Polynomial  stabilizability of a plate  equation in a waveguide  with  dissipation at infinity

Mohamed Malloug

doi:10.5269/bspm.67313

Mohamed Malloug Laboratoire de Mathématique: Modélisation Déteministe et Aléatoire (LAMMDA)

Abstract

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.

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References

L. Aloui, S. Ibrahim, and M. Khenissi. Energy decay for linear dissipative wave equations in exterior domains. J. Differ. Equations., 259(5)(2015), 2061-2079.

L. Aloui and M. Khenissi. Stabilisation pour l’equation des ondes dans un domaine exterieur. Rev. Math. Iberoamericana., 18(2002), 1-16.

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.

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.

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.

A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Math. Ann., 347(2)(2010), 455-478.

N. Burq and M. Hitrik. Energy decay for damped wave equations on partially rectangular domains. Math. Res. Lett., 14(1)(2007), 35-47.

N. Burq. Semi-classical estimates for the resolvent in nontrapping geometries. Int. Math. Res. Not., 5(2002), 221-241.

N. Burq and M. Zorski. Control for Schrodinger operators on tori. Math. Res. Lett., 19(2)(2012), 309-324.

R. Denk and R. Schnauberlt. Astructurally damped plate equation withe Dirichet-Newman boundary conditions . J. Differentiel Equations ., 259(2015), 1323-1353.

L. Hormander. On the uniquess of the Gauchy problem. Mat.Scand ., 7(1955), 177-190.

P. Gerard and E. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J., 71(1993), 559-607.

S. Jaffard. Controle interne exact des vibration d’un plaque rectangulaire. Portugal.Mat ., 47(4)(1990), 423-429.

Th. Jecko. From classical to semiclassical non-trapping behaviour. C. R., Math., Acad. Sci. Paris., 338(7)(2004), 545-548.

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.

M. Khenissi. Equation des ondes amorties dans un domaine exterieur. Bull. Soc. Math. France., 131(2)(2003), 211–228.

M. Malloug and J. Royer. Energy decay in a wave guide with dissipation at infinity. ESAIM. COCV., 24(2018), 519-549.

S. Nouira. polynomial and analytic boundary feedbacks stabilizability of square plate. Bol. Soc. Parana. mat ., 27(2)(2009), 23-34.

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.

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.

J. Royer. Local energy decay and diffusive phenomenon in a dissipative wave guide. Jourrnal of Specral theory., 8(3)(2018), 769-841.

J. Royer. Exponential decay for the Schrodinger equation on a dissipative wave guide. Ann. Henri Poincare., 16(8)(2015), 1807-1836.

M. Zworski. Semiclassical Analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, 2012.