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

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

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.

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

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.
Paper Structure (13 sections, 12 theorems, 107 equations)

This paper contains 13 sections, 12 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reformulation of the free-boundary problem
  4. Well-posedness of the two-phase free-boundary problem
  5. Approximate controllability of a linearized system
  6. An improved observability inequality
  7. Approximate controllability problem with linear constraint
  8. Controllability of the two-phase Stefan problem
  9. A regularity property
  10. A fixed-point argument
  11. Additional comments
  12. Lack of controllability with only one control
  13. Boundary controllability and other extensions

Key Result

Theorem 1

Let $\ell_T\in (\ell_l,\ell_r)$. Then there exists $\delta > 0$ such that, for any $u_0\in H_0^1(0,\ell_0)$ with $u_0 \geq 0$, any $v_0\in H_0^1(\ell_0,L)$ with $v_0 \leq 0$ and any $\ell_0\in (\ell_l,\ell_r)$ satisfying there exist controls $(h_l,h_r)\in L^2(\mathcal{O}_l)\times L^2(\mathcal{O}_r)$ and associated states $(u,v,\ell)$ with such that

Theorems & Definitions (17)

  • Theorem 1
  • Remark 1.1
  • Remark 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Remark 3.1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • ...and 7 more