A complete boundary-control criterion for coupled heat systems in higher dimensions
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded connected smooth domain, $\Gamma \subset \partial\Omega$ a nonempty relatively open set, $T > 0$, and $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$ constant matrices with $m < n$. Consider $$y_t - \Delta y - A y = 0 \ \text{in } \Omega \times (0,T), \qquad y = B f \mathbf{1}_\Gamma \ \text{on } \partial\Omega \times (0,T), \qquad y(0) = y_0 \in L^2(\Omega)^n,$$ with controls $f \in L^\infty(\Gamma \times (0,T))^m$ and solutions defined by transposition. The system is null controllable if every $y_0$ admits a control with $y(T) = 0$. Let $\lambda_1 \le \lambda_2 \le \cdots$ be the Dirichlet eigenvalues of $-\Delta$ on $\Omega$, and for $P \in \mathbb{R}^{p \times p}$ and $R \in \mathbb{R}^{p \times r}$ write $[P; R] = [R \mid PR \mid \cdots \mid P^{p-1} R]$.
With a distributed control $B k \mathbf{1}_\omega$ in the equation instead, null controllability holds if and only if $\operatorname{rank}[A; B] = n$ (Ammar-Khodja, Benabdallah, Dupaix and González-Burgos, 2009). In dimension one, for $\Omega = (0,\pi)$ with the control at the endpoint $x = 0$, Ammar-Khodja, Benabdallah, González-Burgos and de Teresa proved that the boundary system is null controllable if and only if $\operatorname{rank}[L_k; B_k] = nk$ for every $k \ge 1$, where $L_k = \operatorname{diag}(\lambda_1 I - A, \dots, \lambda_k I - A)$ and $B_k$ stacks $k$ copies of $B$. Benabdallah, Boyer, González-Burgos and Olive characterized boundary null controllability of such systems on cylinders $(0,\pi) \times \Omega_2$ with control on $\{0\} \times \omega_2$ (2014).
Problem. Find a necessary and sufficient spectral or algebraic condition for null controllability, in terms of how $A$ and $B$ interact with the Dirichlet spectral data of $\Omega$ and with $\Gamma$. Restating null controllability as an observability inequality for the adjoint system is not such a characterization.
Source: Fernández-Cara (2025), Problem 1, for approximate and null controllability. Stated by AIM — Open Applied Problems for null controllability.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, boundary controllability, coupled parabolic systems
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{ammarkhodja2009,
author = {Ammar-Khodja, Farid and Benabdallah, Assia and Dupaix, C{\'e}dric and Gonz{\'a}lez-Burgos, Manuel},
title = {A {K}alman rank condition for the localized distributed controllability of a class of linear parabolic systems},
journal = {J. Evol. Equ.},
volume = {9},
number = {2},
year = {2009},
pages = {267--291},
doi = {10.1007/s00028-009-0008-8},
}
@article{ammarkhodja2011,
author = {Ammar-Khodja, F. and Benabdallah, A. and Gonz{\'a}lez-Burgos, M. and de Teresa, L.},
title = {The {K}alman condition for the boundary controllability of coupled parabolic systems. {B}ounds on biorthogonal families to complex matrix exponentials},
journal = {J. Math. Pures Appl. (9)},
volume = {96},
number = {6},
year = {2011},
pages = {555--590},
doi = {10.1016/j.matpur.2011.06.005},
}
@article{benabdallah2014,
author = {Benabdallah, Assia and Boyer, Franck and Gonz{\'a}lez-Burgos, Manuel and Olive, Guillaume},
title = {Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the {$N$}-dimensional boundary null controllability in cylindrical domains},
journal = {SIAM J. Control Optim.},
volume = {52},
number = {5},
year = {2014},
pages = {2970--3001},
doi = {10.1137/130929680},
}
@misc{aim-boundary-coupled-heat-control-criterion,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A complete boundary-control criterion for coupled heat systems in higher dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/257-boundary-coupled-heat-control-criterion.md},
note = {Entry 257 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00505, "A complete boundary-control criterion for coupled heat systems in higher dimensions", clopen.solutions.