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Exact controllability to the trajectories of the one-phase Stefan problem

Jon Asier Bárcena-Petisco, Enrique Fernández-Cara, Diego A. Souza

Abstract

This paper deals with the exact controllability to the trajectories of the one--phase Stefan problem in one spatial dimension. This is a free-boundary problem that models solidification and melting processes. It is assumed that the physical domain is filled by a medium whose state is liquid on the left and solid, with constant temperature, on the right. In between we find a free-boundary (the interface that separates the liquid from the solid). In the liquid domain, a parabolic equation completed with initial and boundary conditions must be satisfied by the temperature. On the interface, an additional free-boundary requirement, called the {\it Stefan condition,} is imposed. We prove the local exact controllability to the (smooth) trajectories. To this purpose, we first reformulate the problem as the local null controllability of a coupled PDE-ODE system with distributed controls. Then, a new Carleman inequality for the adjoint of the linearized PDE-ODE system, coupled on the boundary through nonlocal in space and memory terms, is presented. This leads to the null controllability of an appropriate linear system. Finally, a local result is obtained via local inversion, by using {\it Liusternik-Graves' Theorem}. As a byproduct of our approach, we find that some parabolic equations which contains memory terms localized on the boundary are null-controllable.

Exact controllability to the trajectories of the one-phase Stefan problem

Abstract

This paper deals with the exact controllability to the trajectories of the one--phase Stefan problem in one spatial dimension. This is a free-boundary problem that models solidification and melting processes. It is assumed that the physical domain is filled by a medium whose state is liquid on the left and solid, with constant temperature, on the right. In between we find a free-boundary (the interface that separates the liquid from the solid). In the liquid domain, a parabolic equation completed with initial and boundary conditions must be satisfied by the temperature. On the interface, an additional free-boundary requirement, called the {\it Stefan condition,} is imposed. We prove the local exact controllability to the (smooth) trajectories. To this purpose, we first reformulate the problem as the local null controllability of a coupled PDE-ODE system with distributed controls. Then, a new Carleman inequality for the adjoint of the linearized PDE-ODE system, coupled on the boundary through nonlocal in space and memory terms, is presented. This leads to the null controllability of an appropriate linear system. Finally, a local result is obtained via local inversion, by using {\it Liusternik-Graves' Theorem}. As a byproduct of our approach, we find that some parabolic equations which contains memory terms localized on the boundary are null-controllable.
Paper Structure (18 sections, 13 theorems, 126 equations)

This paper contains 18 sections, 13 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reformulation of the free-boundary problem in a cylindrical domain
  4. Reformulation as a null controllability problem
  5. Reformulation as a distributed control problem
  6. Linearization
  7. Well-posedness of the adjoint system
  8. Carleman estimates for parabolic equations with nonlocal boundary conditions
  9. Well-posedness of the linearized system
  10. Exact controllability to the trajectories
  11. Controllability of the linearized problem
  12. A Carleman inequality
  13. Null controllability with nonhomogeneities
  14. Controllability of the nonlinear system
  15. Acknowledgements.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(\bar{u},\bar{\ell}, \bar{v})$ be a trajectory of eq:conStefanwith $(\bar{u},\bar{\ell}) \in \mathcal{T}$ and $\bar{v}(t)>0$ for all $t\in[0,T]$. Then, there exists $\delta > 0$ with the following property: for any $\ell_0\in(\ell_*,+\infty)$ and any $u_0\in H_0^1(0,\ell_0)$ with $u_0(x) \geq 0 there exist nonnegative controls $v\in H^{3/4}(0,T)$ and associated states $(u,\ell)$ with such

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • ...and 16 more