Abstract
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.].
Reference53 articles.
1. Fursikov A.V. and Imanuvilov O.Yu., Controllability of Evolution Equations, Vol. 34 of Lecture Notes Series. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996).
2. Contróle Exact De Léquation De La Chaleur
3. Lions J.-L., Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Vol. 9 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris (1988).
4. Exact controllability theorems for linear parabolic equations in one space dimension
5. Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions