Global \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(t^{-\alpha })$$\end{document}O(t-α) stabilization of fractional-order memristive neural networks with time delays

This article is concerned with the global \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(t^{-\alpha })$$\end{document}O(t-α) stabilization for a class of fractional-order memristive neural networks with time delays (FMDNNs). Two kinds of control scheme (i.e., state feedback control law and output feedback control law) are employed to stabilize a class of FMDNNs. Several stabilization conditions in form of algebraic criteria are presented based on a new fractional-order Lyapunov function method and Leibniz rule. Some examples are given to substantiate the effectiveness of the presented theoretical results.

and store a great quantity of information. For its excellent properties about memory, we can build a new model if the conventional resistors are replaced by the memristors in neural networks, which is called memristive neural networks. Some representative works studied on the properties of the memristive systems display its applicability in several interdisciplinary areas (see Bao and Zeng 2013;Guo et al. 2015;Wang et al. 2003;Wen and Zeng 2012;Zhao et al. 2015). From the description of memristive neural networks, combining memristors with infinite memory is extremely interesting. An advantage of fractional-order systems in comparison to integer-order systems is that fractional-order systems can generate infinite memory. Therefore, merging the memristors into a class of fractional-order neural networks is pretty anticipated. Although stability analysis of fractional-order memristive or memristor-based neural networks has been gradually carried out (see Chen et al. 2014Rakkiyappan et al. 2014Rakkiyappan et al. , 2015b, it is worth noting that fractional-order memristive neural networks can exhibit complicated dynamics or chaotic behaviors if the network's parameters and time delays are appropriately specified. Noticed that many static or dynamic control laws have been designed to stabilize nonlinear control systems, for instance, Chandrasekar and Rakkiyappan (2016), , Guo et al. (2013), Huang et al. (2009), Lou et al. (2013), Mathiyalagan et al. (2015), Rakkiyappan et al. (2015a), , Yang and Tong (2016). In allusion to different system structures and actual control requirements, lots of stabilization criteria are established, for example, periodic intermittent stabilization (Huang et al. 2009), robust stabilization (Yang and Tong 2016), finite-time stabilization (Zhang et al. 2016), impulsive stabilization (Chandrasekar and Rakkiyappan 2016;Huang 2010;Lou et al. 2013). Despite these fruitful achievements, some stabilization approaches can hardly be widely applied in practical problems due to high gain. In addition, an undeniable fact is that stabilization control schemes of fractional-order systems is little studied. Hence, it is necessary to investigate some appropriate controllers for stabilization of fractional-order systems.
Inspired by the above discussion, in this article, we will study the global O(t −α ) stabilization problem for a class of fractional-order memristive neural networks with time delays. We first introduce the concepts about fractional calculation and global stabilization of fractional-order systems. Secondly, for exploring some simple useful controllers, linear state feedback control law and linear output feedback control law are designed to stabilize the fractional-order systems. In addition, stabilization criteria in form of algebraic inequalities are derived by utilizing a new fractional Lyapunov method instead of classical Gronwall inequality. The conditions can be easily verified.

Fractional calculation concepts
First of all, some basics of fractional calculation are given which will be used in the later.

Model description
Consider the fractional-order memristive neural networks with time delays (FMDNNs) described by the following fractional-order equations: for i = 1, 2, . . . , n, where 0 < α < 1, n is the number of neurons in the networks, x i (t) is the state variable of the ith neuron, g j (·), f j (·) denotes the output of the jth unit at time t and t − τ (t), respectively, and g j (0) = f j (0) = 0. τ (t) corresponds to the transmission delay at time t and 0 ≤ τ (t) ≤ τ. u i (t) denotes the external input, a ij (x j (t)) and b ij (x j (t)) represent memristive weights, which are defined as: where the switching jumps T j > 0, â ij , ǎ ij , b ij , and b ij are constants.
Remark 1 Note that a ij (x j (t)) and b ij (x j (t)) are discontinuous in system (1), then the classical definition of solution for differential equations cannot be applied to (1). To deal with this issue, we introduce the concept of Filippov solution.
Definition 4 (Rakkiyappan et al. 2014) For system C t 0 D α t x(t) = g(x), 0 < α < 1, x ∈ R n , with a discontinuous right-hand side, a set-valued map is defined as where co [E] is the closure of convex hull of set E, B(x, δ) = {y : �y − x� ≤ δ}, and µ(N ) is a Lebesgue measure of set N. If x(t), t ∈ [t 0 , T ], is called the solution in Filippov sense of the Cauchy problem for system C t 0 D α t x(t) = g(x), 0 < α < 1, x ∈ R n , with initial condition x(t 0 ) = x 0 , when it is absolutely continuous, and satisfies the differential inclusion as follows: For FMDNNs (1), define the set-value maps (1) for i, j = 1, 2, . . . , n, where co{â ij ,ǎ ij } denotes the closure of convex hull generated by real numbers â ij and ǎ ij , co{b ij ,b ij } denotes the closure of convex hull generated by real numbers b ij and b ij . Throughout this article we denote a m ij = max 1≤i,j≤n Throughout this article, let us suppose: the activation functions g i , f i , i = 1, 2, . . . , n, are global Lipschitz, that is, for all u, v ∈ R, there exist positive constants G i , F i such that The objective of this article is to investigate the global O(t −α ) stabilization problem for system (1). Therefore, the stabilization problem will be converted to find the suit controller u i (t) (i = 1, 2, . . . , n) such that zero solution of the closed-loop system of (1) is globally O(t −α ) stable.
From the theories of differential inclusions and set-valued maps, the Filippov solution of FMDNNs (1) can be defined in the following form. (3) Remark 2 Based on the definitions of Filippov solution and fractional-order differential inclusion, we know that FMDNNs (1) is equivalent to the fractional-order differential inclusion (3) in the Filippov framework.
Next, definitions of global O(t −α ) stability and global O(t −α ) stabilization are given.
stabilized if there exists an appropriate feedback control law such that the closed-loop system of (1) is globally O(t −α ) stable.

State feedback control law
Two kinds of linear controller about state feedback are given, i.e., the linear controller without or with time delays. Firstly, we propose the following state control rule without time delays: for i = 1, 2, . . . , n.
Proof Define two Lyapunov functions as follows: and let From Leibniz rule for fractional differentiation, we have for t ≥ t 0 .
From (6), it follows that for all t ≥ t 0 .
On the basis of Definition 2 and Lemma 1, the following inequality holds It yields for t ≥ t 0 . Hence for i = 1, 2, . . . , n, (1) can be achieved global O(t −α ) stabilization under the designed control law (5).
In the following, we propose the following state control rule with time delays: for i = 1, 2, . . . , n.
for given t ≥ t 0 .

Output feedback control law
Two kinds of linear controller about output feedback are given, i.e., the linear output feedback controller without or with time delays. Firstly, we propose the following output feedback control rule without time delays: for i = 1, 2, . . . , n.
Proof Define two Lyapunov functions as follows: and let for t ≥ t 0 .
Through Theorem 1, we have It is obvious that there exists a k ∈ {1, 2, . . . , n} such that for given t ≥ t 0 .
In the following, we propose the following output feedback control rule with time delays: for i = 1, 2, . . . , n.
Proof Define two Lyapunov functions as follows: and let for t ≥ t 0 .
Remark 3 It needs to point out that fractional-order systems can be said rarely exponential stability. While, global Mittag-Leffler stability or global O(t −α ) stability can be used to describe asymptotic stability of fractional-order systems. In consideration of the complex and rich nonlinear behaviors of fractional-order systems, especially, for the fractional-order systems with time delays, we employ global O(t −α ) stabilization for a class of FMDNNs in Theorems 1-4.

Numerical examples
In this section, two numerical examples are given to show the effectiveness of the proposed theoretical results.
Similarly, select the output feedback controller with time delays designed as follows: Then it follows from Theorem 4 that system (56) can be achieved global O(t −α ) stabilization. From Fig. 6, we can get that the state trajectory of the resulting closed-loop system of (56) with the designed control law (59) is globally O(t −α ) stable.

Concluding remarks
In this article, we exploit the global O(t −α ) stabilization for a class of fractional-order memristive neural networks with time delays. The main theoretical results of this article are that the linear state feedback control law and the output feedback control law are (59) u(t) = −7 sin(x) − tanh(x − 1). constructed to stabilize the fractional systems. In addition, some sufficient conditions ensuring to stabilize fractional-order systems are also given in terms of algebraic inequalities according to a new fractional Lyapunov function and a fractional-order differential inequality skill. The article provides a novel way to construct a Lyapunov function and a new method to deal with fractional-order inequalities, which may be applied to discuss other properties or analyze other more complex systems such as the fractionalorder form of the model explored in the literatures Chandrasekar and Rakkiyappan (2016), Lou et al. (2013), Shang (2014Shang ( , 2015Shang ( , 2016, Wang et al. (2003), Yang and Tong (2016) and so on. Future research will focus on these issues.