The geometrical quantity in damped wave equations on a square

Hébrard, Pascal; Humbert, Emmanuel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 636-661.

The energy in a square membrane $\Omega$ subject to constant viscous damping on a subset $\omega \subset \Omega$ decays exponentially in time as soon as $\omega$ satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau \left(\omega \right)$ of this decay satisfies $\tau \left(\omega \right)=2min(-\mu (\omega ),g(\omega \left)\right)$ (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here $\mu \left(\omega \right)$ denotes the spectral abscissa of the damped wave equation operator and $g\left(\omega \right)$ is a number called the geometrical quantity of $\omega$ and defined as follows. A ray in $\Omega$ is the trajectory generated by the free motion of a mass-point in $\Omega$ subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g\left(\omega \right)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g\left(\omega \right)$ when $\omega$ is a finite union of squares.

[1] M. Asch and G. Lebeau, The spectrum of the damped wave operator for a bounded domain in ${ℝ}^2$. Experiment. Math. 12 (2003) 227-241. | Zbl

[2] 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 (1992) 1024-1065. | Zbl

[3] A. Benaddi and B. Rao, Energy decay rate of wave equations with infinite damping. J. Diff. Equ. 161 (2000) 337-357. | Zbl

[4] C. Carlos and S. Cox, Achieving arbitrarily large decay in the damped wave equation. SIAM J. Control Optim. 39 (2001) 1748-1755. | Zbl

[5] A. Chambert-Loir, S. Fermigier and V. Maillot, Exercices de mathématiques pour l'agrégation. Analyse I. Masson (1995). | Zbl

[6] S. Cox and E. Zuazua, The rate at which energy decays in a damping string. Comm. Partial Diff. Equ. 19 (1994) 213-243. | Zbl

[7] P. Hébrard, Étude de la géométrie optimale des zones de contrôle dans des problèmes de stabilisation. Ph.D. Thesis, University of Nancy 1 (2002).

[8] P. Hébrard and A. Henrot, Optimal shape and position of the actuators for the stabilization of a string. Syst. Control Lett. 48 (2003) 119-209. | Zbl

[9] G. Lebeau, Équation des ondes amorties, in Algebraic and Geometric Methods in Mathematical Physics. Kluwer Acad. Publ., Math. Phys. Stud. 19 (1996) 73-109. | Zbl

[10] J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds. Comm. Pure Appl. Math. 28 (1975) 501-523. | Zbl

[11] J. Sjostrand, Asymptotic distribution of eigenfrequencies for damped wave equations. Publ. Res. Inst. Sci. 36 (2000) 573-611. | Zbl

[12] S. Tabachnikov, Billiards mathématiques. SMF collection Panoramas et synthèses (1995). | MR | Zbl

[13] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping. Comm. Partial Equ. 15 (1990) 205-235. | Zbl