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| - Quantitative Rapid and Finite Time Stabilization of the Heat Equation
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| - 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 Lineaire40 (2023) 1487-1511.].
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| - © The authors. Published by EDP Sciences, SMAI 2024
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| - The authors. Published by EDP Sciences, SMAI
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