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Sliding Mode Control for a Generalization of the Caginalp Phase-Field System

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Abstract

In the present paper, we present and solve the sliding mode control (SMC) problem for a second-order generalization of the Caginalp phase-field system. This generalization, inspired by the theories developed by Green and Naghdi on one side, and Podio-Guidugli on the other, deals with the concept of thermal displacement, i.e., a primitive with respect to the time of the temperature. Two control laws are considered: the former forces the solution to reach a sliding manifold described by a linear constraint between the temperature and the phase variable; the latter forces the phase variable to reach a prescribed distribution \(\varphi ^*\). We prove existence, uniqueness as well as continuous dependence of the solutions for both problems; two regularity results are also given. We also prove that, under suitable conditions, the solutions reach the sliding manifold within finite time.

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Notes

  1. Please note that in this paper we will use both the notations \(p_t\) and \(\partial _tp\) to denote the derivative of a function p.

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Acknowledgements

The current contribution originated from the work done by Davide Manini for the preparation of his master thesis, which has been discussed at the University of Pavia on July 2019. Actually, the paper turns out to offer some extension to the results there contained. The research of Pierluigi Colli is supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022)—Dept. of Mathematics “F. Casorati”, University of Pavia. In addition, PC gratefully acknowledges some other support from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI–C.N.R. Pavia, Italy. Last but not least, the authors are very grateful to the referees for their careful work and for a number of suggestions that led to an improvement of the paper.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata, 1, 27100, Pavia, Italy

    Pierluigi Colli

  2. Scuola Internazionale Superiore di Studi Avanzati, via Bonomea, 265, 34136, Trieste, Italy

    Davide Manini

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  1. Pierluigi Colli
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  2. Davide Manini
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Correspondence to Davide Manini.

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Colli, P., Manini, D. Sliding Mode Control for a Generalization of the Caginalp Phase-Field System. Appl Math Optim 84, 1395–1433 (2021). https://doi.org/10.1007/s00245-020-09682-3

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