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Approximate and null controllability for the multidimensional Coleman–Gurtin model

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Abstract

This paper is devoted to a studying of the controllability properties for the Coleman–Gurtin-type equation, which is a class of multidimensional integral–differential equations. The goal is to prove the existence of a control function which steers the state variable and the integral term to the neighborhood of two given final configurations at the same time, respectively. This new approximate controllability is defined by imposing some additional integral-type constraints on the usual approximate controllability, ensuring that the whole process reaches the neighborhood of the equilibrium. We also provide a characterization of the initial values, which can be driven to zero by a distributed control. The later is a supplement of non-null controllability for the Coleman–Gurtin model in the square integrable space.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, 524048, China

    Xiuxiang Zhou

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  1. Xiuxiang Zhou
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Correspondence to Xiuxiang Zhou.

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This work was partially supported by the National Natural Science Foundation of China under Grants 11926331, 11926337 and 11601213, by the Natural Science Foundation of Guangdong Province under Grant 2018A0303070012, and by the Foundation for Talents of Lingnan Normal University under Grant ZL1612.

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Zhou, X. Approximate and null controllability for the multidimensional Coleman–Gurtin model. Math. Control Signals Syst. 33, 279–295 (2021). https://doi.org/10.1007/s00498-021-00281-3

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  • DOI: https://doi.org/10.1007/s00498-021-00281-3

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