Uniform Boundary Stabilization of a Nonlinear Shallow and Thin Elastic Spherical Cap

Abstract

A coupled nonlinear system describing the vibrations of a shallow thin spherical shell is considered. It is shown that if the “thickness” parameterh/ρ0is large with respect to the “shallowness” parameterQ0=ρ0/R, then the dissipative feedback applied at the boundary of the shell causes the energy of the system to decay to zero at an uniform rate. The limitations on the values of these physical parameters are confirmed by the static nonlinear theory which predicts, for small values of the thickness parameter, the existence of multiple equilibrium states (everted states).