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Liu P, Wang J, Zeng Z. An Overview of the Stability Analysis of Recurrent Neural Networks With Multiple Equilibria. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:1098-1111. [PMID: 34449396 DOI: 10.1109/tnnls.2021.3105519] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
The stability analysis of recurrent neural networks (RNNs) with multiple equilibria has received extensive interest since it is a prerequisite for successful applications of RNNs. With the increasing theoretical results on this topic, it is desirable to review the results for a systematical understanding of the state of the art. This article provides an overview of the stability results of RNNs with multiple equilibria including complete stability and multistability. First, preliminaries on the complete stability and multistability analysis of RNNs are introduced. Second, the complete stability results of RNNs are summarized. Third, the multistability results of various RNNs are reviewed in detail. Finally, future directions in these interesting topics are suggested.
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2
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Multiple asymptotical ω-periodicity of fractional-order delayed neural networks under state-dependent switching. Neural Netw 2023; 157:11-25. [DOI: 10.1016/j.neunet.2022.09.034] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2021] [Revised: 09/28/2022] [Accepted: 09/29/2022] [Indexed: 11/06/2022]
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3
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Chouhan SS, kumar R, Sarkar S, Das S. Multistability Analysis of Octonion-Valued Neural Networks with Time-Varying Delays. Inf Sci (N Y) 2022. [DOI: 10.1016/j.ins.2022.07.123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
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4
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Hao B, Zhang T. Stability analysis and synchronized control of fuzzy Mittag-Leffler discrete-time genetic regulatory networks with time delays. JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2022. [DOI: 10.3233/jifs-212361] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. In allusion to fuzzy genetic regulatory networks of fractional order, this paper firstly establishes a novel difference method called Mittag-Leffler Euler difference, which includes the exponential Euler difference. In the second place, the existence of a unique global bounded solution and equilibrium point, global exponential stability and synchronization of the derived difference models are investigated. Compared with the classical fractional Euler differences, fuzzy Mittag-Leffler discrete-time genetic regulatory networks can better depict and retain the dynamic characteristics of the corresponding continuous-time models. What’s more important is that it starts a new avenue for studying discrete-time fractional-order systems and a set of theories and methods is constructed in studying Mittag-Leffler discrete models.
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Affiliation(s)
- Bing Hao
- Oxbridge College, Kunming University of Science and Technology, Kunming, China
| | - Tianwei Zhang
- City College, Kunming University of Science and Technology, Kunming, China
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5
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Novel global polynomial stability criteria of impulsive complex-valued neural networks with multi-proportional delays. Neural Comput Appl 2021. [DOI: 10.1007/s00521-021-06555-w] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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6
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Multi-periodicity of switched neural networks with time delays and periodic external inputs under stochastic disturbances. Neural Netw 2021; 141:107-119. [PMID: 33887601 DOI: 10.1016/j.neunet.2021.03.039] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2020] [Revised: 03/11/2021] [Accepted: 03/29/2021] [Indexed: 11/21/2022]
Abstract
This paper presents new theoretical results on the multi-periodicity of recurrent neural networks with time delays evoked by periodic inputs under stochastic disturbances and state-dependent switching. Based on the geometric properties of activation function and switching threshold, the neuronal state space is partitioned into 5n regions in which 3n ones are shown to be positively invariant with probability one. Furthermore, by using Itô's formula, Lyapunov functional method, and the contraction mapping theorem, two criteria are proposed to ascertain the existence and mean-square exponential stability of a periodic orbit in every positive invariant set. As a result, the number of mean-square exponentially stable periodic orbits increases to 3n from 2n in a neural network without switching. Two illustrative examples are elaborated to substantiate the efficacy and characteristics of the theoretical results.
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7
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Global exponential stability of delayed complex-valued neural networks with discontinuous activation functions. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.02.006] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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8
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Ganesan S, Gnaneswaran N. State-Feedback Filtering for Delayed Discrete-Time Complex-Valued Neural Networks. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2020; 31:4726-4736. [PMID: 31905153 DOI: 10.1109/tnnls.2019.2957304] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
This article explores a new filtering problem for the class of delayed discrete-time complex-valued neural networks (CVNNs) via state-feedback control design. The novelty of this article comes from the consideration of the newly developed complex-valued reciprocal convex matrix inequality as well as the complex-valued Jensen-based summation inequalities (JSIs). By employing an appropriate Lyapunov-Krasovskii functional (LKF) and by using newly proposed complex-valued inequalities, attention is concentrated on the design of a state-feedback filter such that the associated filtering error system is asymptotically stable with prescribed filter and control gain matrices. The proposed theoretical results are presented in terms of complex-valued linear matrix inequalities (LMIs) that can be solved numerically by using the YALMIP toolbox in MATLAB software. Additionally, one numerical example is given to confirm the validity of the resulting sufficient conditions with the availability of the suitable control and filter design.
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9
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Discrete-Time Stochastic Quaternion-Valued Neural Networks with Time Delays: An Asymptotic Stability Analysis. Symmetry (Basel) 2020. [DOI: 10.3390/sym12060936] [Citation(s) in RCA: 29] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
Stochastic disturbances often cause undesirable characteristics in real-world system modeling. As a result, investigations on stochastic disturbances in neural network (NN) modeling are important. In this study, stochastic disturbances are considered for the formulation of a new class of NN models; i.e., the discrete-time stochastic quaternion-valued neural networks (DSQVNNs). In addition, the mean-square asymptotic stability issue in DSQVNNs is studied. Firstly, we decompose the original DSQVNN model into four real-valued models using the real-imaginary separation method, in order to avoid difficulties caused by non-commutative quaternion multiplication. Secondly, some new sufficient conditions for the mean-square asymptotic stability criterion with respect to the considered DSQVNN model are obtained via the linear matrix inequality (LMI) approach, based on the Lyapunov functional and stochastic analysis. Finally, examples are presented to ascertain the usefulness of the obtained theoretical results.
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10
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Robust Exponential Stability for Discrete-Time Quaternion-Valued Neural Networks with Time Delays and Parameter Uncertainties. Neural Process Lett 2020. [DOI: 10.1007/s11063-020-10196-w] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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11
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Robust Output Feedback Stabilization for Uncertain Discrete-Time Stochastic Neural Networks with Time-Varying Delay. Neural Process Lett 2020. [DOI: 10.1007/s11063-019-10077-x] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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12
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Stepanov-Like Pseudo Almost Periodic Solution of Quaternion-Valued for Fuzzy Recurrent Neural Networks with Mixed Delays. Neural Process Lett 2020. [DOI: 10.1007/s11063-020-10193-z] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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13
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Wang X, Wang Z, Song Q, Shen H, Huang X. A waiting-time-based event-triggered scheme for stabilization of complex-valued neural networks. Neural Netw 2020; 121:329-338. [DOI: 10.1016/j.neunet.2019.09.032] [Citation(s) in RCA: 34] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2019] [Revised: 09/11/2019] [Accepted: 09/22/2019] [Indexed: 10/25/2022]
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14
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Wang Z, Cao J, Cai Z, Huang L. Periodicity and finite-time periodic synchronization of discontinuous complex-valued neural networks. Neural Netw 2019; 119:249-260. [PMID: 31472291 DOI: 10.1016/j.neunet.2019.08.021] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2019] [Revised: 06/28/2019] [Accepted: 08/19/2019] [Indexed: 11/18/2022]
Abstract
This paper discusses the issue of periodicity and finite-time periodic synchronization of discontinuous complex-valued neural networks (CVNNs). Based on a modified version of Kakutani's fixed point theorem, general conditions are obtained to guarantee the periodicity of discontinuous CVNNs. Next, several criteria for finite-time periodic synchronization (FTPS) are given by using a new proposed finite-time convergence theorem. Different from the traditional convergence lemma, the estimated upper bound of the derivative of the Lyapunov function (LF) is allowed to be indefinite or even positive. In order to achieve FTPS, novel discontinuous control algorithms, including state-feedback control algorithm and generalized pinning control algorithm, are designed. In the generalized pinning control algorithm, a guideline is proposed to select neurons to pin the designed controller. Finally, two simulations are given to substantiate the main results.
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Affiliation(s)
- Zengyun Wang
- Department of Mathematics, Hunan First Normal University, Changsha 410205, China; School of Mathematics, Southeast University, Nanjing, 210096, China; The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, Southeast University, Nanjing 210096, China; Changsha University of Science and Technology, Changsha, 410114, China
| | - Jinde Cao
- School of Mathematics, Southeast University, Nanjing, 210096, China; The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, Southeast University, Nanjing 210096, China.
| | - Zuowei Cai
- Department of Technology, Hunan Women's University, Changsha, Hunan 410002, China; Changsha University of Science and Technology, Changsha, 410114, China.
| | - Lihong Huang
- Changsha University of Science and Technology, Changsha, 410114, China
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15
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Li X, Zhang W, Fang JA, Li H. Finite-time synchronization of memristive neural networks with discontinuous activation functions and mixed time-varying delays. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2019.02.051] [Citation(s) in RCA: 25] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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16
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Zhang XM, Han QL, Ge X, Ding D. An overview of recent developments in Lyapunov–Krasovskii functionals and stability criteria for recurrent neural networks with time-varying delays. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2018.06.038] [Citation(s) in RCA: 160] [Impact Index Per Article: 26.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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17
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Popa CA. Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2018.05.004] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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18
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Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays. Neural Netw 2018; 105:277-293. [DOI: 10.1016/j.neunet.2018.05.006] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2017] [Revised: 03/12/2018] [Accepted: 05/04/2018] [Indexed: 11/19/2022]
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19
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Anti-synchronization of complex-valued memristor-based delayed neural networks. Neural Netw 2018; 105:1-13. [DOI: 10.1016/j.neunet.2018.04.008] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2017] [Revised: 03/28/2018] [Accepted: 04/12/2018] [Indexed: 11/23/2022]
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20
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Zhang XM, Han QL, Zeng Z. Hierarchical Type Stability Criteria for Delayed Neural Networks via Canonical Bessel-Legendre Inequalities. IEEE TRANSACTIONS ON CYBERNETICS 2018; 48:1660-1671. [PMID: 29621005 DOI: 10.1109/tcyb.2017.2776283] [Citation(s) in RCA: 42] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
This paper is concerned with global asymptotic stability of delayed neural networks. Notice that a Bessel-Legendre inequality plays a key role in deriving less conservative stability criteria for delayed neural networks. However, this inequality is in the form of Legendre polynomials and the integral interval is fixed on . As a result, the application scope of the Bessel-Legendre inequality is limited. This paper aims to develop the Bessel-Legendre inequality method so that less conservative stability criteria are expected. First, by introducing a canonical orthogonal polynomial sequel, a canonical Bessel-Legendre inequality and its affine version are established, which are not explicitly in the form of Legendre polynomials. Moreover, the integral interval is shifted to a general one . Second, by introducing a proper augmented Lyapunov-Krasovskii functional, which is tailored for the canonical Bessel-Legendre inequality, some sufficient conditions on global asymptotic stability are formulated for neural networks with constant delays and neural networks with time-varying delays, respectively. These conditions are proven to have a hierarchical feature: the higher level of hierarchy, the less conservatism of the stability criterion. Finally, three numerical examples are given to illustrate the efficiency of the proposed stability criteria.
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21
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Wang J, Jiang H, Ma T, Hu C. Delay-dependent dynamical analysis of complex-valued memristive neural networks: Continuous-time and discrete-time cases. Neural Netw 2018; 101:33-46. [DOI: 10.1016/j.neunet.2018.01.015] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2017] [Revised: 12/21/2017] [Accepted: 01/30/2018] [Indexed: 11/29/2022]
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22
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Global Exponential Synchronization of Complex-Valued Neural Networks with Time Delays via Matrix Measure Method. Neural Process Lett 2018. [DOI: 10.1007/s11063-018-9805-9] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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23
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24
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Yao Q, Wang L, Wang Y. Existence–uniqueness and stability of reaction–diffusion stochastic Hopfield neural networks with S-type distributed time delays. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2017.08.060] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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25
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Boundedness and periodicity for linear threshold discrete-time quaternion-valued neural network with time-delays. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2017.06.047] [Citation(s) in RCA: 47] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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26
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Master–slave exponential synchronization of delayed complex-valued memristor-based neural networks via impulsive control. Neural Netw 2017; 93:165-175. [DOI: 10.1016/j.neunet.2017.05.008] [Citation(s) in RCA: 72] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/26/2016] [Revised: 04/05/2017] [Accepted: 05/10/2017] [Indexed: 11/19/2022]
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27
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Bao G, Zeng Z. Region stability analysis for switched discrete-time recurrent neural network with multiple equilibria. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2017.03.065] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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28
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Velmurugan G, Rakkiyappan R, Vembarasan V, Cao J, Alsaedi A. Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay. Neural Netw 2017; 86:42-53. [DOI: 10.1016/j.neunet.2016.10.010] [Citation(s) in RCA: 65] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2016] [Revised: 09/03/2016] [Accepted: 10/27/2016] [Indexed: 10/20/2022]
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29
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Hu J, Zeng C. Adaptive exponential synchronization of complex-valued Cohen–Grossberg neural networks with known and unknown parameters. Neural Netw 2017; 86:90-101. [DOI: 10.1016/j.neunet.2016.11.001] [Citation(s) in RCA: 54] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2016] [Revised: 09/24/2016] [Accepted: 11/04/2016] [Indexed: 11/25/2022]
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30
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Dynamical Behavior of Complex-Valued Hopfield Neural Networks with Discontinuous Activation Functions. Neural Process Lett 2016. [DOI: 10.1007/s11063-016-9563-5] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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31
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Huang C, Cao J, Cao J. Stability analysis of switched cellular neural networks: A mode-dependent average dwell time approach. Neural Netw 2016; 82:84-99. [DOI: 10.1016/j.neunet.2016.07.009] [Citation(s) in RCA: 54] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2016] [Revised: 06/07/2016] [Accepted: 07/18/2016] [Indexed: 11/29/2022]
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32
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Xie D, Jiang Y. Global exponential stability of periodic solution for delayed complex-valued neural networks with impulses. Neurocomputing 2016. [DOI: 10.1016/j.neucom.2016.04.054] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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33
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Synchronization of fractional-order complex-valued neural networks with time delay. Neural Netw 2016; 81:16-28. [DOI: 10.1016/j.neunet.2016.05.003] [Citation(s) in RCA: 191] [Impact Index Per Article: 23.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2015] [Revised: 03/20/2016] [Accepted: 05/09/2016] [Indexed: 11/23/2022]
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34
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Global exponential stability of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays. Neural Netw 2016; 81:1-10. [DOI: 10.1016/j.neunet.2016.04.012] [Citation(s) in RCA: 140] [Impact Index Per Article: 17.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2016] [Revised: 04/25/2016] [Accepted: 04/29/2016] [Indexed: 11/22/2022]
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35
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Chen X, Song Q, Zhao Z, Liu Y. Global μ -stability analysis of discrete-time complex-valued neural networks with leakage delay and mixed delays. Neurocomputing 2016. [DOI: 10.1016/j.neucom.2015.10.120] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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36
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Gong W, Liang J, Zhang C, Cao J. Nonlinear Measure Approach for the Stability Analysis of Complex-Valued Neural Networks. Neural Process Lett 2015. [DOI: 10.1007/s11063-015-9475-9] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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