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Insensitizing Controls of a 1D Stefan Problem for the Semilinear Heat Equation

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Abstract

This paper deals with the existence of insensitizing controls for a 1D free-boundary problem of the Stefan kind for a semilinear PDE. The insensitizing problem consists in finding a control function such that some energy functional of the system is locally insensitive to a perturbation of the initial data. As usual, this problem can be reduced to a nonstandard null controllability problem of some nonlinear coupled system governed by a semilinear parabolic equation with a free-boundary and a linear parabolic equation. Nevertheless, in order to solve the later Stefan problem by the fixed point technique, we need to establish the null controllability of the linear coupled system in a non-cylindrical domain. An observability estimate for the corresponding coupled system in a non-cylindrical domain is established, whose proof relies on a new global Carleman estimate.

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Acknowledgements

The research was partially supported by the general project of Guangdong Teachers College of Foreign Language and Arts (Grant No. 2022G10), NSFC (Grants Nos. 11861038 and 11971179), and the NSF of Guangdong Province (Grant No. S2013010015800).

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Authors and Affiliations

  1. School of Basic Education, Guangdong Teachers College of Foreign Language and Arts, Guangzhou, 510507, China

    Lili Wang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Lili Wang & Peidong Lei

  3. Normal School, Shenyang University, Shenyang, 110044, China

    Qingzhe Wu

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  1. Lili Wang
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  2. Peidong Lei
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  3. Qingzhe Wu
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Correspondence to Lili Wang.

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Wang, L., Lei, P. & Wu, Q. Insensitizing Controls of a 1D Stefan Problem for the Semilinear Heat Equation. Bull Braz Math Soc, New Series 53, 1351–1375 (2022). https://doi.org/10.1007/s00574-022-00308-6

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  • DOI: https://doi.org/10.1007/s00574-022-00308-6

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