Uniform null-controllability times for the Coron--Guerrero problems are $2$ and 2 + 2 √2
Abstract
We study the uniform null-controllability, in the vanishing viscosity limit, of the transport--diffusion equation ut + M ux = ε uxx on [0,L] with a Dirichlet boundary control at x=0. Our results concern the uniform null-controllability time Tunif, defined as the infimum of the times for which the null-controllability cost remains bounded as ε → 0. We determine this time exactly for both signs of the transport velocity: Tunif = 2 L / M in the positive-speed case M>0, and Tunif = (2 + 2 √2) L / |M| in the negative-speed case M<0. The positive-speed result disproves the conjecture Tunif=L/M suggested by Coron and Guerrero. For negative speed, the conjectured threshold had already been disproved by Lissy. Our result determines the exact threshold. We establish the lower bounds by constructing adjoint solutions that violate uniform observability below the respective thresholds. In the positive-speed case, the construction exploits the asymptotic structure of the Blaschke products and model spaces associated with the exponential family generated by the adjoint spectrum. The same model-space structure is used to reduce the upper-bound problems for both signs of M to infinite-time observability inequalities, which are proved through a representation of the boundary-to-interior map and estimates of its Hilbert--Schmidt norm.
Related
- Density of almost squares of horospherical orbits in non-uniform quotients of SL2(ℝ)×SL2(ℝ)
- Almost-Uniform Bayesian Convergence to the Truth Is Not Characterized by Countable Additivity on Conditional Hitting Times
- Turán problems with bounded matching number in k-uniform hypergraphs
- Carleman estimates for a nonconservative Timoshenko beam involving Dirichlet-Neumann boundary conditions and application to exact controllability
- Controllability of time-varying lumped semilinear systems
BibTeX
@misc{koike2026uniform,
title = {{Uniform null-controllability times for the Coron--Guerrero problems are $2$ and $2 + 2 \sqrt2$}},
author = {Kai Koike and Vincent Laheurte},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35355},
eprint = {2609.35355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35355},
}
RIS
TY - UNPB
TI - Uniform null-controllability times for the Coron--Guerrero problems are $2$ and 2 + 2 √2
AU - Kai Koike
AU - Vincent Laheurte
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35355
UR - https://arxiv.org/abs/2609.35355
AB - We study the uniform null-controllability, in the vanishing viscosity limit, of the transport--diffusion equation u_t + M u_x = ε u_xx on [0,L] with a Dirichlet boundary control at x=0. Our results concern the uniform null-controllability time T_unif, defined as the infimum of the times for which the null-controllability cost remains bounded as ε → 0. We determine this time exactly for both signs of the transport velocity: T_unif = 2 L / M in the positive-speed case M>0, and T_unif = (2 + 2 √2) L / |M| in the negative-speed case M<0. The positive-speed result disproves the conjecture T_unif=L/M suggested by Coron and Guerrero. For negative speed, the conjectured threshold had already been disproved by Lissy. Our result determines the exact threshold. We establish the lower bounds by constructing adjoint solutions that violate uniform observability below the respective thresholds. In the positive-speed case, the construction exploits the asymptotic structure of the Blaschke products and model spaces associated with the exponential family generated by the adjoint spectrum. The same model-space structure is used to reduce the upper-bound problems for both signs of M to infinite-time observability inequalities, which are proved through a representation of the boundary-to-interior map and estimates of its Hilbert--Schmidt norm.
ER -