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Sharp Observability Costs for Strongly Coupled Parabolic Systems with Vanishing Principal Coupling
Abstract
We study the loss of observability for a class of strongly coupled parabolic systems as the coupling parameter tends to zero. The system is observed through a fixed matrix B on a measurable subset of space-time with positive measure. Under the Kalman rank condition, we determine the exact blow-up rates of the optimal observability cost. Different components of the state, as well as different combinations of them, may have different blow-up rates, and these rates are determined by the algebraic structure of the pair (B,K). The full-state observability cost is determined by the largest of these rates. We also show that the worst-case minimum L∞-norm of null controls has the same asymptotic behavior.
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BibTeX
@misc{zhu2026sharp,
title = {{Sharp Observability Costs for Strongly Coupled Parabolic Systems with Vanishing Principal Coupling}},
author = {Xiaomin Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35087},
eprint = {2609.35087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35087},
}
RIS
TY - UNPB
TI - Sharp Observability Costs for Strongly Coupled Parabolic Systems with Vanishing Principal Coupling
AU - Xiaomin Zhu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35087
UR - https://arxiv.org/abs/2609.35087
AB - We study the loss of observability for a class of strongly coupled parabolic systems as the coupling parameter tends to zero. The system is observed through a fixed matrix B on a measurable subset of space-time with positive measure. Under the Kalman rank condition, we determine the exact blow-up rates of the optimal observability cost. Different components of the state, as well as different combinations of them, may have different blow-up rates, and these rates are determined by the algebraic structure of the pair (B,K). The full-state observability cost is determined by the largest of these rates. We also show that the worst-case minimum L^∞-norm of null controls has the same asymptotic behavior.
ER -