Further results on Mittag-Leffler synchronization of fractional-order coupled neural networks

In this paper, we focus on the synchronization of fractional-order coupled neural networks (FCNNs). First, by taking information on activation functions into account, we construct a convex Lur’e–Postnikov Lyapunov function. Based on the convex Lyapunov function and a general convex quadratic function, we derive a novel Mittag-Leffler synchronization criterion for the FCNNs with symmetrical coupled matrix in the form of linear matrix inequalities (LMIs). Then we present a robust Mittag-Leffler synchronization criterion for the FCNNs with uncertain parameters. These two Mittag-Leffler synchronization criteria can be solved easily by LMI tools in Matlab. Moreover, we present a novel Lyapunov synchronization criterion for the FCNNs with unsymmetrical coupled matrix in the form of LMIs, which can be easily solved by YALMIP tools in Matlab. The feasibilities of the criteria obtained in this paper are shown by four numerical examples.


Introduction
The rapid development of modern network science and technology makes humans be aware of the importance and universality of the research on complex networks. Qualitative and quantitative analysis on various artificial and real complex network systems need to be conducted. As a typical complex network system, a neural network has a strong intelligence characteristic and learning functions, which plays an important role in artificial intelligence-related fields over the past few decades [1].
In the 1980s, Mandelbrot [2] found that fractional-order phenomena appeared in real life and engineering, which caused great repercussion both in engineering and academic circles. Different from classical calculus, fractional-order calculus has unique memorization and heredity. Thus fractional-order systems can more reasonably reflect the dynamical response process of the model [3]. In fact, since memory exists in the fractional-order derivatives and biological neurons, neurons in the human brain can be more accurately described and simulated by fractional-order neural networks. Nowadays, fractional-order neural networks play an important role in artificial intelligence-related fields, such as automatic control, intelligent robots, pattern recognition, and so on. An increasing number of scholars turn to the study on dynamical behaviors of fractional-order neural networks, and many excellent results are reported (see, e.g., [4][5][6][7][8] and references therein).
Synchronization of fractional-order neural networks, as a collective behavior, has received much attention and is extensively investigated [9][10][11][12][13][14]. In [9] the authors study the synchronization and robust synchronization issues for the FCNNs based on the general convex quadratic Lyapunov function. However, the synchronization criteria are in the form of nonlinear matrix inequalities (NLMIs), which brings certain difficulty in the solving process. In [10] the authors investigate the hybrid synchronization problem of two coupled complex networks with fractional-order dynamical nodes based on the general convex quadratic Lyapunov function by the fractional-order Lyapunov stability theorem. In [11] a Lyapunov function including fractional-order integral term is constructed to derive the synchronization stability conditions for Riemann-Liouville fractional-order delay-coupled complex neural networks. However, the relationship between node networks and coupling matrix is neglected.
Due to the complexity of fractional-order calculus theory, some existing synchronization criteria for fractional-order neural networks have not fully considered the information of FCNNs, such as activation functions and coupled matrix. Based on the fractionalorder Lyapunov stability theory, Lyapunov functions play a key role in the synchronization discrimination of the fractional-order system and directly affect the conservatism of synchronization discrimination conditions. Constructing some Lyapunov functions including the information of activation function may be more reasonable and effective. Recently, the global Mittag-Leffler group consensus and group consensus in finite time for fractional-order multiagent systems are concerned in [15]. Under the fractional-order Filippov differential inclusion framework, by applying the Lur'e-Postnikov-type convex Lyapunov functional approach and Clarke's nonsmooth analysis technique, some sufficient conditions are provided in terms of LMIs. However, the Lur'e-Postnikov Lyapunov function constructed in [15] requires that the activation function should be nondecreasing. In this paper, we remove this constraint. The activation function just needs to satisfy the general Lipschitz condition. Since the existing research methods have some limitations, further research is needed.
Motivated by the discussion above, in this paper, we study the synchronization of FC-NNs. The results obtained enrich the theory of synchronization of FCNNs. The major contributions can be highlighted as follows: • The Lur'e-Postnikov Lyapunov function is extended to a general case suitable for the activation functions satisfying the generalized Lipschitz condition; • A novel Mittag-Leffler synchronization criterion is derived for the FCNNs with symmetrical coupled matrix in the form of LMIs. Then a robust Mittag-Leffler synchronization criterion is given for the FCNNs with uncertain parameters; • A novel Lyapunov synchronization criterion is derived for the FCNNs with unsymmetrical coupled matrix in the form of LMIs, which can be easily solved by YALMIP tools in Matlab; • The information of network node and coupling matrix is adequately considered in the synchronization criteria. The paper is structured as follows. In Sect. 2, we present some definitions and main properties of Caputo fractional-order calculus. In Sect. 3, we present three novel synchro-nization criteria for different types of FCNNs in the form of LMIs. Simulation examples are given in Sect. 4. Conclusions are given in Sect. 5.

Preliminaries
To study the synchronization of FCNNs, we first provide some definitions of Caputo fractional-order calculus and some useful lemmas.
is called the Caputo fractional-order differential of order α.
is convex over and V (0) = 0. Then, for any time instant t ≥ 0, Lemma 1 allows us to construct some more useful convex Lyapunov functions. Based on Lemma 1, the following propositions obviously hold. Proposition 1 ([18]) Let x : [0, ∞) → be a continuous differentiable function, where ⊂ R n . Let P ∈ R n×n be a positive definite matrix. Then, for any time instant t ≥ 0, Moreover, we introduce Lemmas 2 and 3 for the discussion that follows.

Lemma 2
Let α and β be real column vectors of dimensions of n 1 and n 2 , respectively. For real positive symmetric matrices 1 ∈ R n 1 ×n 1 and 2 ∈ R n 2 ×n 2 , we have the following inequality for any matrix S ∈ R n 1 ×n 2 satisfying 1 S Proof The conclusion is obvious, and we omit the proof.
This lemma is equivalent to the Moon inequality [21]. In this paper, we change its form for more convenient applications.

Main results
In this paper, we consider the synchronization of the following FCNNs: where i = 1, 2, . . . , N , α ∈ (0, 1), N is the number of nodes, y i (t) = (y i1 (t), y i2 (t), . . . , y in (t)) T ∈ R n is the state vector of node i, g(y i (t)) = [g 1 (y i1 (t)), g 2 (y i2 (t)), . . . , g n (y in (t))] T with g i (0) = 0 is the activation function of node i, J(t) denotes the external input, B = (B ij ) n×n and 0 < C = diag{C 1 , C 2 , . . . , C n } are real matrices, 0 < d ∈ R denotes the overall coupling strength, A = (A ij ) N×N is the coupled matrix of network, and = ( ij ) n×n corresponds to the inner coupling matrix. Throughout this paper, we make the following assumption on the FCNNs.

Assumption 1 ([23, 24])
The activation function g is continuous and bounded. For any x 1 , x 2 ∈ R, x 1 = x 2 , the activation functions g j satisfy where k + j and kj are known constant values. For convenience, we denote K + = diag{k + j }, Remark 1 Assumption 1 for activation functions is introduced to investigate the Mittag-Leffler synchronization of the FCNNs to guarantee the existence and uniqueness of the equilibrium point. Most of the common activation functions satisfy the generalized Lipschitz condition, such as A linear excitation function, threshold or step excitation function, and so on.
Next, for easier understanding, we give the definition of synchronization.

Mittag-Leffler synchronization analysis for FCNNs with symmetrical coupled matrix
In this section, we consider FCNNs (9) with symmetrical coupled matrix, that is, there is a connection between nodes i and j, otherwise, A ij = 0, i, j = 1, 2, . . . , N .

Theorem 1 FCNNs
where To prove the synchronization of FCNNs (9) with symmetrical coupled matrix, the process involves three steps.
Step 1: We prove that V (e(t)) is positive.
Obviously, V 1 (e(t)) is positive by the positive definiteness of P. Next, we prove that V 2 (e(t)) is nonnegative.
From Assumption 1 we have Then which leads to Step 2: We prove that there exist positive constants α 1 and α 2 satisfying Step 3: We prove that there exists a positive constant For V 1 (e(t)), by Proposition 1 we have ), by Lemma 2 there exist a positive definite matrix 1 and a positive definite diagonal matrix 2 such that For V 2 (e(t)), based on Proposition 3, we have A ij e j (t) A ij e j (t) A ij e j (t) where D k = diag{d k1 , d k2 , . . . , d kn }, k = 1, 2, 3, 4. By simple calculation we get A ij e j (t) Similarly to (20), there exist a positive definite matrix 1 and a positive definite diagonal A ij e j (t) there exist a positive definite matrix 1 and a positive definite diagonal matrix 2 such that and 1 [ we have a positive definite matrix 1 and a positive definite diagonal matrix 2 such that and 1 [- )g(y j (t))], we have a positive definite matrix 1 and a positive definite diagonal matrix 2 such that and 1 [- A ij e j (t), we have a positive definite matrix 1 and a positive definite diagonal matrix 2 such that where ∈ R Nn×Nn .
The proof is completed.

Mittag-Leffler synchronization analysis for FCNNs with uncertain parameters
FCNNs may contain uncertain parameters due to the existence of environmental noises or model errors in many circumstances. In this section, we consider the following FCNNs with uncertain parameters: where A = (A ij ) N×N is the coupled matrix of network that satisfies A ij = A ji > 0 if i = j and there is a connection between node i and node j, otherwise, A ij = 0, and For the uncertain parameters C(t) and B(t) in (32), we make the following assumption.

Assumption 2
The parametric uncertainties C(t) and B(t) are of the form where M, H C , and H B ∈ R n×n are known real constant matrices, and the uncertain matrix F(t) is unknown real time-varying matrix satisfying F T (t)F(t) ≤ I n .
To study the synchronization of (32), we need the following lemma.

Lemma 4 ([26]) For given matrices Y , D, and E of proper dimensions, assume that Y satisfies Y T = Y . Then
for any F satisfying F T F ≤ I if and only if there is a real number ε > 0 such that For the synchronization of FCNNs (32), we have the following result.

Moreover, for * , we have * = -I N ⊗ PMF(t)H C + MF(t)H C T P
- Thus by the Schur complement theorem, * < 0 is equivalent to for some ε 6  The proof is completed.

Synchronization analysis for FCNNs with unsymmetrical coupled matrix
In this section, we consider FCNNs (9) with unsymmetrical coupled matrix, that is, A = (A ij ) N×N is an unsymmetrical coupled matrix representing the coupling strength and topological structure of the networks. Letȳ(t) = 1 N N i=1 y i (t) and define the error vector e i (t) = y i (t) -ȳ(t). From FCNNs (9) we have The synchronization of FCNNs (9) with unsymmetrical coupled matrix is equivalent to the stability of system (48).
1, 2, and matrices 1 ∈ R Nn×Nn such that the following linear matrix inequalities hold: where Proposition 1 implies where By Lemma 2 there exist a positive definite matrix 1 and a positive semidefinite matrix Noting that e(t) = (I Nn -E N ⊗I n N )y(t), we have For V 2 (e(t)), by Proposition 3 we have where D k = diag{d k1 , d k2 , . . . , d kn }, k = 1, 2, 3, 4.
Similarly, there are a positive definite matrix 1 and a positive semi-definite matrix 2 such that There exist a positive definite matrix 1 and a positive semidefinite matrix 2 such that and 1 [ Similarly, we have a positive definite matrix 1 and a positive semidefinite matrix 2 such that = y T (t) I Nn -E N ⊗ I n N I N ⊗ (K 1 K) Combining (58) with (60)-(63) gives So D α V (e(t)) ≤ 0 if y T (t) y(t) ≤ 0. Therefore by Lemma 3 FCNNs (9) with unsymmetrical coupled matrix realize synchronization in the sense of Lyapunov under condition (49). The proof is completed.
The Lyapunov synchronization of the FCNNs with unsymmetrical coupled matrix in Theorem 3 is weaker than the Mittag-Leffler synchronization.
Remark 2 Each sufficient synchronization condition proposed in Theorems 1-3 includes several LMIs. The forms of sufficient synchronization conditions seem to be complicated, but they can be easily solved by Matlab. The sufficient conditions in Theorems 1-3 require (0.5N 2 + 2.5)n 2 + (14.5 + 0.5N)n, (0.5N 2 + 2.5)n 2 + (14.5 + 0.5N)n + 7 and (0.5N 2 + 2.5)n 2 + (14.5 + 0.5N)n decision variables, respectively, where N stands for the number of nodes, and n is the dimension of the state vector for each node.

Numerical examples
In this section, we provide four numerical examples to confirm the correctness of the obtained synchronization criteria.
It is clear that f j satisy Assumption 1 with kj = -0.5 and k + j = 0.5. Solving LMIs in Theorem 1 by using LMI tools in Matlab, we get t min = -0.3441. So the FCNNs in this example realize synchronization.
Remark 3 This example was studied in [9]. The fractional-order neural network is synchronized under a pinning controller U(t) = Ke(t) with K = diag{0.8I, 1.6I, O, O, O}. However, the synchronization criteria are in the form of NLMIs, which brings certain difficulty in the solving process. Differently from the method used in [9], Lemma 2 is adopted to get a synchronization criterion in the form of LMIs. Moreover, a novel convex Lyapunov function V 2 is constructed to take into account the information of the activation functions.
It is clear that f j satisfy Assumption 1 with kj = -0.5 and k + j = 0.5. Solving LMIs in Theorem 2 by LMI tools in Matlab, we get t min = -0.2642. So FCNNs with uncertain parameters realize synchronization. Take It is clear that f j satisfy Assumption 1 with kj = -0.5 and k + j = 0.5. Solving LMIs in Theorem 3 by YALMIP tools in Matlab, we get It is clear that f j satisfy Assumption 1 with kj = -0.5 and k + j = 0.5. Since all conditions in Theorem 1 are satisfied, FCNNs in this example can realize synchronization.
The trajectories of the state system are shown in Fig. 7. From Fig. 7 we see that FCNN system is unstable. The trajectories of the error system are shown in Fig. 8. Figure 8 shows that FCNNs with coefficient (69) realize synchronization.

Conclusions
In this paper, we study the synchronization of FCNNs. We construct some novel convex Lyapunov functions containing the activation function information, based on which, we present several novel Mittag-Leffler synchronization criteria for FCNNs with and without uncertain parameters. Then we establish a novel synchronization criterion in the sense of Lyapunov for FCNNs with unsymmetrical coupled matrix. The benefits of the synchronization criteria obtained in this paper are illustrated by four numerical examples. The global synchronization for stochastic dynamic networks [27,28] and the synchronization in fixed or finite time for fractional-order network [29,30] have attracted considerable attention in the past few decades. In the future, we will study the global synchronization of fractional-order stochastic neural networks basing on the event-triggered strategy.