A data-independent time for global control of the defocusing cubic wave
Let $\Omega \subset \mathbb{R}^3$ be a bounded smooth domain and $\omega \subset \Omega$ an open set satisfying the geometric control condition: for some $T_0 < \infty$, every unit-speed generalized ray of the Dirichlet wave equation meets $\omega$ within time $T_0$. Consider the defocusing cubic wave equation $$u_{tt} - \Delta u + u^3 = \mathbf{1}_\omega f, \qquad u|_{\partial\Omega} = 0,$$ with controls $f \in L^2(\omega \times (0,T))$ and energy-class solutions, $(u, u_t) \in C([0,T]; H_0^1(\Omega) \times L^2(\Omega))$.
When $\omega$ is a neighbourhood of $\partial\Omega$, Dehman, Lebeau and Zuazua proved, for a class of defocusing subcritical nonlinearities including $u^3$, that for every $E_0 > 0$ there is a time $T$ in which any two states of norm at most $E_0$ in $H_0^1(\Omega) \times L^2(\Omega)$ can be joined (2003). Joly and Laurent proved the same for connected $\Omega$ under the geometric control condition (2013). Laurent and Loyola proved observability estimates for such equations with analytic nonlinearity, which includes $u^3$, in the optimal time of the geometric control condition, with constants depending on a bound on the data (2026).
Problem. Is there a time $T_* = T_*(\Omega, \omega)$ such that any two states in $H_0^1(\Omega) \times L^2(\Omega)$ can be joined in time $T_*$ by an energy-class solution with some control $f \in L^2(\omega \times (0,T_*))$? The time may depend on $\Omega$ and $\omega$ but not on the sizes of the two states.
Source: Dehman, Lebeau and Zuazua (2003), remark after Corollary 4, for defocusing subcritical nonlinearities with control near the boundary; for the cubic equation, Coron, Krieger and Xiang (2025). Stated by AIM — Open Applied Problems under the geometric control condition.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, exact controllability, semilinear wave equations
References
@article{dehman2003,
author = {Dehman, Belhassen and Lebeau, Gilles and Zuazua, Enrique},
title = {Stabilization and control for the subcritical semilinear wave equation},
journal = {Ann. Sci. {\'E}cole Norm. Sup. (4)},
volume = {36},
number = {4},
year = {2003},
pages = {525--551},
doi = {10.1016/S0012-9593(03)00021-1},
}
@misc{coron2025,
author = {Coron, Jean-Michel and Krieger, Joachim and Xiang, Shengquan},
title = {Wave maps from circle to {R}iemannian manifold: global controllability is equivalent to homotopy},
year = {2025},
eprint = {2509.12779},
archivePrefix = {arXiv},
}
@article{joly2013,
author = {Joly, Romain and Laurent, Camille},
title = {Stabilization for the semilinear wave equation with geometric control condition},
journal = {Anal. PDE},
volume = {6},
number = {5},
year = {2013},
pages = {1089--1119},
doi = {10.2140/apde.2013.6.1089},
eprint = {1205.2459},
archivePrefix = {arXiv},
}
@article{laurent2026,
author = {Laurent, Camille and Loyola, Crist{\'o}bal},
title = {Global propagation of analyticity and unique continuation for semilinear waves},
journal = {J. {\'E}c. polytech. Math.},
volume = {13},
year = {2026},
pages = {1323--1387},
doi = {10.5802/jep.348},
eprint = {2407.02866},
archivePrefix = {arXiv},
}
@misc{aim-uniform-time-cubic-wave-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A data-independent time for global control of the defocusing cubic wave}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/256-uniform-time-cubic-wave-control.md},
note = {Entry 256 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-00569, "A data-independent time for global control of the defocusing cubic wave", clopen.solutions.