Rapid stabilization for Stokes and Navier-Stokes systems with localized disturbance
Abstract
In this study, we analyze the rapid stabilization of Stokes and the two-and threedimensional Navier-Stokes systems in the presence of an unknown spatially localized disturbance. A multivalued feedback control strategy is proposed which combines the Lyapunov frequency method with the multivalued sign operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator provides robustness against disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach guarantees global rapid exponential stabilization for the Stokes system and local rapid exponential stabilization for the twoand three-dimensional Navier-Stokes systems, with arbitrarily large disturbance bounds. In three dimensions, we obtain global strong solutions and rapid exponential stabilization for initial data sufficiently small in the L 2 -norm with a prescribed, arbitrarily large bound in the H 1 -norm.
L 2 (Ω) n ,
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BibTeX
@misc{calle2026rapid,
title = {{Rapid stabilization for Stokes and Navier-Stokes systems with localized disturbance}},
author = {Christian Calle and Patricio Guzmán and Hugo Parada and Jefferson Prada},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31790},
eprint = {2610.31790},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05839066v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rapid stabilization for Stokes and Navier-Stokes systems with localized disturbance
AU - Christian Calle
AU - Patricio Guzmán
AU - Hugo Parada
AU - Jefferson Prada
PY - 2026
DA - 2026/10/22
PB - HAL
DO - 10.48550/arxiv.2610.31790
UR - https://hal.science/hal-05839066v1
AB - In this study, we analyze the rapid stabilization of Stokes and the two-and threedimensional Navier-Stokes systems in the presence of an unknown spatially localized disturbance. A multivalued feedback control strategy is proposed which combines the Lyapunov frequency method with the multivalued sign operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator provides robustness against disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach guarantees global rapid exponential stabilization for the Stokes system and local rapid exponential stabilization for the twoand three-dimensional Navier-Stokes systems, with arbitrarily large disturbance bounds. In three dimensions, we obtain global strong solutions and rapid exponential stabilization for initial data sufficiently small in the L 2 -norm with a prescribed, arbitrarily large bound in the H 1 -norm. L 2 (Ω) n ,
ER -