Synchronization of Caputo fractional neural networks with bounded time variable delays

1 Introduction

Nowadays, the synchronization of fractional-order delayed neural networks has attracted more and more attention. Some synchronization results have been obtained, for instance, in [1,2,3], where the authors studied the synchronization of fractional-order memristor-based neural networks with delay. The problem of synchronization of the fractional neural network with delay is studied in [4]. In [5,6], the synchronization of fractional-order neural networks with multiple time delays is investigated. Note that in most of the known literature models, the connection weights are constants and the controllers are proportional to the error with a constant.

Motivated by the above discussions, the main goal of this paper is to study synchronization of neural networks with delay and with the Caputo fractional derivative. Note that the studies in this paper are more general than the existing ones in the literature. We study the general case of time-varying self-regulating parameters of all units and also time-varying functions of the connection between two neurons in the network. For example, in [4] the model is considered in the case all the connection weights are constants, the self-inhibition rate is described by a constant, and for a very special output depending on the Lipschitz constants of the activation functions. Also, we are applying Mittag-Leffler function with one parameter, which is deeply connected with the applied fractional derivative, and on the other side is a generalization of the exponential function applied in [1,5,7]. The study in this paper is based on the application of Lyapunov functions. To deal with the presence of the delay, we apply the Razumikhin method. By constructing an appropriate Lyapunov function, applying a special type of its derivative (as introduced in [8]), and a comparison principle with time delay, we obtain some sufficient conditions to ensure synchronization in the model. The theoretical results are illustrated in an example.

The main contributions of this paper could be summarized as follows:

• We include in the model the Mittag-Leffler function in both the self-inhibition rate and the connection weights, which is more appropriate when we are modeling with a Caputo fractional derivative.

• We consider the variable coefficients in both the self-inhibition rate and the connection weights.

• We study the bounded variable in time delay.

• We consider output coupling controller depending on the delay and with variable in time input matrices.

• We apply the Razumikhin method and Dini fractional derivatives of Lyapunov functions, which are connected with both the presence of the delay and Caputo fractional derivative in the system.

2 System description

We will consider the model based on an analog circuit, consisting of capacitors, resistors, and amplifiers. The input–output characteristics of the amplifiers in the circuits are modeled by some functions known as activation functions. In the literature, these functions are assumed to satisfy a wide variety of assumptions, mainly they are Lipschitz functions.

Initially, we will give some basic definitions from fractional calculus. We fix a fractional order q ∈ ( 0 , 1 ) everywhere in the paper, as it is usually done in many applications in science and engineering (for some physical interpretation of the fractional order see, for example, [9]).

1. The Riemann-Liouville fractional derivative of order q ∈ ( 0 , 1 ) of m ( ⋅ ) is given by ([10])

   D t q 0 R L m ( t ) = 1 Γ ( 1 − q ) d d t ∫ 0 t ( t − s ) − q m ( s ) d s , t ≥ t 0 ,

   where Γ ( . ) denotes the Gamma function.

2. The Caputo fractional derivative of order q ∈ ( 0 , 1 ) is defined by ([10])

   D t q 0 C m ( t ) = 1 Γ ( 1 − q ) ∫ 0 t ( t − s ) − q m ′ ( s ) d s , t ≥ 0 .

   We recall that the Mittag-Leffler function with one parameter is defined as

   (1) E q ( z ) = ∑ i = 0 ∞ z i Γ ( 1 + q i ) , q > 0 , z ∈ C .

In our further investigations, we will use the following notations: ∥ x ∥ = ∑ i = 1 n x i 2 is the Euclidean norm of the vector x = ( x 1 , x 2 , … , x n ) , the set C ( J , R n ) of all continuous functions u : J → R n , J ⊂ R + and for any function ϕ ∈ C ( [ − r , 0 ] , R n ) , we define ∥ ϕ ∥ 0 = sup s ∈ [ − r , 0 ] ∥ ϕ ( s ) ∥ , where r > 0 is a given number (it will be connected with the delay).

Consider the general model of neural networks with time variable bounded delay involving the Mittag-Leffler function with one parameter E q ( ⋅ ) defined above:

for i = 1 , 2 , … , n , where

• n represents the number of neurons in the network,

• x i ( t ) is the pseudostate variable denoting the average membrane potential of the i th neuron at time t ,

• x ( t ) = ( x 1 ( t ) , x 2 ( t ) , … , x n ( t ) ) ∈ R n , and p > 0 , q ∈ ( 0 , 1 ) are constants,

• τ ∈ C ( R + , [ − r , ∞ ) ) , − r ≤ τ ( t ) ≤ t for t ≥ 0 denotes the communication delay of the neuron,

• f j ( x j ( t ) ) , g j ( x j ( τ ( t ) ) ) denote the activation functions of the neurons at time t and τ ( t ) , respectively, and they represent the response of the j th neuron to its membrane potential and define f ( x ) = ( f 1 ( x 1 ) , f 2 ( x 2 ) , … , f n ( x n ) ) and g ( x ) = ( g 1 ( x 1 ) , g 2 ( x 2 ) , … , g n ( x n ) ) ,

• I = ( I 1 , I 2 , … , I n ) is an external bias vector, C ( t ) = diag ( c 1 ( t ) , c 2 ( t ) , … , c n ( t ) ) , and c i ( t ) E q ( p t q ) is the self-inhibition rate of i th neuron (the rate at which the i th neuron returns to a resting state without being connected to the neural network),

• matrix A ( t ) = { a i j ( t ) } n × n and a i j ( t ) E q ( p t q ) denote the connection weights (the influential strength of the j th neuron to the i th neuron) at the current time t ),

• matrix B ( t ) = { b i j ( t ) } n × n and b i j ( t ) E q ( p t q ) indicate the delayed connection weights (the influential strength of the j th neuron to the i th neuron at delay time τ ( t ) ).

We set up the following initial condition to model (2):

where ϕ i ∈ C ( [ − r , 0 ] , R n ) , and let ϕ = ( ϕ 1 , ϕ 2 , … , ϕ n ) .

Note that model (2) is more general than the existing ones in the literature. For example, the synchronization of model (2) is studied in [4] in the case all the connection weights are constants, the self-inhibition rate is described by a constant, and for a very special output depending on the Lipschitz constants of the activation functions.

A similar model is studied in [13], but the connection weights are bounded, which is not satisfied in this case because of the presence of Mittag-Leffler function.

System (2) is considered as the driven system, and the corresponding response system (slave system) is described by the following fractional-order differential equations:

with initial condition

for i = 1 , 2 , … , n , where y = ( y 1 , y 2 , … , y n ) denotes the state variable of the response system (4) and u = ( u 1 , u 2 , … , u n ) indicates the synchronous controller to be designed.

The main goal of the paper is to implement appropriate controllers u i ( t ) , i = 1 , 2 , … , n for the response system, such that the controlled response system (2) and (3) could be synchronized with the drive system (4).

One approach to study synchronization problems is based on applying Lyapunov-like functions. Thus, we start by defining what is a Lyapunov function and then its derivative among the fractional equation.

In connection with the Caputo fractional derivative, it is necessary to define in an appropriate way the derivative of the Lyapunov functions among a nonlinear Caputo fractional differential equation

where the function G : R + × R n × R n → R n , G = ( G 1 , G 2 , … , G n ) .

The studied model is a partial case of the nonlinear system (6). Some existence and uniqueness results of global solutions of Caputo fractional differential equations are given in [12].

In this paper, we will use Dini fractional derivative of the Lyapunov function V ∈ Λ ( J , R n ) among the fractional system (6), introduced in [8] by

where ψ ∈ C ( [ − r , 0 ] , R n ) , t h denotes the integer part of the fraction t h , and C s q = q ( q − 1 ) … ( q − s + 1 ) s ! .

We will illustrate the application of the above defined Dini fractional derivative among the fractional system (6) to a particular Lyapunov function.

Let ψ ∈ C ( [ − r , 0 ] , R n ) with ψ = ( ψ 1 , ψ 2 , … , ψ n ) . Then, the Dini fractional derivative of the Lyapunov function V among the fractional system (6) is

Using the formula

in (8), we obtain

In the sequel, we will use the following comparison result:

3 Synchronization with output coupling controller depending on delay

The controller with time delay in the response system can be taken as output coupling in the following way:

where E q ( p t q ) Ξ ( t ) = { E q ( p t q ) ξ i j ( t ) } and E q ( p t q ) ϒ ( t ) = { E q ( p t q ) η i j ( t ) } denote the control income matrices at current time t and delay time τ ( t ) , respectively.

Define the error function e ( t ) = x ( t ) − y ( t ) , where x ( ⋅ ) , y ( ⋅ ) are solutions of the initial value problems for the driven system (2), (3) and for the response system (4), respectively, with controller defined by (11). Then, the synchronization of system (4) is equivalent to asymptotic stability of the zero solution of the system for the error functions given by

where

We assume the following:

Assumption A1. The neuron activation functions are Lipschitz, i.e., there exist positive numbers L i , M i , i = 1 , 2 , … , n , such that

for i = 1 , 2 , … , n , and for all u , v ∈ R .

Assumption A2. Functions a i , j , b i , j , ξ i , j , η i , j ∈ C ( R + , R ) and there exist positive constants H i j , K i j , P i j , Q i j , i , j = 1 , 2 , … , n , such that

and

Assumption A3. Functions c i ∈ C ( R + , ( 0 , ∞ ) ) , i = 1 , 2 , … , n , are such that c i ( t ) ≥ C ( t ) > 0 , for t ≥ 0 , where C ∈ C ( [ 0 , ∞ ) , ( 0 , ∞ ) ) and

where the constants L i , M i , i = 1 , 2 , … , n , and H i j , K i j , i , j = 1 , 2 , … , n are defined in Assumptions A1 and A2, respectively.