Inertial manifolds and finite dimensional exponential stabilization of nonlinear beam equations

The objective of this paper is to show a new approach to solve the spillover problem for certain nonlinear beam equations. The spillover problem is whether or not one can find a finite dimensional feedback control to stabilize the beam system exponentially. If yes, then how. From a different angle, one can investigate such a nonlinear beam without control as an infinite dimensional dynamical system. In view of the weak damping, one may wonder whether or not there exist some finite dimensional permanent patterns of dynamics. Indeed, the author can prove that to these questions the answers are all yes. However, since the purpose of this paper is to show the connection between the existence of inertial manifolds and the existence of a solution to the spillover problem, the author does not address the issue of the global attractor