  The numerical approximation of an optimal control problem with $L^1$-control
of a Timoshenko beam is considered and analyzed by using the finite element
method. From the practical point of view, inclusion of the $L^1$--norm in the
cost functional is interesting in the case of beam vibration model, since the
sparsity enforced by the $L^1$--norm is very useful for localizing actuators or
control devices. The discretization of the control variables is performed by
using piecewise constant functions. The states and the adjoint states are
approximated by a \emph{locking free scheme} of linear finite elements.
Analogously to the purely $L^2$--norm penalized optimal control, it is proved
that this approximation have optimal convergence order, which do not depend on
the thickness of the beam.
