A locking-free optimal control problem with $L^1$ cost for optimal placement of control devices in Timoshenko beam

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.