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Global exponential stability of Cliffordvalued neural networks with timevarying delays and impulsive effects
Advances in Difference Equations volume 2021, Article number: 208 (2021)
Abstract
In this study, we investigate the global exponential stability of Cliffordvalued neural network (NN) models with impulsive effects and timevarying delays. By taking impulsive effects into consideration, we firstly establish a Cliffordvalued NN model with timevarying delays. The considered model encompasses realvalued, complexvalued, and quaternionvalued NNs as special cases. In order to avoid the issue of noncommutativity of the multiplication of Clifford numbers, we divide the original ndimensional Cliffordvalued model into \(2^{m}n\)dimensional realvalued 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.
Introduction
Dynamic analysis of neural network (NN) models has gained tremendous research interest in recent decades, since NN models play a significant role in various applications. In particular, the stability theory of NN models has been extensively studied to solve many problems related to science and engineering, including associative memory, image and signal processing, optimization, pattern recognition and others [1–8]. On the other hand, complex signals exist in most NN applications in the real world. As such, multidimensional problems such as color night vision, radar imaging, polarized signal classification, 3D wind forecasting can be solved more favorably by implementing complexvalued and quaternionvalued NN models [9–26]. To date, several important results regarding various dynamics of complexvalued and quaternionvalued NN models have been published, some of them stability analysis [13, 14, 20–22, 26], synchronization analysis [11], stabilizability and instabilizability [12], controllability and observability [23], optimization [24, 25] and so on.
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 [27–31]. Cliffordvalued NN models stand as a generalization of realvalued, complexvalued, and quaternionvalued NNs. In this respect, Cliffordvalued NN models are superior to realvalued, complexvalued, and quaternionvalued NNs for undertaking spatial geometric transformation and highdimensional data problems [29–32]. Theoretical and applied studies of Cliffordvalued 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 realvalued and complexvalued NN models. As such, studies on Cliffordvalued NN dynamics are still limited due to those utilizing the principle of noncommutativity of the product of Clifford numbers [33–42].
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 Cliffordvalued NN models with time delays, their global asymptotic stability issues were examined in [34] by decomposing the ndimensional Cliffordvalued NN model into \(2^{m}n\)dimensional realvalued models. Leveraging on a direct method, the existence and global exponential stability of almost periodic solutions were derived for Cliffordvalued neutral highorder 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 Cliffordvalued 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 antiperiodic solutions corresponding to a class of Cliffordvalued inertial Cohen–Grossberg NN models by constructing suitable LKFs.
Due to the limited speed of signal propagation, time delays (either constant or timevarying) are often encountered in NN models operating in realworld applications [3, 5–8]. Time delays are the main source of various dynamics such as chaos, divergence, poor functionality and instability [11–13, 36, 39, 41]. Hence it is necessary to analyze the NN dynamics that incorporate either constant or timevarying delays. In this respect, NN models with various time delays have been extensively studied, and many significant results have been obtained [3, 5, 11–13, 33–46]. On the other hand, due to noise, change in frequency, or switching phenomenon, the impulsive effects occur in realworld systems [47]. The resulting dynamical behaviors of the systems are more complex than those of classical dynamical systems [47, 48]. Generally, impulses can affect the dynamic behaviors of networks. As such, many papers dealing with dynamical systems with impulsive effects have appeared in recent years [43–50].
To the best of our knowledge, there are hardly any papers that deal with the issue of global exponential stability of Cliffordvalued NNs with timevarying 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 Cliffordvalued NNs with impulsive effects. In order to achieve our main results, the ndimensional Cliffordvalued NN model is firstly decomposed into \(2^{m}n\)dimensional realvalued 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 Cliffordvalued 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 Cliffordvalued NN models with timevarying 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 Cliffordvalued NN model has been decomposed into real, complex, and quaternionvalued NN models; (3) our proposed method can be easily employed for other dynamic behaviors with respect to different types of Cliffordvalued NN models.
The organization of this article is as follows. The proposed Cliffordvalued 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.
Mathematical Fundamentals and Problem Formulation
Notations
Let \(\mathbb{R}^{n}\) and \(\mathbb{A}^{n}\) denote the ndim real vector space and ndim real Clifford vector space, respectively. \(\mathbb{R}^{n \times n}\) and \(\mathbb{A}^{n \times n}\) denote the set of all \(n \times n\) real matrices and the set of all \(n \times n\) real Clifford matrices, respectively. \(\mathbb{A}\) is defined as the Clifford algebra with m generators over the real number \(\mathbb{R}\). Superscripts T and ∗, respectively, indicate matrix transposition and matrix involution transposition. A matrix (<0) denotes a positive (negative) definite matrix. We define the norm of \(\mathbb{R}^{n}\) as \(\p\=\sum_{i=1}^{n}p_{i}\). Besides that, , denote \(\B\=\max_{1\leq i\leq n}\{\sum_{j=1}^{n}b_{ij} \}\), while \(p=\sum_{A}p^{A}e_{A}\in \mathbb{A}\) denote \(\p\_{\mathbb{A}}=\sum_{A}p^{A}\) and for , denote \(\B\_{\mathbb{A}}=\max_{1\leq i\leq n}\{\sum_{j=1}^{n}b_{ij}_{ \mathbb{A}}\}\). For \(\varphi \in \mathcal{C}([\tau , 0], \mathbb{A}^{n})\), we introduce the norm \(\\varphi \_{\tau }\leq \sup_{\tau \leq s\leq 0}\\varphi (t+s) \\). and , respectively, denote the maximum and minimum eigenvalues of matrix .
Clifford Algebra
The Clifford real algebra over \(\mathbb{R}^{m}\) is defined as
where \(e_{A}=e_{h_{1}}e_{h_{2}}\ldots e_{h_{\nu }}\) with \(A=\{h_{1},h_{2},\ldots ,h_{\nu }\}\), \(1 \leq h_{1} < h_{2} <\cdots < h_{\nu }\leq m\).
Moreover \(e_{\emptyset }=e_{0}=1\) and \(e_{h}=e_{\{h\}}\), \(h=1,2,\ldots ,m\) 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. \(e_{4}e_{5}e_{6}e_{7}=e_{4567}\).
Let \(\Lambda =\{\emptyset ,1,2,\ldots , A,\ldots , 12\dots m\}\), and we have
where \(\sum_{A}\) denotes \(\sum_{A\in \Lambda }\) and \(\mathbb{A}\) is isomorphic to \(\mathbb{R}^{2^{m}}\).
For any Clifford number \(p=\sum_{A}p^{A}e_{A}\), the involution of p is defined by
where \(\bar{e}_{A}=(1)^{\frac{\sigma [A](\sigma [A]+1)}{2}}e_{A}\), and
From the definition, we can directly deduce that \(e_{A}\bar{e}_{A}=\bar{e}_{A}e_{A}=1\). For a Cliffordvalued function \(p=\sum_{A}p^{A}e_{A}:\mathbb{R}\rightarrow \mathbb{A}\), where \(p^{A}:\mathbb{R}\rightarrow \mathbb{R}\), \(A\in \Lambda \), and its derivative is represented by \(\frac{dp(t)}{dt}=\sum_{A}\frac{dp^{A}(t)}{dt}e_{A}\).
Since \(e_{B}\bar{e}_{A}=(1)^{\frac{\sigma [A](\sigma [A]+1)}{2}}e_{B}e_{A}\), we can write \(e_{B}\bar{e}_{A}=e_{C}\) or \(e_{B}\bar{e}_{A}=e_{C}\), where \(e_{C}\) is a basis of Clifford algebra \(\mathbb{A}\). As an example, \(e_{h_{1}h_{2}}\bar{e}_{h_{2}h_{3}}=e_{h_{1}h_{2}}e_{h_{2}h_{3}} =e_{h_{1}}e_{h_{2}}e_{h_{2}}e_{h_{3}}=e_{h_{1}}(1)e_{h_{3}}=e_{h_{1}}e_{h_{3}} = e_{h_{1}h_{3}}\). Therefore, it is possible to identify a unique corresponding basis \(e_{C}\) for a given \(e_{B}\bar{e}_{A}\). Define
and then \(e_{B}\bar{e}_{A} =(1)^{\sigma [B.\bar{A}]}e_{C}\).
Moreover, for any \(\mathscr{G}\in \mathbb{A}\), there is a unique \(\mathscr{G}^{C}\) that satisfies \(\mathscr{G}^{B.\bar{A}}=(1)^{\sigma [B.\bar{A}]}\mathscr{G}^{C}\) for \(e_{B}\bar{e}_{A}=(1)^{\sigma [B.\bar{A}]}e_{C}\). Therefore
and \(\mathscr{G}= \sum_{C}\mathscr{G}^{C}e_{C} \in \mathbb{A}\).
Problem Definition
Consider a Cliffordvalued NN model with timevarying delays, as follows:
where \(i,j= 1,2,\ldots ,n\), and n corresponds to the number of neurons; \(p_{i}(t)\in \mathbb{A}\) represents the state vector of the ith unit; \(d_{i}\in \mathbb{R}^{+}\) 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; \(a_{ij}, b_{ij}\in \mathbb{A}\) indicate the strengths of connection weights without and with timevarying delays between cells i and j, respectively; \(u_{i}\in \mathbb{A}\) is an external input of the ith unit; \(g_{j}(\cdot ):\mathbb{A}^{n} \rightarrow \mathbb{A}^{n}\) is the activation functions of signal transmission; \(\tau (t)\in \mathbb{R}^{+}\) is the transmission delay at time t. Furthermore, \(\varphi _{i}\in \mathcal{C}([\tau , 0],\mathbb{A}^{n})\) is the initial condition for the considered NN model (1).
The impulsive Cliffordvalued NN model with timevarying delays can be described as follows:
where \(\triangle p_{i}(t_{k})= p_{i}(t_{k}^{+})p_{i}(t_{k}^{})\) is the jump in state variable \(p_{i}\). In addition, the impulsive moment \(t_{k}\) (\(k\in \mathbb{N}\)) fulfills \(0< t_{1}< t_{2}<\cdots t_{k}<\cdots \) , which is a strictly increasing sequence such that \(\lim_{k\rightarrow +\infty }t_{k}=+\infty \); \(p_{i}(t_{k}^{+})\) and \(p_{i}(t_{k}^{})\) are the right and left limits of \(p_{i}(t_{k})\), respectively, while \(I_{ik}\in \mathcal{C}[\mathbb{R}^{n}, \mathbb{R}^{n}]\) denotes the incremental change of the state at time \(t_{k}\), while \(I_{ik}(0)=0\) for all \(k\in \mathbb{N}\).
For the convenience of discussion, we rewrite (2) in the vector form
where \(p(t)=(p_{1}(t),p_{2}(t),\ldots,p_{n}(t))^{T}\in \mathbb{A}^{n}\); with \(d_{i}>0\), \(i=1,2,\ldots,n\); and ; ; \(u=(u_{1},u_{2},\ldots,u_{n})^{T}\in \mathbb{A}^{n}\); \(g(p(t))=(g_{1}(p_{1}(t)),g_{2}(p_{2}(t)),\ldots,g_{n}(p_{n}(t)))^{T} \in \mathbb{A}^{n}\); \(g(p(t\tau (t)))=(g_{1}(p_{1}(t\tau (t))),g_{2}(p_{2}(t\tau (t))),\ldots,g_{n}(p_{n}(t \tau (t))))^{T}\in \mathbb{A}^{n}\) and .
(H1) Function \(g_{j}(\cdot )\) fulfills the Lipschitz continuity condition with respect to the ndimensional Clifford vector. For each \(j=1,2,\ldots,n\), there exists a positive constant \(k_{j}\) such that, for any \(x, y\in \mathbb{A}\),
where \(k_{j}\) is known as the Lipschitz constant and \(g_{j}(0)=0\). And there exist positive constants \(k_{j}\) such that \(g_{j}(x)_{\mathbb{A}}\leq k_{j}\) (\(j=1,2,\ldots,n\)).
Remark 2.1
There exists a constant \(k_{j}>0\) (\(j=1,2,\ldots,n\)) such that \(\forall x=(x_{1}, x_{2},\ldots, x_{2^{m}})^{T}\in \mathbb{R}^{2^{m}}\), and \(\forall y=(y_{1}, y_{2},\ldots, y_{2^{m}})^{T}\in \mathbb{R}^{2^{m}}\)
By means of assumption (H1), it is clear that
where .
(H2) Timevarying delay \(\tau (t)\) is differential and it fulfills the following conditions:
where τ and μ are known real constants.
Remark 2.2
It is worth noting that Cliffordvalued NN models are the generalized form of real, complex, and quaternionvalued NN models. For example, when we take into account \(m=0\) in NN model (3), then the model can be reduced to the realvalued NN model proposed in [6]. Suppose, we take \(m=1\) in NN model (3), then the model can be reduced to the complexvalued NN model proposed in [14]. If we choose \(m=2\) in NN model, then the model can be reduced to the quaternionvalued 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].
Definition 2.3
([34])
With reference to the Cliffordvalued NN model (3), the vector \(p^{*}=(p_{1}^{*},p_{2}^{*},\ldots,p_{n}^{*})^{T}\in \mathbb{A}^{n}\) is an equilibrium point, if and only if \(p^{*}\) is a solution of the following equation:
and the impulsive jump is able to satisfy , \(k\in \mathbb{N}\).
Definition 2.4
([39])
Pertinent to the Cliffordvalued NN model (3), its global exponential stable is guaranteed if for any solution \(p(t, t_{0},\varphi )\) with the initial condition \(\varphi \in \mathcal{C}([\tau , 0], \mathbb{A}^{n})\), there exist constants \(\alpha >0\) and such that
Main results
In order to avoid the issues of noncommutativity of multiplication of Clifford numbers, we transform the Cliffordvalued NN model (3) into the realvalued NN models. This can be achieved with the help of \(e_{A}\bar{e}_{A}=\bar{e}_{A}e_{A}=1\) and \(e_{B}\bar{e}_{A}e_{A}=e_{B}\). Given any \(\mathscr{G}\in \mathbb{A}\), a unique \(\mathscr{G}^{C}\) that is able to satisfy \(\mathscr{G}^{C}e_{C}g^{A}e_{A}=(1)^{\sigma [B.\bar{A}]}\mathscr{G}^{C}g^{A}e_{B}= \mathscr{G}^{B.\bar{A}}g^{A}e_{B}\) can be identified. By decomposing (3) into \(\dot{p}=\sum_{A}\dot{p}^{A}e_{A}\), we have
where
Corresponding to the basis of Clifford algebra, we reformulate the Cliffordvalued NN model to novel realvalued ones. Define
The NN model (8) can be written as
with the initial value
where \(\bar{\varphi }(s)=((\varphi ^{0}(s))^{T},(\varphi ^{1}(s))^{T},\ldots,( \varphi ^{A}(s))^{T},\ldots,(\varphi ^{12\ldots m}(s))^{T})^{T}\in \mathbb{R}^{2^{m}n}\).
In addition, we can express (6) as the following inequality:
where
From Definition 2.3 there exists a unique equilibrium point \(w^{*}\) for Cliffordvalued NN model (9). Next, we prove the global exponential stability of (9). Firstly, we shift the equilibrium point of (9) to the origin using the transformation \(\tilde{w}=ww^{*}\). Then we rewrite Cliffordvalued NN model (9) as
where \(\tilde{g}(\tilde{w})=\bar{g}(\tilde{w}+w^{*})\bar{g}(w^{*})\) and \(\tilde{\varphi }(s)=\bar{\varphi }(s)w^{*}\) is the initial condition. Obviously, the equilibrium point of NN (3) is also the equilibrium point of (9), and the stability for NN (3) is equivalent to that for (9). As such, in the sequel, we focus our study on the realvalued NNs.
Lemma 3.1
([46])
Let , then
for any \(\tilde{w}\in \mathbb{R}^{2^{m}}\) if is a symmetric matrix.
Exponential stability
Theorem 3.2
Given that (H1) and (H2) hold, the following conditions are fulfilled for \(k\in \mathbb{N}\):

(i)
For given scalars \(\epsilon : 0<\epsilon < \min d_{i}\) (\(i=1,\ldots,n\)), \(\tau \geq 0\) and μ, if there exist positive definite matrices and , and a positive diagonal matrix such that
(13)where Θ is negative definite.

(ii)
For \(\gamma _{k}\geq 0\), \(k\in \mathbb{N}\), one can have .

(iii)
For \(\eta >1\), one can have \(\eta \tau \leq \inf \{t_{k}t_{k1}\}\).

(iv)
, \(k\in \mathbb{N}\), where , where is a constant.
Then the equilibrium point of NN model (12) is globally exponentially stable. It has an exponential convergence rate ϵ. Moreover,
Proof
Construct a LKF for the NN model (12) as follows:
where \(\mathfrak{V}(t, \tilde{w}(t))\) is radially unbounded and are positive definite matrices.
Based on Lemma (3.1), we have
When \(t\neq t_{k}\), \(k\in \mathbb{N}\), the time derivative of \(\mathfrak{V}(t, \tilde{w}(t))\) can be computed along the solutions of NN model (12), one obtains
From assumptions (H1) and (H2), one can get
From assumption (H2), one can obtain
Substituting (18) in (17), we have
where \(\zeta (t)=(\tilde{w}^{T}(t), \tilde{g}^{T}(\tilde{w}(t)), \tilde{g}^{T}( \tilde{w}(t\tau (t))))^{T}\) and Θ is defined in (13).
When \(t=t_{k}\), \(k\in \mathbb{N}\) by using (ii), (iii), we have
that is,
where .
For any solution \(\tilde{w}(t, t_{0}, \tilde{w}_{0})\) of (3), by (19) and (20), we have
Since \(\eta \tau \leq \inf \{t_{k}t_{k1}\}\), one has \(k1\leq \frac{(t_{k1}t_{0})}{\eta \tau }\), which implies
That is,
On the other hand
From (16), (23), (24), we have
that is,
By Definition 2.4, the NN model (12) is globally exponentially stable. The proof of Theorem 3.2 is completed. □
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).
Corollary 3.4
Suppose (H1) and (H2) hold, then the following conditions are satisfied:

(i)
For given scalars \(\epsilon : 0<\epsilon < \min d_{i}\) (\(i=1,\ldots,n\)), \(\tau \geq 0\) and μ, if there exist positive definite matrices and , and a positive diagonal matrix such that
(27)where Θ̃ 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 timevarying 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).
Corollary 3.6
Given (H1), the following conditions are satisfied:

(i)
For given scalars \(\epsilon : 0<\epsilon < \min d_{i}\) (\(i=1,\ldots,n\)), if there exist a positive definite matrix and a positive diagonal matrix such that
(30)where Θ̄ is negative definite. As such, the equilibrium point pertaining to NN model (29) is globally exponentially stable. Moreover,
(31)
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 Cliffordvalued NNs. As such, from the theoretical and practical point of view, studying the dynamical behaviors of Cliffordvalued NNs are difficult tasks. On the other hand, the decomposition approach has proven to be very effective to overcome the difficulty of the noncommutativity of Clifford number multiplication. By decomposition approach, the original ndimensional Cliffordvalued system is decomposed into \(2^{m}n\)dimensional realvalued system and the coefficient and activation functions of networks are explicitly expressed. Therefore, the decomposition approach is highly favorable to analyze the dynamics of Cliffordvalued NNs. Most of authors have recently obtained sufficient criteria for Cliffordvalued NN models using the decomposition method [34, 36, 38, 42].
Remark 3.8
In [35], fuzzy operations are incorporated into the Cliffordvalued cellular NN model to investigate its \(S^{p}\)almost periodic solutions. The effects of discrete delays in Cliffordvalued recurrent NNs are considered in [36], and the associated globally asymptotic almost automorphic synchronization criteria are obtained. The leakage delay is introduced into Cliffordvalued highorder 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 Cliffordvalued NN models [33–42], in this paper, by considering a class of impulsive Cliffordvalued 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 [33–42].
Remark 3.9
When we consider the exponential convergence rate \(\epsilon =0\) 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.
Example 1
For \(m=2\) and \(n=2\), we have the following Cliffordvalued impulsive NN model:
The number of the multiplication generators, \(e_{1}^{2}=e_{2}^{2}=e_{12}^{2}=e_{1}e_{2}e_{12}=1\), \(e_{1}e_{2}=e_{2}e_{1}=e_{12}\), \(e_{1}e_{12}=e_{12}e_{1}=e_{2}\), \(e_{2}e_{12}=e_{12}e_{2}=e_{1}\), \(\dot{p}_{1}(t)=\dot{p}^{0}_{1}(t)e_{0}+\dot{p}^{1}_{1}(t)e_{1}+ \dot{p}^{2}_{1}(t)e_{2}+\dot{p}^{12}_{1}(t)e_{12}\), \(\dot{p}_{2}(t)=\dot{p}^{0}_{2}(t)e_{0}+\dot{p}^{1}_{2}(t)e_{1}+ \dot{p}^{2}_{2}(t)e_{2}+\dot{p}^{12}_{2}(t)e_{12}\). Furthermore, we take
According to their definitions, we have
and
It can be checked that when \(\epsilon =0.324\), \(\tau =0.8\), \(\mu =0.5\), \(\eta =2\) and \(\varsigma _{k}=(1)^{k} (\frac{(e^{0.324}+2.6)}{2} )^{ \frac{1}{2}}\), \(k\in \mathbb{N}\), by using MATLAB, the LMI condition of (13) in Theorem 3.2 is true with \(t_{\min }=0.0022\). The feasible solutions of the existing positive definite matrices , and positive diagonal matrix are
As we know,
that is, \(\gamma _{k}=\varsigma _{k}1\). Therefore,
Hence, when , we have, , \(k\in \mathbb{N}\).
On the other hand, it is straightforward to ascertain that all conditions of Theorem 3.2 are fulfilled. Therefore, the unique equilibrium of NN model (12) is globally exponentially stable, which is verified by the numerical simulation. Under the initial condition \(\varphi _{1}(t)=0.5e_{0}+0.9e_{1}0.3e_{2}0.7e_{12}\) for \(t\in [\tau , 0]\), and \(\varphi _{2}(t)=0.6e_{0}0.4e_{1}+0.2e_{2}+0.8e_{12}\) for \(t\in [\tau , 0]\), the time responses of the state of NN model (33) \(p_{1}^{0}(t)\), \(p_{2}^{0}(t)\), \(p_{1}^{1}(t)\), \(p_{2}^{1}(t)\), \(p_{1}^{2}(t)\), \(p_{2}^{2}(t)\), \(p_{1}^{12}(t)\), \(p_{2}^{12}(t)\) with impulsive effects are presented in Figs. 1–5, respectively. When the impulse effects are absent in the NN model (33), the time responses of the states of NN model (33) \(p_{1}^{0}(t)\), \(p_{2}^{0}(t)\), \(p_{1}^{1}(t)\), \(p_{2}^{1}(t)\), \(p_{1}^{2}(t)\), \(p_{2}^{2}(t)\), \(p_{1}^{12}(t)\) are given in Figs. 6–10, respectively. It is obvious from Figs. 1–10 that the model converges to an equilibrium point, which means that model (33) is globally exponentially stable.
Conclusion
In this study, the global exponential stability analysis pertaining to the Cliffordvalued NN models with timevarying delays and impulsive effects has been comprehensively investigated. To tackle the problem, we firstly decomposed the considered ndimensional Cliffordvalued NN model into \(2^{m}n\)dimensional realvalued 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 Cliffordvalued 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 Cliffordvalued NNs with the help of different controller schemes. The corresponding results will be presented in the near future.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
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Acknowledgements
This research is made possible through financial support from the Rajamangala University of Technology Suvarnabhumi, Thailand. The authors are grateful to the Rajamangala University of Technology Suvarnabhumi, Thailand for supporting this research.
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The research is supported by the Rajamangala University of Technology Suvarnabhumi, Thailand.
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Funding acquisition, NB; Conceptualization, GR; Software, GR; Formal analysis, GR and NB; Methodology, GR and NB; Supervision, GR, PA, RS, PH and CPL; Writing–original draft, GR; Validation, GR and NB; Writing–review and editing, GR and NB. All authors have read and agreed to the published version of the manuscript.
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Rajchakit, G., Sriraman, R., Boonsatit, N. et al. Global exponential stability of Cliffordvalued neural networks with timevarying delays and impulsive effects. Adv Differ Equ 2021, 208 (2021). https://doi.org/10.1186/s1366202103367z
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Keywords
 Cliffordvalued neural network
 Exponential stability
 Lyapunov–Krasovskii functional
 Impulsive effects