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A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances

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Abstract

In this paper, we introduce a weak maximum principle-based approach to input-to-state stability (ISS) analysis for certain nonlinear partial differential equations (PDEs) with certain boundary disturbances. Based on the weak maximum principle, a classical result on the maximum estimate of solutions to linear parabolic PDEs has been extended, which enables the ISS analysis for certain nonlinear parabolic PDEs with certain boundary disturbances. To illustrate the application of this method, we establish ISS estimates for a linear reaction–diffusion PDE and a generalized Ginzburg–Landau equation with mixed boundary disturbances. Compared to some existing methods, the scheme proposed in this paper involves less intensive computations and can be applied to the ISS analysis for a wide class of nonlinear PDEs with boundary disturbances.

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Acknowledgements

The authors would like to thank the reviewers for providing us with valuable comments and suggestions, which helped us to improve the quality of presentation of this paper. This work is supported in part by NSFC under Grant NSFC-11901482 and in part by NSERC under Grant RGPIN-2018-04571.

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, Sichuan, China

    Jun Zheng

  2. Department of Electrical Engineering, Polytechnique Montréal, P.O. Box 6079, Station Centre-Ville, Montreal, QC, H3T 1J4, Canada

    Guchuan Zhu

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  1. Jun Zheng
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  2. Guchuan Zhu
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Correspondence to Jun Zheng.

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Zheng, J., Zhu, G. A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances. Math. Control Signals Syst. 32, 157–176 (2020). https://doi.org/10.1007/s00498-020-00258-8

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  • DOI: https://doi.org/10.1007/s00498-020-00258-8

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