We intend to conduct a fairly complete study on
Timoshenko beams with pointwise feedback controls and seek to obtain information
about the eigenvalues, eigenfunctions, Riesz-Basis-Property,
spectrum-determined-growth-condition, energy decay rate and various stabilities
for the beams. One major difficulty of the present problem is the non-simplicity
of the eigenvalues.
In fact, we shall indicate in this paper situations where the multiplicity of
the eigenvalues is at least two. We build all the above-mentioned results from an effective
asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude
with the Riesz-Basis-Property and the
spectrum-determined-growth-condition. Finally, these results are used to examine
the stability effects on the system by the location of the pointwise control
relative to the length of the whole beam.