Exponential stability in the Lagrange sense for Clifford-valued recurrent neural networks with time delays

This paper considers the Clifford-valued recurrent neural network (RNN) models, as an augmentation of real-valued, complex-valued, and quaternion-valued neural network models, and investigates their global exponential stability in the Lagrange sense. In order to address the issue of non-commutative multiplication with respect to Clifford numbers, we divide the original n-dimensional Clifford-valued RNN 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} real-valued models. On the basis of Lyapunov stability theory and some analytical techniques, several sufficient conditions are obtained for the considered Clifford-valued RNN models to achieve global exponential stability according to the Lagrange sense. Two examples are presented to illustrate the applicability of the main results, along with a discussion on the implications.

Clifford algebra provides a powerful theory for solving problems in geometry problems. In addition, Clifford algebra has been applied to numerous areas of science and engineering such as neural computation, computer and robot vision, and control problems [24][25][26][27][28]. In dealing with high-dimensional data and spatial geometric transformation, Cliffordvalued NN models are superior to real-valued, complex-valued, and quaternion-valued NN models [26][27][28][29]. Recently, theoretical and applied studies on Clifford-valued NN models have become a new research subject. However, the dynamic properties of Cliffordvalued NN models are usually more complex than those of real-valued, complex-valued and quaternion-valued NN models. Due to the issue of non-commutativity of multiplication with respect to Clifford numbers, studies on Clifford-valued NN dynamics are still limited [30][31][32][33][34][35][36][37][38][39].
By using the linear matrix inequality approach, the authors in [30] derived the global exponential stability criteria for delayed Clifford recurrent NN models. For Clifford-valued NN models with time delays, their global asymptotic stability problems were investigated in [31] by a decomposition method. Using a direct approach, the presence and global exponential stability conditions pertaining to almost periodic solutions were extracted for Clifford-valued neutral high-order Hopfield NN models with leakage delays in [34]. Using the Banach fixed point theorem and the Lyapunov functional, the global asymptotic almost periodic synchronization problems for the Clifford-valued NN cellular models were examined in [35]. The weighted pseudo-almost automorphic solutions for neutral type fuzzy cellular Clifford NN models were discussed in [37]. In [39], the authors explored the presence of an anti-periodic solution to the problem of Clifford-valued inertial Cohen-Grossberg NN models by using suitable Lyapunov functional. Recently, the effects of neutral delay and discrete delays have been considered in a class of Clifford-valued NNs [40], and the associated existence, uniqueness and global stability criteria have been obtained.
It is worth noting that many previous NNs studies focus mostly on the global stability in the Lyapunov sense. In many real physical systems, however, the equilibrium point may not exist. Besides, computationally restrictive and multi-stability dynamics have been found to be necessary for dealing with critical neural computation in applications [41,42]. As an example, when employing NN models for associative memory or pattern recognition, multiple equilibrium points are often required. In these cases, NN models are no longer globally stable in the Lyapunov sense. In dealing with multi-stable models, more realistic concepts of stability are necessary [43][44][45][46]. It should be noticed that in Lyapunov stability of NN models, the existence and uniqueness of the equilibrium points are required. In contrast, the Lagrange stability, which is concerned with stability of the total model, does not require the information of equilibrium points. Therefore, the Lagrange stability or the attractive sets of NN models is useful, and many studies have been published [45][46][47][48].
Motivated by the above discussions, we investigate the exponential stability in the Lagrange sense for Clifford-valued RNNs with time delays in this paper. To the best of our knowledge, there is no result on the topic of Lagrange stability of Clifford-valued NNs. Undoubtedly, this interesting topic is still an open challenge. Therefore, our main objective of this paper is to derive new sufficient conditions to ensure the Lagrange exponential stability for Clifford-valued RNNs. The main contributions of this work are as follows: (1) This is the first paper to study such a problem for global exponential stability of Clifford-valued RNN models in the Lagrange sense, which encompasses real-valued, complex-valued, and quaternion-valued NN models as special cases. (2) By using the Lyapunov functional and some analytical methods, new sufficient conditions for ascertaining the exponential stability of the considered Clifford-valued RNN models in the Lagrange sense are derived. This is achieved by decomposing the n-dimensional Clifford-valued RNN model into 2 m n-dimensional real-valued RNN models. (3) The results obtained can be used for the study of monostable, multi-stable, and other complex NN models. In addition, the obtained results are new and different compared with those in the existing literature. The usefulness of the obtained results is validated with two numerical examples.
We organize this paper as follows. The proposed Clifford-valued RNN model is formally defined in Sect.

Mathematical fundamentals and problem formulation 2.1 Notations
A is defined as the Clifford algebra and has m generators over the real number R. Let R n and A n be an n-dimensional real vector space and an n-dimensional real Clifford vector space, respectively; while 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. We define the norm of R n as r = n i=1 |r i |, and for A = (a ij ) n×n ∈ R n×n , denote A = max 1≤i≤n { n j=1 |a ij |}. While r = A r A e A ∈ A, denote |r| A = A |r A |, and for A = (a ij ) n×n ∈ A n×n , denote A A = max 1≤i≤n { n j=1 |a ij | A }. Superscripts T and * , respectively, denote the matrix transposition and matrix involution transposition. For any matrix, P > 0 (< 0) denotes a positive (negative) definite matrix. Let C C F + be the set of all nonnegative continuous functionals L : C → [0, ∞). For ϕ ∈ C([-τ , 0], A n ), ϕ τ ≤ sup -τ ≤s≤0 ϕ(t + s) .
Moreover, e ∅ = e 0 = 1 and e l = e {l} , l = 1, 2, . . . , m, are denoted as the Clifford generators, and they fulfill the following relations: For 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 r = A r A e A : R → A, where r A : R → R, A ∈ , its derivative is represented by 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, e l 1 l 2ē l 2 l 3 = -e l 1 l 2 e l 2 l 3 = -e l 1 e l 2 e l 2 e l 3 = -e l 1 (-1)e l 3 = e l 1 e l 3 = e l 1 l 3 . Therefore, it is possible to identify a unique corresponding basis e C for a given e BēA . Define Moreover, for

Problem definition
The following Clifford-valued RNN model with discrete time-varying delays is considered: where i, j = 1, 2, . . . , n, and n corresponds to the number of neurons; r i (t) ∈ A represents the state vector of the ith unit; d i ∈ R + indicates the rate with which the ith unit will reset 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 on the ith unit; h j (·) : A n → A n is the activation function of signal transmission; τ j (t) corresponds to the transmission delay which satisfies 0 ≤ τ j (t) ≤ τ , where τ is a positive constant and For convenience of discussion, we rewrite (1) in the vector form where (A1) Function h 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 (j = 1, 2, . . . , n) is known as the Lipschitz constant and h j (0) = 0. In addition, there exists a positive constant k j such that |h(x)| A ≤ k j for any x ∈ A. According to Assumption (A1), it is clear that where K = diag{k 1 , k 2 , . . . , k n }.
Remark 2.1 There exists a constant k j > 0 (j = 1, 2, . . . , n) such that ∀x = (x 1 , x 2 , . . . , x 2 m ) T ∈ R 2 m and ∀y = (y 1 , y 2 , . . . , Remark 2.2 The proposed system model in this paper is more general than the system model proposed in previous works [4,6,19]. For example, when we consider m = 0 in NN model (3), then the model can be reduced to the real-valued NN model proposed in [4]. If we take m = 1 in NN model (3), then the model can be reduced to the complex-valued NN model proposed in [6]. If we choose m = 2 in NN model (3), then the model can be reduced to the quaternion-valued NN model proposed in [19].

Main results
In order to overcome the issue of non-commutativity of multiplication of Clifford numbers, we transform the original Clifford-valued model (3) into multidimensional realvalued model. This can be achieved with the help of e AēA =ē A e A = 1 and e BēA e A = e B . Given any G ∈ A, unique G C that is able to satisfy where Corresponding to the basis of Clifford algebra, the Clifford-valued NN model can be reformulated into novel real-valued ones. Definẽ  (6) can be written aṡ with the initial valuẽ whereφ(s) = ((ϕ 0 (s)) T , (ϕ 1 (s)) T , . . . , (ϕ A (s)) T , . . . , (ϕ 12...m (s)) T ) T ∈ R 2 m n .
In addition, notice that (4) can be expressed as the following inequality: , and two constants ρ > 0 and υ > 0 such that, for any solutionr(t) of NN model (7), for any t ≥ 0 andφ ∈ C β . Then NN (7) is said to be globally exponentially attractive (GEA) with respect to V(t), and the compact set = {r(t) ∈ R 2 m n |V(t) ≤ ρ} is said to be a GEA set of NN model (7). (7) is globally exponentially stable in the Lagrange sense if it is both uniformly stable in the Lagrange sense and is globally exponentially attractive.

Lemma 3.7 ([47])
Given matrices E ∈ R 2 m n×2 m n , F ∈ R 2 m n×2 m n , G ∈ R 2 m n×2 m n , H ∈ R 2 m n×2 m n , and appropriate reversible matrices X , Y, then

Exponential stability
Theorem 3.9 Under Assumption (A1), if there exist positive definite matrices P, R, S ∈ R 2 m n×2 m n , positive diagonal matrices C, Q ∈ R 2 m n×2 m n such that the following LMI hold: where = P + CK -PD -D T P +KQK, then NN model (7) is globally exponentially stable in the Lagrange sense. Moreover, the set = r ∈ R 2 m n r T (t)Pr(t) ≤Ũ T SŨ (12) is a globally exponentially attractive set of NN model (7), where is a proper positive constant.
Proof Consider the following Lyapunov function which is positive definite and radially unbounded Then, the Dini derivative D + V(t) can be computed along the solutions of NN model (7), we get There exists a positive diagonal matrix Q, and through Assumption (A1) we have Using Assumption (A1) and Lemma 3.4, there exist positive defined matrices R and S such that the following inequalities are true: and 2 r T (t)P +h T r(t) C Ũ ≤ r T (t)P +h T r(t) C S -1 r T (t)P +h T r(t) C Through Lemma 3.7 and (14)- (17), we have There exists > 0, and from (10) we have where˜ = (1 + )(P + CK) -PD -D T P +KQK. According to Lemma 3.5, we have This indicates that By combining (11), (18), and (20), we have
As such, NN model (7) is globally exponentially stable in the Lagrange sense, and the set = r ∈ R 2 m n r T (t)Pr(t) ≤Ũ T SŨ is a globally exponentially attractive set of NN model (7). The proof of Theorem 3.9 is completed. Now we have the following Corollary 3.10.

Corollary 3.10
Under (A1), if there exist positive definite matrices P, R, S ∈ R 2 m n×2 m n , positive diagonal matrix Q ∈ R 2 m n×2 m n such that the following LMI hold: where = P -PD -D T P +KQK. As such, NN model (7) is globally exponentially stable in the Lagrange sense. Moreover, the set = r ∈ R 2 m n r T (t)Pr(t) ≤Ũ T SŨ is a globally exponentially attractive set of NN model (7).
Proof Consider the following Lyapunov function which is positive definite and radially unbounded: Corollary 3.10 can be proven by applying a similar approach pertaining to Theorem 3.9. So, the proof is omitted here. 1, 2, . . . , n) is a positive constant, then NN (6) is globally exponentially stable in the Lagrange sense. Moreover, the compact set

Corollary 3.11 Suppose that the activation function h i
is the globally exponentially attractive set of (6), where Proof First of all, we prove that NN model (6) is uniformly stable in the Lagrange sense. Consider the following Lyapunov function which is positive definite and radially unbounded: Let 0 < μ i < d i (i = 1, 2, . . . , n). Then the Dini derivative D + V(t) can be computed along the positive half trajectory of NN model (6), and we have Through Lemma 3.8, for any t ≥ 0, we have This ensures that the solution of NN model (6) is uniformly bounded. Hence, NN model (6) is uniformly stable in the Lagrange sense. Observe that Then L ∈ C C F + , and (31) implies that, for any t ≥ 0, Through Definition 3.2, NN model (6) is globally exponentially stable and˜ 1 is a globally exponentially attractive set. This proves the global exponential stability in the Lagrange sense of NN model (6).
is a globally exponentially attractive set of NN model (6), where Proof We first prove that˜ 2 is a globally exponentially attractive set. We employ the positive definite and radially unbounded Lyapunov function Then the Dini derivative D + V(t) can be computed along the positive half trajectory of NN model (6), and we have Similarly, we infer that˜ 2 is a globally exponentially attractive set by Lemma 3.8 and Definition 3.2. Thus, the proof procedure is omitted.
Remark 3.13 It is worth mentioning that the Clifford number multiplication does not satisfy commutativity, which complicates the investigation of Clifford-valued NNs' dynamics. On the other hand, the method of decomposition is very successful in overcoming the difficulty of non-commutative Clifford numbers multiplication. Recent authors have obtained sufficient conditions for global stability and periodic solutions of Clifford-valued NN models by using the decomposition method [31,33,35,39,40].

Numerical examples
Two numerical examples are presented to demonstrate the feasibility and effectiveness of the results established in Sect. 3.

Conclusions
In this paper, the global exponential stability problem for Clifford-valued RNN models with time-varying delays in the Lagrange sense has been examined with the use of Lyapunov functional and LMI methods. The sufficient conditions have been obtained by decomposing the n-dimensional Clifford-valued RNN model into a 2 m n-dimensional realvalued RNN model to ensure the global exponential stability of the considered NN model in the Lagrange sense. Furthermore, the estimated global exponentially attractive set has been obtained. In addition, the validity and feasibility of the results obtained have been demonstrated with two numerical examples. The results obtained in this paper can be further extended to other complex systems. We will be exploring the stabilizability and instabilizability analysis of Clifford-valued NN models with the help of various control systems. The corresponding results will be carried out in the near future.