Adaptive Fixed-Time Synchronization of Delayed Memristor-Based Neural Networks with Discontinuous Activations

1  Introduction

Memristor is nonlinear resistance having memory function, which represents relationship between magnetic flux and charge. Based on the connections between circuits, Chua proposed the existence of it in 1971 [1]. The first physical memristor is made by Hewlett-Packard Laboratory in 2008 [2]. Unlike ordinary resistors, memristors retain data after a power failure. At present, there are two main applications of memristors: data storage and class brain calculation. Due to the characteristics of small size, low power consumption and fast operation, memristor can be combined with artificial neural networks(NNs) to produce MNNs with more diverse functions and more complex structures. Research on MNNs or NNs have been going on, such as stability [3,4], H∞ synchronization [5], FTS [6], Mittag-Leffler synchronization [7] and so on.

Due to the chaotic characteristics of MNNs, it appeared in many fields, such as image protection [8] and random number generation [9]. Therefore, the study about synchronization of MNNs is particularly important. Exponential synchronization of MNNs with impulsive and perturbations were put forward by [10], and given a impulsive controller. Li et al. [11] proposed synchronization within finite-time of BAM MNNs and provided a switching control scheme. FTS of MNNs with fractional order was realized in [12] by using a fractional sliding mode control scheme with no singular terms.

In engineering practice, people always expect the system to realize synchronization as soon as possible. Finite time synchronization attracted people’s attention because of its fast error convergence rate and robustness [13,14]. Finite-time synchronization can make the error go to zero for a short time. So many of research on finite-time synchronization. Hua et al. [15] used delay-independent controller to settle synchronization of MNNs with inertial item in finite time without using variable transformation while studing from MNNs themselves. Shi et al. [16] put forward finite-time synchronization of Cohen-Grossberg MNNs and divide error into two states. What needs to be noted is that finite-time synchronization is connected with initial value but it is difficult to acquire for large or complex systems. So this is very limited for applications of finite-time synchronization. While, the limitation was solved by FTS which proposed by [17]. Different from finite-time synchronization, the ST of FTS is not connected with initial values, which greatly improves the application range of FTS in complex dynamicalnetworks [18] and MNNs. So more and more scholars began to study the FTS of MNNs or NNs in recent years. Based on a new proposed lemma, Li et al. [19] realized the fixed time synchronization problem of coupled NNs, and simultaneously the discontinuity and parameters mismatch of the system were considered. And Ren et al. [20] put forward the FTS of stochastic NNs and got the synchronization criteria. Besides, many achievements about FTS of complex-valued and quaternion-valued MNNs or NNs had also been made in [21,22].

The connection weights of MNNs are switchable, so it belongs to a switching system, which is of great significance to the research of switching systems [23]. Meanwhile, it is known that valid control tactics can make the synchronization faster, more stable and easier to meet the needs of practical production applications. Sliding mode control is an effective control method to suppress interference and ensure the stable operation of the nonlinear system [24]. Under the sliding model control method, Xiao et al. [25] realized the FTS of NNs. In addition, with the aid of event-triggered algorithm, Bao et al. [26] studied the synchronization of MNNs. Adaptive control is a resultful control schemes [27,28], it has good robustness and has effective suppression of external interference. Especially when the system has unknown parameters and external disturbance, the adaptive control algorithm can track them, so as to ensure the stable operation of the system. Time delay is universal for nonlinear systems. Especially in engineering practice, when the system signal transmission volume is large, it will lead to channel congestion, resulting in discrete time delay, and when there are parallel channels in the signal transmission path, it will lead to distributed time delay. Time delay will affect the stability of the system and hinder the synchronization of the system. In addition, systems with time delay have complex chaotic characteristics and show complex dynamic behaviors, which are very worthy of study. While some previous literatures [6,12,20,25,26] only considered the constant delay or discrete delay but ignored the distributed delay. For NNs, the activation function realizes the nonlinear transformation of data and solves some problems that the linear model cannot solve. Activation functions are mainly divided into continuous and discontinuous types, but discontinuous activation functions are seldom considered by researchers due to the complexity of analysis.

From above discussion, we find that few people have studied FTS of MNNs, especially the case with mixed delays and discontinuous activation functions. Hence, inspired by above conditions, the adaptive FTS of delayed MNNs with discontinuous activations are studied in this study. And our contributions are enumerated in following aspects. (1) Complex MNNs model with mixed delay and discontinuous activations is considered. (2) A feedback control scheme and an adaptive control algorithm are proposed for continuous and discontinuous activations, respectively, and the FTS criterion is obtained. Besides, the results are extended to finite-time synchronization. (3) The ST is not affected by the initial value of system and can be adjusted by the controller and system parameters.

2  Model Formulations and Preliminaries

The drive system of MNNs is

ẋi(t)=-zixi(t)+∑j=1nkij(xi(t))fj(xj(t))+∑j=1nlij(xi(t))gj(xj(t-Δ(t)))+∑j=1nmij(xi(t))∫t-w(t)tfj(xj(s))ds+Ii.(1)

where x(t)=(x1(t),...,xn(t))T∈Rn, zi > 0 means rate for ith neuron returns to its idle state and i, j = 1, 2, ..., n . kij(⋅),lij(⋅),mij(⋅) are connection weights. fj(⋅) and gj(⋅) are activation functions with or without time delay respectively. Δ(t) delegates discrete time delay and 0≤Δ(t)≤Δ1,0≤Δ°(t)≤Δ2<1. w(t) is distributed time delay and 0≤w(t)≤w1, where Δ1,Δ2,w1 are constants. Ii(t) is external input vector. The starting condition of (1) is xi(s)=φi(s)∈C([-τ,0],Rn),τ=max{Δ1,w1}. And kij(xi(t))=PijCi×signij,lij(xi(t))=P′ijCi×signij,mij(xi(t))=P″ijCi×signij where signij = 1 when i = j , signij = −1 when i≠j, and Ci denotes capacitor which voltage is xi(t). We define the the memristors between fj(xj(t)) and xi(t), gj(xj(t-Δ(t))) and xi(t) , ∫t-w(t)tfj(xj(s))ds and xi(t) are Qij, Q′ij, Q″ij severally and which resistances are Pij, P′ij, P″ij, respectively.

There exist constants k′ij, k″ij, l′ij, l″ij, m′ij, m″ij, such that

kij(xi(t))={k′ij,|xi(t)|≤Lj,k″ij,|xi(t)|>Lj,

lij(xi(t))={l′ij,|xi(t)|≤Lj,l″ij,|xi(t)|>Lj,

mij(xi(t))={m′ij,|xi(t)|≤Lj,m″ij,|xi(t)|>Lj,

where K[A] denotes the convex closure of the sets A, co¯[ε1,ε2] is the convex closure generated by real numbers ε1 and ε2, L j indicates switching threshold and L j > 0. Due to the solutions to MNNs are in the sense of Filippov, so set-valued mappings and differential inclusions [29] are used to deal with the discontinuity of MNNs.

Recur to set-valued mappings, it acquires

K[kij(xi(t))]={k′ij,|xi(t)|≤Lj,co¯{k̄ij,kij*},|xi(t)|=Lj,k″ij,|xi(t)|>Lj,

K[lij(xi(t))]={l′ij,|xi(t)|≤Lj,co¯{l̄ij,lij*},|xi(t)|=Lj,l″ij,|xi(t)|>Lj,

K[mij(xi(t))]={m̄ij,|xi(t)|≤Lj,co¯{m̄ij,mij*},|xi(t)|=Lj,m̃ij,|xi(t)|>Lj,

where co¯ delegates convex closure, we set k̄ij= max{k′ij,k″ij},kij*= min{k′ij,k″ij}, k̄ij= max{|k′ij|,|k″ij|}, l̄ij= max{l′ij,l″ij}, lij*= min{l′ij,l″},l̄ij= max{|l′ij|,|l″ij|}, m̄ij= max{m′ij,m″ij}, mij*= min{m′ij,m″ij}, m̄ij= max{|m′ij|,|m″ij|}.

If the activation functions are continuous, one can obtain

ẋi(t)∈-zixi(t)+∑j=1nK[kij(xi(t))]fj(xj(t))+∑j=1nK[lij(xi(t))]gj(xj(t-Δ(t)))+∑j=1nK[mij(xi(t))]∫t-w(t)tfj(xj(s))ds+Ii,(2)

then we set k̃ij(xi(t))∈K[kij(xi(t))],l̃ij(xi(t))∈K[lij(xi(t))],m̃ij(xi(t))∈K[mij(xi(t))], so (2) is rewritten to

ẋi(t)=-zixi(t)+∑j=1nk̃ij(xi(t))fj(xj(t))+∑j=1nl̃ij(xi(t))gj(xj(t-Δ(t)))+∑j=1nm̃ij(xi(t))∫t-w(t)tfj(xj(s))ds+Ii.(3)

There also exist k^ij(yi(t))∈K[kij(yi(t))],l^ij(yi(t))∈K[lij(yi(t))],m^ij(yi(t))∈K[mij(yi(t))] for response system, such that

ẏi(t)=-ziyi(t)+∑j=1nk^ij(yi(t))fj(yj(t))+∑j=1nl^ij(yi(t))gj(yj(t-Δ(t)))+∑j=1nm^ij(yi(t))∫t-w(t)tfj(yj(s))ds+Ii+ui(t).(4)

where the ui(t) will be introduced below and starting value of (4) is yi(s)=ϕi(s)∈C([-τ,0],Rn).

The error is ei(t)=yi(t)-xi(t), so we have

ėi(t)=-ziei(t)+∑j=1nF^ij(t)+∑j=1nĜij(t)+∑j=1nF̃ij(t),(5)

where F^ij(t)=k^ij(yi(t))fj(yj(t))-k̃ij(xi(t))fj(xj(t)), Ĝij(t)=l^ij(yi(t))gj(yj(t-Δ(t)))-l̃ij(xi(t))gj(xj(t-Δ(t))), F̃ij(t)=m^ij(yi(t))∫t-w(t)tfj(yj(s))-m̃ij(xi(t))∫t-w(t)tfj(xj(s)).

If activation functions are discontinuous, we set f⌣j(xj)∈K[ fj(xj) ],g⌣j(xj)∈K[ gj(xj) ],f⌣j(yj)∈K[ fj(yj) ],g⌣j(yj)∈K[ gj(yj) ].

The error is calculated as

ėi(t)=-ziei(t)+∑j=1nFij*(t)+∑j=1nGij*(t)+∑j=1nFij**(t),(6)

where Fij*(t)=k^ij(yi(t))f⌢j(yj(t))−k˜ij(xi(t))f⌣j(xj(t)), Gij*(t)=l^ij(yi(t))g⌢j(yj(t−Δ(t)))−l˜ij(xi(t)) g⌣j(xj(t−Δ(t))),Fij**(t)=m^ij(yi(t))∫t−w(t)tf⌢j(yj(s))ds−m˜ij(xi(t))∫t−w(t)tf⌣j(xj(s))ds.

Definition 1. [30] For any original value, the FTS of MNNs will be attained if the ST function T(e0(s)) satifies

limt→T(e0(s))∥e(t)∥=0;∥e(t)∥=0,∀t≥T(e0(s));

T(e0(s))≤Tmax.

where Tmax is a constant.

Lemma 1. [31] (1) and (4) will attain FTS if a continuous radically unbounded function V(⋅): Rn→R+∪{0} satifies

(i)   V(c)=0⇔c=0,

(ii)   V°(c(t))≤-d1Vr1(c(t))-d2Vr2(c(t))

for any solution c(t).

where d1 > 0, d2 > 0, 0 < r1 < 1, r2 > 1. And the ST is calculated as T(c0)≤Tmax=1d1(1-r1)+1d2(r2-1),∀c0∈Rn.

Lemma 2. [32] There are some inequalities as follows: ∑i=1npiϑ1≥(∑i=1npi)ϑ1,∑i=1npiϑ2≥n1-ϑ2(∑i=1npi)ϑ2, where pi>0,0<ϑ1≤1,ϑ2>1.

3  Main Results

The FTS of MNNs with continuous and discontinuous activations will be studied in this section.

3.1 The Continuous Activations for FTS of MNNs

Lemma 3. [33] If ∀k̃ij(x)∈K[kij(x)],∀l̃ij(x)∈K[lij(x)], ∀k^ij(y)∈K[kij(y)],∀l^ij(y)∈K[kij(y)], the following inequalities are obtained

|k^ij(y)fj(y)-k̃ij(x)fj(x)|≤←kijhj|y-x|,

|l^ij(y)gj(y(t-Δ(t)))-l̃ij(x)gj(x(t-Δ(t)))|≤←lijρj|y(t-Δ(t))-x(t-Δ(t))|.

where hj,ρj are constants.

To realize FTS, control algorithm is given as

ui(t)=-μ1iei(t)-sign(ei(t))(μ2i+μ3i|ei(t)|α+μ4i|ei(t)|β+∑j=1nμ5i|ej(t-Δ(t))|),(7)

where μ1i to μ5i are constants, sign(⋅) is standard sign function and 0<α<1,β>1.

Theorem 1. Under the above assumptions and control scheme (7), the FTS of (1) and (4) will be carried out if the following conditions hold

{ μ1i≥∑j=1n(k←ij+k←ijhj22)−zi,μ2i≥2∑j=1nm←ijw1f¯j,μ5i≥max1≤j≤n(l←ijρj).(8)

and it can obtain

T1≤2q1(1-α)+2(β-1)q2,(9)

where q1= min1≤i≤n{μ3i}2α+12,q2= min1≤i≤n{μ4i}n1-β22β+12.

Proof. Construct Lyapunov function as

V(t)=12∑i=1neT(t)e(t).(10)

Along the error system (5), the derivative of (10) can be obtained as

V°(t)=∑i=1neT(t)[-ziei(t)+∑j=1nF^ij(t)+∑j=1nĜij(t)+∑j=1nF̃ij(t)-sign(ei(t))(μ1i+μ2i|ei(t)|+μ3i|ei(t)|α+μ4i|ei(t)|β+∑j=1nμ5i|ej(t-Δ(t))|)](11)

By means of Assumption 1 and Lemma 3, we yield

∑i=1neT(t)∑j=1nF^ij(t)≤∑i=1n| ei(t) |∑j=1nk←ijhj| ej(t) |       ≤∑i=1n∑j=1nk←ij| ei(t) |22+∑i=1n∑j=1nk←ijhj2| ej(t) |22,(12)

and

∑i=1neT(t)∑j=1nG^ij(t)≤∑i=1n| ei(t) |∑j=1nl←ijρj| ej(t−Δ(t)) |       ≤∑i=1n∑j=1nl←ijρj| ei(t) || ej(t−Δ(t)) |,(13)

One can also get

∑i=1neT(t)∑j=1nF˜ij(t)≤2∑i=1n∑j=1nm←ijw1f¯j| ei(t) |.(14)

Substitute (12)–(14) into (11), we have

V˙(t)≤∑j=1n{ [ −zi+∑j=1nk←ij+k←ijhj22−μ1i ]ei2(t)  +( 2∑j=1nm←ijw1f¯j− μ2i )| ei(t) |+∑j=1n(l←ijρj−μ5i)| ei(t) || ej(t−Δ(t)) |  −μ3i| ei(t) |α+1−μ4i| ei(t) |β+1 }.(15)

According to Throrem 1 and Lemma 2, it has

V°(t)≤-∑i=1nμ3i|ei(t)|α+1-∑i=1nμ4i|ei(t)|β+1≤-min1≤i≤n{μ3i}2α+12(V(t))α+12-min1≤i≤n{μ4i}2β+12n1-β2(V(t))β+12=-q1(V(t))α+12-q2(V(t))β+12,(16)

where q1= min1≤i≤n{μ3i}2α+12,q2= min1≤i≤n{μ4i}n1-β22β+12, and it yields

T1≤2q1(1-α)+2(β-1)q2.(17)

Remark 1. Especially, when α=1-12δ,β=1+12δ, the ST is expressed as

T2≤Tmax=πδq3q4n1-β22α+β+22,(18)

where q3= min1≤i≤n{μ3i},q4= min1≤i≤n{μ4i} and δ>1.

Corollary 1. Under the below control algorithm, (1) and (4) can reach to finite-time synchronization when Assumption 1 and following conditions hold:

ui(t)=-κ1iei(t)-sign(ei(t))(κ2i+κ3i|ei(t)|α̃+∑j=1nκ4i|ej(t-Δ(t))|),(19)

and

{ κ1i≥∑j=1n(k←ij+k←ijhj22)−zi,κ2i≥2∑j=1nm←ijw1f¯j,κ4i≥max1≤j≤n(l←ijρj),(20)

where κ1i to κ4i are constants and 0<α̃<1, and it can obtain

T3=(V(0))1-α̃2min1≤i≤n{κ3i}2α̃+121-α̃2.(21)

Remark 2. Unlike the exponential or asymptotic [7–9,27,28] which time to synchronization is long or infinite, the FTS has faster convergence. In addition, the finite-time synchronization[14,26,30] has a fast convergence rate but relies on initial values of the system greatly. However, the starting values of some large and complex systems are hard to know, but FTS overcomes this problem and it can be used widely.

3.2 The Discontinuous Activations for FTS of MNNs

Assumption 1. fv,gv:R→R are continuous apart from a countable set of ioslate points {θε}. The left limit fv{θε-} and right limit fv{θε+} of fv exists. Meanwhile, the left limit gv{θε-} and right limit gv{θε+} of gv exists, and the discontinuous activations fv(λ̃),gv(λ̃) saify K[fv(λ̃)]=[min{fv(λ̃-),fv(λ̃+)},max{fv(λ̃-),fv(λ̃+)}], K[gv(λ̃)]=[min{gv(λ̃-),gv(λ̃+)},max{gv(λ̃-),gv(λ̃+)}].

Assumption 2. There are constants γv,γ′v,ςv,ς′v, such that sup| f⌢v(χ˜)−f⌣v(λ˜) |≤γv| χ˜−λ˜ |+ςv, sup| g⌢v(χ˜)−g⌣v(λ˜) |≤γ′v| χ˜−λ˜ |+ς′v, where f⌣v(λ˜)∈K[ fv(λ˜) ],f⌢v(χ˜)∈K[ fv(χ˜) ],g⌣v(λ˜)∈K[ gv(λ˜) ],g⌢v(χ˜)∈K[ gv(χ˜) ].

According to (6), one can get

ėi(t)=-ziei(t)+∑j=1nk^ij(yi(t))[f⌢j(yj(t))-f⌣j(xj(t))]+∑j=1n[k^ij(yi(t))-k̃ij(xi(t))]f⌣j(xj(t))+∑j=1nl^ij(yi(t))[g⌢j(yj(t-Δ(t)))-g⌣j(xj(t-Δ(t)))]+∑j=1n[l^ij(yi(t))-l̃ij(xi(t))]g⌣j(xj(t-Δ(t)))+∑j=1n[m^ij(yi(t))∫t-w(t)tf⌢j(yj(s))ds-m̃ij(xi(t))∫t-w(t)tf⌣j(xj(s))ds]+ui(t). (22)