Finite-time Mittag-Leffler synchronization of fractional-order memristive BAM neural networks with time delays

Author’s Accepted Manuscript Finite-time Mittag-Leffler synchronization of fractional-order memristive BAM neural networks with time delays Jianying Xiao, Shouming Zhong, Yongtao Li, Fang Xu www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)31084-0 http://dx.doi.org/10.1016/j.neucom.2016.09.049 NEUCOM17578

To appear in: Neurocomputing Received date: 16 March 2016 Revised date: 12 September 2016 Accepted date: 22 September 2016 Cite this article as: Jianying Xiao, Shouming Zhong, Yongtao Li and Fang Xu, Finite-time Mittag-Leffler synchronization of fractional-order memristive BAM neural networks with time delays, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.09.049 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Finite-time Mittag-Leffler synchronization of fractional-order memristive BAM neural networks with time delays✩ Jianying Xiaoa,b,∗, Shouming Zhongb, Yongtao Lic , Fang Xua a School of Sciences, Southwest Petroleum University, Chengdu, 610050, PR China of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, PR China c College of Chemistry and Chemical Egineering, Southwest Petroleum University, Chengdu, 610050, PR China

b School

Abstract This paper investigates the finite-time synchronization of delayed fractional-order memristive bidirectional associative memory neural networks (FMBAMNNs). Based on Lyapunov theory, fractional-order differential inequalities, norm properties and linear feedback controller, the separated criteria are obtained to ensure the finite-time synchronization of studied FMBAMNNs with different fractional order α, respectively. Moreover, the derived criteria are easily checked and they contribute to the reduction of amount of calculation. Finally, two numerical examples are given to demonstrate the effectiveness of the theoretical results. Keywords: Memristor, Fractional-order neural networks, Synchronization, Bidirectional associative memory.

1. Introduction Fractional calculus as an extension of usual calculus dates from the late seventeenth century and it is regarded as a generalization of derivation and integration of arbitrary order ([1]). Compared with integer-order systems, fractional-order ones have apparent advantages in describing the memory and hereditary properties of different kinds of materials and processes. Moreover, fractional-order systems are more likely to be used to describe most of the real-world behaviors than integer-order ones because they can provide more practical value and accurate results. Therefore, many researchers have paid close attention to studying the dynamical behaviors of fractional-order systems and have drawn some wonderful results in the literature ([2-15,20-27,33-38,56-57]). In the past several decades, bidirectional associative memory neural networks (BAMNNs) have been successfully applied in the filed of artificial intelligence and pattern recognition because they generalize the single-layer auto-associative Hebbian circuit to a two-layer pattern matched hetero-associative correlation ([47-51]). The two-layer pattern matched heteroassociative circuit can help BAMNNs store and recall pattern pairs which are regarded as the bidirectionally stable states ([47]). Accordingly, the reverberating interfered message between the two-layer circuit would be unavoidably involved in the information which is derived by decoding in BAM. That way, some hereditary properties present in this kind process of intelligent information. Given that fact, it is very important ✩ This work was supported by National Natural Science Foundation of China (61202045 and 11401494) ∗ Corresponding author: School of Sciences, Southwest Petroleum University, Chengdu, 610050, PR China. Email address: [email protected] (Jianying Xiao)

Preprint submitted to Neurocomputing

to take into account merging infinite memory into BAMNNs as the authors suggested in ([19]). On one hand, BAMNNs can be realized in a circuit where the self-feedback connection weights and the common connection weights are established by memristors. The memristor which was found by Chua ([28]) is well-known as the fourth fundamental nonlinear circuit element. The other three basic nonlinear circuit elements are capacitor, inductor and resistor. Almost forty years later, a group of scientists from Hewlett-Packard library identified the memristor and employed it to build the prototype with the nanometer size ([29]). Since then, the memristor has been substituted for resistor to emulate neural synapses in analog circuits because it owns memory and performances more like biological synapses than the resistor. There are many papers which verify that the memristor possesses the characteristic of pinched hysteresis. Owe to this characteristic, some researchers establish a new class of NNs named memristive or memristor-based neural networks (MNNs) by replacing resistors with memristors ([39-42,44,50,51,54]). Very recently, based on nonsmooth analysis ([30,31]), some authors studied memristive or memristorbased BAM neural networks (MBAMNNs) and investigated the various dynamical behaviors of BAMMNNs in order to realize their better applications in associative memories, neural learning and pattern recognition ([45,50,51]). On the other hand, as we shared at the beginning, fractional-order systems possess infinite memory and hereditary properties. In addition, time delays inevitably exist in dynamical systems and they may initiate undesirable dynamical behaviors such as divergence, oscillation, even or instability. Therefore, many researches about several dynamical behaviors of NNs with time delays become more and more significant and popular. From the above three perspectives, it is necessary to develop some economical and practical systems for fractional-order memristive bidirectional September 26, 2016

2. Preliminaries and model description

associative memory neural networks (FMBAMNNs) with time delays. Moreover, synchronization has been an active topic in the area of nonlinear science since Pecora and Carroll introduced chaos synchronization ([46]). Very recently, many researchers showed more interest in the study of synchronization of fractional-order neural networks(FNNs). Up to now, there are many kinds of synchronization of FNNs such as global Mittag-Leffler synchronization, projective synchronization and so on ([7,8,20]). The global Mittag-Leffler stability and synchronization was discussed for FMNNs by introducing a generalized Gronwall-like inequality of Caputo fractional derivative in ([8]). In ([20]), authors dealt with the global Mittag-Leffler synchronization for a class of FNNs considering discontinuous activations by employing a singular Gronwall inequality and applying Filippov solution theorem. Some new sufficient criteria were derived for achieving the projective synchronization of FMNNs by using a new hybrid controller in ([7]). The above mentioned kinds of synchronization show that the trajectories of the slave system can catch the trajectories of the master system in infinite-time. In practice, it is extremely important that the synchronization should be reached in finite-time. Recently, many researchers have paid their close attention to the investigation of finite-time synchronization of NNs and many better results have been exhibited in the literature ([52,53,54]). Especially, the finite-time synchronization was studied for NNs considering arbitrary time delays in ([52]). In ([53]), on the basis of two different time-delayed feedback controllers, authors derived some good criteria for the finite-time synchronization for NNs. Very recently, authors in ([54]) investigated the finitetime for MNNs by empolying analysis method, finite-time stability theorem and applying a proper feedback controller. The problem of finite-time synchronization was settled for FMNNs with time delays by applying Laplace transform, Mittag-Leffler function, the generalized Gronwall’s inequality and linear feedback control method in ([21]). Motivated by the above discussion, this paper is concerned with the finite-time synchronization for FMBAMNNs with time delays. The main contributions of this paper are the following aspects: (1) As the authors in ([43]) proposed the idea of circuit design, an economical class of FMBAMNNs is constructed by just using one pair of memristors to store multiple states of the input message while the snapes of FMNNs were designed with multiple memristors in earlier publications ([39-42,50,51,54]). (2) For the fractional-order α satisfying 0 < α < 1, based on Lyapunov theory, some easily checked criteria are obtained to achieve the finite-time synchronization of FMBAMNNs by mainly emoloying a new inequality about the Caputo derivative of absolute value function and a generalized Gronwall-like inequality. (3) Given that fractional-order α satisfying 1 < α < 2, based on theory of vector norm, matrix norm and comparison theorem, a relaxed judgment condition is derived to realize the finite-time synchronization of FMBAMNNs by mainly applying the generalized Gronwall’s inequality. (4) The obtained criteria are simple in form rather than linear matrix inequalities, they are more easily checked and contribute to reducing the computational burden.

Notation. Throughout this paper, the Caputo fractional derivative operator Dαt0 ,t is chosen for fractional-order derivative with order α. Denote D−α t0 ,t as fractional integral with fractional order α. For vector υ = (υ1 , υ2 , . . . , υn )T , its p-norm is defined as 1  ||υ|| p = ( ni=1 |υi | p ) p , p ≥ 1. C q ([t0 , +∞), Rn ) is denoted as the space of q-order continuous and differentiable functions from [t0 , +∞) into Rn . Matrices, if their dimensions are not explicitly stated, are supposed to be compatible for algebraic operations. The notation || · || refers to the p-norm of vector. Sometimes, when no confusion would arise, the dimensions of a function or a matrix will be omitted for convenience. 2.1. Some preliminaries on fractional-order systems In this subsection, we recall some basics of fractional calculus, definitions and lemmas which will be required later. Definition 1. (Podlubny (1999)) The fractional integral for a function Θ(t) is defined as  t 1 α I Θ(t) = (t − υ)α−1 Θ(υ)dυ, Γ(α) t0 where t ≥ t0 , α > 0 and Γ(·) is the Euler’s gamma function, that ∞ is, Γ(α) = 0 tα−1 e−α dt. Definition 2. (Podlubny (1999)) Caputo fractional derivative for a function Θ(t) ∈ C n ([t0 , +∞), Rn ) is defined as  t 1 Θ(n) (υ) Dαt0 ,t = dυ, Γ(n − α) t0 (t − υ)α−n+1 where t ≥ t0 , α > 0 and n is an integer such that 0 < n − 1 < α < n. Moreover, when 0 < α < 1,  t  Θ (υ) 1 α Dt0 ,t = dυ. Γ(1 − α) t0 (t − υ)α Definition 3. (Podlubny (1999)) The one-parameter MittagLeffler function is defined as Eα (z) =

+∞ 

zm , Γ(mα + 1) m=0

where α > 0 and z ∈ C. The two-parameter Mittag-Leffler function is defined as Eα,β (z) =

+∞ 

zm , Γ(mα + β) m=0

where α > 0, β > 0 and z ∈ C. When β = 1, one has Eα (z) = Eα,1 (z). Further, E1,1 (z) = ez . Definition 4. (Podlubny (1999)) The Laplace transform of Mittag-Leffler function is L(tβ−1 Eα,β (ψtα )) = 2

sα−β , t ≥ 0, sα − ψ

and j-th memristor M j , and

Wi (xi (t)) =

⎧ ⎪ ˆ i , |xi (t)| ≤ S i , ⎪ W ˜ j (y j (t)) = ⎨ W ⎪ ⎪ ˇ i , |xi (t)| > S i , ⎩ W

ˆ˜ , |y (t)| ≤ T W j j j ˇ ˜ W j , |y j (t)| > T j ,

where switching jumps S i > 0 and T j > 0, i = 1, 2, . . . , n, j = 1, 2, . . . , m. Moreover, throughout this paper, the neuron activation functions are assumed to satisfy the following assumption: Assumption 1. For i = 1, 2, . . . , n, and j = 1, 2, . . . , m, the neuron activation function f j and f˜i are Lipschitz continuous and for all a, b, c, d ∈ R, there exists a real number h j and h˜ i such that

Figure 1: Schematic diagram of memristive neuronal cell (Wen-shiping et al.2015).

f j (a) − f j (b) ≤ h j , f j (0) = 0, a  b, a−b f˜i (c) − f˜i (d) ˜ ˜ ≤ hi , fi (0) = 0, c  d. 0≤ c−d

where t and s are the variables in the time domain and Laplace domain, respectively; ψ is a real number; the real part Re(s) of 1 s is satisfied the following condition:Re(s) > |ψ| α .

0≤

Then, equation (1) can be rewritten as follows:

2.2. Model

Dα0,t xi (t)

In this subsection, following the work in [43], which implemented the neuronal cell of MNNs in an MNN shown in Figure 1. Based on the Kirchoff’s current law, the i-th and j-th subsystems of FMBAMNNs with time delays are described by the following equations:

= − ci (xi (t))xi (t) +

m  sign ji f j (y j (t)) j=1

R ji

+

m  sign ji f j (y j (t − τ)) j=1

F ji

+

i=1

m 

b ji f j (y j (t − τ)) + Ii (t),

j=1

Dα0,t y j (t) = − c˜ j (y j (t))y j (t) +

n 

a˜ i j f˜i (xi (t))

i=1

+ Ii (t),

+

n 

b˜ i j f˜i (xi (t − τ)) + I˜j (t),

i=1

C˜ j Dα0,t y j (t) n   1  ˜ 1 + W j (y j (t)) y j (t) = − + ˜ ˜ Fi j i=1 Ri j n n  signi j f˜i (xi (t))  signi j f˜i (xi (t − τ)) + + I˜ j (t), + ˜ ij R F˜i j

t ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . , m,

(2)

where Ii (t) Ci R ji Ci F ji Ci m     1 1 1 + Wi (xi (t)) ci (xi (t)) = + Ci j=1 R ji F ji a ji =

i=1

t ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . , m,

a ji f j (y j (t))

j=1

Ci Dα0,t xi (t) m   1 1  + Wi (xi (t)) xi (t) = − + R ji F ji j=1 +

m 

(1)

where Dα0,t is the Caputo derivative with order α satisfying 0 < α < 1 and 1 < α < 2, respectively; Ci and C˜ j denote the capacitor of xi (t) and y j (t), respectively; τ expresses the transmission delay; Ii (t) and I˜ j (t) denote the external inputs; R ji stands for the resistor combining the continuous activation function f j (y j (t)) with xi (t); F ji represents the resistor combining the continuous activation function f j (y j (t − τ)) with xi (t); R˜ i j corresponds to the resistor between the continuous activation function f˜i (xi (t)) and y j (t); F˜i j describes the resistor be˜ tween the continuous

activation function fi (xi (t − τ)) with y j (t); 1, i  j; is the sign function; Wi (xi (t)) signi j = sign ji = −1, i = j, and W j (y j (t)) denote the memductance of the i-th memristor Mi

sign ji

, b ji =

sign ji

, Ii (t) =

signi j signi j I˜ j (t) , b˜ i j = , I˜j (t) = C˜ j C˜ j R˜ i j C˜ j F˜i j n   1 1 1  + W j (y j (t)) c˜ j (y j (t)) = + ˜ ˜ ˜ C j i=1 Ri j Fi j a˜ i j =

According to the feature of Wi (xi (t)) and W j (y j (t)) in this paper, for convenience, denote ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨cˆ i , |xi (t)| ≤ S i , ⎨cˆ˜ j , |y j (t)| ≤ T j , ci (xi (t)) = ⎪ c˜ j (y j (t)) = ⎪ ⎪ ⎪ ⎩cˇ i , |xi (t)| > S i , ⎩cˇ˜ j , |y j (t)| > T j , 3

in which switching jumps S i > 0 and T j > 0, cˆi , cˇ i , cˆ˜ j , and cˇ˜ j (i = 1, 2, . . . , n, j = 1, 2, . . . , m) are all constants. The initial conditions of (2) are supposed to be:

Definition 6. (Aubin & Frankowska (2009)) A set-valued map N with nonempty values is said to be upper semi-continuous at x0 ∈ M ⊆ Rn , if for any open set G containing N(x0 ), there exists a neighborhood H of x0 such that N(H) ⊆ G. N(x) is said to have a closed (convex, compact) image if, for each x ∈ M, N(x) is closed (convex, compact).

xi (0) = xi0 , i = 1, 2, . . . , n, y j (0) = y j0 , j = 1, 2, . . . , m. Then, we have Dα0,t xi (t) = − ci (xi (t))xi (t) +

m 

a ji f j (y j (t))

j=1

+

m 

b ji f j (y j (t − τ)) + Ii (t),

Definition 7. (Filippov (1988)) For differential system dx dt = s(t, x), where s(t, x) is discontinuous in x. The set-valued map of s(t, x) is defined as:   N(t, x) = co[s(B(x, ι)) \ G)],

j=1

Dα0,t y j (t) = − c˜ j (y j (t))y j (t) +

n 

a˜ i j f˜i (xi (t))

ι>0 μ(G)=0

i=1

+

n 

b˜ i j f˜i (xi (t − τ)) + I˜j (t),

where B(x, ι) = {y : ||y− x|| ≤ ι} is the ball of center x and radius ι; Intersection is taken over all sets G of measure zero and over all ι > 0; and μ(G) is the Lebesgue measure of set G.

i=1

xi (0) = xi0 , i = 1, 2, . . . , n, y j (0) = y j0 , j = 1, 2, . . . , m, t ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . , m,

(3)

Remark 1. In earlier publications of FMNNs ([3942,50,51,54]), the synapses of the NNs were designed with multiple memristors, but this design are dropped here. In our circuit design, only one pair of memristors is used for saving multiple states of the input message for a neuron cell, which will be helpful to reduce the cost to establish the corresponding FMNNs in future. Therefore, our circuit design is more economical and practical.

A Filippov solution of system dx dt = s(t, x) with initial condition x(0) = x0 is absolutely continuous on any subinterval t ∈ [t1 , t2 ] of [0, T ], which satisfies x(0) = x0 , and the differential inclusion:

Remark 2. Recently, several results about the dynamical behaviors have been investigated for FNNs, in which the fractional-order is only considered to satisfy 0 < α < 1. Very recently, authors in ([21]) derived new sufficient criteria for the finite-time synchronization of FMNNs with fractional-order α satisfying 0 < α < 1 and 1 < α < 2. In this paper, the corresponding conditions will be investigated for the finite-time synchronization of FMBAMNNs with α satisfying 0 < α < 1 as well as 1 < α < 2 based on different methods, respectively. So our studied NNs are more general and practical.

dx ∈ N(t, x). dt

f or

a.a. t ∈ [0, T ].

Let Co[A] denote the convex closure of given set A. Then, we can get that ⎧ ⎪ ⎪ |xi (t)| < S i , cˆ i , ⎪ ⎪ ⎪ ⎨ Co[ci (xi (t))] = ⎪ [ci , ci ], |xi (t)| = S i , ⎪ ⎪ ⎪ ⎪ ⎩cˇ i , |xi (t)| > S i , ⎧ ⎪ ⎪ |y j (t)| < T j , cˆ˜ j , ⎪ ⎪ ⎪ ⎨ Co[˜c j (y j (t))] = ⎪ [˜ c , c ˜ ], |y j (t)| = T j , ⎪ i i ⎪ ⎪ ⎪ ⎩cˇ˜ j , |y j (t)| > T j ,

According to the above discussion, the values of ci (xi (t)) and c˜ j (y j (t)) change as the states xi (t) and y j (t) vary. It is obvious that ci (xi (t)) and c˜ j (y j (t)) are piecewise continuous even or discontinuous functions at most two points of discontinuity such as ±S i and ±T j , respectively. The next definitions are helpful to convert the above equation into the conventional one.

where ci = min{ˆci , cˇ i }, ci = max{ˆci , cˇ i }, c˜ i = min{cˆ˜ i , cˇ˜ i }, c˜ i = max{cˆ˜ i , cˇ˜ i }, for i = 1, 2, . . . , n, j = 1, 2, . . . , m. Obviously, Co[ci (xi (t))] and Co[˜c j (y j (t))] are all closed, convex and compact in xi (t) and y j (t), respectively. Based on the theory of

Definition 5. (Aubin & Frankowska (2009)) Let M ⊆ R , x → N(x) is called a set-valued map from M → Rn , if for each point x of a set M ⊆ Rn , there corresponds a nonempty set N(x) ⊆ Rn . n

4

constructed as following:

differential inclusions, from (3), as follows: Dα0,t xi (t)

Dα0,t x˘i (t)

∈ − Co[ci (xi (t))]xi (t) +

m 

= − ci x˘i (t) +

a ji f j (y j (t))

j=1

+

m 

+

m 

b ji f j (˘y j (t − τ)) + Ii (t) + ui (t),

j=1

j=1

Dα0,t y j (t)

Dα0,t y˘ j (t) n 

= − c˜ j y˘ j (t) +

a˜ i j f˜i (xi (t))

i=1

+

n 

a ji f j (˘y j (t))

j=1

b ji f j (y j (t − τ)) + Ii (t),

∈ − Co[˜c j (y j (t))]y j (t) +

m 

n 

a˜ i j f˜i ( x˘i (t))

i=1

+

b˜ i j f˜i (xi (t − τ)) + I˜j (t),

i=1

n 

b˜ i j f˜i ( x˘i (t − τ)) + I˜j (t) + u˜ j (t),

i=1

xi (0) = xi0 , i = 1, 2, . . . , n, y j (0) = y j0 , j = 1, 2, . . . , m, t ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . , m,

x˘i (0) = x˘i0 , i = 1, 2, . . . , n, y˘ j (0) = y˘ j0 , j = 1, 2, . . . , m, t ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . , m,

(4)

or equivalently, for i = 1, 2, . . . , n, j = 1, 2, . . . , m, t ≥ 0, there exists measurable functions ci ∈ Co[ci (xi (t))] and c˜ j ∈ Co[˜c j (y j (t))] such that

Meanwhile, the linear feedback controllers are chosen to be ui (t) = ki ( x˘i (t) − xi (t)) and u˜ j (t) = k˘ j (˘y j (t) − y j (t)). Define e xi (t) = x˘i (t) − xi (t) and ey j (t) = y˘ j (t) − y j (t). Then, the fractional-order error system between the master system (5) and the slave system (6) are easily derived as following:

Dα0,t xi (t) = − ci xi (t) +

m 

Dα0,t e xi (t)

a ji f j (y j (t))

j=1

+

m 

= − (ci − ki )e xi (t) +

b ji f j (y j (t − τ)) + Ii (t),

j=1

= − c˜ j y j (t) +

m 

+

m 

b ji f j (ey j (t − τ)),

j=1

Dα0,t ey j (t)

a˜ i j f˜i (xi (t))

i=1

+

n 

a ji f j (ey j (t))

j=1

Dα0,t y j (t) n 

(6)

= − (˜c j − k˜ j )ey j (t) +

b˜ i j f˜i (xi (t − τ)) + I˜j (t),

a˜ i j f˜i (e xi (t))

i=1

i=1

xi (0) = xi0 , i = 1, 2, . . . , n, y j (0) = y j0 , j = 1, 2, . . . , m, t ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . , m,

n 

+

n 

b˜ i j f˜i (e xi (t − τ)),

i=1

e xi (0) = e xi0 , i = 1, 2, . . . , n, ey j (0) = ey j0 , j = 1, 2, . . . , m,

(5)

t ≥ 0, i = 1, 2, . . . , n, j = 1, 2, . . . , m,

(7)

where f j (ey j (t)) = f j (˘y j (t)) − f j (y j (t)) and f˜i (e xi (t)) = f˜i ( x˘i (t)) − f˜i (xi (t)). 2.3. Properties Definition 8. The master system (5) is said to be synchronized with the slave system (6) in finite-time for proper established linear feedback controllers. That is, the state of fractionalorder error system (7) is said to be Mittag-Leffler stalbe in finite-time, if there exist positive constants {ε, δ, T }, δ > ε, such that ||e x0 || + ||ey0|| < ε then ||e x (t)|| + ||ey(t)|| < δ, t ∈ F where F is the interval [0, T ), e xi (t) = x˘i (t) − xi (t) and ey j (t) = y˘ j (t) − y j (t).

In this paper, we use the master-slave synchronization approach to drive the corresponding conditions. In order to obtain the synchronization behavior for the considered FMBAMNNs (1), the slave system corresponding to the master system (5) is 5

n max1≤i≤n,1≤ j≤m {μi |a ji |d j + μi |b ji |d j } < 0, Q˜ j = − min {˜c j − k˜ j } + 1≤ j≤m νj (9)

Remark 3. Very recently, authors in ([21]) gave the definition about global Mittag-Leffler stabilization for FBAMNNs and authors in ([21]) introduced the definition of finite-time synchronization for FMNNs. Based on the above definitions, this paper draws the definition about finite-time Mittage-Leffler synchronization for FMBAMNNs with time delays.

Eα (−σtα ) <

2.4. Main results

Proof. Consider the following function V(t) =



t

t ≥ t0 ,

t0

d|Q(t)| ds

(t − s)α

ds.

Remark 4. The close relationship is established between the Caputo fractional-order derivative of the absolute value function of Q(t) and the Caputo fractional-order derivative of Q(t) by Lemma 1, which is very useful for the next discussion. Lemma 2.(Wu & Zeng et al., 2015) Let 0 < α < 1 and Q(t) be a continuous function on [0, +∞], if there exist constants a > 0 and b ≥ 0 such that Dαt0 ,t Q(t) ≤ −aQ(t) + b,

t ≥ 0,

then Q(t) ≤ Q(0)Eα (−atα ) + btα Eα,α+1 (−atα ),

t ≥ 0,

where Eα (·) is one-parameter Mittag-Leffler function and Eα,α+1 (·) is two-parameter Mittag-Leffler function.

Dα0,t V(t) ≤

ν j |ey j (t)|.

(11)

j=1

 1 sgn(μi |e xi (t)|) Γ(1 − α) i=1 n

+

Theorem 1. Under Assumption 1, for i = 1, 2, . . . , n, j = 1, 2, . . . , m, if there exist constants μi > 0 and ν j > 0 such that

1≤i≤n

m 

Applying Lemma 1 in (12), it is easy to get

Remark 5. Lemma 2 is so useful that it can be employed to settle the convergence problem of fractional-order systems and construct the corresponding criteria of dynamical behaviors for some fractional-order systems.

Qi = − min {ci − ki } +

μi |e xi (t)| +

According to the Caputo fractional-order derivative definition, we have  t ˙ V(θ) 1 Dα0,t V(t) = dθ Γ(1 − α) 0 (t − θ)α  t d [n μ |e (t)| + m ν |e (t)|] 1 i=1 i xi j=1 j y j dθ dθ = α Γ(1 − α) 0 (t − θ)  t d n 1 dθ [ i=1 μi |e xi (t)|] = dθ Γ(1 − α) 0 (t − θ)α  t d [m ν |e (t)|] 1 j=1 j y j dθ dθ + Γ(1 − α) 0 (t − θ)α  t n d 1 i=1 dθ [μi |e xi (t)|] = dθ Γ(1 − α) 0 (t − θ)α   t m d [ν |e (t)|] 1 j=1 dθ j y j dθ + Γ(1 − α) 0 (t − θ)α n  t d  1 dθ [μi |e xi (t)|] = dθ Γ(1 − α) i=1 0 (t − θ)α m  t d [ν |e (t)|]  1 dθ j y j dθ. (12) + Γ(1 − α) j=1 0 (t − θ)α

where 1 = Γ(1 − α)

n  i=1

Lemma 1.(Chen & Chen, 2015) Let 0 < α < 1. If Q(t) ∈ C 1 ([t0 , +∞), Rn ), then

Dαt0 ,t |Q(t)|

(10)

then the master system (5) is synchronized with the slave system (6) in finite-time under the linear feedback controllers.

In this subsection, given to the separated fractional order α satisfying 0 < α < 1 and 1 < α < 2, the corresponding sufficient criteria are established to assure the finite-time MittagLeffler synchronization for FMBAMNNs with time delays by employing different lemmas. First, for given fractional order α satisfying 0 < α < 1, the following fundamental lemmas will be useful for the next theoretical analysis.

Dαt0 ,t |Q(t)| ≤ sgn(Q(t))Dαt0 ,t Q(t),

δ (−σ = max {Qi , Q˜ j }), 1≤i≤n,1≤ j≤m ε

=

1 Γ(1 − α)

n 

m 

 0

t d [μ e (t)] dθ i xi dθ (t − θ)α



sgn(ν j |ey j (t)|)

j=1

0

t d [ν j ey (t)] j dθ dθ (t − θ)α

sgn(μi |e xi (t)|)Dα0,t (μi e xi (t))

i=1

m max1≤i≤n,1≤ j≤m {ν j |˜ai j |d˜i + ν j |b˜ i j |d˜i } < 0, μi (8)

+

m  j=1

6

sgn(ν j |ey j (t)|)Dα0,t (ν j ey j (t)).

(13)

According to the error system (7), for t ≥ 0, it follows that Dα0,t V(t) ≤

that

n m    μi − (ci − ki )|e xi (t)| + |a ji || f j (ey j (t))| i=1

+ +

m  j=1 n 



|b ji || f j (ey j (t − τ))| +

j=1 m 

−σ =

|˜ai j || f˜i (e xi (t))| +

i=1

ν j − (˜c j − k˜ j )|ey j (t)|

Dα0,t V(t) ≤ − σV(t), t ≥ 0.

 |b˜ i j || f˜i (e xi (t − τ))| . (14)

≤

n 

i=1

|e xi (t)|[−μi (ci − ki )] +

V(t) ≤ V(0)Eα (−σtα ), t ≥ 0.

+

n 

μi |a ji ||d j||ey j (t)|

i=1 j=1

m n  

≤

μi |b ji ||d j||ey j (t − τ)|

i=1 n 

+

  |ey j (t)|[−ν j (˜c j − k˜ j ) + ν j |˜ai j ||d˜i ||e xi (t)|

j=1

+

n

m

≤ −M

ν j |b˜ i j ||d˜i ||e xi (t − τ)|

μi |e xi (t)| + nPmax

m 

i=1

˜ −M

˜ max ν j |ey j (t)| + mP

j=1

=

n  i=1 m 

≤ ρ(

|ey j (t)|

n 

≤ ρ(

|e xi (t)|

ν j |ey j (0)|Eα (−σtα ).

(20)

n  i=1 n 

|e xi (t)| + |e xi (0)| +

m  j=1 m 

ν j |ey j (t)| |ey j (t)|)

j=1 m 

|ey j (0)|)Eα (−σtα ).

(21)

j=1

i=1

˜ + nPmax ]. ν j |ey j (t)|[−M νj j=1

Because the condition (10) holds in Theorem 1, we can get ρ(

n 

(15)

(

{μi |a ji |d j + μi |b ji |d j },

1≤ j≤m

max

(22)

n 

|e xi (t)| +

m 

|ey j (t)|) ≤ εEα (−σtα ) ≤ δ.

(23)

j=1

By Definition 8 and inequality (23), for the choosen linear feedback controllers, the master system (5) is said to be synchronized with the slave system (6) in finite-time if there exists ||e x0 || + ||ey0 || < ε then it implies ||e x (t)|| + ||ey (t)|| < δ. The proof is completed.

˜ = min {˜c j − k˜ j }, M 1≤i≤n,1≤ j≤m

|ey j (t)|) ≤ ρεEα (−σtα ) ≤ ρδ.

j=1

i=1

1≤i≤n

max

m 

That is

M = min {ci − ki }, 1≤i≤n,1≤ j≤m

|e xi (t)| +

i=1

where

˜ max = P

μi |e xi (t)| +

i=1

˜ max mP μi |e xi (t)|[−M + ] μi

Pmax =

ν j |ey j (t)|

j=1 m  j=1

i=1

j=1

m 

μi |e xi (0)| +

n 

j=1 i=1 n 

m 

Let ρ = max1≤i≤n,1≤ j≤m {μi , ν j }, we can obtain

j=1 i=1

n m  

μi |e xi (t)| +

i=1

i=1 j=1 m 

(19)

That is, for t ≥ 0,

m n  

i=1

(18)

Applying Lemma 2 in (18), it is easy to obtain

Based on Assumption 1, one can get Dα0,t V(t)

(17)

From (12) to (17), we can easily get



j=1 n 

{Qi , Q˜ j } < 0.

max

1≤i≤n,1≤ j≤m

{ν j |˜ai j |d˜i + ν j |b˜ i j |d˜i }.

Since the conditions (8) and (9) in Theorem 1 hold, it is not difficult to obtain

Next, for given fractional order α satisfying 1 < α < 2, the following basic lemmas are necessary for the subsequent theoretical discussion.

˜ max mP < 0, μi ˜ + nPmax < 0 Q˜ j = − M νj

Lemma 3.(De la Sen (2011)) The following properties hold. (1). There exist constants η1 ≥ 1, η2 ≥ 1 such that for any 0 < α < 1,

Qi = − M +

(16)

||Eα,1 (Λtα )|| ≤ η1 ||eΛt ||, ||Eα,α (Λtα )|| ≤ η2 ||eΛt ||,

According to (16), there exists a positive constant σ > 0 such 7

where Λ denotes matrix, and ||·|| describes any vector or induced matrix norm. (2). If α ≥ 1, then for β = 1, 2, α

where A = (ai j )n×m , A˜ = (˜ai j )m×n ,

||Eα,β (Λtα )|| ≤ ||eΛt ||.

B = (bi j )n×m , B˜ = (b˜ i j )m×n ,

C = diag{c1 , c2 , . . . , cn },

If Λ is a diagonal stability matrix, then there exists a constant  > 0 such that for t ≥ 0

K = diag{k1 , k2 , . . . , kn }, C˜ = diag{˜c1 , c˜ 2 , . . . , c˜ n }, K˜ = diag{k˜ 1 , k˜ 2 , . . . , k˜ n },

||Eα,β (Λtα )|| ≤ ||e−ςt ||, 0 < α < 1; ||Eα,β (Λtα )|| ≤ ||e−ςt ||, 1 ≤ α < 2,

e x (t) = [e x1 (t), e x2 (t), . . . , e xn (t)]T , ey (t) = [ey 1 (t), ey 2 (t), . . . , ey m (t)]T ,

where ς is the largest eigenvalue of the diagonal matrix Λ.

e x (0) = [e x1 (0), e x2 (0), . . . , e xn (0)]T , ey (0) = [ey 1 (0), ey 2 (0), . . . , ey m (0)]T ,

Remark 6. Obviously, Lemma 3 can be not only employed for general fractional-order α satisfying 0 < α < 1 but also applied in the problem of fractional-order systems with α satisfying 1 < α < 2.

f (ey (t)) = [ f (ey 1 (t)), f (ey 2 (t)), . . . , f (ey m (t))]T , f (ey (t − τ)) = [ f (ey 1 (t − τ)), f (ey 2 (t − τ)), . . . , f (ey m (t − τ))]T , f˜(e x (t)) = [ f˜(e x1 (t)), f˜(e x2 (t)), . . . , f˜(e xn (t))]T , f˜(e x (t − τ)) = [ f˜(e x1 (t − τ)), f˜(e x2 (t − τ)), . . . , f˜(e xn (t − τ))]T .

Lemma 4.(Ye, Gao & Ding (2007)) (Generalized GronwallBellman inequality). Let φ(t), ϕ(t) be nonnegative and locally integrable on 0 ≤ t ≤ T (some T ≤ +∞) and let ψ(t) be a nonnegative, nondecreasing continuous function defined on 0 ≤ t ≤ T , ψ(t) ≤ N, and let N be a real constant, α > 0 with  t φ(t) ≤ ϕ(t) + ψ(t) (t − ω)(α−1) φ(ω)dω, t ∈ [0, T ),

According to Assumption 1, it is easy to obtain Dα0,t e x (t) ≤ − (C − K)e x (t) + AD|ey (t)| + BD|ey (t − τ)|, (26) ˜ y (t) + A˜ D|e ˜ x (t)|) + B˜ D|e ˜ x (t − τ)|, (27) Dα0,t ey (t) ≤ − (C˜ − K)e where

0

D = diag{d1 , d2 , . . . , dn }, D˜ = diag{d˜1 , d˜2 , . . . , d˜n }.

and then 

+∞ t 

φ(t) ≤ ϕ(t) +

[

0

n=1

By applying the Laplace transform and the inverse Laplace transform in the above inequalities, one can get

(ψ(t)Γ(α))n (t − ω)(nα−1) ϕ(ω)]dω, t ∈ [0, T ), Γ(nα)

e x (t) ≤ Eα,1 ((K − C)tα )e x0 (0) + tEα,2 ((K − C)tα )e x1 (0)  t + (t − υ)α−1 Eα,α ((K − C)(t − υ)α )

Moreover, if ϕ(t) is a nondecreasing function on [0, T ), then φ(t) ≤ ϕ(t)E α (ψ(t)Γ(α)tα ), t ∈ [0, T ),

0

(28) {AD|ey (υ)| + BD|ey (υ − τ)|}dυ ˜ α )ey (0) + tEα,2 ((K˜ − C)t ˜ α )ey (0) ey (t) ≤ Eα,1 ((K˜ − C)t 0 1  t ˜ − s)α ) + (t − s)α−1 Eα,α ((K˜ − C)(t

where Eα (·) is one-parameter Mittag-Leffler function. Theorem 2. Under Assumption 1, for i = 1, 2, . . . , n, j = 1, 2, . . . , m, the master system (5) is synchronized with the slave system (6) in finite-time under the linear feedback controllers and the following inequality holds e−ξt (1 + t)εEα (ζΓ(α)tα ) ≤

δ , t ∈ [0, T ). ε

0

˜ x (s − τ)|}ds. ˜ x (s)| + B˜ D|e {A˜ D|e By employing Lemma 3, we obtain α

0

{||AD|ey (υ)| + BD|ey (υ − τ)|||}dυ,

= − (C − K)e x (t) + A f (ey (t)) + B f (ey (t − τ)), ˜ y (t) + A˜ f˜(e x (t)) + B˜ f˜(e x (t − τ)), = − (C˜ − K)e

˜

˜

α

˜

˜

(30)

α

ey (t) ≤ ||e(K−C)t ||||ey 0 (0)|| + t||e(K−C)t ||||ey 1 (0)||  t α ˜ ˜ + (t − s)α−1 ||e(K−C)(t−s) ||

e x (0) = e x0 , ey (0) = ey 0 , t ≥ 0,

α

e x (t) ≤ ||e(K−C)t ||||e x0 (0)|| + t||e(K−C)t ||||e x1 (0)||  t α + (t − υ)α−1 ||e(K−C)(t−υ) ||

(24)

Proof. After simple calculation, we can obtain the following equivalent vector form of (7) Dα0,t e x (t) Dα0,t ey (t)

(29)

0

˜ x (s − τ)|||}ds. ˜ x (s)| + B˜ D|e {||A˜ D|e

(25) 8

(31)

Since the matrices K − C and K˜ − C˜ are diagonal stability matrices, one can get the following inequalities according to Lemma 3 α

||Eα,β((K − C)tα )|| ≤ ||e(K−C)t || ≤ e−γt ,

(32)

˜ C)t ˜ α ˜ α )|| ≤ ||e(K− || ≤ e−ϑt , ||Eα,β ((K˜ − C)t

(33)

From (38) and (39), we have eγt ||e x (t)|| + eϑt ||ey (t)|| ≤ eξt (||e x (t)|| + ||ey (t)||) ≤ (||e x0 (0)|| + t||e x1 (0)||)  t + (t − υ)α−1 eξυ {||A||D||ey(υ)|| + ||B||D||ey(υ − τ)||}dυ 0

+ (||ey0 (0)|| + t||ey 1 (0)||)  t ˜ D||e ˜ x (s)|| + || B|| ˜ D||e ˜ x (s − τ)||}ds. (40) + (t − s)α−1 eϑs {||A||

where γ is the largest eigenvalue of the diagonal matrix K − C ˜ and ϑ is the largest eigenvalue of the diagonal matrix K˜ − C.

0

Denote ω(t) = supt−τ≤t¯≤t (||e x(t¯)||eξt ) and (t) supt−τ≤t˜≤t (||ey(t˜)||eξt ). Then, it is obvious that

Applying the inequalities (32) and (33) in (30) and (31), respectively, we have

||e x (s)||eξs ≤ ω(s), ||e x (s − τ)||eξ(s−τ) ≤ ω(s),

||e x (t)|| ≤ e−γt (||e x0 (0)|| + t||e x1 (0)||)  t + e−γt (t − υ)α−1 eγυ

||ey (υ)||eξυ ≤ (υ), ||ey(υ − τ)||eξ(υ−τ) ≤ (υ).

0

{||A||D||ey(υ)|| + ||B||D||ey(υ − τ)||}dυ,

From (40) to (41), we have ω(t) + (t) ≤ (||e x0 (0)|| + ||ey 0 (0)||) + t(||e x1 (0)|| + ||ey 1 (0)||)  t + (t − υ)α−1 eξυ {||A||D||ey(υ)|| + ||B||D||ey(υ − τ)||}dυ 0  t ˜ D||e ˜ x (s)|| + || B|| ˜ D||e ˜ x (s − τ)||}ds. (42) + (t − s)α−1 eϑs {||A||

0

0

(35)

˜ D˜ + eξτ || B|| ˜ D}. ˜ Let ζ = max1≤i≤n,1≤ j≤m {||A||D + eξτ ||B||D, ||A|| Then, from (42), we get

Multiplying on both sides of (34) and (35) by eγt and eϑt , respectively, we can obtain

ω(t) + (t) ≤ (||e x0 (0)|| + ||ey 0 (0)||) + t(||e x1 (0)|| + ||ey 1 (0)||)  t + (t − )α−1 ζ(ω() + ())d. (43)

eγt ||e x (t)|| ≤ (||e x0 (0)|| + t||e x1 (0)||)  t + (t − υ)α−1 eγυ

0

Denote φ = ω(t)+(t), ϕ(t) = ||e x0 (0)||+||ey0 (0)||+t(||e x1 (0)||+ ||ey 1 (0)||), ψ(t) = ζ. Then, by applying Lemma 4 in (43), we have  t φ(t) ≤ ϕ(t) + ψ(t) (t − ω)(α−1) φ(ω)dω, t ∈ [0, T ),

0

{||A||D||ey(υ)|| + ||B||D||ey(υ − τ)||}dυ,

(36)

eϑt ||ey (t)|| ≤ (||ey 0 (0)|| + t||ey 1 (0)||)  t + (t − s)α−1 eϑs

0

and then

0

˜ D||e ˜ x (s)|| + || B|| ˜ D||e ˜ x (s − τ)||}ds. {||A||



(37)

+∞ t 

φ(t) ≤ ϕ(t) +

[

0

Let ξ = max1≤i≤n,1≤ j≤m {γ, ϑ}. Then, we get

n=1

(ψ(t)Γ(α))n (t − ω)(nα−1) ϕ(ω)]dω, t ∈ [0, T ), Γ(nα)

Moreover, if ϕ(t) is a nondecreasing function on [0, T ), then we have

eγt ||e x (t)|| ≤ eξt ||e x (t)|| ≤ (||e x0 (0)|| + t||e x1 (0)||)  t + (t − υ)α−1 eξυ

φ(t) ≤ ϕ(t)E α (ψ(t)Γ(α)tα )

0

{||A||D||ey(υ)|| + ||B||D||ey(υ − τ)||}dυ,

(41)

(34)

||ey (t)|| ≤ e−ϑt (||ey 0 (0)|| + t||ey 1 (0)||)  t + e−ϑt (t − s)α−1 eϑs ˜ D||e ˜ x (s)|| + || B|| ˜ D||e ˜ x (s − τ)||}ds. {||A||

=

= ((||e x0 (0)|| + ||ey 0 (0)||) + t(||e x1 (0)|| + ||ey 1 (0)||))Eα(ζΓ(α)tα ).

(38)

That is eϑt ||ey (t)|| ≤ eξt ||ey (t)|| ≤ (||ey 0 (0)|| + t||ey 1 (0)||)  t + (t − s)α−1 eϑs

eξt (||e x (t)|| + ||ey (t)||) ≤ ((||e x0 (0)|| + ||ey 0 (0)||) + t(||e x1 (0)|| + ||ey 1 (0)||))Eα(ζΓ(α)tα ).

0

˜ D||e ˜ x (s)|| + || B|| ˜ D||e ˜ x (s − τ)||}ds. {||A||

(39)

Since the initial condition ||e x (t)|| + ||ey(t)|| < ε and the condi9

3. Numerical Examples

tion (24) in Theorem 2 hold, we obtain ||e x (t)|| + ||ey (t)|| ≤ e−ξt (1 + t)εEα (ζΓ(α)tα ) ≤ δ.

(44)

According to Definition 8 and the inequality (44), we can conclude that the master system (5) is said to be synchronized with the slave system (6) in finite-time for the given linear feedback controllers. The proof is completed.

In this section, two numerical examples are given to demonstrate the derived results. Example 1. Consider the two-dimensional master system (5) with     −0.4 0.5 0.5 −0.5 A= ,B= , 0.4 −2.5 0.6 0.8     −0.4 −0.8 0.51 −0.19 A˜ = , B˜ = , 0.5 −1 0.51 1.3  1 |y j + 1| − |y j − 1| , j = 1, 2, f j (y j ) = 20  1 |xi + 1| − |xi − 1| , i = 1, 2, f˜i (xi ) = 10 α = 0.98, τ = 1.

Remark 7. As authors in ([21]) proved that Mittag-Leffler synchronization implied asymptotic synchronization, the master system (5) is asymptotically synchronized with the slave system (6) in finite-time under the feedback controller by directly employing Lemma 3 in ([21]). Thus, the proof of the above relationship is omitted here. Remark 8. The choice of the connection weights of the master system (5) and the slave system (6) depends on their states, respectively. Actually, the values for the connection weights are not unique. Therefore, the master system (5) and the slave system (6) are called the state-dependent switching systems. If the connection weights cˆ i = cˇ i and cˆ˜ j = cˇ˜ j , the systems (5) and (6) are reduced to be FBAMNNs and the obtained criteria of this paper are also applicable for the finite-time Mittag-Leffler synchronization of FBAMNNs with time delays.

Let ⎧ ⎪ ⎪ ⎨0.99, ci (xi (t)) = ⎪ ⎪ ⎩1.01, ⎧ ⎪ ⎪ ⎨1.01, c˜ j (y j (t)) = ⎪ ⎪ ⎩0.99,

Remark 9. For the fractional-order α satisfying 0 < α < 1, based on Lyapunov theory, some criteria are obtained to achieve the finite-time synchronization of FMBAMNNs by mainly emoloying a new inequality about the Caputo derivative of absolute value function and a generalized Gronwall-like inequality.

|xi (t)| ≤ 1, i = 1, 2, |xi (t)| > 1, |y j (t)| ≤ 1, j = 1, 2. |y j (t)| > 1,

Consider the two-dimensional slave system (6) with the same parameters as the master system (5). By choosing μ1 = μ1 = 10, ν1 = ν2 = 8 and k1 = k2 = −14, k˜ 1 = k˜ 2 = −16 and letting ci (xi (t)) = c˜ j (y j (t)) = 1, j = 1, 2, from (8), we have

Remark 10. Given that fractional-order α satisfying 1 < α < 2, based on theory of vector norm, matrix norm and comparison theorem, a relaxed judgment condition is derived to realize the finite-time synchronization of FMBAMNNs by mainly applying the generalized Gronwall’s inequality.

Q1 = −14.5808 < 0, Q2 = −14.2640 < 0, Q˜ 1 = −16.7500 < 0, Q˜ 2 = −16.1750 < 0,

Remark 11. In this paper, based on Lyapunov method, some sufficient criteria in Theorem 1 are derived to tackle the problem of the finite-time Mittag-Leffler synchronization of FBAMNNs with fractional-order α satisfying 0 < α < 1. Because the obtained criteria are simple in form rather than linear matrix inequalities, they are more easily checked and contribute to reducing the computational burden.

Obviously, the conditions (8) and (9) of Theorem 1 hold. Next, under different initial conditions satisfying ||e x0 || + ||ey0 || < ε, the condition (10) of Theorem 1 still holds. Thus, the master system (5) is synchronized with the slave system (6) in finitetime for the given linear feedback controllers. Fig.2 describes the synchronization behaviors of different states under the linear feedback controllers. The state trajectories of the master system (5) and the slave system (6) are shown in Fig.3.

Remark 12. For fractional-order α satisfying 1 < α < 2, this paper provides a judgment condition for the problem of the finite-time Mittag-Leffler synchronization of FBAMNNs with time delays by employing some properties of the vector norm and matrix norm and applying comparison theorem in the corresponding norm. The derived condition in Theorem 2 is also easily verified and fast to be calculated.

Example 2. Consider the two-dimensional master system (5) with     0.5 0 0.5 0 , ,B= A= 0 0.5 0 0.5     0.5 0 0.5 0 , , B˜ = A˜ = 0 0.5 0 0.5

Remark 13. In this paper, for given different fractional-order α, different theoretical methods are applied in dealing with the corresponding problems of the finite-time Mittag-Leffler synchronization of FBAMNNs with time delays. The above applied methods can be popularized to settle other dynamical behaviors of fractional-order systems.

f j (y j ) = tanh(y j ), j = 1, 2, f˜i (xi ) = tanh(xi ), i = 1, 2, α = 1.5, τ = 0.5. 10

Let

The error states 50 e (t) x

|xi (t)| ≤ 1, i = 1, 2, |xi (t)| > 1,

Amplitude

⎧ ⎪ ⎪ ⎨0.90, ci (xi (t)) = ⎪ ⎪ ⎩1.11, ⎧ ⎪ ⎪ ⎨1.11, c˜ j (y j (t)) = ⎪ ⎪ ⎩0.90,

|y j (t)| ≤ 1, j = 1, 2. |y j (t)| > 1,

1

e (t) x

2

0

−50

0.5

0

2

1.5

1 Time t The error states

50 ey (t)

Consider the two-dimensional slave system (6) with the same parameters as the master system (5). We know τ(1.5) = 0.8862. By selecting k1 = k2 = −0.8, k˜ 1 = k˜ 2 = −0.8 and letting ci (xi (t)) = c˜ j (y j (t)) = 1, j = 1, 2, we verify that the condition (24) of Theorem 2 holds under different initial conditions satisfying ||e x0 || + ||ey0 || < ε. Thus, the master system (5) is synchronized with the slave system (6) in finite-time for the given linear feedback controllers. Fig.4 shows the synchronization behaviors of different states under the linear feedback controllers. The state trajectories of the master system (5) and the slave system (6) are demonstrated in Fig.5.

Amplitude

1

ey (t) 2

0

−50

0

0.5

1 Time t

1.5

2

Figure 3: State trajectories of the master system (5) and the slave system (6) with the feedback controller. The states of x1 and x11 0 Amplitude

x1 The states of x1 and x11

Amplitude

80 x1

60

x

−5

11

−10

x

11

−15

40

0

2

20

6 4 Time t The states of x and x 2

0.5

1 Time t The states of x and x 2

x

2

1.5

2

12

80 x Amplitude

12

0 0

Amplitude

0

10

8

2

60

x

−5

12

−10

x

12

−15

40

0

2

4

0

6

8

10

Time t

20 0

0.5

1 Time t

1.5

The states of y and y

2

1

11

0 Amplitude

y1 The states of y and y 1

11

Amplitude

80 y1

60

y

−5

11

−10

y

11

−15

40

0

2

20

4 6 Time t The states of y and y 2

0.5

1 Time t The states of y and y 2

1.5

Amplitude

12

y2

2

12

80 y2

60

y12

−5 −10

y12 −15

40

0

2

4

6

8

10

Time t

20 0

10

0 0

Amplitude

0

8

0

0.5

1 Time t

1.5

2

Figure 4: The synchronization behaviors of different states under the linear feedback controllers. Figure 2: The synchronization behaviors of different states under the linear feedback controllers.

to settle the corresponding problems of the finite-time MittagLeffler synchronization of FBAMNNs with time delays. The above applied methods can be popularized to settle other dynamical behaviors of fractional-order systems. Meanwhile, the obtained results can also be extended to some recently relative works such as networked control systems.

4. Conclusion This paper studies the problem of finite-time synchronization of a class of FMBAMNNs with time delays. For given different fractional-order α, different theoretical methods are employed 11

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The error states 5 e (t) Amplitude

x

1

e (t)

0

x

2

−5

−10

0

2

6 Time t The error states 4

10

8

Amplitude

2 ey (t) 1

0

ey (t) 2

−2 −4 −6

0

2

4

6

8

10

Time t

Figure 5: State trajectories of the master system (5) and the slave system (6) with the feedback controller.

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