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Stamov G, Stamova I, Spirova C. Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach. ENTROPY (BASEL, SWITZERLAND) 2021; 23:1631. [PMID: 34945937 PMCID: PMC8700440 DOI: 10.3390/e23121631] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 10/22/2021] [Revised: 11/27/2021] [Accepted: 11/30/2021] [Indexed: 11/16/2022]
Abstract
In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds' existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology-the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.
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Affiliation(s)
- Gani Stamov
- Department of Mathematical Physics, Technical University of Sofia, 8800 Sliven, Bulgaria; (G.S.); (C.S.)
| | - Ivanka Stamova
- Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
| | - Cvetelina Spirova
- Department of Mathematical Physics, Technical University of Sofia, 8800 Sliven, Bulgaria; (G.S.); (C.S.)
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Stability Analysis for Nonlinear Impulsive Control System with Uncertainty Factors. COMPUTATIONAL INTELLIGENCE AND NEUROSCIENCE 2020; 2020:8818794. [PMID: 33273901 PMCID: PMC7700033 DOI: 10.1155/2020/8818794] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/13/2020] [Revised: 09/09/2020] [Accepted: 11/06/2020] [Indexed: 11/23/2022]
Abstract
Considering the limitation of machine and technology, we study the stability for nonlinear impulsive control system with some uncertainty factors, such as the bounded gain error and the parameter uncertainty. A new sufficient condition for this system is established based on the generalized Cauchy–Schwarz inequality in this paper. Compared with some existing results, the proposed method is more practically applicable. The effectiveness of the proposed method is shown by a numerical example.
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Global Stability of Integral Manifolds for Reaction–Diffusion Delayed Neural Networks of Cohen–Grossberg-Type under Variable Impulsive Perturbations. MATHEMATICS 2020. [DOI: 10.3390/math8071082] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The present paper introduces the concept of integral manifolds for a class of delayed impulsive neural networks of Cohen–Grossberg-type with reaction–diffusion terms. We establish new existence and boundedness results for general types of integral manifolds with respect to the system under consideration. Based on the Lyapunov functions technique and Poincarѐ-type inequality some new global stability criteria are also proposed in our research. In addition, we consider the case when the impulsive jumps are not realized at fixed instants. Instead, we investigate a system under variable impulsive perturbations. Finally, examples are given to demonstrate the efficiency and applicability of the obtained results.
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Suriguga M, Kao Y, Hyder AA. Uniform stability of delayed impulsive reaction–diffusion systems. APPLIED MATHEMATICS AND COMPUTATION 2020; 372:124954. [DOI: 10.1016/j.amc.2019.124954] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/02/2023]
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5
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Stability of Sets Criteria for Impulsive Cohen-Grossberg Delayed Neural Networks with Reaction-Diffusion Terms. MATHEMATICS 2019. [DOI: 10.3390/math8010027] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The paper proposes an extension of stability analysis methods for a class of impulsive reaction-diffusion Cohen-Grossberg delayed neural networks by addressing a challenge namely stability of sets. Such extended concept is of considerable interest to numerous systems capable of approaching not only one equilibrium state. Results on uniform global asymptotic stability and uniform global exponential stability with respect to sets for the model under consideration are established. The main tools are expansions of the Lyapunov method and the comparison principle. In addition, the obtained results for the uncertain case contributed to the development of the stability theory of uncertain reaction-diffusion Cohen-Grossberg delayed neural networks and their applications. Moreover, examples are given to demonstrate the feasibility of our results.
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Yang Z, Zhou W, Huang T. Input-to-state stability of delayed reaction-diffusion neural networks with impulsive effects. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2018.12.019] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Li Y, Xiang J. Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen–Grossberg neural networks with delays. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2018.12.064] [Citation(s) in RCA: 37] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Wang Q, Wang JL, Ren SY, Huang YL. Analysis and adaptive control for lag H∞synchronization of coupled reaction–diffusion neural networks. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2018.08.058] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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Tang HA, Duan S, Hu X, Wang L. Passivity and synchronization of coupled reaction–diffusion neural networks with multiple time-varying delays via impulsive control. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2018.08.005] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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Stability of Inertial Neural Network with Time-Varying Delays Via Sampled-Data Control. Neural Process Lett 2018. [DOI: 10.1007/s11063-018-9905-6] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Passivity and Synchronization of Coupled Reaction–Diffusion Cohen–Grossberg Neural Networks with Fixed and Switching Topologies. Neural Process Lett 2018. [DOI: 10.1007/s11063-018-9879-4] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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Liu L, Chen WH, Lu X. Impulsive H∞ synchronization for reaction–diffusion neural networks with mixed delays. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2017.07.023] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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13
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Wang SX, Huang YL, Xu BB. Pinning synchronization of spatial diffusion coupled reaction-diffusion neural networks with and without multiple time-varying delays. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2016.09.096] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Sun M, He X, Wang T, Tan J, Xia D. Circuit implementation of digitally programmable transconductance amplifier in analog simulation of reaction-diffusion neural model. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2016.07.062] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Li J, He C, Zhang W, Chen M. Adaptive synchronization of delayed reaction-diffusion neural networks with unknown non-identical time-varying coupling strengths. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2016.09.006] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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A short-term power load forecasting model based on the generalized regression neural network with decreasing step fruit fly optimization algorithm. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2016.09.027] [Citation(s) in RCA: 156] [Impact Index Per Article: 22.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Zhang W, Li J, Ding C, Xing K. $${\varvec{p}}$$ p th Moment Exponential Stability of Hybrid Delayed Reaction–Diffusion Cohen–Grossberg Neural Networks. Neural Process Lett 2016. [DOI: 10.1007/s11063-016-9572-4] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Chen WH, Luo S, Zheng WX. Impulsive Synchronization of Reaction-Diffusion Neural Networks With Mixed Delays and Its Application to Image Encryption. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2016; 27:2696-2710. [PMID: 26812737 DOI: 10.1109/tnnls.2015.2512849] [Citation(s) in RCA: 87] [Impact Index Per Article: 10.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
This paper presents a new impulsive synchronization criterion of two identical reaction-diffusion neural networks with discrete and unbounded distributed delays. The new criterion is established by applying an impulse-time-dependent Lyapunov functional combined with the use of a new type of integral inequality for treating the reaction-diffusion terms. The impulse-time-dependent feature of the proposed Lyapunov functional can capture more hybrid dynamical behaviors of the impulsive reaction-diffusion neural networks than the conventional impulse-time-independent Lyapunov functions/functionals, while the new integral inequality, which is derived from Wirtinger's inequality, overcomes the conservatism introduced by the integral inequality used in the previous results. Numerical examples demonstrate the effectiveness of the proposed method. Later, the developed impulsive synchronization method is applied to build a spatiotemporal chaotic cryptosystem that can transmit an encrypted image. The experimental results verify that the proposed image-encrypting cryptosystem has the advantages of large key space and high security against some traditional attacks.
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Impulsive control for the synchronization of coupled neural networks with reaction–diffusion terms. Neurocomputing 2016. [DOI: 10.1016/j.neucom.2016.05.034] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Tian X, Xu R. Stability and Hopf Bifurcation of Time Fractional Cohen–Grossberg Neural Networks with Diffusion and Time Delays in Leakage Terms. Neural Process Lett 2016. [DOI: 10.1007/s11063-016-9544-8] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Wang H, Yu Y, Wen G, Zhang S. Stability Analysis of Fractional-Order Neural Networks with Time Delay. Neural Process Lett 2014. [DOI: 10.1007/s11063-014-9368-3] [Citation(s) in RCA: 53] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Rakkiyappan R, Chandrasekar A, Lakshmanan S, Park JH. Exponential stability of Markovian jumping stochastic Cohen–Grossberg neural networks with mode-dependent probabilistic time-varying delays and impulses. Neurocomputing 2014. [DOI: 10.1016/j.neucom.2013.10.018] [Citation(s) in RCA: 41] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Wang X, Qi H. New LMI-based Criteria for Lagrange Stability of Cohen-Grossberg Neural Networks with General Activation Functions and Mixed Delays. INT J COMPUT INT SYS 2013. [DOI: 10.1080/18756891.2013.805587] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022] Open
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