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Media type:
E-Article
Title:
Quantitative Rapid and Finite Time Stabilization of the Heat Equation
Contributor:
Xiang, Shengquan
Published:
EDP Sciences, 2024
Published in:
ESAIM: Control, Optimisation and Calculus of Variations, 30 (2024), Seite 40
Language:
Not determined
DOI:
10.1051/cocv/2024029
ISSN:
1262-3377;
1292-8119
Origination:
Footnote:
Description:
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.].