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

TL;DR

This paper presents a finite element method for an $L^1$-penalized optimal control problem in Timoshenko beams, enabling sparse actuator placement with proven optimal convergence independent of beam thickness.

Contribution

It introduces a locking-free finite element scheme for $L^1$-control problems in Timoshenko beams, demonstrating optimal convergence rates.

Findings

The scheme achieves optimal convergence order.

The method effectively localizes control devices.

Convergence is independent of beam thickness.

Abstract

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.