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arXiv · 2402.06710

Remarks on the control of two-phase Stefan free-boundary problems

Raul K. C. Araújo·Enrique Fernández-Cara·Juan Límaco·Diego A. Souza
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Abstract

This paper concerns the null controllability of the two-phase 1D Stefan problem with distributed controls. This is a free-boundary problem that models solidification or melting processes. In each phase, a parabolic equation, completed with initial and boundary conditions must be satisfied; the phases are separated by a phase-change interface where an additional free-boundary condition is imposed. We assume that two localized sources of heating/cooling controls act on the system (one in each phase). We prove a local null controllability result: the temperatures and the interface can be respectively steered to zero and to a prescribed location provided the initial data and interface position are sufficiently close to the targets. The ingredients of the proofs are a compactness-uniqueness argument (to deduce appropriate observability estimates adapted to constraints) and a fixed-point formulation and resolution of the controllability problem (to deduce the result for the nonlinear system). We also prove a negative result corresponding to the case where only one control acts on the system and the interface does not collapse to the boundary.

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Raul K. C. Araújo, Enrique Fernández-Cara, Juan Límaco, Diego A. Souza. 2024-02-09. Remarks on the control of two-phase Stefan free-boundary problems. https://doi.org/10.1137/21m1402261

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