Global Mittag–Leffler Stabilization of Fractional-Order BAM Neural Networks with Linear State Feedback Controllers

In this paper, the global Mittag–Leffler stabilization of fractional-order BAMneural networks is investigated. First, a new lemma is proposed by using basic inequality to broaden the selection of Lyapunov function. Second, linear state feedback control strategies are designed to induce the stability of fractional-order BAM neural networks. .ird, based on constructed Lyapunov function, generalized Gronwall-like inequality, and control strategies, several sufficient conditions for the global Mittag–Leffler stabilization of fractional-order BAM neural networks are established. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.


Introduction
Bidirectional associative memory (BAM) neural networks are a type of extended unidirectional auto-associator of Hopfield neural networks. ey are composed of two layers: the X-layer and the Y-layer, which can store and recall pattern pairs [1]. And they are widely applied in pattern recognition, signal processing, and associative memories. It is of great importance to investigate the dynamic stability behaviors of such networks to meet application requirements.
e fractional-order model is developed quickly because of its memory and genetic characteristics [2][3][4][5]. When the fractional order is in the interval [0, 1], a new lemma for the Caputo fractional derivatives is proposed in [2]. Some properties of the Lyapunov direct method for noninteger order systems are presented in [3]. In [4], the authors extend Lyapunov direct method for noninteger order systems. In [5], the finite-time stability of fractional-order impulsive switched systems is considered.
ere are many research studies on fractional-order BAM neural networks [6][7][8][9][10][11]. A novel result about finite-time impulsive stability of fractional-order memristive BAM neural networks is obtained in [6]. Quasi-pinning synchronization and β-exponential pinning stabilization for a class of fractional-order BAM neural networks with time-varying delays and discontinuous neuron activations are considered in [8]. In [10], sufficient conditions for the existence, uniqueness, and global Mittag-Leffler stability for the solutions of the fractional difference model of BAM neural networks are provided. In [11], based on Cauchy-Schwartz inequality and Burkholder-Davis-Gundy inequality, some sufficient conditions are derived to ensure the uniform stability of stochastic fractional-order memristor fuzzy BAM neural networks. e definitions of Mittag-Leffler stability and generalized Mittag-Leffler stability are proposed in [12,13]. Subsequently, some investigations focus on Mittag-Leffler stability [14][15][16][17][18][19][20][21]. Global Mittag-Leffler stability and synchronization analysis of discrete fractional-order complexvalued neural networks with time delay are given in [14]. In [15], the global Mittag-Leffler stability of multiple equilibrium points for the impulsive fractional-order quaternionvalued neural networks is investigated by employing the Lyapunov method. In [16], sufficient conditions ensuring the existence, uniqueness, and global Mittag-Leffler stability of the solutions of the fractional-order coupled system on a network without strong connections are derived. In [17], a new criterion is proposed to ensure the Mittag-Leffler stability for fractional-order neural networks in the quaternion field. e finite-time Mittag-Leffler stability for fractional-order quaternion-valued memristive neural networks with impulsive effect is investigated in [20]. Researchers not only study stability but also introduce control strategies to improve the stability. Different controllers have been applied to postpone Hopf bifurcation and broaden the stability domain on fractional-order systems [22][23][24][25]. Adaptive control approaches are adopted to induce Mittag-Leffler stabilization and synchronization for delayed fractional-order BAM neural networks in [26]. With linear and partial state feedback controls, global Mittag-Leffler stability of fractional-order BAM neural network is analyzed using Caputo fractional derivative and generalized Gronwall inequality [27]. In [28], state feedback stabilizing control and output feedback stabilizing control are designed for the stabilization of fractional-order memristive neural networks. However, the influence of whole state feedback controllers on stability of fractional-order BAM neural networks is not considered.
Motivated by the above discussion, we investigate the global Mittag-Leffler stability of fractional-order BAM neural networks with linear feedback controllers, including single and whole state feedback controllers. e main contributions include the following: First, a novel lemma is proposed using basic inequality to broaden the choice of the Lyapunov function. Second, linear state feedback controllers are designed to stabilize the systems.
ird, some sufficient conditions for global Mittag-Leffler stability are given by using the fractional Lyapunov method and introducing feedback controllers. Finally, numerical simulations are performed to show the effect of different state feedback controllers on the selected system. e paper is organized as follows. e preliminaries and the model descriptions are given in Section 2. Some sufficient conditions for global Mittag-Leffler stability of fractional-order BAM neural networks with two types of feedback controls are given in Section 3. In Section 4, a numerical simulation using the Adams-type forecast correction method is presented to illustrate the effectiveness of the theoretical results. Conclusions are given in Section 5.

Preliminaries and Model Description
In this section, some relevant definitions and lemmas about fractional calculus are introduced and the fractional-order BAM neural networks are described. Caputo fractional derivative is adopted.
Definition 1 (see [27]). e Caputo fractional derivative of order α for a function f(t) ∈ C n ([0, +∞), R) is defined by where n − 1 < α < n, n ∈ N + , Γ(α) is Euler's gamma function, and Γ(α) (2) Definition 2 (see [27]). One-parameter Mittag-Leffler function is defined as and two-parameter Mittag-Leffler function is defined as where the real part Re(α) of the complex number α is Re(α) > 0, z and β are both complex numbers, and Γ(·) is Euler's gamma function. Obviously, In this paper, we consider the fractional-order BAM neural networks given by the following fractional differential equations: where C t D α 0 is the Caputo derivative of order 0 < α < 1, i � 1, 2, . . . , n, j � 1, 2, . . . , m, x i and y j are the neural states, a i and c j > 0 are the self-inhibitions, b ij and d ji are the synaptic connection strengths, f j and g i are the activation functions satisfying f j (0) � 0 and g i (0) � 0, and u i (t) and v j (t) denote the external inputs.
To ensure the existence and uniqueness of solution of system (5), the following assumption is given.
e neuron activation functions f j and g i satisfy Lipschitz condition with the Lipschitz constants Now, we give the definition of the globally Mittag-Leffler stability for system (5) and some relevant lemmas.
Definition 3 (see [12]) (global Mittag-Leffler stability). Under the condition of u i (t) � 0 and v j (t) � 0 (i � 1, 2, . . . , n, j � 1, 2, . . . , m), the zero solution of system (5) is globally Mittag-Leffler stable if there exist two positive constants η 1 , η 2 > 0 such that for any trajectories x(t) and y(t) of system (5) with initial values x 0 and y 0 , satisfying Remark 1. If the equilibrium point is not at the origin, it can be shifted to the origin by coordinate transformation, so we just considered the case of zero equilibrium point.
Lemma 1 (see [29]). If are continuous and differentiable functions, for all i � 1, 2, . . . , n, and P ∈ R n×n is a positive definite matrix, then for general quadratic form function en, where 0 < α < 1 and E α (·) and E α,α+1 (·) are one-parameter Mittag-Leffler function and two-parameter Mittag-Leffler function, respectively. Next, we propose a new lemma by applying basic in- , a, b ≥ 0 to broaden the choice of the Lyapunov function for analyzing the stability.
and two positive definite matrices N 1 n×n and N 2 m×m , and all eigenvalues of N 1 and N 2 are greater than or equal to 1, the following inequality is satisfied: en, where 0 < α < 1, E α (·) is one-parameter Mittag-Leffler function and x(0) and y(0) are the initial value.

Main Results
In this section, linear state feedback controls are designed and some sufficient conditions are derived to ensure global Mittag-Leffler stability of fractional-order BAM neural networks.

A Single Linear State Feedback Control.
e external inputs u i (t) and v j (t) in system (5), which only depend on a single linear state feedback control, are designed as follows: is a (n + m) × (n + m) negative definite matrix, where A n×n � diag − a i + h i and B m×m � diag − c j + k j are negative definite matrices and C � (|b ij |F j + |d ji |G i ) n×m , then system (5) is globally Mittag-Leffler stable under designed control law (16).
Proof. Since H 1 is a negative definite matrix, there exist 0 < l 1i , l 1j < 1 such that H 1 is a negative semidefinite matrix, Consider a Lyapunov function as follows: Using Lemma 1, we have Let According to the conditions of eorem 1, obviously c 1 > 0. Combining (20) and (21), we obtain Using Lemma 2, we obtain at is, According to Lemma 3, we have By Definitions 3 and 4, system (5) is globally Mittag-Leffler stable under designed control law (16).

e Whole Linear State Feedback Control.
Meanwhile, we consider the following the external inputs u i (t) and v j (t) in system (5) that depend on the whole linear state feedback control: In the proof of eorem 1, it is a little strict to construct a negative definite matrix. Now, we will explore the stability of system (5) by establishing a nonzero matrix with zero diagonal elements and control law (26).
are satisfied and H 2 is a (n + m) × (n + m) nonzero matrix with zero diagonal elements, where C � (|b ij |F j + |d ji |G i ) n×m and λ max H 2 is the largest eigenvalue of matrix H 2 ; then, system (5) is globally Mittag-Leffler stable under designed control law (26).
Proof. Since the diagonal elements of H 2 are 0, the trace of N is 0, and obviously, λ max H 2 > 0.
Consider the following Lyapunov function: By Lemma 1, we obtain due to Substituting (31) into (30), we have Mathematical Problems in Engineering 5 where at is, Using Lemma 2, we obtain at is, According to Lemma 3, we have By Definitions 3 and 4, system (5) is globally Mittag-Leffler stable under designed control law (26).

Remark 2.
In the proof of eorems 1 and 2, by constructing a negative definite matrix or a nonzero matrix with zero diagonal elements, we enlarge the inequality and have . en, based on Lemma 2, the condition of Lemma 3 is satisfied n i�1 x 2 , where N 1 and N 2 of Lemma 3 are set to be unit matrices. Finally, using Lemma 3 and definition of Mittag-Leffler stability, we determine that system (5) is globally Mittag-Leffler stable under different control laws.

Numerical Simulation
In this section, a numerical example is given to show the effectiveness of our proposed theoretical results by Adama-Bashforth-Moulton predictor-corrector algorithm [30].
Consider the following four-dimensional fractional-order BAM neural networks: As depicted in Figure 1, the state trajectories of system (38) without external controllers u i (t) and v j (t) cannot converge to the origin.
Based on selected activation functions f j and g i , we have Lipschitz constants F j � G i � 1. For single state feedback controller (16), setting u 1 (t) � − 0.1x 1 (t), u 2 (t) � − 0.2x 2 (t), v 1 (t) � − 0.1y 1 (t), and v 2 (t) � − 0.2y 2 (t), we obtain a negative definite matrix H 1 : erefore, it follows from eorem 1 that system (38) can achieve global Mittag-Leffler stability under the designed single linear state feedback control laws and v 2 (t) � − 0.2y 2 (t). Figure 2 shows state trajectories of system (38) with the designed single linear state feedback control law. As shown in Figure 2, state trajectories of system (38) converge to the origin.

Conclusion
In this paper, the global Mittag-Leffler stabilization problem for a class of fractional-order BAM neural networks is analyzed. Linear state feedback control is designed to induce the stability of fractional-order BAM neural networks, including a single linear state feedback control and the whole linear state feedback control. A novel lemma is proposed to broaden the selection of Lyapunov function. Some sufficient conditions of global Mittag-Leffler stability are given by using fractional Lyapunov stability theory, generalized Gronwall-like inequality, and the designed controls. In addition, a numerical example is given to show the influence of different state feedback controllers on the selected system. In the future research, we will try to investigate the effect of external disturbances on the stability of fractional-order BAM neural networks under linear state feedback controls. And we will explore in depth the stability of incommensurating fractional-order BAM neural networks with time delay.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.