Global exponential stability of Clifford-valued neural networks with time-varying delays and impulsive effects

In this study, we investigate the global exponential stability of Clifford-valued neural network (NN) models with impulsive effects and time-varying delays. By taking impulsive effects into consideration, we firstly establish a Clifford-valued NN model with time-varying delays. The considered model encompasses real-valued, complex-valued, and quaternion-valued NNs as special cases. In order to avoid the issue of non-commutativity of the multiplication of Clifford numbers, we divide the original n-dimensional Clifford-valued model into 2mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{m}n$\end{document}-dimensional real-valued models. Then we adopt the Lyapunov–Krasovskii functional and linear matrix inequality techniques to formulate new sufficient conditions pertaining to the global exponential stability of the considered NN model. Through numerical simulation, we show the applicability of the results, along with the associated analysis and discussion.

Clifford algebra offers a powerful framework for solving geometrical problems. This discipline has shown useful results in various science and engineering areas, including control and robotic related problems [27][28][29][30][31]. Clifford-valued NN models stand as a generalization of real-valued, complex-valued, and quaternion-valued NNs. In this respect, Clifford-valued NN models are superior to real-valued, complex-valued, and quaternionvalued NNs for undertaking spatial geometric transformation and high-dimensional data problems [29][30][31][32]. Theoretical and applied studies of Clifford-valued NN models have recently become a new topic of research. However, the dynamical properties of Cliffordvalued NN models are typically more complex than those of real-valued and complexvalued NN models. As such, studies on Clifford-valued NN dynamics are still limited due to those utilizing the principle of non-commutativity of the product of Clifford numbers [33][34][35][36][37][38][39][40][41][42].
In [33], the authors derived the global exponential stability criteria for delayed Cliffordvalued recurrent NN models in terms of linear matrix inequalities (LMIs). Pertaining to Clifford-valued NN models with time delays, their global asymptotic stability issues were examined in [34] by decomposing the n-dimensional Clifford-valued NN model into 2 m ndimensional real-valued models. Leveraging on a direct method, the existence and global exponential stability of almost periodic solutions were derived for Clifford-valued neutral high-order Hopfield NN models with leakage delays in [37]. In addition, the use of Banach fixed point theorem and Lyapunov-Krasovskii functional (LKF) technique for addressing the global asymptotic almost periodic synchronization issues for Clifford-valued cellular NN models was conducted in [38]. A study of the weighted pseudo almost automorphic solutions pertaining to neutral type fuzzy cellular NN models with mixed delays and D operator in Clifford algebra was presented in [40]. In [42], the authors investigated the existence of anti-periodic solutions corresponding to a class of Clifford-valued inertial Cohen-Grossberg NN models by constructing suitable LKFs.
To the best of our knowledge, there are hardly any papers that deal with the issue of global exponential stability of Clifford-valued NNs with time-varying delays and impulsive effects. Indeed, this interesting subject remains an open challenge. Motivated by the above facts, our research focuses on to derive the sufficient conditions of global exponential stability of Clifford-valued NNs with impulsive effects. In order to achieve our main results, the n-dimensional Clifford-valued NN model is firstly decomposed into 2 m ndimensional real-valued NN models. This avoids the issues related to the principle of noncommutativity of multiplication with respect to Clifford numbers. Based on the LKF and LMI techniques and some mathematical concepts, we derive new sufficient conditions for ascertaining the exponential stability of the considered Clifford-valued NN model. The conditions obtained in this study are expressed in LMIs, and the associated feasible solutions are verified by using the MATLAB software. The obtained results are validated with a numerical simulation.
The main contributions of our study are as follows: (1) we for the first time analyze the global exponential stability of Clifford-valued NN models with time-varying delays as well as impulsive effects; (2) in comparison with other results, the results of our study is new and more general even when the considered Clifford-valued NN model has been decomposed into real-, complex-, and quaternion-valued NN models; (3) our proposed method can be easily employed for other dynamic behaviors with respect to different types of Clifford-valued NN models.
The organization of this article is as follows. The proposed Clifford-valued NN model is formally defined in Sect. 2. We then explain the new stability criterion and present the numerical example and the associated simulation results in Sects. 3 and 4, respectively. A summary of the findings is given in Sect. 5.

Notations
Let R n and A n denote the n-dim real vector space and n-dim real Clifford vector space, respectively. R n×n and A n×n denote the set of all n × n real matrices and the set of all n × n real Clifford matrices, respectively. A is defined as the Clifford algebra with m generators over the real number R. Superscripts T and * , respectively, indicate matrix transposition and matrix involution transposition. A matrix P > 0 (< 0) denotes a positive (negative) definite matrix. We define the norm of R n as p = n i=1 |p i |. Besides that, For ϕ ∈ C([-τ , 0], A n ), we introduce the norm ϕ τ ≤ sup -τ ≤s≤0 ϕ(t +s) . λ M (P) and λ m (P), respectively, denote the maximum and minimum eigenvalues of matrix P.

Clifford Algebra
The Clifford real algebra over R m is defined as Moreover e ∅ = e 0 = 1 and e h = e {h} , h = 1, 2, . . . , m are denoted as the Clifford generators, and they fulfill the following relationship: For the sake of simplicity, when an element is the product of multiple Clifford generators, its subscripts are incorporated together, e.g. e 4 e 5 e 6 e 7 = e 4567 .
From the definition, we can directly deduce that e AēA =ē A e A = 1. For a Clifford-valued function p = A p A e A : R → A, where p A : R → R, A ∈ , and its derivative is represented e B e A , we can write e BēA = e C or e BēA = -e C , where e C is a basis of Clifford algebra A. As an example, Moreover, for any G ∈ A, there is a unique G C that satisfies

Problem Definition
Consider a Clifford-valued NN model with time-varying delays, as follows: where i, j = 1, 2, . . . , n, and n corresponds to the number of neurons; p i (t) ∈ A represents the state vector of the ith unit; d i ∈ R + indicates the rate with which the ith unit resets its potential to the resting state in isolation when it is disconnected from the network and external inputs; a ij , b ij ∈ A indicate the strengths of connection weights without and with time-varying delays between cells i and j, respectively; u i ∈ A is an external input of the ith unit; g j (·) : A n → A n is the activation functions of signal transmission; τ (t) ∈ R + is the transmission delay at time t. Furthermore, ϕ i ∈ C([-τ , 0], A n ) is the initial condition for the considered NN model (1).
(H1) Function g j (·) fulfills the Lipschitz continuity condition with respect to the ndimensional Clifford vector. For each j = 1, 2, . . . , n, there exists a positive constant k j such that, for any x, y ∈ A, where k j is known as the Lipschitz constant and g j (0) = 0. And there exist positive constants k j such that |g j (x)| A ≤ k j (j = 1, 2, . . . , n).
(H2) Time-varying delay τ (t) is differential and it fulfills the following conditions: where τ and μ are known real constants.
Remark 2.2 It is worth noting that Clifford-valued NN models are the generalized form of real-, complex-, and quaternion-valued NN models. For example, when we take into account m = 0 in NN model (3), then the model can be reduced to the real-valued NN model proposed in [6]. Suppose, we take m = 1 in NN model (3), then the model can be reduced to the complex-valued NN model proposed in [14]. If we choose m = 2 in NN model, then the model can be reduced to the quaternion-valued NN model proposed in [26]. Therefore, the proposed system model in this paper is more general than the system model proposed in [6,14,26]. (3), the vector p * = (p * 1 , p * 2 , . . . , p * n ) T ∈ A n is an equilibrium point, if and only if p * is a solution of the following equation:

Exponential stability
where is negative definite.
Remark 3.3 When the impulse effects are absent in the NN model (12), By applying a similar approach to the one proposed in Theorem 3.2, the following global exponential stability criteria can be obtained for the NN model (26). (H1) and (H2) hold, then the following conditions are satisfied: (i) For given scalars : 0 < < min d i (i = 1, . . . , n), τ ≥ 0 and μ, if there exist positive definite matrices P ∈ R 2 m n×2 m n and Q ∈ R 2 m n×2 m n , and a positive diagonal matrix M = diag{ 1 , . . . , n } ∈ R 2 m n×2 m n such that

Corollary 3.4 Suppose
where˜ is negative definite.
As such, the equilibrium point of NN model (26) is globally exponentially stable. Further, Proof Construct the same LKF (15). The remaining proof follows the same procedure used for Theorem 3.2. Therefore, it is omitted here.
Remark 3.5 With the absence of the time-varying delays and impulsive effects, NN model (12) reduces to the following model: By applying a similar approach to the one proposed in Theorem 3.2, the following global exponential stability criteria can be obtained for NN model (29).
where¯ is negative definite. As such, the equilibrium point pertaining to NN model (29) is globally exponentially stable. Moreover, Proof Construct the following LKF: The remaining proof follows the one used for Theorem 3.2. Therefore, it is omitted here.
Remark 3.7 The multiplication of Clifford numbers does not satisfy commutativity, which complicates the study of Clifford-valued NNs. As such, from the theoretical and practical point of view, studying the dynamical behaviors of Clifford-valued NNs are difficult tasks.
On the other hand, the decomposition approach has proven to be very effective to overcome the difficulty of the non-commutativity of Clifford number multiplication. By decomposition approach, the original n-dimensional Clifford-valued system is decomposed into 2 m n-dimensional real-valued system and the coefficient and activation functions of networks are explicitly expressed. Therefore, the decomposition approach is highly favor-able to analyze the dynamics of Clifford-valued NNs. Most of authors have recently obtained sufficient criteria for Clifford-valued NN models using the decomposition method [34,36,38,42].
Remark 3.8 In [35], fuzzy operations are incorporated into the Clifford-valued cellular NN model to investigate its S p -almost periodic solutions. The effects of discrete delays in Clifford-valued recurrent NNs are considered in [36], and the associated globally asymptotic almost automorphic synchronization criteria are obtained. The leakage delay is introduced into Clifford-valued high-order Hopfield NN models in [39] to explore its existence and global exponential stability of almost automorphic solutions. To be compared with some previous studies of Clifford-valued NN models [33][34][35][36][37][38][39][40][41][42], in this paper, by considering a class of impulsive Clifford-valued NN system model, we have studied more realistic dynamical behavior. Therefore, the proposed results in this paper is different and new compared with those in the existing literature [33][34][35][36][37][38][39][40][41][42].
Remark 3.9 When we consider the exponential convergence rate = 0 in the main results, the exponential stability criteria can be converted to asymptotically stability criteria.

Numerical Example
We present numerical simulation to ascertain the feasibility and effectiveness of the results established in Sect. 3.

Conclusion
In this study, the global exponential stability analysis pertaining to the Clifford-valued NN models with time-varying delays and impulsive effects has been comprehensively investigated. To tackle the problem, we firstly decomposed the considered n-dimensional Clifford-valued NN model into 2 m n-dimensional real-valued ones. By using an appropriate Lyapunov functional and some inequality techniques, we have established new LMIbased sufficient conditions. These conditions guarantee the global exponential stability of the equilibrium point pertaining to the considered Clifford-valued NN model. To ascertain the validity of the main results, a standard numerical example has been provided. It is worth mentioning that the results achieved in this paper can further be extended to various complex systems. We will shortly attempt to explore the stabilization analysis of Clifford-valued NNs with the help of different controller schemes. The corresponding results will be presented in the near future.