Global constrained null control of a two-phase Stefan interface
Let $L, d_l, d_r, \lambda, T > 0$ and $L_0, L_T \in (0, L)$. On a rod $(0, L)$ two phases with temperatures $y$ and $z$ are separated by an interface $x = \ell(t)$ and controlled by the temperatures $v_l, v_r$ at the two ends: $$ \begin{gathered} y_t - d_l y_{xx} = 0 \quad (0 < x < \ell(t)), \qquad z_t - d_r z_{xx} = 0 \quad (\ell(t) < x < L),\\ y(\ell(t), t) = z(\ell(t), t) = 0, \qquad y(0, t) = v_l(t), \qquad z(L, t) = v_r(t),\\ d_l y_x(\ell(t)^-, t) - d_r z_x(\ell(t)^+, t) = -\lambda \ell'(t), \end{gathered} $$ with initial data $y(\cdot, 0) = y_0$, $z(\cdot, 0) = z_0$, $\ell(0) = L_0$. Araújo, Fernández-Cara, Límaco and Souza proved local null controllability for the version with zero end temperatures and two distributed controls, one in each phase: for initial temperatures that are nonnegative in the first phase and nonpositive in the second, small enough, and an initial interface close enough to $L_T$, both temperatures can be driven to zero and the interface to $L_T$ at time $T$. With one of the two controls removed and nonzero initial temperature in the uncontrolled phase, null controllability fails whenever $0 < \ell(T) < L$, and they note a local result of the same kind for small data with boundary controls.
Problem. For arbitrary $y_0 \in C_c^\infty(0, L_0)$ and $z_0 \in C_c^\infty(L_0, L)$, are there controls $v_l, v_r \in L^\infty(0, T)$ and a solution of this system on the whole interval $[0, T]$, with $\ell \in C^1([0, T])$, $0 < \ell(t) < L$ throughout and temperatures continuous in $L^2$ after extension by zero to $(0, L)$, such that $y(\cdot, T) = z(\cdot, T) = 0$ and $\ell(T) = L_T$? No sign restrictions or smallness assumptions are imposed on the data or the controls.
Source: Stated by AIM — Open Applied Problems, after Fernández-Cara (2025), §3, Problem 5, who asks for this boundary-controlled model whether global control or a data-dependent minimal time holds; the local theory is Araújo, Fernández-Cara, Límaco and Souza (2022).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, free boundary problems, null controllability, Stefan problem
References
@article{fernandezcara2025,
author = {Fern{\'a}ndez-Cara, Enrique},
title = {Remarks on control and inverse problems for {PDE}s},
journal = {SeMA J.},
volume = {82},
number = {3},
year = {2025},
pages = {267--288},
doi = {10.1007/s40324-024-00363-7},
}
@article{araujo2022,
author = {Ara{\'u}jo, Raul K. C. and Fern{\'a}ndez-Cara, Enrique and L{\'i}maco, Juan and Souza, Diego A.},
title = {Remarks on the control of two-phase {S}tefan free-boundary problems},
journal = {SIAM J. Control Optim.},
volume = {60},
number = {5},
year = {2022},
pages = {3078--3099},
doi = {10.1137/21M1402261},
eprint = {2402.06710},
archivePrefix = {arXiv},
}
@misc{aim-global-two-phase-stefan-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global constrained null control of a two-phase Stefan interface}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/258-global-two-phase-stefan-control.md},
note = {Entry 258 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove that two Dirichlet boundary controls can drive a one-dimensional two-phase Stefan system to zero temperature and any prescribed interior interface position in every positive time. The initial temperatures may be arbitrary real smooth functions compactly supported in their phases, and the interface remains strictly inside the rod throughout. A short uncontrolled evolution gives analytic interface traces without a …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 258-signed-stefan-control/stefan_signed_control.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00752, "Global constrained null control of a two-phase Stefan interface", clopen.solutions.