Polynomial stabilizability of a plate equation in a waveguide with dissipation at infinity
DOI:
https://doi.org/10.5269/bspm.67313Abstract
We consider the dissipative plate equation in a waveguide, in the case where the usual Geometric Control Condition of Bardos, Lebeau and Rauch [5] is not necessarily satisfied. More precisely, assuming that the damping is concentrated close to infinity. We prove polynomial energy decay with respect to a stronger norm of the initial data.
References
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12. P. Gerard and E. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J., 71(1993), 559-607.
13. S. Jaffard. Controle interne exact des vibration d’un plaque rectangulaire. Portugal.Mat ., 47(4)(1990), 423-429.
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16. M. Khenissi. Equation des ondes amorties dans un domaine exterieur. Bull. Soc. Math. France., 131(2)(2003), 211–228.
17. M. Malloug and J. Royer. Energy decay in a wave guide with dissipation at infinity. ESAIM. COCV., 24(2018), 519-549.
18. S. Nouira. polynomial and analytic boundary feedbacks stabilizability of square plate. Bol. Soc. Parana. mat ., 27(2)(2009), 23-34.
19. J. Le Rousseau and L. Robbiano. Spectral Inequality and Resolvent estimate for the Bi-Laplace operator. Jourrnal of European Mathematical Society., 22(4)(2019), 1003-1094.
20. K. Ramdani, T . Takahashi, and M. Tucsnak. Internal Stabilisation of plate equation in a square: The continuous and the semi discretized problems . J. Math. Pures App ., 85(1)(2006), 17-37.
21. J. Royer. Local energy decay and diffusive phenomenon in a dissipative wave guide. Jourrnal of Specral theory., 8(3)(2018), 769-841.
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23. M. Zworski. Semiclassical Analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, 2012.
2. L. Aloui and M. Khenissi. Stabilisation pour l’equation des ondes dans un domaine exterieur. Rev. Math. Iberoamericana., 18(2002), 1-16.
3. K. Ammari, M. Tucsnak and G . Tenenbaum. A sharp geometric condition for the boundary exponential stabilizability of a square plate by moment feedbacks only, Control of coupled partial differential equation . Internat. Ser. Numer., vol 155, Birkhauser, Basel ., 155(2007), 1-11.
4. C. J. K. Batty, R. Chill, and Y. Tomilov. Fine scales of decay of operator semigroups.Jourrnal of European Mathematical Society., 18(4)(2016), 853-929.
5. C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim., 30(5)(2007), 1024-1065.
6. A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Math. Ann., 347(2)(2010), 455-478.
7. N. Burq and M. Hitrik. Energy decay for damped wave equations on partially rectangular domains. Math. Res. Lett., 14(1)(2007), 35-47.
8. N. Burq. Semi-classical estimates for the resolvent in nontrapping geometries. Int. Math. Res. Not., 5(2002), 221-241.
9. N. Burq and M. Zorski. Control for Schrodinger operators on tori. Math. Res. Lett., 19(2)(2012), 309-324.
10. R. Denk and R. Schnauberlt. Astructurally damped plate equation withe Dirichet-Newman boundary conditions . J. Differentiel Equations ., 259(2015), 1323-1353.
11. L. Hormander. On the uniquess of the Gauchy problem. Mat.Scand ., 7(1955), 177-190.
12. P. Gerard and E. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J., 71(1993), 559-607.
13. S. Jaffard. Controle interne exact des vibration d’un plaque rectangulaire. Portugal.Mat ., 47(4)(1990), 423-429.
14. Th. Jecko. From classical to semiclassical non-trapping behaviour. C. R., Math., Acad. Sci. Paris., 338(7)(2004), 545-548.
15. G. Lebeau. Equation des ondes amorties. In : A. Boutet de Monvel and V. Marchenko (editors), Algebraic and geometric methods in mathematical physics, Kluwer Academic Publishers., 1996, 73-109.
16. M. Khenissi. Equation des ondes amorties dans un domaine exterieur. Bull. Soc. Math. France., 131(2)(2003), 211–228.
17. M. Malloug and J. Royer. Energy decay in a wave guide with dissipation at infinity. ESAIM. COCV., 24(2018), 519-549.
18. S. Nouira. polynomial and analytic boundary feedbacks stabilizability of square plate. Bol. Soc. Parana. mat ., 27(2)(2009), 23-34.
19. J. Le Rousseau and L. Robbiano. Spectral Inequality and Resolvent estimate for the Bi-Laplace operator. Jourrnal of European Mathematical Society., 22(4)(2019), 1003-1094.
20. K. Ramdani, T . Takahashi, and M. Tucsnak. Internal Stabilisation of plate equation in a square: The continuous and the semi discretized problems . J. Math. Pures App ., 85(1)(2006), 17-37.
21. J. Royer. Local energy decay and diffusive phenomenon in a dissipative wave guide. Jourrnal of Specral theory., 8(3)(2018), 769-841.
22. J. Royer. Exponential decay for the Schrodinger equation on a dissipative wave guide. Ann. Henri Poincare., 16(8)(2015), 1807-1836.
23. M. Zworski. Semiclassical Analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, 2012.
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2025-04-30
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