Impulsive Fractional Cohen-Grossberg Neural Networks: Almost Periodicity Analysis

: In this paper, a fractional-order Cohen–Grossberg-type neural network with Caputo fractional derivatives is investigated. The notion of almost periodicity is adapted to the impulsive generalization of the model. General types of impulsive perturbations not necessarily at ﬁxed moments are considered. Criteria for the existence and uniqueness of almost periodic waves are proposed. Furthermore, the global perfect Mittag–Lefﬂer stability notion for the almost periodic solution is deﬁned and studied. In addition, a robust global perfect Mittag–Lefﬂer stability analysis is proposed. Lyapunov-type functions and fractional inequalities are applied in the proof. Since the type of Cohen–Grossberg neural networks generalizes several basic neural network models, this research contributes to the development of the investigations on numerous fractional neural network models.


Introduction
Recently, fractional-order differential systems have attracted a lot of attention in research since fractional-order derivatives are distinguished by its substantial degree of reliability and accuracy. In fact, the mathematical models with fractional-order derivatives are mostly applied in the description of universal laws. This is due to the fact that compared with the classical integer-order derivatives most fractional-order derivatives are non-local and possess memory effects and hereditary properties [1][2][3].
Among the numerous proposed fractional operators [4,5], one of the most commonly used fractional-order derivative is the Caputo-type derivative. The main reason of such intensive implementation in mathematical modeling is that it has all the advantages of fractional-order derivatives, and in addition, the initial conditions of fractional-order models with Caputo fractional operators can be physically interpreted as in the integer-order models. Hence, there are also advantages related to the geometric interpretations [6][7][8].
Since more and more experimental results show that real-world models follow fractional calculus dynamics, very recently fractional-order differential systems are successfully applied in various fields of science and engineering [9,10], including COVID-19 models [11].
One of the implications of fractional-order systems is in the neural network modeling. Due to the fact that fractional models are more effective than integer-order models in numerous applications, the existing theory of integer-order neural networks and related models have been extended and improved to the fractional-order case. See, for example, some very resent results in [12][13][14][15].
However, the research on the theory and applications of the Cohen-Grossberg-type neural networks with fractional-order derivatives still needs more development. Recently, proposed. Section 4 is devoted to examples through which we demonstrate the obtained results. The conclusion remarks are stated in Section 5.
Notations. In this paper, Z stands for the set of all integers; R n denotes the ndimensional Euclidean space; R + = [0, ∞); for a q ∈ R n , q = (q 1 , q 2 , . . . , q n ) T , we will consider the following norm ||q|| = ∑ n i=1 q 2 i .

Preliminaries
In this section, we will give some basic definitions and lemmas. The fractional-order neural network model under consideration will be also formulated.

Fractional Calculus Notes
First, we will recall nomenclatures related to fractional calculus.
Definition 1 ( [2,3]). Let t 0 ∈ R. The Caputo fractional derivative of order ν, 0 < ν < 1 with a lower limit t 0 for a continuously differentiable in R real function λ(t) is defined by In [27], the authors considered an impulsive control strategy with impulsive perturbations at fixed instants, applied to the following fractional-order Cohen-Grossberg neural network model where 0 < ν < 1, i = 1, 2, . . . , n, n denotes the number of units in the neural network, q i (t) is the state of the ith unit at time t, a i (q i (t)) denotes a standard amplification function, b i (q i (t)) stands for a well-behaved function, f j (.) stands for the activation function, c ij is the connection weight between the jth neuron and ith neuron, and I i is the input from outside of the network.
In this paper, we will extend the model proposed in [27], considering variable impulsive perturbations. Some of the parameters will also be generalized. Let θ 0 (q) = t 0 for q ∈ R n , and the continuous functions θ k : R n → R, k = ±1, ±2, . . . are such that In this manuscript, we consider the following fractional-order Cohen-Grossberg neural network model with variable impulsive perturbations: where the model functions a i ∈ C[R, R + ], c ij , f j , I i , p ik ∈ C[R, R] and b i ∈ C[R 2 , R + ], i, j = 1, . . . , n.
The second condition in Equation (2) is the impulsive condition. The impulsive functions p ik can be used to control the qualitative behavior of the model. Their choice determines the controlled outputs q i (t + ), i = 1, 2, . . . , n.
We denote by τ k : t = θ k (q), q ∈ R n , k = ±1, ±2, . . . and by t l k , the moment when the integral curve (t, q(t)) of the solution q(t) of Equation (2), Equation (3) meets the hypersurfaces τ k , i.e., each of the points t l k is a solution of one of the equations t = θ k (q(t)). The impulsive points t l k are the points of discontinuity of the solution q(t) at which where the matrices p k = diag(p 1k , p 2k , . . . , p nk ), k = ±1, ±2, . . . . It is also known [25,38,39,42,62,64] that, in general, k = l k , k, l k = ±1, ±2, . . . or it is possible for the integral curve (t, q(t)) of Equation (2) to not meet the hypersurface τ k at the moment t k .
Denote q(t) = (q 1 (t), q 2 (t), . . . , q n (t)) T , F(t, q) = (F 1 (t, q), F 2 (t, q), . . . , F n (t, q)), where i = 1, 2, . . . , n, and Equation (2) can be represented as It is well known that considering variable impulsive perturbations is more general and leads to numerous difficulties in existence, uniqueness, and continuability of solutions, such as phenomenon "beating" of solutions, merging solutions after an impulsive perturbations, bifurcation, etc. [25,38,39,42,62]. In order to avoid such complications and, also, to establish our main results on the almost periodicity of solutions, we will assume that t l k < t l k+1 < . . . for l k ∈ Z. In addition, we assume that the integral curves of Equation (2) meet each hypersurface τ k at most once.

Almost Periodicity Definitions
In this subsection, we will adopt the almost periodicity definitions from [52,53] to the impulsive fractional neural network model (Equation (2)).
First, we will state the following classical definition [49,53].

Definition 4.
A continuous function g : R → R is said to be almost periodic on R in the sense of Bohr if for every ε > 0 and for every t ∈ R, the set Consider the set [38,52,53] of all unbounded and strictly increasing sequences of impulsive points of the type {t l k } } . For any two given T,T ∈ T denote by s(T ∪T) : T → T the map for which the set s(T ∪T) forms a strictly increasing sequence. Let . is a sequence of real numbers, will be denoted by Φ r .
We will also apply the following almost periodicity definitions.

Definition 5. Consider a set of sequences
The set (5) is said to be uniformly almost periodic if from each infinite sequence of shifts {t l k − s r }, l k = ±1, ±2, . . . , r = 1, 2, . . . , s r ∈ R, we can choose a convergent subsequence in T .

Definition 7.
The function φ ∈ PC[R, R] is said to be an almost periodic piecewise continuous function with points of discontinuity of the first kind t l k , {t l k } ∈ T if for every sequence of real numbers {s m }, there exists a subsequence {s r }, s r = s m r , such that Φ r is compact in We will also introduce the following assumptions: A1. The functions a i , i = 1, 2, . . . , n are continuous on R, almost periodic in the sense of Bohr, and there exist positive constants a i and a i such that 1 < a i ≤ a i (χ) ≤ a i for χ ∈ R.
A3. The functions f i are continuous on R, f i (0) = 0, and there exist constants A4. The functions c ij and I i are almost periodic in the sense of Bohr for all i, j = 1, 2, . . . , n.
A5. The continuous functions p ik are almost periodic in the sense of Bohr and

Remark 2.
For more details about the assumptions A1-A6, we refer to the investigations on the almost periodicity in integer-order Cohen-Grossberg and related neural network models [38,40,41,45,46,53].
It is very well known [25,48,53] that the assumptions A1-A6 imply the existence of a subsequence {s r }, s r = s m r of an arbitrary sequence of real numbers {s m } that "moves" Equation (2) to the following model Following the almost periodicity theory [52,53] for impulsive systems, we will denote the set of all systems of Equation (6) by H (2). For the vector representation of Equation (6), we will need the notation

Lyapunov-Type Functions Definitions and Lemmas
Here, we will recall some Lyapunov functions-related definitions and lemmas [25,35]. Define the sets V is defined and continuous on Ω, V has nonnegative values, and V(t, 0, 0) = 0 for t ∈ R; 2. V is differentiable in t and locally Lipschitz continuous with respect to its second and third arguments on each of the sets Ω k ; 3. For any (t * , q * ), (t * ,q * ) ∈ τ k , and each k = ±1, ±2, . . . , there exist the finite limits and V(t * − , q * ,q * ) = V(t * , q * ,q * ).
We will use the the following derivative of order ν, 0 < ν < 1 of a function V ∈ V 0 [25]: The following results follows from Lemma 1.5 in [25].
Lemma 1. Assume that the function V ∈ V 0 satisfies for t ≥ t 0 and q,q ∈ R n : Then, for t ∈ [t 0 , ∞), we have

Remark 3.
For the case d = 0, we have which is basically the result in Corollary 1.3 from [25].

Main Almost Periodicity Results
In this section, we will state our main almost periodicity and global perfect Mittag-Leffler stability criteria for the impulsive fractional-order Cohen-Grossberg neural network model in Equation (2). These results are the first contributions to the almost periodicity theory for such fractional models and extend and generalize the results in [38,40,45,46] for integer-order models to the fractional-order case.
Since lim and for t = θ s k (q), which imply that the function β(t) is a solution of Equation (2). Finally, in order to prove the almost periodicity of the solution β(t), we will use again the sequence {s r } that moves Equation (2) to H(2) and the function Applying similar arguments as above, we obtain V((β(t + s r ), β(t + s l )) < Aε 2 and, hence, for l ≥ r ≥ r 0 (ε), we obtain Therefore, Equations (19) and (20) imply that the sequence of functions β(t + s r ) converges uniformly to the solution β(t) as r → ∞.
The properties (a) and (b) follow directly. The proof of Theorem 1 is completed.
One of the main qualitative properties of the solutions of neural network models, including almost periodic solutions, is their stability. That is why there exists numerous stability results for different types of solutions of integer-order Cohen-Grossberg neural networks [21][22][23][24]39,42,43]. The most important stability concept for neural network models is that of the exponential stability [46]. For fractional-order systems, the corresponding stability notion is that of Mittag-Leffler stability introduced in [69] (see also [19,25,[28][29][30][31]). By the next definition, we will introduce the notion of global perfect Mittag-Leffler stability for a solution of the fractional neural network model (Equation (2)). (2) with an initial value q 0 ∈ R n is:
(c) globally perfectly Mittag-Leffler stable if it is globally Mittag-Leffler stable and the number R in a) is independent of t 0 ∈ R. Theorem 2. Suppose that conditions of Theorem 1 hold. Then, the almost periodic solution β(t) of Equation (2) is globally perfectly Mittag-Leffler stable.
Proof. Let β(t) be the almost periodic solution of Equation (2), and β 1 (t) be an arbitrary solution of Equation (6).
and consider the model If we take the Lyapunov function to be W(t,β) = V(t, β, β +β), then apply Lemma 1, we conclude that the zero solutionβ(t) = 0 of Equation (21) is globally perfectly Mittag-Leffler stable. Hence, the solution β(t) of Equation (2) is globally perfectly Mittag-Leffler stable. This proves Theorem 2.
A10. There exist constantsL i > 0,H The proof of the next result is similar to the proof of Theorem 1.
Proof. It follows from the condition 1 of Theorem 3, there exists a globally perfectly Mittag-Leffler stable almost periodic solution β(t) of Equation (2). The proof of its global perfect robust Mittag-Leffler stability follows directly from Definition 10. Thus, the proof of Theorem 3 is completed.

Examples
Example 1. In order to demonstrate our almost periodicity results, we consider Equation (2) for n = 2, or we consider the following fractional impulsive Cohen-Grossberg neural network model where i = 1, 2, q(t) = (q 1 (t), q 2 (t)) T , t ∈ R, f i (q i ) = 1 2 From the choice of the system parameters for t = θ k (q), k = ±1, ±2, . . . we conclude that all assumptions A1-A4 are satisfied for a 1 = 0.7, a 1 = 1.3, a 2 = 0.2, a 2 = 0.6, and the greatest eigenvalue of the symmetric matrix is negative.
In addition, we assume that the impulsive functions p ik (q i ) = γ ik q i at t = θ k (q) are such that for any k = ±1, ±2, . . . and assumption A6 is satisfied. Thus, all conditions of Theorem 1 are satisfied. Therefore, there exists a unique almost periodic solution β(t) of Equation (23). Furthermore, since the conditions of Theorem 2 hold, the almost periodic solution is globally perfectly Mittag-Leffler stable.
If the uncertain parameters satisfy conditions 2 and 3 of Theorem 3, then the almost periodic solution of Equation (23) is globally, perfectly, and robustly Mittag-Leffler stable. We can also mention, that if, for example, condition A11 is not satisfied, we can not make any conclusion about the global perfect robust Mittag-Leffler stability behavior of the almost periodic state.

Remark 6.
As we can see from the presented examples, the proposed impulsive control technique may be efficiently applied in the global perfect Mittag-Leffler stability analysis of the almost periodic solutions.

Conclusions
In this paper, we investigate an impulsive fractional-order Cohen-Grossberg neural network model. The notion of almost periodicity is extended to the model under consideration. Using Lyapunov-type functions, criteria for the existence and uniqueness of almost periodic waves for the proposed fractional-order neural network model are established. The concept of global perfect Mittag-Leffler stability of the almost periodic solution is also introduced and studied. In addition, the almost periodicity of the model under uncertain parameters is investigated. Our qualitative criteria generalize and complement some existing almost periodicity results for fractional Cohen-Grossberg neural network models to the impulsive case. We propose an impulsive control technique via variable impulsive perturbations. The proposed technique can be applied in the investigation of qualitative properties of different types of fractional-order impulsive neural network models. Funding: This research was funded in part by the European Regional Development Fund through the Operational Program "Science anatiod Educn for Smart Growth" under contract UNITe № BG05M2OP001-1.001-0004 (2018-2023).