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On exponential convergence of nonlinear gradient dynamics system with application to square root finding

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Abstract

Gradient dynamics systems and their exponential convergence theories are investigated in this paper. Differing from widely considered linear gradient dynamics system (LGDS), a class of nonlinear gradient dynamics system (NGDS) is investigated with the exponential convergence analyzed. As an application to scalar square root finding, by defining six different square-based nonnegative error-monitoring functions (i.e., energy functions), six different NGDSs are theoretically designed and proposed in the form of first-order differential equations. Moreover, inspired by the exponential convergence theory of the LGDS, for each of the six proposed NGDSs, the corresponding exponential convergence theory is proved rigorously based on Lyapunov theory. Numerical verification and comparison further illustrate the efficacy of the proposed six NGDSs, in which the main differences and respective usages, as well as the application background and condition, are discussed in detail.

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Acknowledgments

This work is supported by the 2012 Scholarship Award for Excellent Doctoral Student Granted by Ministry of Education of China (under grant 3191004) and by the Foundation of Key Laboratory of Autonomous Systems and Networked Control of Ministry of Education of China (with project number 2013A07). Besides, the authors would like to thank the editors and reviewers sincerely for their time and effort spent in handling the paper, as well as many detailed and constructive comments provided for improving much further the presentation and quality of this paper.

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Authors and Affiliations

  1. School of Information Science and Technology, Sun Yat-sen University (SYSU), Guangzhou, 510006, China

    Yunong Zhang, Dechao Chen, Dongsheng Guo, Bolin Liao & Ying Wang

  2. The SYSU-CMU Shunde International Joint Research Institute, Shunde, 528300, China

    Yunong Zhang, Dechao Chen, Dongsheng Guo, Bolin Liao & Ying Wang

  3. Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, Guangzhou, 510640, China

    Yunong Zhang, Dechao Chen, Dongsheng Guo, Bolin Liao & Ying Wang

  4. College of Information Science and Engineering, Jishou University, Jishou, 416000, China

    Bolin Liao

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  1. Yunong Zhang
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  2. Dechao Chen
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  3. Dongsheng Guo
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Correspondence to Yunong Zhang.

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Zhang, Y., Chen, D., Guo, D. et al. On exponential convergence of nonlinear gradient dynamics system with application to square root finding. Nonlinear Dyn 79, 983–1003 (2015). https://doi.org/10.1007/s11071-014-1716-3

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  • DOI: https://doi.org/10.1007/s11071-014-1716-3

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