Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks


We consider a dynamic linear shallow shell model, subject to nonlinear dissipation active on a portion of its boundary in physical boundary conditions. Our main result is a uniform stabilization theorem which states a uniform decay rate of the resulting solutions. Mathematically, the motion of a shell is described by a system of two coupled partial differential equations, both of hyperbolic type: (i) an elastic wave in the 2-d in-plane displacement, and (ii) a Kirchhoff plate in the scalar normal displacement. These PDEs are defined on a 2-d Riemann manifold. Solution of the uniform stabilization problem for the shell model combines a Riemann geometric approach with microlocal analysis techniques. The former provides an intrinsic, coordinate-free model, as well as a preliminary observability-type inequality. The latter yield sharp trace estimates for the elastic wave-critical for the very solution of the stabilization problem-as well as sharp trace estimates for the Kirchhoff plate-which permit the elimination of geometrical conditions on the controlled portion of the boundary. © 2002 Elsevier Science (USA). All rights reserved.

Publication Title

Journal of Mathematical Analysis and Applications