
n X j=1
|pij | · lj |ej (t)|
|pij | lj + |pji | li (ei (t))2 . 2
ξ+1
ki2 |ei (t)|
Therefore,
−
n X i=1
η+1
ki3 |ei (t)|
If ki1 >
n P
j=1
i=1
|pij |lj +|pji |li 2
V˙ (e(t)) ≤ −
n X i=1
where ρi = ki1 −
j=1
−
i=1
η+1
ki3 |ei (t)|
.
(34)
(35)
.
− σi , i = 1, 2, ..., n, we can obtain that
ρi (ei (t))2 −
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n P
ξ+1
ki2 |ei (t)|
n X
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−
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n n X X |p | l + |p | l ij j ji i − σi − ki1 (ei (t))2 V˙ (e(t)) ≤ 2 i=1 j=1 n X
(33)
pro of
n X
|pij |lj +|pji |li 2
n X i=1
ξ+1
ki2 |ei (t)|
−
n X i=1
η+1
ki3 |ei (t)|
,
(36)
+ σi > 0.
Let ρ = min{ρi }, k2 = min{ki2 }, k3 = min{ki3 }, we have i
V˙ (e(t)) ≤ −ρ ξ+1 2
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Since 0 <
i
n X i=1
(ei (t))2 − k2
< 1 and
η+1 2
i
n X
2
(|ei (t)| )
i=1
ξ+1 2
− k3
n X
2
(|ei (t)| )
η+1 2
.
(37)
i=1
> 1, we can derive from Lemma 1 that
n n n n X X X 1−η X 2 ξ+1 2 ξ+1 2 η+1 2 η+1 (|ei (t)| ) 2 ≥ ( |ei (t)| ) 2 , (|ei (t)| ) 2 ≥ n 2 ( |ei (t)| ) 2 . i=1
i=1
i=1
i=1
(38)
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Then we can obtain that n n n X X 1−η X 2 η+1 2 ξ+1 V˙ (e(t)) ≤ −ρ (ei (t))2 − k2 ( |ei (t)| ) 2 |ei (t)| ) 2 − k3 · n 2 ( i=1
ξ+1 2
, b = k3 · n
ξ+1 2
1−η 2
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= −2ρ · V (e(t)) − k2 · 2 Let a = k2 · 2
i=1
i=1
(V (e(t)))
·2
η+1 2
ξ+1 2
− k3 · n
1−η 2
·2
η+1 2
(V (e(t)))
η+1 2
. (39)
, c = 2ρ, we have
η+1 ξ+1 V˙ (e(t)) ≤ −a(V (e(t))) 2 − b(V (e(t))) 2 − cV (e(t)).
(40)
According to Theorem 1, the origin of system (5) can achieve fixed-time stability. Moreover,
1 c 1 ln(1 + ) + η+1 ln(1 + ξ+1 a c( 2 − 1) c(1 − 2 ) 2 c 2 c = ln(1 + ) + ln(1 + ). c(1 − ξ) a c(η − 1) b
re-
5 Tmax =
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Corollary 1. If Assumption 1 holds and ki1 >
n P
j=1
c ) b
|pij |lj +|pji |li 2
− σi , i =
1, 2, ..., n, systems (2) and (4) can achieve fixed-time synchronization under controller (28). Furthermore,
6 Tmax =
ξ+1 2
, b = min{ki3 } · n
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where a = min{ki2 } · 2
2 2 + , a(1 − ξ) b(η − 1)
i
1−η 2
i
·2
η+1 2
(41)
.
Proof. Similarly, it can be proved that ξ+1 η+1 V˙ (e(t)) ≤ −a(V (e(t))) 2 − b(V (e(t))) 2 − cV (e(t)),
where a = min{ki2 } · 2 i
ξ+1 2
, b = min{ki3 } · n
1−η 2
i
·2
η+1 2
, c = min{2ki1 − i
|pji | li ) + 2σi }.
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Since the Lyapunov function V (e(t)) = D+ V (e(t)) = V˙ (e(t)). Therefore, D+ V (e(t)) ≤ −a(V (e(t))) ≤ −a(V (e(t)))
ξ+1 2 ξ+1 2
12
1 2
n P
(42) n P
(|pij | lj +
j=1
(ei (t))2 is derivable, we have
i=1
− b(V (e(t))) − b(V (e(t)))
η+1 2 η+1 2
− cV (e(t)) .
(43)
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According to Lemma 2, the origin of system (5) can achieve fixed-time stability. Moreover, 2 2 + . a(1 − ξ) b(η − 1)
pro of
6 Tmax =
Corollary 2. If Assumption 1 holds and ki1 >
n P
j=1
|pij |lj +|pji |li 2
− σi , i =
1, 2, ..., n, systems (2) and (4) can achieve fixed-time synchronization under controller (28). Furthermore,
1−ξ 1 a η−ξ 2 2 = · + , a b 1−ξ η−1
7 Tmax ξ+1 2
, b = min{ki3 } · n
1−η 2
i
i
·2
η+1 2
.
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where a = min{ki2 } · 2
(44)
Proof. Similarly, it can be proved that
ξ+1 η+1 V˙ (e(t)) ≤ −a(V (e(t))) 2 − b(V (e(t))) 2 ,
where a = min{ki2 } · 2
ξ+1 2
, b = min{ki3 } · n i
1−η 2
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i
·2
η+1 2
(45)
.
According to Lemma 3, the origin of system (5) can achieve fixed-time stability. Moreover,
1−ξ 2 2 1 a η−ξ · + . a b 1−ξ η−1
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7 Tmax =
Next consider the following controller: ξ
η
ui (t) = −ki1 ei (t) − ki2 sign(ei (t)) |ei (t)| − ki3 sign(ei (t)) |ei (t)| , i = 1, 2, ..., n, where positive constants
ki1 ,
and η = 1 + β1 , β > 1.
ki2 ,
(46)
ki3 ,
i = 1, 2, ..., n are control gains, ξ = 1 −
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Corollary 3. If Assumption 1 holds and ki1 >
n P
j=1
|pij |lj +|pji |li 2
1 β
− σi , i =
1, 2, ..., n, systems (2) and (4) can achieve fixed-time synchronization under controller (46). Furthermore, πβ 8 Tmax =√ , ab 13
(47)
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where a = min{ki2 } · 2
ξ+1 2
i
, b = min{ki3 } · n
1−η 2
i
·2
η+1 2
.
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Proof. Similarly, it can be proved that ξ+1 η+1 V˙ (e(t)) ≤ −a(V (e(t))) 2 − b(V (e(t))) 2 ,
where a = min{ki2 } · 2 i
Since
ξ+1 2
=1−
ξ+1 2
, b = min{ki3 } · n
1−η 2
i
η+1 1 2β , 2
=1+
1 2β ,
·2
η+1 2
(48)
.
we have
1
1
D+ V (e(t)) = V˙ (e(t)) ≤ −a(V (e(t)))1− 2β − b(V (e(t)))1+ 2β , where β > 1.
(49)
bility. Moreover,
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According to Lemma 4, the origin of system (5) can achieve fixed-time staπβ 8 Tmax =√ . ab
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4. Numerical simulations
In this section, we propose a secure communication scheme based on the fixed-time synchronization of neural networks, and numerical simulations are
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provided to verify the obtained theoretical results. Example 1. Sender generates the following neural network (master system): x˙ i (t) = −σi xi (t) +
2 X j=1
pij fj (xj (t)) + Ii , t ≥ 0,
(50)
i = 1, 2, where σ1 = 0.9, σ2 = 1.1, p11 = 2, p12 = −0.1, p21 = −5, p22 = 4.5, I1 = sin t, I2 = cos t. Without loss of generality, suppose pij , i, j = 1, 2 are the secret keys in this secure communication scheme, then sender transmits pij ,
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i, j = 1, 2 to receiver via transmission channel. Let f1 (α) = f2 (α) = 0.5(|α + 1| − |α − 1|), then we have l1 = l2 = 1. The
initial values of system (50) are x1 (ν) = 3, x2 (ν) = −2, ν ∈ [−1, 0]. The state trajectories of x1 (t) and x2 (t) are shown in Fig.2. 14
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10 x1(t) x2(t)
1
2
0 -5 -10 -15 0
4
8
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x (t), x (t)
5
12
t/s
16
20
Fig. 2. The state trajectories of x1 (t) and x2 (t), which are generated by system
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(50). The initial values of system (50) are x1 (ν) = 3, x2 (ν) = −2, ν ∈ [−1, 0]. By using plaintext signals m1 (t) = 1.6 sin t + 0.7 cos(0.3t) and m2 (t) =
(51)
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− sin(1.2t) + 2 cos(3t), sender generates the following signals: ri (t), 0 ≤ t < 3, Mi (t) = mi (t − 3), t ≥ 3,
i = 1, 2, where ri (t) denotes random signal. Sender calculates the encrypted signals Ci (t) = Mi (t) + xi (t), i = 1, 2, and transmits Ci (t), i = 1, 2 to receiver via transmission channel. Figs.3-5 illustrate the state trajectories of mi (t), Mi (t)
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and Ci (t), i = 1, 2. It should be pointed out that, the initial values of system (50), ri (t), mi (t) and Mi (t) can only be known by sender, i = 1, 2. Receiver receives the secret keys pij , i, j = 1, 2, and the encrypted signals Ci (t), i = 1, 2. Then receiver generates the following slave system: y˙ i (t) = −σi yi (t) +
2 X j=1
pij fj (yj (t)) + Ii + ui (t), t ≥ 0,
(52)
i = 1, 2, where ui (t) denotes controller. The initial values of system (52) are
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y1 (ν) = −1, y2 (ν) = 3, ν ∈ [−1, 0].
Receiver chooses k11 = 6, k21 = 8, k12 = k22 = 1, k13 = k23 = 1, ξ = 0.5 and 2 P |pij |lj +|pji |li η = 1.5. We can verify that ki1 satisfies ki1 > − σi , i = 1, 2. 2 j=1
15
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6 m2(t)
3 m1(t), m2(t)
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m1(t)
0
-3
-6 0
4
8
12
t/s
16
20
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Fig. 3. The state trajectories of plaintext signals m1 (t) = 1.6 sin t + 0.7 cos(0.3t)
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and m2 (t) = − sin(1.2t) + 2 cos(3t).
6
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M1(t), M2(t)
3
M1(t) M2(t)
0
-3
-6
0
4
8
12
16
20
t/s
Fig. 4. The state trajectories of M1 (t) and M2 (t). If 0 ≤ t < 3, Mi (t) =
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ri (t), i = 1, 2, where ri (t) denotes random signal; if t ≥ 3, Mi (t) = mi (t − 3), i = 1, 2.
16
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10 C1(t) C2(t)
0 -5 -10 -15 0
4
8
pro of
C 1(t), C 2(t)
5
12
t/s
16
20
Fig. 5. The state trajectories of the encrypted signals C1 (t) and C2 (t), where
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Ci (t) = Mi (t) + xi (t), i = 1, 2.
It can be calculated that a = 20.75 , b = 2, c = 3.7. According to Theorem 2, systems (50) and (52) can realize fixed-time synchronization under controller 5 (28), and Tmax ≈ 2.3897.
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5 ≈ 2.3897 < 3, we have xi (t) = yi (t), i = 1, 2, t ≥ 3. Receiver can Since Tmax
decrypt the encrypted signals by calculating m ˆ i (t) = Ci (t+3)−yi (t+3), i = 1, 2, t ≥ 0, and it is obvious that
m ˆ i (t) = Ci (t + 3) − yi (t + 3) = Mi (t + 3) + xi (t + 3) − yi (t + 3)
(53)
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= Mi (t + 3) = mi (t), t ≥ 0.
Fig.6 is the schematic diagram of the proposed secure communication scheme. According to Corollary 1 and Corollary 2, systems (50) and (52) can re6 alize fixed-time synchronization under controller (28), and Tmax ≈ 4.3784,
7 Tmax ≈ 4.3620. In fact, systems (50) and (52) can also realize fixed-time
synchronization under controller (46) according to Corollary 3 (β = 2), and
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8 6 7 8 Tmax ≈ 3.4259. Obviously, Tmax , Tmax and Tmax are larger than 3, so it seems
that xi (t) = yi (t), i = 1, 2, t ≥ 3 is not necessarily true. Therefore, according to Corollary 1, Corollary 2 and Corollary 3, plaintext signals can’t be recovered successfully. However, according to Theorem 2, plaintext signals can be 17
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Plaintext signal
mi t
The recovered plaintext signal
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Prefixed by random signal
mˆ i t
M i t Ci t Encryption
Transmission channel
xi t
Decryption
yi t
Master system
Fixed-time synchronization control
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Slave system
Fig. 6. The schematic diagram of the proposed secure communication scheme.
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recovered successfully.
Fig.7 and Fig.8 illustrate the state trajectories of yi (t) and ei (t) under controller (28) (controller (46)), i = 1, 2. The state trajectories of −e1 (t) and e2 (t) under controller (28) (controller (46)) are illustrated in Fig.9, which is presented
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with a log axis on the y-axis.
Remark 3. In this secure communication scheme, systems (50) and (52) must realize fixed-time synchronization, and Tmax should satisfy Tmax < 3. If systems (50) and (52) only realize asymptotic synchronization, synchronization will be reached when time tends to infinity, so xi (t) = yi (t), i = 1, 2, t ≥ 3 can’t be satisfied, that means plaintext signals can’t be recovered successfully. If systems (50) and (52) only realize finite-time synchronization, synchronization will be reached within the settling time, which depends on the initial values of systems
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(50) and (52). However, since receiver don’t know the initial values of system (50), the settling time will be impossible to be estimated, so xi (t) = yi (t), i = 1, 2, t ≥ 3 is not necessarily true, that means plaintext signals can’t be recovered successfully.
18
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10 y1(t)
y1(t), y2(t)
0 -5 -10 -15 0
4
8
pro of
y2(t)
5
12
t/s
16
20
Fig. 7. Under controller (28) (controller (46)), the state trajectories of y1 (t)
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and y2 (t), which are generated by system (52). The initial values of system (52) are y1 (ν) = −1, y2 (ν) = 3, ν ∈ [−1, 0]. The controller parameters are k11 = 6,
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k21 = 8, k12 = k22 = 1, k13 = k23 = 1, ξ = 0.5 and η = 1.5.
6
2
e (t), e (t)
3
e2(t)
1
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0
e1(t)
-3
T5max
-6
0
1
T7max
T8max 0
2
0
3
T6max 00
4
5
t/s
Fig. 8. The state trajectories of the synchronization errors e1 (t) and e2 (t) under i controller (28) (controller (46)), where ei (t) = yi (t) − xi (t), i = 1, 2. Tmax
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denotes the upper bound estimate for the settling time, i = 5, 6, 7, 8, where
5 6 7 Tmax is derived by Theorem 2, Tmax is derived by Corollary 1, Tmax is derived 8 is derived by Corollary 3. by Corollary 2, Tmax
19
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10 1 - e 1(t) e2(t)
10 -2
10 -4
T5max
10 -6
T8max
0
0
1
pro of
-e1(t), e 2(t)
10 0
2
T7max
0
3
t/s
T6max
00
4
5
Fig. 9. The state trajectories of −e1 (t) and e2 (t) under controller (28) (controller
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(46)), where the y-axis is presented with a log axis.
5 6 7 8 . Table 1. The comparisons among Tmax , Tmax , Tmax and Tmax 6 Tmax
7 Tmax
8 Tmax
2.3897
4.3784
4.3620
3.4259
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Example 1
5 Tmax
Remark 4. Any upper bound estimate formula for the settling time is derived by strict theoretical analysis. Due to the difficulty and conservativeness of strict theoretical analysis, the upper bound estimate for the settling time will
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be larger than the synchronization time illustrated in numerical simulations (see Fig.8 and Fig.9). Example 1 shows that, with the same controller parameters, 6 7 8 5 (see Table 1, Fig.8 and Fig.9). and Tmax Tmax is much smaller than Tmax , Tmax 6 5 is derived by As we know, Tmax is derived by Theorem 2 and Theorem 1, Tmax 7 8 Corollary 1 and Lemma 2, Tmax is derived by Corollary 2 and Lemma 3, Tmax
is derived by Corollary 3 and Lemma 4. Therefore, compared with Lemmas 2, 3, 4, Theorem 1 provides a tighter upper estimate formula for the settling time.
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Remark 5. In Example 1, if we choose some larger k11 , k21 , k31 , and keep the 5 other controller parameters unchanged, then the obtained Tmax can be smaller. 6 7 8 However, Tmax , Tmax and Tmax still remain unchanged in this case.
20
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5. Conclusions
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In this paper, we give a novel fixed-time stability theorem, and the corresponding stability criterion, proof, upper bound estimate formula for the settling time are all different from those in the existing fixed-time stability theorems. By utilizing the new fixed-time stability theorem, the fixed-time synchronization of neural networks is studied via feedback control. Finally, a secure communication scheme based on the fixed-time synchronization of neural networks is proposed. Numerical simulations illustrate that, compared with the existing fixed-time stability theorems, the new fixed-time stability theorem can provide
re-
a tighter upper bound estimate formula for the settling time. What is more, the plaintext signals can be recovered according to the new fixed-time stability theorem, while the plaintext signals can not be recovered according to the existing fixed-time stability theorems. Because the new fixed-time stability theorem has a tighter upper bound estimate formula for the settling time, it not only can be
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used to design secure communication schemes with lower communication cost (see Example 1), but also can be used to realize the fast cooperative control of intelligent robots. In the future, we will study the application of the new
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fixed-time stability theorem in the fast cooperative control of intelligent robots.
Acknowledgements
The work is supported by the National Key R&D Program of China (Grant Nos. 2016YFB0800604 and 2016YFB0800602), the National Natural Science Foundation of China (Grant Nos. 61972051, 61771071, 11771196, 61872203 and 61802212), and the Shandong Province Natural Science Foundation (Grant No.
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ZR2018BF023).
21
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References
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Conflict of interest
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The authors declare no conflict of interest concerning the publication of this manuscript.
PII: DOI: Reference:
S0893-6080(19)30430-7 https://doi.org/10.1016/j.neunet.2019.12.028 NN 4360
To appear in:
Neural Networks
Received date : 16 April 2019 Revised date : 1 November 2019 Accepted date : 27 December 2019 Please cite this article as: C. Chen, L. Li, H. Peng et al., A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks. Neural Networks (2020), doi: https://doi.org/10.1016/j.neunet.2019.12.028. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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.
© 2020 Elsevier Ltd. All rights reserved.
Journal Pre-proof A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks First author:
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Chuan Chen 1. School of Cyber Security, Qilu University of Technology, Jinan 250353, China 2. Shandong Provincial Key Laboratory of Computer Networks, Jinan, 250353, China [email protected]
Corresponding author:
Chuan Chen 1. School of Cyber Security, Qilu University of Technology, Jinan 250353, China 2. Shandong Provincial Key Laboratory of Computer Networks, Jinan, 250353, China [email protected]
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Lixiang Li Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [email protected]
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Ling Mi School of Mathematics and Statistics, Qilu University of Technology, Jinan 250353, China [email protected]
Other authors:
Haipeng Peng, Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [email protected]
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Yixian Yang Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [email protected] Hui Zhao
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School of Information Science and Engineering, University of Jinan, Jinan 250022, China [email protected]
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Manuscript
Click here to view linked References
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A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks Chuan Chena,b,∗, Lixiang Lic,∗, Haipeng Pengc , Yixian Yangc , Ling Mid,∗, Hui Zhaoe a School
of Cyber Security, Qilu University of Technology, Jinan 250353, China Provincial Key Laboratory of Computer Networks, Jinan, 250353, China c Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China d School of Mathematics and Statistics, Qilu University of Technology, Jinan 250353, China e School of Information Science and Engineering, University of Jinan, Jinan 250022, China b Shandong
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Abstract
In this paper, we derive a new fixed-time stability theorem based on definite integral, variable substitution and some inequality techniques. The fixed-time stability criterion and the upper bound estimate formula for the settling time
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are different from those in the existing fixed-time stability theorems. Based on the new fixed-time stability theorem, the fixed-time synchronization of neural networks is investigated by designing feedback controller, and sufficient conditions are derived to guarantee the fixed-time synchronization of neural networks. To show the usability and superiority of the obtained theoretical results, we pro-
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pose a secure communication scheme based on the fixed-time synchronization of neural networks. Numerical simulations illustrate that the new upper bound estimate formula for the settling time is much tighter than those in the existing fixed-time stability theorems. What is more, the plaintext signals can be recovered according to the new fixed-time stability theorem, while the plaintext signals can not be recovered according to the existing fixed-time stability theorems.
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Keywords: Fixed-time stability, Fixed-time synchronization, Neural networks
∗ Corresponding
author Email addresses: [email protected] (Chuan Chen), li [email protected] (Lixiang Li), [email protected] (Ling Mi)
Preprint submitted to Journal of LATEX Templates
November 1, 2019
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1. Introduction Stability [1, 2, 3] is an important dynamical behavior of nonlinear system,
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and the stability analysis of nonlinear systems has been paid much attention in recent years [4, 5, 6, 7]. Especially, asymptotic stability [8, 9, 10, 11, 12] and exponential stability [13, 14, 15] have been investigated intensively. Asymptotic stability and exponential stability belong to infinite-time stability, which means stability can only be reached when time tends to infinity. Compared with asymptotic stability and exponential stability, finite-time stability [16, 17, 18] has an obvious advantage in practical applications. For finite-time stability, stability can be reached within a finite time (the settling time), which can be
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estimated based on the initial value of the studied system. However, if the accurate initial value of the studied system is unknown beforehand, the settling time can’t be estimated.
In 2012, fixed-time stability was put forward [19]. For fixed-time stability
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[19, 20, 21], stability can also be reached within the settling time. What is more, the settling time has a fixed upper bound for any initial value of the studied system. Although we know the settling time of fixed-time stability has a fixed upper bound, deriving the tightest upper bound estimate formula is impossible due to the difficulty of theoretical analysis. However, referring to the upper
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bound estimate formulas in some existing fixed-time stability theorems, can we derive a tighter upper bound estimate formula for the settling time? Considering the current over-dependence on the existing fixed-time stability theorems in [19, 20, 21], developing a new fixed-time stability theorem with a tighter upper bound estimate formula for the settling time will be very meaningful. On the other hand, since the synchronization problem of nonlinear systems can be converted into the stability problem of error system, some fixed-time
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stability theorems have been used to study the fixed-time synchronization of nonlinear systems [20, 22, 23, 24, 25, 26, 27, 28, 29]. In [24], the fixed-time synchronization of complex-valued neural networks was studied. By designing non-chattering controllers, the fixed-time synchronization of complex networks
2
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with impulsive effects was considered in [29]. As we know, the stability of neural network has important applications in
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associative memory [30, 31, 32] and optimization [33]. Take associative memory for example, memory mode is designed as a stable state (i.e. attractor) of neural network. A neural network can have more than one attractor, and the memory capacity of neural network is the number of attractors. Each attractor has an attraction basin, which is the domain that the attractor can attract. If a given input belongs to the attraction basin of an attractor, neural network can stabilize to this attractor, that is, the stored memory mode is recalled. Meanwhile, the synchronization of neural networks also has important applications in many
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fields, such as secure communication [34], image encryption [35], the cooperative control of intelligent robots, etc. If synchronization time and robustness are taken into account, we must consider the fixed-time synchronization of neural networks.
In this paper, we establish a novel fixed-time stability theorem. The fixed-
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time stability criterion and the upper bound estimate formula for the settling time are different from those in [19, 20, 21]. Theoretical analysis shows that the new upper bound estimate formula for the settling time in this paper is tighter than that in [19]. In the proof of the new fixed-time stability theorem,
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we depend heavily on definite integral, variable substitution and some inequality techniques. The proof of the new fixed-time stability theorem is a strict mathematical proof, and it is very general. By adopting the proof techniques in the new fixed-time stability theorem, the existing finite/fixed-time stability theorems can also be given strict mathematical proofs. To demonstrate the usability of the new fixed-time stability theorem, we investigate the fixed-time synchronization of neural networks by means of the new fixed-time stability theorem. In fact, the new fixed-time stability theorem and the analysis techniques
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in this paper can also be used to study the fixed-time synchronization/stability of other nonlinear systems. To show the application of the obtained theoretical results, we propose a secure communication scheme based on the fixed-time synchronization of neural networks. Numerical simulations illustrate that the new 3
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R ij ,i,j=1,2,...,n
... ...
R1n
R 2n
...
...
...
R12
R 22 R 21 R11
Rn2
...
R n1
Cn
...
x2 t
xn t
R2
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R1
...
f 2
...
Rn
f n
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f1
R nn
C2
C1
x1 t
Jn
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J2
J1
Fig. 1. The circuit implementation of a neural network without time delay. upper bound estimate formula for the settling time is much tighter than those
urn a
in [19, 20, 21]. What is more, the plaintext signals can be recovered according to the new fixed-time stability theorem, while the plaintext signals can not be recovered according to the existing fixed-time stability theorems. The rest of this article is arranged as follows. Some necessary preliminaries are included in Section 2, and theoretical results are obtained in Section 3. In Section 4, a secure communication scheme is proposed, and numerical simulations are provided to verify the theoretical results. Conclusion is given in
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Section 5.
4
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2. Preliminaries A neural network without time delay can be implemented by a very large
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scale integration (VLSI) circuit as illustrated in Fig.1. By Kirchoff’s current law, the equation of the ith subsystem can be written as follows: n
Ci x˙ i (t) +
xi (t) X fj (xj (t))signij − xi (t) = + Ji , Ri Rij j=1
(1)
i = 1, 2, ..., n, where xi (t) denotes the voltage of the ith capacitor; fj (·) represents the activation function; Ci represents the capacitance of the ith capacitor, and Ci is the input capacitance of fi (xi (t)); Ri represents the resistance of the
re-
ith parallel-resistor, and Ri is the input resistance of fi (xi (t)); Rij expresses the resistance between fj (xj (t)) and xi (t); Ji denotes external input current; signij = −1 if i = j, otherwise signij = 1. From the perspective of a biological neural network, xi (t) denotes the instantaneous input to the ith neuron; fj (·) is a nonlinear amplifier; fj (xj (t)) denotes the instantaneous output of the jth
refer to Ref.[36].
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neuron. For more details about the circuit of neural network, the readers can
Equation (1) can be rewritten as
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x˙ i (t) = −σi xi (t) +
n X
pij fj (xj (t)) + Ii ,
(2)
j=1
i = 1, 2, ..., n, where σi > 0 denotes the rate of neuron self-inhibition; pij expresses connection weight; Ii denotes external input. σi , pij and Ii are given by n X 1 1 1 signij Ji σi = + , pij = , Ii = . Ci Ri j=1 Rij Ci Rij Ci
(3)
System (2) is called the master system, and this is the corresponding slave
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system:
y˙ i (t) = −σi yi (t) +
n X
pij fj (yj (t)) + Ii + ui (t),
j=1
i = 1, 2, ..., n, where ui (t) denotes controller.
5
(4)
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The synchronization errors are defined as ei (t) = yi (t) − xi (t), i = 1, 2, ..., n.
e˙ i (t) = −σi ei (t) +
n X j=1
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Then error system can be described by pij [fj (yj (t)) − fj (xj (t))] + ui (t),
(5)
i = 1, 2, ..., n. In this paper, let e(t) = (e1 (t), e2 (t), ..., en (t))T .
Assumption 1. Activation functions are Lipschitz continuous, i.e., for ∀µ, ν ∈ R, there exist constants lj > 0 such that
(6)
|fj (µ) − fj (ν)| ≤ lj |µ − ν| , j = 1, 2, ..., n.
Lemma 1 [37]. If α1 , α2 , ..., αn ≥ 0, 0 < ξ ≤ 1, η > 1, we have
i=1
αiξ
≥
n X
αi
i=1
!ξ
n X
n X
re-
n X
,
αiη
i=1
≥n
1−η
αi
i=1
!η
.
(7)
Definition 1. The origin of system (5) can achieve finite-time stability, if there exists a constant T (e(0)) > 0 such that
lim
t→T (e(0))
ke(t)k1 = 0 and ke(t)k1 = 0
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for ∀t > T (e(0)), where T (e(0)) is called the settling time. Definition 2. The origin of system (5) can achieve fixed-time stability, if two conditions can be satisfied: (i) The origin of system (5) can achieve finite-time stability; (ii) For any e(0), there exists a fixed constant Tmax > 0 such that
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T (e(0)) ≤ Tmax .
Lemma 2 [19]. Suppose V (·) : Rn → R+ ∪ {0} is a continuous radially unbounded function, and the following two conditions hold: (i) V (e(t)) = 0 ⇔ e(t) = 0;
(ii) Any solution e(t) of system (5) satisfies D+ V (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)),
(8)
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for some a, b > 0, 0 < ξ < 1 and η > 1, where D+ V (e(t)) denotes the upper right-hand Dini derivative of V (e(t)). Then the origin of system (5) can achieve fixed-time stability, and 1 Tmax =
1 1 + . a(1 − ξ) b(η − 1) 6
(9)
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Lemma 3 [20]. Suppose V (·) : Rn → R+ ∪ {0} is a continuous radially un(i) V (e(t)) = 0 ⇔ e(t) = 0;
pro of
bounded function, and the following two conditions hold:
(ii) Any solution e(t) of system (5) satisfies
V˙ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)), for some a, b > 0, 0 < ξ < 1 and η > 1.
Then the origin of system (5) can achieve fixed-time stability, and 1−ξ 1 1 a η−ξ 1 2 + Tmax = · . a b 1−ξ η−1
(10)
(11)
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Lemma 4 [21]. Suppose V (·) : Rn → R+ ∪ {0} is a continuous radially unbounded function, and the following two conditions hold: (i) V (e(t)) = 0 ⇔ e(t) = 0;
(ii) Any solution e(t) of system (5) satisfies
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D+ V (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)), for some a, b > 0, ξ = 1 −
1 2β
and η = 1 +
1 2β ,
(12)
where β > 1.
Then the origin of system (5) can achieve fixed-time stability, and (13)
urn a
πβ 3 Tmax =√ . ab
3. Main results
First, by using definite integral, variable substitution and some inequality techniques, we derive a novel fixed-time stability theorem. Theorem 1. Suppose V (·) : Rn → R+ ∪{0} is a continuous radially unbounded function, and the following two conditions hold:
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(i) V (e(t)) = 0 ⇔ e(t) = 0;
(ii) Any solution e(t) of system (5) satisfies V˙ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)) − cV (e(t)), 7
(14)
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for some a, b, c > 0, 0 < ξ < 1 and η > 1. Then the origin of system (5) can achieve fixed-time stability, and 1 c 1 c ln(1 + ) + ln(1 + ). c(1 − ξ) a c(η − 1) b
Proof. Let W (s) = V 1−ξ (s), then we have V˙ (s) =
(15)
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4 Tmax =
1 ˙ W (s)V ξ (s). 1−ξ
(16)
It follows from V˙ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)) − cV (e(t)) that
1 ˙ W (e(t))V ξ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)) − cV (e(t)). 1−ξ 1
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Since V (s) = W 1−ξ (s), we can obtain
˙ (e(t)) ≤ (1 − ξ) −a − bV η−ξ (e(t)) − cV 1−ξ (e(t)) ≤ −a(1 − ξ). W
(17)
(18)
˙ (e(t)) ≤ −a(1 − ξ), there always exists a constant T (e(0)) = Because W > 0 such that
lim
t→T (e(0))
W (e(t)) = 0 and W (e(t)) = 0 for ∀t > T (e(0)).
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W (e(0)) a(1−ξ)
In view of W (e(t)) = 0 ⇔ V (e(t)) = 0 ⇔ e(t) = 0, we can say that there exists a constant T (e(0)) > 0 such that
lim
t→T (e(0))
ke(t)k1 = 0 and ke(t)k1 = 0 for
∀t > T (e(0)). According to Definition 1, the origin of system (5) can achieve finite-time stability.
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Since V˙ (e(t)) ≤ −aV ξ (e(t)) − bV η (e(t)) − cV (e(t)), it follows that dt 1 ≥ , dV (e(t)) −aV ξ (e(t)) − bV η (e(t)) − cV (e(t))
(19)
then we have
T (e(0)) ≤
Z
0
V (e(0))
1 ds = ξ −as − bsη − cs
Z
V (e(0))
asξ
0
1 ds. + bsη + cs
(20)
Next, two cases are considered separately.
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(i) If 0 ≤ V (e(0)) ≤ 1, T (e(0)) ≤
Z
0
1
1 ds ≤ asξ + bsη + cs
8
Z
0
1
1 ds. asξ + cs
(21)
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Therefore, T (e(0)) ≤ (ii) If V (e(0)) > 1, Z 1 T (e(0)) ≤ 0
1
0
1 c ln(1 + ). c(1 − ξ) a
(23)
Z V (e(0)) 1 1 ds + ds ξ η ξ as + bs + cs as + bsη + cs 1 Z +∞ 1 1 ds + ds. ξ η as + cs bs + cs 1
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≤
Z
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Let z = s1−η , we have dz = (1 − η)s−η ds, thus Z +∞ Z 0 Z 1 1 1 sη 1 1 ds = dz = dz bsη + cs 1 − η 1 bsη + cs η − 1 0 b + cs1−η 1 Z 1 1 1 1 c dz = ln(1 + ). = η − 1 0 b + cz c(η − 1) b Therefore,
T (e(0)) ≤
(22)
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Let w = s1−ξ , we have dw = (1 − ξ)s−ξ ds, thus Z 1 Z 1 Z 1 1 sξ 1 1 1 dw ds = dw = ξ ξ 1 − ξ 0 as + cs 1 − ξ 0 a + cs1−ξ 0 as + cs Z 1 1 1 c 1 = dw = ln(1 + ). 1 − ξ 0 a + cw c(1 − ξ) a
1 c 1 c ln(1 + ) + ln(1 + ). c(1 − ξ) a c(η − 1) b
(24)
(25)
(26)
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According to Definition 2, the origin of system (5) can achieve fixed-time stability, and
4 Tmax =
1 c 1 c ln(1 + ) + ln(1 + ). c(1 − ξ) a c(η − 1) b
Remark 1. As we know, if λ > 0, ln(1 + λ) < λ. Since ln(1 +
c a)
<
c a,
ln(1 +
c b)
<
c b,
c a
> 0,
c b
> 0, we have
then
1 c 1 c 1 1 ln(1 + ) + ln(1 + ) < + . c(1 − ξ) a c(η − 1) b a(1 − ξ) b(η − 1)
(27)
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Therefore, compared with Lemma 2, Theorem 1 can give Tmax a more accurate estimate.
Remark 2. In Lemma 2, Lemma 3 and Theorem 1, ξ and η are only required to 1 satisfy 0 < ξ < 1 and η > 1, while in Lemma 4, ξ and η must satisfy ξ = 1 − 2β
9
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1 2β ,
and η = 1 +
where β > 1. Therefore, in terms of applicability, Lemma 2,
Lemma 3 and Theorem 1 have some advantages over Lemma 4.
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To study the fixed-time synchronization of systems (2) and (4), we design the following controller: ξ
η
ui (t) = −ki1 ei (t) − ki2 sign(ei (t)) |ei (t)| − ki3 sign(ei (t)) |ei (t)| , i = 1, 2, ..., n,
(28)
where 0 < ξ < 1, η > 1, positive constants ki1 , ki2 , ki3 are control gains, i = 1, 2, ..., n.
Theorem 2. If Assumption 1 holds and ki1 >
n P
j=1
|pij |lj +|pji |li −σi , 2
i = 1, 2, ..., n,
Furthermore, 5 Tmax =
c 2 c 2 ln(1 + ) + ln(1 + ), c(1 − ξ) a c(η − 1) b
ξ+1 2
i
|pji | li ) + 2σi }.
, b = min{ki3 } · n i
1−η 2
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where a = min{ki2 } · 2
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systems (2) and (4) can achieve fixed-time synchronization under controller (28).
·2
η+1 2
, c = min{2ki1 − i
(29) n P
(|pij | lj +
j=1
Proof. Choose the following Lyapunov function: n
1 T 1X e (t)e(t) = (ei (t))2 . 2 2 i=1
urn a
V (e(t)) =
(30)
We calculate the derivative of V (e(t)) along system (5): V˙ (e(t)) = =
n X
i=1 n X i=1
ei (t)e˙ i (t) n
ei (t) − σi ei (t) +
n X j=1
o (31) pij [fj (yj (t)) − fj (xj (t))] + ui (t) .
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It follows from Assumption 1 that |fj (yj (t)) − fj (xj (t))| ≤ lj |yj (t) − xj (t)| = lj |ej (t)| ,
10
(32)
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then we have
≤
n X
ei (t)
i=1 j=1 n X n X i=1 j=1
|pij | lj
(ej (t))2 (ei (t))2 + = 2 2
i=1
ei (t)ui (t) = −
n X i=1
ki1 (ei (t))2 −
n X i=1
|ei (t)|
i=1 n X n X i=1 j=1
It is obvious that n X
n X
pij [fj (yj (t)) − fj (xj (t))] ≤
n X j=1
|pij | · lj |ej (t)|
|pij | lj + |pji | li (ei (t))2 . 2
ξ+1
ki2 |ei (t)|
Therefore,
−
n X i=1
η+1
ki3 |ei (t)|
If ki1 >
n P
j=1
i=1
|pij |lj +|pji |li 2
V˙ (e(t)) ≤ −
n X i=1
where ρi = ki1 −
j=1
−
i=1
η+1
ki3 |ei (t)|
.
(34)
(35)
.
− σi , i = 1, 2, ..., n, we can obtain that
ρi (ei (t))2 −
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n P
ξ+1
ki2 |ei (t)|
n X
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−
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n n X X |p | l + |p | l ij j ji i − σi − ki1 (ei (t))2 V˙ (e(t)) ≤ 2 i=1 j=1 n X
(33)
pro of
n X
|pij |lj +|pji |li 2
n X i=1
ξ+1
ki2 |ei (t)|
−
n X i=1
η+1
ki3 |ei (t)|
,
(36)
+ σi > 0.
Let ρ = min{ρi }, k2 = min{ki2 }, k3 = min{ki3 }, we have i
V˙ (e(t)) ≤ −ρ ξ+1 2
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Since 0 <
i
n X i=1
(ei (t))2 − k2
< 1 and
η+1 2
i
n X
2
(|ei (t)| )
i=1
ξ+1 2
− k3
n X
2
(|ei (t)| )
η+1 2
.
(37)
i=1
> 1, we can derive from Lemma 1 that
n n n n X X X 1−η X 2 ξ+1 2 ξ+1 2 η+1 2 η+1 (|ei (t)| ) 2 ≥ ( |ei (t)| ) 2 , (|ei (t)| ) 2 ≥ n 2 ( |ei (t)| ) 2 . i=1
i=1
i=1
i=1
(38)
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Then we can obtain that n n n X X 1−η X 2 η+1 2 ξ+1 V˙ (e(t)) ≤ −ρ (ei (t))2 − k2 ( |ei (t)| ) 2 |ei (t)| ) 2 − k3 · n 2 ( i=1
ξ+1 2
, b = k3 · n
ξ+1 2
1−η 2
pro of
= −2ρ · V (e(t)) − k2 · 2 Let a = k2 · 2
i=1
i=1
(V (e(t)))
·2
η+1 2
ξ+1 2
− k3 · n
1−η 2
·2
η+1 2
(V (e(t)))
η+1 2
. (39)
, c = 2ρ, we have
η+1 ξ+1 V˙ (e(t)) ≤ −a(V (e(t))) 2 − b(V (e(t))) 2 − cV (e(t)).
(40)
According to Theorem 1, the origin of system (5) can achieve fixed-time stability. Moreover,
1 c 1 ln(1 + ) + η+1 ln(1 + ξ+1 a c( 2 − 1) c(1 − 2 ) 2 c 2 c = ln(1 + ) + ln(1 + ). c(1 − ξ) a c(η − 1) b
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5 Tmax =
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Corollary 1. If Assumption 1 holds and ki1 >
n P
j=1
c ) b
|pij |lj +|pji |li 2
− σi , i =
1, 2, ..., n, systems (2) and (4) can achieve fixed-time synchronization under controller (28). Furthermore,
6 Tmax =
ξ+1 2
, b = min{ki3 } · n
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where a = min{ki2 } · 2
2 2 + , a(1 − ξ) b(η − 1)
i
1−η 2
i
·2
η+1 2
(41)
.
Proof. Similarly, it can be proved that ξ+1 η+1 V˙ (e(t)) ≤ −a(V (e(t))) 2 − b(V (e(t))) 2 − cV (e(t)),
where a = min{ki2 } · 2 i
ξ+1 2
, b = min{ki3 } · n
1−η 2
i
·2
η+1 2
, c = min{2ki1 − i
|pji | li ) + 2σi }.
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Since the Lyapunov function V (e(t)) = D+ V (e(t)) = V˙ (e(t)). Therefore, D+ V (e(t)) ≤ −a(V (e(t))) ≤ −a(V (e(t)))
ξ+1 2 ξ+1 2
12
1 2
n P
(42) n P
(|pij | lj +
j=1
(ei (t))2 is derivable, we have
i=1
− b(V (e(t))) − b(V (e(t)))
η+1 2 η+1 2
− cV (e(t)) .
(43)
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According to Lemma 2, the origin of system (5) can achieve fixed-time stability. Moreover, 2 2 + . a(1 − ξ) b(η − 1)
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6 Tmax =
Corollary 2. If Assumption 1 holds and ki1 >
n P
j=1
|pij |lj +|pji |li 2
− σi , i =
1, 2, ..., n, systems (2) and (4) can achieve fixed-time synchronization under controller (28). Furthermore,
1−ξ 1 a η−ξ 2 2 = · + , a b 1−ξ η−1
7 Tmax ξ+1 2
, b = min{ki3 } · n
1−η 2
i
i
·2
η+1 2
.
re-
where a = min{ki2 } · 2
(44)
Proof. Similarly, it can be proved that
ξ+1 η+1 V˙ (e(t)) ≤ −a(V (e(t))) 2 − b(V (e(t))) 2 ,
where a = min{ki2 } · 2
ξ+1 2
, b = min{ki3 } · n i
1−η 2
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i
·2
η+1 2
(45)
.
According to Lemma 3, the origin of system (5) can achieve fixed-time stability. Moreover,
1−ξ 2 2 1 a η−ξ · + . a b 1−ξ η−1
urn a
7 Tmax =
Next consider the following controller: ξ
η
ui (t) = −ki1 ei (t) − ki2 sign(ei (t)) |ei (t)| − ki3 sign(ei (t)) |ei (t)| , i = 1, 2, ..., n, where positive constants
ki1 ,
and η = 1 + β1 , β > 1.
ki2 ,
(46)
ki3 ,
i = 1, 2, ..., n are control gains, ξ = 1 −
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Corollary 3. If Assumption 1 holds and ki1 >
n P
j=1
|pij |lj +|pji |li 2
1 β
− σi , i =
1, 2, ..., n, systems (2) and (4) can achieve fixed-time synchronization under controller (46). Furthermore, πβ 8 Tmax =√ , ab 13
(47)
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where a = min{ki2 } · 2
ξ+1 2
i
, b = min{ki3 } · n
1−η 2
i
·2
η+1 2
.
pro of
Proof. Similarly, it can be proved that ξ+1 η+1 V˙ (e(t)) ≤ −a(V (e(t))) 2 − b(V (e(t))) 2 ,
where a = min{ki2 } · 2 i
Since
ξ+1 2
=1−
ξ+1 2
, b = min{ki3 } · n
1−η 2
i
η+1 1 2β , 2
=1+
1 2β ,
·2
η+1 2
(48)
.
we have
1
1
D+ V (e(t)) = V˙ (e(t)) ≤ −a(V (e(t)))1− 2β − b(V (e(t)))1+ 2β , where β > 1.
(49)
bility. Moreover,
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According to Lemma 4, the origin of system (5) can achieve fixed-time staπβ 8 Tmax =√ . ab
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4. Numerical simulations
In this section, we propose a secure communication scheme based on the fixed-time synchronization of neural networks, and numerical simulations are
urn a
provided to verify the obtained theoretical results. Example 1. Sender generates the following neural network (master system): x˙ i (t) = −σi xi (t) +
2 X j=1
pij fj (xj (t)) + Ii , t ≥ 0,
(50)
i = 1, 2, where σ1 = 0.9, σ2 = 1.1, p11 = 2, p12 = −0.1, p21 = −5, p22 = 4.5, I1 = sin t, I2 = cos t. Without loss of generality, suppose pij , i, j = 1, 2 are the secret keys in this secure communication scheme, then sender transmits pij ,
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i, j = 1, 2 to receiver via transmission channel. Let f1 (α) = f2 (α) = 0.5(|α + 1| − |α − 1|), then we have l1 = l2 = 1. The
initial values of system (50) are x1 (ν) = 3, x2 (ν) = −2, ν ∈ [−1, 0]. The state trajectories of x1 (t) and x2 (t) are shown in Fig.2. 14
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10 x1(t) x2(t)
1
2
0 -5 -10 -15 0
4
8
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x (t), x (t)
5
12
t/s
16
20
Fig. 2. The state trajectories of x1 (t) and x2 (t), which are generated by system
re-
(50). The initial values of system (50) are x1 (ν) = 3, x2 (ν) = −2, ν ∈ [−1, 0]. By using plaintext signals m1 (t) = 1.6 sin t + 0.7 cos(0.3t) and m2 (t) =
(51)
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− sin(1.2t) + 2 cos(3t), sender generates the following signals: ri (t), 0 ≤ t < 3, Mi (t) = mi (t − 3), t ≥ 3,
i = 1, 2, where ri (t) denotes random signal. Sender calculates the encrypted signals Ci (t) = Mi (t) + xi (t), i = 1, 2, and transmits Ci (t), i = 1, 2 to receiver via transmission channel. Figs.3-5 illustrate the state trajectories of mi (t), Mi (t)
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and Ci (t), i = 1, 2. It should be pointed out that, the initial values of system (50), ri (t), mi (t) and Mi (t) can only be known by sender, i = 1, 2. Receiver receives the secret keys pij , i, j = 1, 2, and the encrypted signals Ci (t), i = 1, 2. Then receiver generates the following slave system: y˙ i (t) = −σi yi (t) +
2 X j=1
pij fj (yj (t)) + Ii + ui (t), t ≥ 0,
(52)
i = 1, 2, where ui (t) denotes controller. The initial values of system (52) are
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y1 (ν) = −1, y2 (ν) = 3, ν ∈ [−1, 0].
Receiver chooses k11 = 6, k21 = 8, k12 = k22 = 1, k13 = k23 = 1, ξ = 0.5 and 2 P |pij |lj +|pji |li η = 1.5. We can verify that ki1 satisfies ki1 > − σi , i = 1, 2. 2 j=1
15
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6 m2(t)
3 m1(t), m2(t)
pro of
m1(t)
0
-3
-6 0
4
8
12
t/s
16
20
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Fig. 3. The state trajectories of plaintext signals m1 (t) = 1.6 sin t + 0.7 cos(0.3t)
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and m2 (t) = − sin(1.2t) + 2 cos(3t).
6
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M1(t), M2(t)
3
M1(t) M2(t)
0
-3
-6
0
4
8
12
16
20
t/s
Fig. 4. The state trajectories of M1 (t) and M2 (t). If 0 ≤ t < 3, Mi (t) =
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ri (t), i = 1, 2, where ri (t) denotes random signal; if t ≥ 3, Mi (t) = mi (t − 3), i = 1, 2.
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10 C1(t) C2(t)
0 -5 -10 -15 0
4
8
pro of
C 1(t), C 2(t)
5
12
t/s
16
20
Fig. 5. The state trajectories of the encrypted signals C1 (t) and C2 (t), where
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Ci (t) = Mi (t) + xi (t), i = 1, 2.
It can be calculated that a = 20.75 , b = 2, c = 3.7. According to Theorem 2, systems (50) and (52) can realize fixed-time synchronization under controller 5 (28), and Tmax ≈ 2.3897.
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5 ≈ 2.3897 < 3, we have xi (t) = yi (t), i = 1, 2, t ≥ 3. Receiver can Since Tmax
decrypt the encrypted signals by calculating m ˆ i (t) = Ci (t+3)−yi (t+3), i = 1, 2, t ≥ 0, and it is obvious that
m ˆ i (t) = Ci (t + 3) − yi (t + 3) = Mi (t + 3) + xi (t + 3) − yi (t + 3)
(53)
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= Mi (t + 3) = mi (t), t ≥ 0.
Fig.6 is the schematic diagram of the proposed secure communication scheme. According to Corollary 1 and Corollary 2, systems (50) and (52) can re6 alize fixed-time synchronization under controller (28), and Tmax ≈ 4.3784,
7 Tmax ≈ 4.3620. In fact, systems (50) and (52) can also realize fixed-time
synchronization under controller (46) according to Corollary 3 (β = 2), and
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8 6 7 8 Tmax ≈ 3.4259. Obviously, Tmax , Tmax and Tmax are larger than 3, so it seems
that xi (t) = yi (t), i = 1, 2, t ≥ 3 is not necessarily true. Therefore, according to Corollary 1, Corollary 2 and Corollary 3, plaintext signals can’t be recovered successfully. However, according to Theorem 2, plaintext signals can be 17
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Plaintext signal
mi t
The recovered plaintext signal
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Prefixed by random signal
mˆ i t
M i t Ci t Encryption
Transmission channel
xi t
Decryption
yi t
Master system
Fixed-time synchronization control
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Slave system
Fig. 6. The schematic diagram of the proposed secure communication scheme.
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recovered successfully.
Fig.7 and Fig.8 illustrate the state trajectories of yi (t) and ei (t) under controller (28) (controller (46)), i = 1, 2. The state trajectories of −e1 (t) and e2 (t) under controller (28) (controller (46)) are illustrated in Fig.9, which is presented
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with a log axis on the y-axis.
Remark 3. In this secure communication scheme, systems (50) and (52) must realize fixed-time synchronization, and Tmax should satisfy Tmax < 3. If systems (50) and (52) only realize asymptotic synchronization, synchronization will be reached when time tends to infinity, so xi (t) = yi (t), i = 1, 2, t ≥ 3 can’t be satisfied, that means plaintext signals can’t be recovered successfully. If systems (50) and (52) only realize finite-time synchronization, synchronization will be reached within the settling time, which depends on the initial values of systems
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(50) and (52). However, since receiver don’t know the initial values of system (50), the settling time will be impossible to be estimated, so xi (t) = yi (t), i = 1, 2, t ≥ 3 is not necessarily true, that means plaintext signals can’t be recovered successfully.
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10 y1(t)
y1(t), y2(t)
0 -5 -10 -15 0
4
8
pro of
y2(t)
5
12
t/s
16
20
Fig. 7. Under controller (28) (controller (46)), the state trajectories of y1 (t)
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and y2 (t), which are generated by system (52). The initial values of system (52) are y1 (ν) = −1, y2 (ν) = 3, ν ∈ [−1, 0]. The controller parameters are k11 = 6,
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k21 = 8, k12 = k22 = 1, k13 = k23 = 1, ξ = 0.5 and η = 1.5.
6
2
e (t), e (t)
3
e2(t)
1
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0
e1(t)
-3
T5max
-6
0
1
T7max
T8max 0
2
0
3
T6max 00
4
5
t/s
Fig. 8. The state trajectories of the synchronization errors e1 (t) and e2 (t) under i controller (28) (controller (46)), where ei (t) = yi (t) − xi (t), i = 1, 2. Tmax
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denotes the upper bound estimate for the settling time, i = 5, 6, 7, 8, where
5 6 7 Tmax is derived by Theorem 2, Tmax is derived by Corollary 1, Tmax is derived 8 is derived by Corollary 3. by Corollary 2, Tmax
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10 1 - e 1(t) e2(t)
10 -2
10 -4
T5max
10 -6
T8max
0
0
1
pro of
-e1(t), e 2(t)
10 0
2
T7max
0
3
t/s
T6max
00
4
5
Fig. 9. The state trajectories of −e1 (t) and e2 (t) under controller (28) (controller
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(46)), where the y-axis is presented with a log axis.
5 6 7 8 . Table 1. The comparisons among Tmax , Tmax , Tmax and Tmax 6 Tmax
7 Tmax
8 Tmax
2.3897
4.3784
4.3620
3.4259
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Example 1
5 Tmax
Remark 4. Any upper bound estimate formula for the settling time is derived by strict theoretical analysis. Due to the difficulty and conservativeness of strict theoretical analysis, the upper bound estimate for the settling time will
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be larger than the synchronization time illustrated in numerical simulations (see Fig.8 and Fig.9). Example 1 shows that, with the same controller parameters, 6 7 8 5 (see Table 1, Fig.8 and Fig.9). and Tmax Tmax is much smaller than Tmax , Tmax 6 5 is derived by As we know, Tmax is derived by Theorem 2 and Theorem 1, Tmax 7 8 Corollary 1 and Lemma 2, Tmax is derived by Corollary 2 and Lemma 3, Tmax
is derived by Corollary 3 and Lemma 4. Therefore, compared with Lemmas 2, 3, 4, Theorem 1 provides a tighter upper estimate formula for the settling time.
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Remark 5. In Example 1, if we choose some larger k11 , k21 , k31 , and keep the 5 other controller parameters unchanged, then the obtained Tmax can be smaller. 6 7 8 However, Tmax , Tmax and Tmax still remain unchanged in this case.
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5. Conclusions
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In this paper, we give a novel fixed-time stability theorem, and the corresponding stability criterion, proof, upper bound estimate formula for the settling time are all different from those in the existing fixed-time stability theorems. By utilizing the new fixed-time stability theorem, the fixed-time synchronization of neural networks is studied via feedback control. Finally, a secure communication scheme based on the fixed-time synchronization of neural networks is proposed. Numerical simulations illustrate that, compared with the existing fixed-time stability theorems, the new fixed-time stability theorem can provide
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a tighter upper bound estimate formula for the settling time. What is more, the plaintext signals can be recovered according to the new fixed-time stability theorem, while the plaintext signals can not be recovered according to the existing fixed-time stability theorems. Because the new fixed-time stability theorem has a tighter upper bound estimate formula for the settling time, it not only can be
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used to design secure communication schemes with lower communication cost (see Example 1), but also can be used to realize the fast cooperative control of intelligent robots. In the future, we will study the application of the new
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fixed-time stability theorem in the fast cooperative control of intelligent robots.
Acknowledgements
The work is supported by the National Key R&D Program of China (Grant Nos. 2016YFB0800604 and 2016YFB0800602), the National Natural Science Foundation of China (Grant Nos. 61972051, 61771071, 11771196, 61872203 and 61802212), and the Shandong Province Natural Science Foundation (Grant No.
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ZR2018BF023).
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Conflict of interest
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The authors declare no conflict of interest concerning the publication of this manuscript.