System decomposition method-based global stability criteria for T-S fuzzy Cli ff ord-valued delayed neural networks with impulses and leakage term

: This paper investigates the global asymptotic stability problem for a class of Takagi-Sugeno fuzzy Cli ff ord-valued delayed neural networks with impulsive e ff ects and leakage delays using the system decomposition method. By applying Takagi-Sugeno fuzzy theory, we first consider a general form of Takagi-Sugeno fuzzy Cli ff ord-valued delayed neural networks. Then, we decompose the considered n -dimensional Cli ff ord-valued systems into 2 m n -dimensional real-valued systems in order to avoid the inconvenience caused by the non-commutativity of the multiplication of Cli ff ord numbers. By using Lyapunov-Krasovskii functionals and integral inequalities, we derive new su ffi cient criteria to guarantee the global asymptotic stability for the considered neural networks. Further, the results of this paper are presented in terms of real-valued linear matrix inequalities, which can be directly solved using the MATLAB LMI toolbox. Finally, a numerical example is provided with their simulations to demonstrate the validity of the theoretical analysis.


Introduction
The study of neural networks (NNs) has attracted considerable attention among researchers over the past two decades because it plays an important role in a wide range of applications such as associative memory, automatic control, pattern recognition, image processing, secure communication and optimization problems, see for examples [1][2][3][4][5][6][7]. Recently, the extension of real-valued NNs such as complex-valued NNs and quaternion-valued NNs have attracted significant attention due to their ability to solve a variety of engineering problems [8-13, 22, 23]. It is important to point out that all of these applications depend on the stability of the equilibrium of NNs. Thus, the stability analysis is a necessary step for the design and applications of NNs. As a result, a number of theoretical results leakage terms affect the system's dynamics [11,24,34,35,49,50]. Similar to time delays, impulsive perturbations also affect the dynamics of NNs. Therefore, it is important to consider the impulsive effects when analysing the dynamics of NNs [48][49][50][51][52].
By the above discussions, we aim to investigate the global asymptotic stability of T-S fuzzy Cliffordvalued delayed NNs with impulses by applying the system decomposition method. To the best of our knowledge, few studies have investigated the stability analysis of Clifford-valued NNs with time delays by the decomposition method. However, T-S fuzzy Clifford-valued NNs with leakage delays and impulses have not been fully explored and are not receiving much attention, which motivates us to investigate this paper. This paper has the following main merits: 1) To represent more realistic dynamics of Clifford-valued NNs, we present a general form of T-S fuzzy Clifford-valued NNs with time delays and impulsive effects. 2) The system decomposition method is employed to examine the global asymptotically stability of T-S fuzzy Clifford-valued NNs. 3) By considering suitable LKFs that contain double integral terms and by employing integral inequalities, enhanced stability conditions for the concerned NN model are derived in terms of real-valued LMIs, which could be verified directly by the MATLAB LMI toolbox.
The paper is structured as follows: Section 2 provides the problem model. Section 3 gives the main results of this paper. Section 4 discusses a numerical example that demonstrate the feasibility of the derived results. Section 5 shows the conclusion of this paper.

Notations
The Clifford algebra over R is defined as A with m generators. Let R n , A n denote the n-dimensional real and real Clifford vector space, respectively. R n×m , A n×m denote the set of all n × m real and real Clifford matrices, respectively. The superscript T and * denote, respectively, the matrix transposition and involution transposition. The matrix P > 0 (P < 0) means that P is the positive (negative) definite matrix. ⋆ denotes the elements below the main diagonal of a symmetric matrix. I is the identity matrix with appropriate dimensions.

Problem definition
Consider the following Clifford-valued NNs with time-varying delays and leakage terms where u(t) = (u 1 (t), ..., u n (t)) T ∈ A n denotes the neuron state vector; is the Clifford-valued activation function; L = (l 1 , ..., l n ) T ∈ A n is the Clifford-valued external input vector; σ > 0 denotes the leakage delay; τ(t) denotes the time-varying delays satisfies 0 ≤ τ(t) ≤ τ, τ(t) ≤ µ < 1 where τ and µ are real constants; ψ(t) is the initial condition which is continuously differential on t ∈ [−ρ, 0] and ρ = max{σ, τ}. Assumption 1: For all j = 1, ..., n, the neuron activation functions f j (·) is continuous and bounded, there exist positive diagonal matrix K = diag{k 1 , ..., k n } such that It is obvious from Assumption 1 that, Proof: Since the activation function of NNs (2.1) is bounded, there exist constants K j such that, According to the self-feedback connection weight matrix D > 0 that D is invertible. We denote Ω = {u ∈ A n : ∥u∥ ≤ ∥D −1 ∥(∥A∥K + ∥J∥)} and define the map A n −→ A n by Here, H is a continuous map and by applying ∥ f (u)∥ A ≤ K, we obtain that, Thus, H maps Ω into itself. By Brouwers fixed point theorem, it can be derived that there exist a fixed point u * of H, satisfying Pre multiplying by D on two sides, gives which is equivalent to −Du * + A f (u * ) + J = 0. This completes the proof.
Remark 2.2. According to the above Assumption 1, this paper assumes that Clifford-valued activation functions satisfy Lipschitz conditions. Similar to previous results [12,13], we consider the boundedness of activation functions in order to derive the existence of the equilibrium point. Obviously, this assumption of boundedness can lead to limitations in choosing activation functions. Therefore, the boundedness of solutions is one of the most important aspects of the systems that needs to be taken into account, see for examples [14][15][16][17][18][19][20][21].
Conveniently, we transform v(t) = u(t) − u * to shift the equilibrium point. Then, NN (2.1) can be re-written asv where v(t) is the state vector, φ(t) = ψ(t) − u * is the initial condition and the transformed activation Based on [39][40][41][42][43], the T-S fuzzy Clifford-valued NNs can be shown as follows
When the Clifford-valued NNs (2.8) is incorporated with impulse effects, we have is the impulse at moments t k and v(t + k ) and v(t − k ) denotes the right and left hand limits of v(t k ), respectively. In addition, I k = diag{I 1 , ..., I n } ∈ R n×n denotes the impulsive matrix and the impulse time

Main results
First, we use e AēA =ē A e A = 1 to rewrite the original Clifford-valued NNs. Similar to the papers [30,31,35,37], it is simple to obtain a unique G C satisfying which implies the following transformation NNs (3.1).
The second term in NN (2.9) can be defined as Then, we can decompose NN (2.9) into the following real-valued one: According to Clifford algebra, NN (3.1) can be expressed as a new real-valued NNs. Let Then, NN (3.1) can be written as Furthermore, (2.2) can be written in the following form: ) are assumed to satisfy the following conditions Lemma 3.2.
[54] For any constant positive definite matrix M = M T ∈ R 2 m n×2 m n , any constant matrix X ∈ R 2 n ×2 m n , any vector θ 1 , θ 2 ∈ R 2 m n , and ϑ ∈ (0, 1), such that M X X T M > 0, the following condition Global asymptotic stability analysis In this subsection, we will derive the sufficient criteria to assure the global asymptotic stability of the considered NNs (3.2) using the LKFs and LMI method.
any matrix X and scalars ϵ 1 > 0 such that the following LMIs hold for all p = 1, 2, ..., m: Proof: Construct the following LKF for NN model (3.2): Moreover, it follows from (3.4) that Combining (3.7) and (3.8), we have It is easy to verify that When t t k , k ∈ Z + , we can compute the upper right derivative of (3.6) along the trajectories of (3.2), we have where (3.14) The first integral term in (3.14) can be defined as By applying Lemma (3.1) in the following forms 2), the following inequality true: t−τv (s)ds .

(3.19)
By applying Lemma (3.1), the second integral term in (3.14) can be defined as By applying Lemma (3.1), we get There exist positive scalar ϵ 1 > 0. By Assumption 1, we have Using the Schur complement it can be derived from (3.26) that From condition (3.5), we have The remaining proof is similar to that in Theorem (3.3), and so it is omitted.
Remark 3.6. When the leakage term is absent, NN (3.29) decreases as follows: if there exist positive definite symmetric matrices P, Q 1 , Q 2 , R 1 and S 11 S 12 ⋆ S 22 > 0, any matrix X and scalars ϵ 1 > 0 such that the following LMIs hold for all p = 1, 2, ..., m: Proof:Construct the following LKF for NN model (3.31): The remaining proof is similar to that in Theorem (3.3), and so it is omitted.
Remark 3.8. According to our knowledge, there are no studies that have compared the global asymptotic stability criteria for time-varying delays, impulse effects as well as leakage terms among the obtained global asymptotic stability criteria for T-S fuzzy Clifford-valued NNs, which shows the novelty of this paper.

Numerical example
This section provides a numerical example to demonstrate the validity of the obtained results. Example 1: Let p = 1, 2. Consider the following plant rules for T-S fuzzy Clifford-valued NNs.

Conclusions
In this paper, the problem of global asymptotic stability of T-S Clifford-valued fuzzy delayed NNs with impulsive effects and leakage term has been investigated. By applying T-S fuzzy theory, we first considered a general form of T-S fuzzy Clifford-valued NNs with time-varying delays. Then, we decomposed the original Clifford-valued NNs into the 2 m n-dimensional real-valued NNs in order to solve the non-commutativity issue pertaining Clifford numbers. By considering appropriate LKFs and integral inequalities, new sufficient criteria are obtained to guarantee the global asymptotic stability of the considered networks. Furthermore, the results of this paper are presented in the form of LMIs, which can be solved using the MATLAB LMI toolbox. Finally, a numerical example is presented with their simulations to demonstrate the validity of the theoretical analysis.
By applying the main results of this paper, we can analyze various dynamical behaviors of T-S fuzzy Clifford-valued NNs including finite-time stability, passivity, state estimation, synchronization, and others. There are certain advancements worth investigating further in this proposed area of research. We will soon attempt to examine the finite-time dissipativity of T-S fuzzy Clifford-valued NNs with time delays.