Issue |
ESAIM: COCV
Volume 30, 2024
|
|
---|---|---|
Article Number | 40 | |
Number of page(s) | 25 | |
DOI | https://doi.org/10.1051/cocv/2024029 | |
Published online | 03 May 2024 |
Quantitative Rapid and Finite Time Stabilization of the Heat Equation
School of Mathematical Sciences, Peking University,
100871,
Beijing,
PR China
* Corresponding author: shengquan.xiang@math.pku.edu.cn
Received:
6
December
2022
Accepted:
26
March
2024
The finite time stabilizability of the one dimensional heat equation is proved by Coron-Nguyên [J.-M. Coron and H.-M. Nguyen, Arch. Ration. Mech. Anal. 225 (2017) 993–1023], while the same question for multidimensional spaces remained open. Inspired by Coron-Trélat [J.-M. Coron and E. Trélat, SIAM J. Control Optim. 43 (2004) 549–569] we introduce a new method to stabilize multidimensional heat equations quantitatively in finite time and call it Frequency Lyapunov method. This method naturally combines spectral inequality [G. Lebeau and L. Robbiano, Comm. Partial Diff. Equ. 20 (1995) 335–356] and constructive feedback stabilization. As application this approach also yields a constructive proof for null controllability, which gives sharing optimal cost CeC/T with explicit controls and works perfectly for related nonlinear models such as Navier–Stokes equations [S. Xiang, Ann. Inst. H. Poincaré C Anal. Non Lineaire 40 (2023) 1487–1511.].
Mathematics Subject Classification: 35S15 / 35K55 / 93D15
Key words: Finite time stabilization / quantitative / spectral estimate / null controllability
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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