In this paper we study a hinged, extensible, and elastic nonlinear beam equation
with structural damping and Balakrishnan-Taylor damping with the full exponent $2(n+\beta)+ 1$ . 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.