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Zhang JR, Lu JG, Jin XC, Yang XY. Novel results on asymptotic stability and synchronization of fractional-order memristive neural networks with time delays: The 0<δ≤1 case. Neural Netw 2023; 167:680-691. [PMID: 37722271 DOI: 10.1016/j.neunet.2023.09.007] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2022] [Revised: 07/14/2023] [Accepted: 09/04/2023] [Indexed: 09/20/2023]
Abstract
This paper investigates the asymptotic stability and synchronization of fractional-order (FO) memristive neural networks with time delays. Based on the FO comparison principle and inverse Laplace transform method, the novel sufficient conditions for the asymptotic stability of a FO nonlinear system are given. Then, based on the above conclusions, the sufficient conditions for the asymptotic stability and synchronization of FO memristive neural networks with time delays are investigated. The results in this paper have a wider coverage of situations and are more practical than the previous related results. Finally, the validity of the results is checked by two examples.
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Affiliation(s)
- Jia-Rui Zhang
- Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, PR China; Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, PR China; Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai 200240, PR China
| | - Jun-Guo Lu
- Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, PR China; Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, PR China; Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai 200240, PR China.
| | - Xiao-Chuang Jin
- Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, PR China; Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, PR China; Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai 200240, PR China
| | - Xing-Yu Yang
- Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, PR China; Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, PR China; Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai 200240, PR China
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Cao J, Udhayakumar K, Rakkiyappan R, Li X, Lu J. A Comprehensive Review of Continuous-/Discontinuous-Time Fractional-Order Multidimensional Neural Networks. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:5476-5496. [PMID: 34962883 DOI: 10.1109/tnnls.2021.3129829] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
The dynamical study of continuous-/discontinuous-time fractional-order neural networks (FONNs) has been thoroughly explored, and several publications have been made available. This study is designed to give an exhaustive review of the dynamical studies of multidimensional FONNs in continuous/discontinuous time, including Hopfield NNs (HNNs), Cohen-Grossberg NNs, and bidirectional associative memory NNs, and similar models are considered in real ( [Formula: see text]), complex ( [Formula: see text]), quaternion ( [Formula: see text]), and octonion ( [Formula: see text]) fields. Since, in practice, delays are unavoidable, theoretical findings from multidimensional FONNs with various types of delays are thoroughly evaluated. Some required and adequate stability and synchronization requirements are also mentioned for fractional-order NNs without delays.
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LMI-Based Synchronization Conditions to R-L Fractional Time-Varying Delayed Neural Networks with Parametric Uncertainty. Neural Process Lett 2022. [DOI: 10.1007/s11063-022-11026-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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Zhang S, Yang Y, Sui X, Zhang Y. Synchronization of fractional-order memristive recurrent neural networks via aperiodically intermittent control. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:11717-11734. [PMID: 36124610 DOI: 10.3934/mbe.2022545] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
In this paper, synchronization of fractional-order memristive recurrent neural networks via aperiodically intermittent control is investigated. Considering the special properties of memristor neural network, differential inclusion theory is introduced. Similar to the aperiodically strategy of integer order, aperiodically intermittent control strategy of fractional order is proposed. Under the framework of Fillipov's solution, based on the intermittent strategy of fractional order systems and the properties Mittag-Leffler, sufficient criteria of aperiodically intermittent strategy are obtained by constructing appropriate Lyapunov functional. Some comparisons are given to demonstrate the advantages of aperiodically strategy. A simulation example is given to illustrate the derived conclusions.
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Affiliation(s)
- Shuai Zhang
- College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China
- Engineering Lab of Intelligence Business & Internet of Things, Henan Province, Xinxiang 453007, China
- College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
| | - Yongqing Yang
- School of Science, Jiangnan University, Wuxi 214122, China
| | - Xin Sui
- School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China
| | - Yanna Zhang
- College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China
- Engineering Lab of Intelligence Business & Internet of Things, Henan Province, Xinxiang 453007, China
- College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
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Viera-Martin E, Gómez-Aguilar JF, Solís-Pérez JE, Hernández-Pérez JA, Escobar-Jiménez RF. Artificial neural networks: a practical review of applications involving fractional calculus. THE EUROPEAN PHYSICAL JOURNAL. SPECIAL TOPICS 2022; 231:2059-2095. [PMID: 35194484 PMCID: PMC8853315 DOI: 10.1140/epjs/s11734-022-00455-3] [Citation(s) in RCA: 11] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2021] [Accepted: 01/13/2022] [Indexed: 05/13/2023]
Abstract
In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.
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Affiliation(s)
- E. Viera-Martin
- Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos Mexico
| | - J. F. Gómez-Aguilar
- CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos Mexico
| | - J. E. Solís-Pérez
- Escuela Nacional de Estudios Superiores Unidad Juriquilla, Universidad Nacional Autónoma de México, Boulevard Juriquilla 3001, Juriquilla La Mesa, C.P. 76230 Juriquilla, Querétaro Mexico
| | - J. A. Hernández-Pérez
- Universidad Autónoma del Estado de Morelos/Centro de Investigación en Ingeniería y Ciencias Aplicadas, Av. Universidad No. 1001, Col Chamilpa, C.P. 62209 Cuernavaca, Morelos Mexico
| | - R. F. Escobar-Jiménez
- Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos Mexico
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Udhayakumar K, Rihan FA, Rakkiyappan R, Cao J. Fractional-order discontinuous systems with indefinite LKFs: An application to fractional-order neural networks with time delays. Neural Netw 2021; 145:319-330. [PMID: 34798343 DOI: 10.1016/j.neunet.2021.10.027] [Citation(s) in RCA: 15] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2021] [Revised: 09/16/2021] [Accepted: 10/29/2021] [Indexed: 11/18/2022]
Abstract
In this article, we discuss bipartite fixed-time synchronization for fractional-order signed neural networks with discontinuous activation patterns. The Filippov multi-map is used to convert the fixed-time stability of the fractional-order general solution into the zero solution of the fractional-order differential inclusions. On the Caputo fractional-order derivative, Lyapunov-Krasovskii functional is proved to possess the indefinite fractional derivatives for fixed-time stability of fragmentary discontinuous systems. Furthermore, the fixed-time stability of the fractional-order discontinuous system is achieved as well as an estimate of the new settling time.. The discontinuous controller is designed for the delayed fractional-order discontinuous signed neural networks with antagonistic interactions and new conditions for permanent fixed-time synchronization of these networks with antagonistic interactions are also provided, as well as the settling time for permanent fixed-time synchronization. Two numerical simulation results are presented to demonstrate the effectiveness of the main results.
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Affiliation(s)
- K Udhayakumar
- Department of Mathematics, Bharathiar University, Coimbatore 641 046, India; Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al-Ain, 15551, United Arab Emirates
| | - Fathalla A Rihan
- Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al-Ain, 15551, United Arab Emirates.
| | - R Rakkiyappan
- Department of Mathematics, Bharathiar University, Coimbatore 641 046, India
| | - Jinde Cao
- School of Mathematics, Southeast University, Nanjing 210096, China; Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea
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Zheng B, Hu C, Yu J, Jiang H. Synchronization analysis for delayed spatio-temporal neural networks with fractional-order. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.01.128] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Sun Y, Liu Y. Adaptive Synchronization Control and Parameters Identification for Chaotic Fractional Neural Networks with Time-Varying Delays. Neural Process Lett 2021. [DOI: 10.1007/s11063-021-10517-7] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Quasi-Synchronization of Fractional-Order Complex-Valued Memristive Recurrent Neural Networks with Switching Jumps Mismatch. Neural Process Lett 2021. [DOI: 10.1007/s11063-020-10342-4] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Zhang X, Ma Y. LMIs conditions to robust pinning synchronization of uncertain fractional-order neural networks with discontinuous activations. Soft comput 2020. [DOI: 10.1007/s00500-020-05315-7] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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Peng X, Wu H, Cao J. Global Nonfragile Synchronization in Finite Time for Fractional-Order Discontinuous Neural Networks With Nonlinear Growth Activations. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2019; 30:2123-2137. [PMID: 30442618 DOI: 10.1109/tnnls.2018.2876726] [Citation(s) in RCA: 39] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
This paper is concerned with the global nonfragile Mittag-Leffler synchronization and the global synchronization in finite time for fractional-order discontinuous neural networks, where activation functions are discontinuous at 0, or modeled as a local Hölder functions with the nonlinear growth property in a neighborhood of 0. First, two lemmas concerned with the convergence with respect to an absolutely continuous function are developed. Second, a new property, which introduces an inequality of the fractional derivative for the variable upper limit integral with respect to the nonsmooth integrable function, is presented and applied in the synchronization results' analysis. In addition, under the fractional Filippov differential inclusion framework, by utilizing the Lur'e Postnikov-type Lyapunov functional, nonsmooth analysis method, and the convergence properties developed in this paper, the synchronization conditions are derived in the form of linear matrix inequalities. Moreover, the upper bound of the setting time for the global nonfragile synchronization in finite time is calculated accurately. Finally, two illustrations are presented to verify the correctness of the theoretical results.
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Aperiodic intermittent pinning control for exponential synchronization of memristive neural networks with time-varying delays. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2018.12.070] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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Robust Mittag-Leffler Synchronization for Uncertain Fractional-Order Discontinuous Neural Networks via Non-fragile Control Strategy. Neural Process Lett 2018. [DOI: 10.1007/s11063-018-9787-7] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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