Elsevier

Neural Networks

Volume 133, January 2021, Pages 87-100
Neural Networks

Improved approach to the problem of the global Mittag-Leffler synchronization for fractional-order multidimension-valued BAM neural networks based on new inequalities

https://doi.org/10.1016/j.neunet.2020.10.008Get rights and content

Highlights

  • The unified model is first established for the studied systems of FOMVBAMNNs which can be translated into the related multidimension-valued systems as long as the state variables, the connection weights and the AFs of the neural networks are valued to be real, complex, or quaternion. Meanwhile, the corresponding AFs have been supposed to be the general ones, not limited to be linear threshold ones.

  • Different the common decomposition method, the studied systems of FOMVBAMNNs need not any decomposition and the difference in solving the noncommutativity of quaternion multiplication makes the process comprehensive and concise.

  • Inspired by Xiao et al. (2020), we generalize this idea to construct new LKFs. Mainly employing Lemma 2 and Lemma 3, the useful coefficients can be used to establish the more proper conditions in order to make the derived criteria more multiple, more flexible, less computative and less conservative.

Abstract

This paper studies the problem of the global Mittag-Leffler synchronization for fractional-order multidimension-valued BAM neural networks (FOMVBAMNNs) with general activation functions (AFs). First, the unified model is established for the researched systems of FOMVBAMNNs which can be turned into the corresponding multidimension-valued systems as long as the state variables, the connection weights and the AFs of the neural networks are valued to be real, complex, or quaternion. Then, without any decomposition, the criteria in unified form are derived by constructing the new Lyapunov–Krasovskii functionals (LKFs) in vector form, combining two new inequalities and considering the easy controllers. It is worth mentioning that the obtained criteria have many advantages in higher flexibility, more diversity, smaller computation, and lower conservatism. Finally, a simulation example is provided to illustrate the availability and improvements of the acquired results.

Introduction

During the recent few decades, many scholars have paid close attention to neural networks because of their increasing application in various aspects such as combinatorial optimization, pattern recognition, static image treatment, signal processing and so on (Cao and Wang, 2005, Chen et al., 2014, Cheng et al., 2020b, Cheng et al., 2019, Cheng et al., 2020a, Faydasicok and Arik, 2012, Huang et al., 2012a, Huang et al., 2012b, Shan et al., 2020, Shan et al., 2019, Shen and Wang, 2008, Shi et al., 2020, Wang et al., 2019, Wang et al., 2020a, Wang et al., 2020b, Wen et al., 2018a, Wen et al., 2018b, Xiao and Zhong, 2018, Xie et al., 2020, Zeng and Wang, 2006, Zhang et al., 2020, Zhao et al., 2019, Zhao et al., 2020). However, due to the expansion of application fields, such as two-dimensional conversion, three-dimensional conversion, color image processing, satellite’s attitude control and so on, the common real-valued neural networks have been unable to meet the needs. With the development of multidimensional algebra, the neural networks develop rapidly from real-valued neural networks to complex-valued neural networks (Hirose, 2003), even or quaternion-valued neural networks (Isokawa et al., 2009, Isokawa et al., 2007, Isokawa et al., 2012). Particularly, since the early 21st century, owe to the development of modern mathematics and extension of the future applications, some researchers have focused on quaternion and have applied quaternion in neural networks to construct quaternion-valued neural networks because they stand out in encoding of 3D affine transformations directly such as spatial rotation, image impress, computer graphics, color night vision and so on (Isokawa et al., 2009, Matsui et al., 2004).

On one hand, no matter which multidimensional algebra is in neural networks, the corresponding multidimension-valued neural networks have their own advantages, respectively. The real-valued neural networks are the basic ones which have more and more developments. The complex-valued neural networks have advantages in solving the two-dimensional conversion and the quaternion-valued neural networks have absolute superiorities in settling the three-dimensional conversion. In particular, the quaternion-valued neural networks have more advantages in dealing with multidimentional problems than the real-valued ones or the complex-valued ones. The quaternion-valued neural networks were applied in resolving color image compressing by Isokawa, Kusakabe, Matsui, and Peper (2003). Buchholz and Le Bihan (2006) separated the polarized signals optimally and shew the better effects of separation in the quaternion-valued neural networks than in the real-valued ones. Meanwhile, Rishiyur (2006) proved that the sizes of encoding in the quaternion-valued neural networks could be more greatly reduced than in the complex-valued ones. Very recently, the various dynamic behaviors about the quaternion-valued neural networks have been analyzed by some researchers (Chen et al., 2017, Hu et al., 2017, Isokawa et al., 2012, Liu et al., 2017, Liu et al., 2016, Shu et al., 2017, Tu et al., 2017, Wei and Cao, 2019). Particularly, the stability problems of the quaternion-valued neural networks were studied by Liu et al., 2017, Liu et al., 2016 who divided the quaternion-valued neural networks into two complex-valued neural networks or four real-valued neural networks, respectively. Subsequently, Tu et al. (2017) researched the global dissipativity problem for the delayed quaternion-valued neural networks which need not any decomposition. The special linear threshold AFs were employed in Chen et al. (2017) in order to obtain sufficient conditions for the corresponding equilibrium points’ existence, uniqueness and global asymptotical stability problems of the quaternion-valued neural networks. A new fixed-time controller was designed by Wei and Cao (2019) to realize the synchronization of the quaternion-valued neural networks.

On the other hand, based on iterative learning principle, bidirectional associative memory (BAM) is employed to establish a double-deck nonlinear neural network which can own the auto-associative and unidirectional memories and applied in many applications especially in pattern recognition (Anbuvithya et al., 2015, Huang et al., 2012b, Sakthivel et al., 2015). Compared to the single-deck ones, the compositions of neurons in BAM neural networks can be administrated in two layers. The neurons in the layers can interconnect with each other from one layer to the other layer. Specially, as the heteroassociative pattern matchers function, the pattern pairs can be stored and recalled in BAM (Huang et al., 2012b). Besides, compared to integer-order neural networks, fractional-order neural networks can both have the capacity of infinite memory and also describe heredity, so the real-world physical systems can be described more accurately and precisely (Hu et al., 2019, Hu et al., 2018, Pratap et al., 2019, Sakthivel et al., 2019, Selvaraj et al., 2018, Wu et al., 2015, Xiao et al., 2017). It is necessary to construct the type of neural networks which can unite BAM and fractional-order derivatives in order to develop the advantage of memory of neural networks better and better. So far, there have been some achievements about the research for the various dynamics behaviors of fractional-order BAM neural networks (FOBAMNNs). Particularly, Wu et al. (2015) established and studied the system of FOBAMNNs and obtained the criteria for the judgment of Mittag-Leffler stabilization of FOBAMNNs. As the multidimensional algebra develops, complex and quaternion have been extended in some FOBAMNNs to formulate the correlative FOMVBAMNNs such as fractional-order complex-valued BAM neural networks (FOCVBAMNNs) and fractional-order quaternion-valued BAM neural networks (FOQVBAMNNs) (Xiao, Wen, Yang, & Zhong, 2020c). Xiao, Cao, Cheng, Wen, Zhang, and Zhong (2020a) mainly deduced a new fractional-order derivative inequality and constructed three novel Lyapunov–Krasovskii functionals (LKFs) with some coefficients in order to derive the separative criteria for the global Mittag-Leffler synchronization problems of FOQVBAMNNs, FOCVBAMNNs and FORVBAMNNs with the special linear threshold AFs, respectively.

As Liu et al. (2017) proposed, the research of the dynamical behaviors for quaternion-valued neural networks with different technique is still open and challenging. Reviewing the existing literature, such as integer-order quaternion-valued neural networks in (Chen et al., 2017, Hu et al., 2017, Liu et al., 2017, Liu et al., 2016, Shu et al., 2017, Tu et al., 2017, Wei and Cao, 2019) and fractional-order quaternion-valued neural networks in (Pahnehkolaei et al., 2019, Xiao et al., 2020a, Xiao et al., 2020b, Xiao et al., 2020c, Xiao and Zhong, 2019, Yang et al., 2018), we find that there is no unified model for the studied multi-dimension-valued neural networks. Therefore, many results and criteria are short of conciseness and uniformity. Compared with the very recent results about FOQVBAMNNs (Xiao et al., 2020a, Xiao et al., 2020c), the main challenges and improvements in this paper are the unified structure of FOMVBAMNNs, the combination of two new inequalities and the establishment of new LKFs with coefficients in vector form. Hence, the analysis in this paper can be regarded as another new approach and an improvement for the study on the dynamical behaviors of quaternion-valued neural networks and fractional-order neural networks.

Motivated by the above discussion, the intention of this paper is to investigate the problem of the global Mittag-Leffler synchronization for FOMVBAMNNs with general AFs. The main contributions of this paper can be summarized as follows

(1) The unified model is first established for the studied systems of FOMVBAMNNs which can be translated into the related multi-dimension-valued systems as long as the state variables, the connection weights and the AFs of the neural networks are valued to be real, complex, or quaternion. Meanwhile, the corresponding AFs have been supposed to be the general ones, not limited to be linear threshold ones. Compared to Xiao et al. (2020c), the subsequent criteria can be directly applied in the problem of the global Mittag-Leffler synchronization for the different systems of FOMVBAMNNs. By valuing the corresponding algebra such as real, complex, and quaternion in the conditions of Theorem 1 or Corollary 1, the related criteria can be derived for the studied problem of FOMVBAMNNs such as FORVBAMNNs, FOCVBAMNNs and FOQVBAMNNs.

(2) Different the common decomposition method (Xiao et al., 2020a, Xiao et al., 2020b, Xiao and Zhong, 2019, Yang et al., 2018), the studied systems of FOMVBAMNNs need not any decomposition and the difference in solving the noncommutativity of quaternion multiplication makes the process comprehensive and concise.

(3) Inspired by Xiao et al. (2020c), we generalize this idea to construct new LKFs such as Vˆ(t)=|μˆzˆx(t)||μˆzˆx(t)|+|νˆzˆy(t)||νˆzˆy(t)|. Mainly employing Lemma 2, the useful coefficients such as μˆ and νˆ can be used to establish the more proper conditions. Moreover, owing to application of Lemma 3, the additional parameters such as σ and δ can be valued to be different positive numbers in order to make the derived criteria more multiple, more flexible, less computative and less conservative.

The subsequent parts of this paper are structured as follows. In Section 2, some basics of fractional theory are reviewed and the unified system of FOMVBAMNNs is described and some lemmas including two new inequalities are presented. Then, the unified global Mitta-Leffler synchronization criteria for the systems of FOMVBAMNNs are acquired by employing the new inequalities, constructing general Lyapunov–Krasovskii functionals (LKFs) and designing simple linear controllers in Section 3. In Section 4, a numerical example is given to illustrate the availability and improvements of the derived results. Finally, Section 5 concludes this paper.

Notations

Mm×n denotes the corresponding sets of all m×n-dimensional multi-dimension-valued matrix. Particularly, M can be valued to be Q or C or R. That is, Qm×n, Cm×n, and Rm×n denote the corresponding sets of all m×n-dimensional quaternion-valued, complex-valued, and real-valued matrices, respectively. When the multi-dimension-valued function zˆ(t)Qn can be described by zˆ(t)=zˆR(t)+izˆI(t)+jzˆJ(t)+kzˆK(t), where i, j, k are standard imaginary units in Q which satisfies the Hamilton rules: i2=j2=k2=ijk=1, ij=ji=k, jk=kj=i, and ki=ik=j. For any zˆp(t)Q, the modulus of zˆp(t), denoted by |zˆp|, is defined as |zˆp|=zˆpzˆp=(zˆR)2+(zˆI)2+(zˆJ)2+(zˆK)2, where zˆp=zˆpRizˆpIjzˆpJkzˆpK stands for the conjugate transpose of zˆp(t). For any zˆ(t)=(zˆ1(t),zˆ2(t),,zˆn(t))Qn, where zˆp(t)=zˆpR(t)+izˆpI(t)+jzˆpJ(t)+kzˆpK(t), p=1,2,,n, the ϱnorm of zˆ(t), denoted by zˆϱ, is defined as zˆϱ=(|zˆ1|ϱ+|zˆ2|ϱ++|zˆn|ϱ)1ϱ=(p=1n|zˆp|ϱ)1ϱ=(p=1nQ=R,IJ,K|zˆpQ|ϱ)1ϱ=(p=1n(|zˆpR|ϱ+|zˆpI|ϱ+|zˆpJ|ϱ+|zˆpK|ϱ))1ϱ. When the multi-dimension-valued function zˆ(t)Cn can be denoted by zˆ(t)=zˆR(t)+izˆI(t), where i is standard imaginary unit in C which satisfies the rules: i2=1. For any zˆp(t)C, the modulus of zˆp(t), denoted by |zˆp|, is defined as |zˆp|=zˆpzˆp=(zˆR)2+(zˆI)2, where zˆp=zˆpRizˆpI stands for the conjugate transpose of zˆp(t). For any zˆ(t)=(zˆ1(t),zˆ2(t),,zˆn(t))Cn, where zˆp(t)=zˆpR(t)+izˆpI(t), p=1,2,,n, the ϱnorm of zˆ(t), denoted by zˆϱ, is defined as zˆϱ=(|zˆ1|ϱ+|zˆ2|ϱ++|zˆn|ϱ)1ϱ=(p=1n|zˆp|ϱ)1ϱ=(p=1nC=R,I|zˆpC|ϱ)1ϱ=(p=1n(|zˆpR|ϱ+|zˆpI|ϱ))1ϱ.When the multi-dimension-valued function zˆ(t)Rn can be described by zˆ(t)=zˆR(t). For any zˆp(t)R, the modulus of zˆp(t), denoted by |zˆp|, is defined as |zˆp|=zˆpzˆp=(zˆR)2, where zˆp=zˆpR stands for the conjugate transpose of zˆp(t). For any zˆ(t)=(zˆ1(t),zˆ2(t),,zˆn(t))Rn, where zˆp(t)=zˆpR(t), p=1,2,,n, the ϱnorm of zˆ(t), denoted by zˆϱ, is defined as zˆϱ=(|zˆ1|ϱ+|zˆ2|ϱ++|zˆn|ϱ)1ϱ=(p=1n|zˆp|ϱ)1ϱ=(p=1nR=R|zˆpR|ϱ)1ϱ=(p=1n(|zˆpR|ϱ))1ϱ. For any matrix Aˆ=(aˆpq)n×nMn×n, AˆT and Aˆ represent the transpose and the conjugate transpose of matrix Aˆ, respectively. Cς([t0,+),M) is denoted as the space of ς-order continuous and differentiable functions from [t0,+) into M.

Section snippets

Model description and preliminaries

In this section, we recall some basics of fractional calculus, definitions and lemmas which will be required later.

Definition 1

Podlubny, 1999

The fractional integral for a function Ω(t) is defined as IαΩ(t)=1Γ(α)t0t(tβ)α1Ω(β)dβ,where tt0,α>0 and Γ() is the Euler’s gamma function, that is, Γ(α)=0tα1etdt.

Definition 2

Podlubny, 1999

Caputo fractional derivative for a function Ω(t)Cn+1([0,+),R) (the set of all n+1 order continuous differentiable functions on [0,+)) is defined as D0,tαΩ(t)=1Γ(nα)t0tΩ(n)(β)(tβ)αn+1dβ,where tt0,α>0 and n

Global Mittag-Leffler synchronization of FOMVBAMNNs with general AFs

When the considered AFs are assumed to be multi-dimension-valued general ones defined as fˆq(zˆyq)=fˆqR(zˆyqR)+ifˆqI(zˆyqI)+jfˆqJ(zˆyqJ)+kfˆqK(zˆyqK)(M=Q) or fˆq(zˆyq)=fˆqR(zˆyqR)+ifˆqI(zˆyqI)(M=C) or fˆq(zˆyq)=fˆqR(zˆyqR)(M=R), gˆp(zˆxp)=gˆpR(zˆxpR)+igˆpI(zˆxpI)+jgˆpJ(zˆxpJ)+kgˆpK(zˆxpK)(M=Q) or gˆp(zˆxp)=gˆxpR(zˆxpR)+igˆpI(zˆxpI)(M=C) or gˆp(zˆxp)=gˆxpR(zˆxpR)(M=R), the unified global Mittag-Leffler synchronization criteria are easily acquired by the subsequent Theorem.

Theorem 1

For p=1,2,,n, q=1,2,,m

Numerical simulation examples

In this section, a simulation example will be given to demonstrate our theoretical results.

Example 1

Consider the following 2-neuron FOMVBAMNNs(M = Q) D0,tαx(t)=Cˆx(t)+q=1mAˆfˆ(y(t))+Iˆ(t),D0,tαy(t)=Dˆy(t)+p=1nBˆgˆ(x(t))+Jˆ(t), where α=0.92, x(t)=xR+ixI+jxJ+kxK=(x1(t),x2(t))T, y(t)=yR+iyI+jyJ+kyK=(y1(t),y2(t))T, Cˆ=diag(0.2,0.2), Dˆ=diag(0.2,0.2), aˆ11=0.03+0.02i0.01j+0.002k, aˆ12=0.0110.01i+0.002j0.01k, aˆ21=0.051+0.02i0.02j+0.004k, aˆ22=0.020.02i+0.004j+0.01k, bˆ11=0.25+0.45i0.52j+0.21k, bˆ

Conclusions

In this paper, the whole problem of the global Mittag-Leffler synchronization has been settled for the unified systems of FOMVBAMNNs with general AFs or special linear threshold ones. First, on the basis of multidimension algebra, the generalized model has been constructed for the researched systems. Meanwhile, the corresponding AFs have been supposed to be the general ones under the common constraints of Assumption 1. Then, the unified criteria with several superiorities have been acquired

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank the Editor in Chief, the Associate Editor and the anonymous reviewers very much for a number of valuable constructive suggestions and comments which have improved the quality of this paper greatly. This work was supported by National Natural Science Foundation of China (No: 12001452).

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