Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping

Abstract

In this paper we study a hinged, extensible, and elastic nonlinear beam equation with structural damping and Balakrishnan-Taylor damping with the full exponent $2\left(n+\beta \right)+1$. This strongly nonlinear equation, initially proposed by Balakrishnan and Taylor in 1989, is a very general and useful model for large aerospace structures. In this work, the existence of global solutions and the existence of absorbing sets in the energy space are proved. For this equation, the feature is that the exponential rate of the absorbing property is not a global constant, but which is uniform for the family of trajectories starting from any given bounded set in the state space. Then it is proved that there exists an inertial manifold whose exponentially attracting rate is accordingly non-uniform. Finally, the spillover problem with respect to the stabilization of this equation is solved by constructing a linear state feedback control involving only finitely many modes. The obtained results are robust in regard to the uncertainty of the structural parameters.