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Observability Inequality from Measurable Sets for Degenerate Parabolic Equations and its Applications

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Abstract

In this study, we employ the established Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, along with the telescoping series method, to establish an observability inequality for the degenerate parabolic equation over measurable subsets in the time-space domain. As a direct application, we formulate a captivating Stackelberg–Nash game problem and provide a proof of the existence of its equilibrium. Additionally, we characterize the set of Stackelberg–Nash equilibria and delve into the analysis of a norm optimal control problem.

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Acknowledgements

The authors are grateful to editor for many useful comments on presentation. The constructive suggestions from anonymous referees are very helpful to improve the manuscript substantially. The first three authors are supported by the National Natural Science Foundation of China under Grant 11871478, the Science Technology Foundation of Hunan Province. The last author is supported by the National Natural Science Foundation of China under Grant 11971363, and by the Fundamental Research Funds for the Central Universities under Grant 2042023kf0193.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Yuanhang Liu, Weijia Wu & Donghui Yang

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Can Zhang

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  1. Yuanhang Liu
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  3. Donghui Yang
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  4. Can Zhang
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Correspondence to Weijia Wu.

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Communicated by Ebrahim Sarabi.

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Liu, Y., Wu, W., Yang, D. et al. Observability Inequality from Measurable Sets for Degenerate Parabolic Equations and its Applications. J Optim Theory Appl 200, 1017–1055 (2024). https://doi.org/10.1007/s10957-023-02359-1

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  • DOI: https://doi.org/10.1007/s10957-023-02359-1

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