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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.06710 |
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| _version_ | 1866910325663596544 |
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| author | Araújo, Raul K. C. Fernández-Cara, Enrique Límaco, Juan Souza, Diego A. |
| author_facet | Araújo, Raul K. C. Fernández-Cara, Enrique Límaco, Juan Souza, Diego A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on the control of two-phase Stefan free-boundary problems Araújo, Raul K. C. Fernández-Cara, Enrique Límaco, Juan Souza, Diego A. Optimization and Control 35R35, 80A22, 93B05, 93C20 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. |
| title | Remarks on the control of two-phase Stefan free-boundary problems |
| topic | Optimization and Control 35R35, 80A22, 93B05, 93C20 |
| url | https://arxiv.org/abs/2402.06710 |