Delay-coupled fractional order complex Cohen-Grossberg neural networks under parameter uncertainty: Synchronization stability criteria

1 Department of Mathematics, Near East University TRNC, Mersin 10, Turkey 2 Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi-630 004, India 3 Department of Mathematics, Cankaya University, Ankara 06530, Turkey 4 School of Mathematics, Southeast University, Nanjing 210096, China, and Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea 5 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, 410114, China 6 Faculty of Automatic Control, Electronics and Computer Science, Department of Automatic Control, and Robotics, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland


Introduction
In 1695, the foundation of non-integer order calculus become first of all discussed through Guillaume de Leibnitz and Gottfried Wilhelm Leibnitz, and its development were very gradual for long period. In practice, many real-world objects need to be described by fractional order models due to the fact dynamics of fractional order models are more accurate than integer order models [14][15][16][17][18]. Currently, fractional order calculus has been very promising areas of research and thus successfully applied in numerous fields such as circuit systems [9], market dynamics [21], biological models [26], dielectric polarization [28], and so on. Recently, many researchers have investigated asymptotic behavior of fractional-order dynamical systems and some interesting and important results were accounted. In [12], the authors discussed the asymptotic stability of fractional order time delayed systems based on the fractional order Halanay inequality. In [33], the author studied the global asymptotic stability of Caputo-Liouville generalized fractional order electrical RLC circuit model based on the Lyapunov stability theory.
Moreover, fractional order calculus has been integrated into artificial neural networks, and fractional order neural network (FNNs) are a kind of potentially applicable networks. In past few years, there has been an increasing interest in the investigation of dynamical behaviours of neural networks, and some important scientific results were obtained [4-7, 37, 39]. In [1], authors discussed the stability and synchronization of memristive FNNs with multiple delays, comparison principle, set valued maps and the Lyapunov function method were utilized to assure the stability and synchronization of memristive FNNs. In [7], synchronization with discontinuous activation have been discussed. Differential inclusion theory, Lyapunov stability theory and some novel sufficient criteria were applied to ensure the finite-time stabilization for the addressed model. In [42], authors studied the stability of fractional-order multiple time varying delayed competitive type neural networks. By means of Lyapunov method and graph theory techniques, some novel conditions were derived to achieve global asymptotic stability for the addressed model.
In [43], authors studied the synchronization of FNNs in complex field. Based on the fractional comparison theorem, some novel conditions were obtained to achieve asymptotical synchronization for the addressed model. cannot be acquired. Therefore, it is necessary and significant to research the fractional order CNNs with uncertain parameters and very few consequences on studies were paid. For example, by means of Kronecker product, Mittag-Leffler function and Lyapunov stability theory, Shuxue et al. [34] considered several synchronization conditions which can ensure the asymptotical synchronization of complex structure on fractional order CNNs with uncertain parameters and without time delays. On the other hand, time delays are not omitted into the dynamical behavior of neural networks, which can lead to the system oscillation, instability behaviors and divergence owing to the finite switching speed of amplifiers. Therefore, it is much more important to consider time delays in studying in investigating synchronization of fractional order CNNs and some remarkable results on this topic have been paid in existing literature. For example, Zhang et al. [46] researched synchronization stability of fractional order complex CNNs with coupling delays and several conditions to ensure the synchronization stability of complex CNNs were established based on Riemann-Liouville fractional order derivative properties, LMI approach, and Lyapunov theory.
Cohen-Grossberg neural networks (CGNNs) is a standout amongst the most renowned type and its special case of Hopfield type neural networks, which became first of all originated by means of Cohen and Grossberg in 1983 [3]. In recent years, CGNNs have received growing attention due to their widespread application in different areas, such as secure communications, nonlinear optimization problems, image processing, and parallel computation. As a type of FNN, fractional order CGNNs dynamical behavior has been extensively investigated by many researchers and some excellent results have been devoted to fractional order CGNNs, see Ref [30,38]. On the other hand, the result of complex coupled Cohen-Grossberg neural networks is more complicated and unpredictable dynamical behaviors than different forms of CNNs. Owing to the complex structure of CGNNs, there is few works published on synchronization analysis of coupled CGNNs [35,45]. The authors of [35] proposed some criteria to ensure the synchronization criteria in finite time issues of integer order CGNNs with linear coupling delays and nonlinear coupling delays. In [45], the authors derived several criteria which can guarantee the synchronization criteria in fixed time issues for integer order CGNNs with delayed couplings. To the best of our knowledge, nevertheless, asymptotical synchronization stability of fractional order coupled Cohen-Grossberg neural networks with and without parameter uncertainty has not yet been investigated.
Sparked by the above reason and discussion, the main aim is to study the synchronization stability analysis of fractional order coupled complex interconnected Cohen-Grossberg neural networks under with and without parameter uncertainties under linear coupling delays. The main contributions of this work are indexed as pursues: 1). The complex interconnected fractional order coupled Cohen-Grossberg neural networks model with and without parameter uncertainties are presented in the first time. 2). By using fractional-order stability theory, a new fractional-order comparison theorem for multiple delayed fractional order linear system is established and it is improved those in the existing works literature. 3). Several sufficient criteria in voice LMI techniques for synchronization stability and robust synchronization stability are established theoretically via proposed fractional order comparison theorem. 4). Our proposed synchronization stability results are enhancing the present fractional order Cohen-Grossberg time delayed neural networks and integer-order coupled neural networks. 5). Moreover, the presented results in this paper are also still valid for the synchronization stability of delay-coupled integer order complex Cohen-Grossberg neural networks with and without parameter uncertainties, and these results do not discuss in the previous works of literature.
Notations. In this paper, N signify the space of natural numbers from 1 to n, R n stands for the space of n-D Euclidean space, respectively, and R n×n stands for a set of all n × n real matrices. ⊗ means the Kronecker product of two matrices. λ M (·) and λ m (·) denote the maximum and minimum eigenvalues of the corresponding matrix. I n represent the identity matrix with n dimensions. C([−δ, 0], R n ) signifies the set of all continuous functions from [−δ, 0] to R n , where δ > 0. For m(t) = (m 1 (t), ..., m n (t)) T ∈ R n , we denote

Preliminaries and problem model formulation
In this part, some basic knowledge of fractional order calculus, some useful lemma's, problem statement and some necessary assumptions will be given.

Equation (2.3) can be written as follows:
∆(s) represent the characteristic matrix of system (2.2) and det ∆(s) stands for the characteristic polynomial of ∆(s). It's obvious that the stability of system (2.2) is completely determined by the distribution of eigenvalues of ∆(s).

Remark 2.3
If δ 2 kl = 0, system (2.1) is equivalent to the following expression: The characteristic matrix of system (2.4) is denoted by: It is obviously, stability of system (2.4) is completely determined by the distribution values of eigenvalues of∆(s).
It is obviously, stability of system (2.4) is completely determined by the distribution values of eigenvalues of ∆(s).
Theorem 2.5 If 0 < α < 1, all the roots of characteristic equation det ∆(s) = 0 have negative real parts, then the zero solution of system (2.1) is Lyapunov globally asymptotically stable.
Proof. The proof of theorem is almost the similar as those of Theorem 3.1 in [13], so we omit it here.
Lemma 2.8 [8] Let m(t) ∈ R n be a differentiable vector valued function and M ∈ R n×n is constant, symmetric and positive definite matrix. Then the following relationship is holds: Lemma 2.9 [13] Consider the following delayed fractional order differential inequality and delayed fractional order linear system Lemma 2.11 [13] If 0 < α < 1, all the eigenvalues ofÊ satisfy |arg λ > π 2 | and the characteristic equation det ∆ (s) = 0 has no pure imaginary roots for any δ 1 > 0, and −E + F < 0, then the zero solution of system (2.4) is Lyapunov globally asymptotically stable.
Lemma 2.12 [20] Let γ ∈ R, U, V, W, Z be matrices with suitable dimensions. Then the Kronecker product has the following properties: Lemma 2.13 [25] Let m = 0 be the equilibrium point of fractional order differential system D α m(t) = h t, m(t) . Assume that there exists a Lyapunov functional H t, m(t) and k-class function θ l , (l = 1, 2, 3), satisfying: Then the fractional order differential system is asymptotically stable.
Lemma 2.14 [41] For any vectors β 1 , β 2 ∈ R m and any matrix 0 < G ∈ R n×n , then the following relationship holds: In this article, we consider an array of linear coupled fractional order Cohen-Grossberg neural networks (FCCGNNs) consisting of N identical nodes with each isolated node network being an ndimensional dynamical system, which is presented by: with the singe delayed isolated node networks T signifies the state of the neuron at time t; D(·) signifies an amplification function; U(·) signifies an an appropriately behaved function; h(z l (t)) = h 1 (z l1 (t)), ..., h n (z ln (t)) T signifies the activation function of the neurons at time t; V = (v lk ) n×n and W = (w lk ) n×n represents the connection weights of the k-th neuron on l-th neuron; δ 1 > 0 and δ 2 > 0 represents the positive and constant delays, respectively; J = J 1 , ...., J n is the constant external input of the network; β 1 > 0 and β 2 > 0 represents the strengths of constant and delayed coupling weights, respectively. Φ = diag{φ 1 , ..., φ n } > 0 and Ψ = diag{ψ 1 , ..., ψ n } > 0 denotes the inertial coupling between two nodes; A = A lk N×N is the topological structure of the network and coupling strengths, where A lk satisfies the following conditions: for k l, A lk = A kl > 0 if there is link between node l and k, otherwise A lk = A kl = 0; for l = k, the diagonal elements are (2.17) The initial values of system (2.15) are presented by In order to prove our main results, we need the following fundamental assumptions.

Main results
In this section, several synchronization stability results are derived to ensure that FCCGNNs with and without parameter uncertainties is globally asymptotical synchronization stability depending on comparison theorem and LMI techniques, respectively.

Asymptotical synchronization stability of FCCGNNs
Proof. For the FCCGNNs error system (2.18), construct the following Lyapunov functional: Taking the fractional order time derivative of H(t) along the trajectories of (2.18) and, based on Lemma 2.8, one can get By application of positive the diagonal property of matrix, positive definiteness and based on Lemma 2.14, and from (2.17), it is deduced to
If there is no coupling delays and time delays in FCCGNNs (2.15), then the result is given as follows.
where D, U, Λ are same definitions in Theorem 3.2.
Proof. For the error system (2.18) without coupling delays and time delays, take the same Lyapunov functional (3.15) as in Theorem 3.2: The rest of the proof is similar to the proof of Theorem 4.1 in Ref [34], the FCCGNNs (2.15) with no coupling delays and time delays realizes globally asymptotically stable, thus the proof is ended.
If there are no coupling delays in FCCGNNs (3.28) with the ranges of parameters given by (3.29), then the result is given as follows.
Proof. For the error system (3.30) without coupling delays, choose the same Lyapunov functional in (3.15), then one has The rest of the proof for lim t→+∞ m(t) 2 = 0 similar as in Corollary 3.3. Therefore, the FCCGNNs (3.28) with no coupling delays is globally robust asymptotically stable.
If there is no coupling delays and time delays in FCCGNNs (3.28) with the ranges of parameters given by (3.29), then the result is given as follows.
Proof. For the error system (3.30) with no coupling delays and no time delays, choose the same Lyapunov functional in (3.15), then one has The rest of the proof for lim t→+∞ m(t) 2 = 0 similar as in Corollary 3.4. Therefore, the FCCGNNs (3.30) with no coupling delays and no time delays achieves globally robust asymptotically stable.
Remark 3.8 The author of [46] presented the synchronization stability conditions of Riemann Liouville sense fractional-order complex coupled neural networks under coupling delays. By using, Riemann Liouville derivative properties and some inequality techniques, several algebraic sufficient conditions are derived to verify the global asymptotic synchronization stability conditions of the proposed model. While in this paper, Caputo derivative properties, Cohen-Grossberg neural networks type models, Kronecker product and uncertain parameter are taken into consideration. Moreover, our obtained corollaries are new and purely different from those existing works.

Conclusions
In this paper, we have investigated the global asymptotical synchronization stability and global robust asymptotical synchronization stability topic for FCCGNNs under coupling delays. On the one side, via Lyapunov method, LMI technique, and proposed comparison principle theorem, we have derived the several asymptotical synchronization stability results for the considered complex network without parameter uncertainties. On the other side, thanks to some inequality techniques and robust analysis skills, the author's concerns the issues of global robust asymptotical synchronization stability for the considered complex network with parameter uncertainties. In the end, we provide two computer simulations to demonstrate the validity of the proposed analytical methods. Our future work will be extended to stability, stabilization and synchronization of Riemann-Liouville sense delay-coupled fractional order memristive Cohen-Grossberg BAM neural networks with time varying discrete and distributed delays.