Stabilisation frontière de problèmes de Ventcel

Abstract

Abstract. The problem of boundary stabilization for the isotropic linear elastodynamic system and the wave equation with Ventcel's conditions are considered (see [12]). The boundary observability and the exact controllability were etablished in [11]. We prove here the enegy decay to zero for the elastodynamic system with stationary Ventcel's conditions by introducing a nonlinear boundary feedback. We also give a boundary feedback leading to arbitrarily large energy decay rates for the elastodynamic system with evolutive Ventcel's conditions. A spectral study proves, finally, that the natural feedback is not sufficient to assure the exponential decay in the case of the wave equation with Ventcel's conditions. Résumé. On considère le problème de la stabilisation frontière de problèmes de Ventcel pour le système linéaire isotrope de l'élasticité et pour l'équation des ondes (cf. [12]). L'observabilité et la contrôlabilité ontétéétablies dans [11]. On montre ici la décroissance vers zéro de l'énergie pour le système de l'élasticité avec conditions de Ventcel stationnaires par un feedback non linéaire ; on montre aussi la décroissance exponentielle arbitrairement grande de l'énergie de la solution du système de l'élasticité par des feedbacks frontières. On montre enfin, par uneétude spectrale, que le feedback naturel est insuffisant pour assurer la décroissance exponentielle de l'énergie dans le cas de l'équation des ondes. AMS Subject Classification. 93B03, 93B05, 93D15.Reçu le 29 novembre 1999. Révisé le 9 juin, le 17 juillet, le 29 août et le 25 septembre 2000.

On the exponential stabilization of the electromagneto‐elastic system with Wentzell conditions

“…4 The study of some mathematical and numerical aspects of such problems were given in Lemrabet 3 and Goldstein. 5 For a detailed survey on Wentzell boundary conditions, we refer the reader to the previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein, and the list is not exhaustive.…”

“…The issue of stability of solutions of the Wentzell problems has attracted a great deal of attention in the last 3 decades. To this end, see, for instance, previous works 4, [7][8][9]14,15,18,19 and the references therein. In Heminna, 14 the author showed that the natural feedback is not sufficient to guarantee the exponential decay of the energy ((t)) in the case of the wave equation with Wentzell conditions.…”

Polynomial stabiization of the wave equation with Ventcel's boundary conditions

“…As mentioned before, A. Heminna has shown in [7,8] that the feedback law on Γ 1 is actually not sufficient to guarantee the exponential decay of the energy even if the subset Γ 1 satisfy the control geometric condition. More precisely the author showed that if Γ 1 = {(1, y) : 0 < y < 1} ∪ {(x, 1) : 0 < x < 1} then the operator A associated with problem (2.1) has eigenvalues that tend to the imaginary axis.…”

“…Ventcel boundary conditions are characterized by the presence of tangential differential operators of the same order than the interior operator. Such boundary conditions are usually justified by asymptotic methods and appear in mechanics [13], in diffusion processes [12,21] or wave phenomena [2,[7][8][9]14].…”

“…It was shown by A. Heminna in [7,8] that the feedback law of Ventcel type on a part Γ 1 of the boundary is actually not sufficient to guarantee the exponential decay of the energy even if this subset Γ 1 satisfies the control geometric condition. Our goal here is to show that for a specific choice of Γ 1 and of Ω and smooth enough initial data, the solution of our problem admits a polynomial decay.…”