Sparse optimal control of Timoshenko's beam using a locking-free finite element approximation

This paper addresses the optimal control problem with sparse controls of a Timoshenko beam, its numerical approximation using the finite element method, and the numerical solution via nonsmooth methods. Incorporating sparsity-promoting terms in the cost function is practically useful for beam vibration models and results in the localization of the control action that facilitates the placement of actuators or control devices. We consider two types of sparsity-inducing penalizers: the L1$$ {L}<^>1 $$-norm and the L0$$ {L}<^>0 $$-penalizer, which measures function support. We analyze discretized problems utilizing linear finite elements with a locking-free scheme to approximate the states and adjoint states. We confirm that this approximation has the looking-free property required to achieve a linear convergence linear order of approximation for L1$$ {L}<^>1 $$ control case and depending on the set of switching points in the L0$$ {L}<^>0 $$ controls. This is similar to the purely L2$$ {L}<^>2 $$-norm penalized optimal control, where the order of approximation is independent of the thickness of the beam. Sparsity given by L1$$ {L}<^>1 $$ and L0$$ {L}<^>0 $$ penalties is an advantageous control strategy in the Timoshenko model because smaller supports in controls help to place actuator devices. We have confirmed theoretically and numerically the locking-free property of the sparse optimal controls by using a mixed formulation of the beam equations. In addition, we have derived optimality conditions and proposed nonsmooth numerical schemes to solve the associated optimization problems.image