Null-controllable data for cascade heat equations: almost sharp spectral spaces and modal critical times
Abstract
We study one-dimensional cascades of heat equations controlled through their last component. For a fixed time T>0, we consider the Hilbert space 𝒞T of all initial data that can be steered exactly to zero at time T, endowed with the norm of the minimal L2 control. This space can be strongly anisotropic: in explicit examples, it is incomparable with every isotropic spectral Sobolev or sub-Gaussian Gevrey scale associated with an adapted basis. For two equations, under the block Fattorini--Hautus condition, we give an almost sharp spectral description of 𝒞T. More precisely, in an adapted Riesz basis of generalized eigenvectors we define an explicit increasing scale (𝒱τ)τ∈R of Hilbert spaces such that 𝒱T−ε↪𝒞T↪𝒱T for every ε>0. The weights of 𝒱T combine heat dissipation with the loss of observation inside each double generalized eigenspace. When the coupling and control regions are disjoint, this recovers the known minimal null-control time T⋆ and identifies it as a transition time for controllable data: for TT⋆, the natural state space is properly contained in 𝒞T. We extend the analysis to arbitrary finite upper triangular heat cascades. The observation losses of the different directions in each spectral block may have distinct exponential orders, leading to a hierarchy of critical times rather than a single threshold. This hierarchy describes successive gains of null controllability on adapted modal subspaces and accounts for the interactions and cancellations between different coupling paths.
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BibTeX
@misc{lhachemi2026nullcontrollable,
title = {{Null-controllable data for cascade heat equations: almost sharp spectral spaces and modal critical times}},
author = {Hugo Lhachemi and Christophe Prieur and Emmanuel Trélat},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14134},
eprint = {2610.14134},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05788850v1},
}
RIS
TY - UNPB
TI - Null-controllable data for cascade heat equations: almost sharp spectral spaces and modal critical times
AU - Hugo Lhachemi
AU - Christophe Prieur
AU - Emmanuel Trélat
PY - 2026
DA - 2026/10/09
PB - HAL
DO - 10.48550/arxiv.2610.14134
UR - https://hal.science/hal-05788850v1
AB - We study one-dimensional cascades of heat equations controlled through their last component. For a fixed time T>0, we consider the Hilbert space 𝒞_T of all initial data that can be steered exactly to zero at time T, endowed with the norm of the minimal L^2 control. This space can be strongly anisotropic: in explicit examples, it is incomparable with every isotropic spectral Sobolev or sub-Gaussian Gevrey scale associated with an adapted basis. For two equations, under the block Fattorini--Hautus condition, we give an almost sharp spectral description of 𝒞_T. More precisely, in an adapted Riesz basis of generalized eigenvectors we define an explicit increasing scale (𝒱_τ)_τ∈R of Hilbert spaces such that 𝒱_T−ε↪𝒞_T↪𝒱_T for every ε>0. The weights of 𝒱_T combine heat dissipation with the loss of observation inside each double generalized eigenspace. When the coupling and control regions are disjoint, this recovers the known minimal null-control time T^⋆ and identifies it as a transition time for controllable data: for TT^⋆, the natural state space is properly contained in 𝒞_T. We extend the analysis to arbitrary finite upper triangular heat cascades. The observation losses of the different directions in each spectral block may have distinct exponential orders, leading to a hierarchy of critical times rather than a single threshold. This hierarchy describes successive gains of null controllability on adapted modal subspaces and accounts for the interactions and cancellations between different coupling paths.
ER -