Optimal boundary control for the Timoshenko–Ehrenfest truncated model

Abstract

In this work, we investigate the optimal control problem associated with a truncated version of the Timoshenko–Ehrenfest beam model, which captures essential features of transverse vibrations in elastic structures. We begin by establishing the well-posedness of the system through the Faedo–Galerkin approximation method, ensuring existence and uniqueness of solutions. The associated optimal control problem is then formulated, and the Pontryagin maximum principle is employed to characterize the optimality conditions. To obtain the analytical solution aiming numerical issues, we apply a Fourier series expansion, which allows for the explicit representation of both the state and the adjoint variables. Finally, we present numerical simulations that demonstrate the efficiency of the proposed control strategy in suppressing unwanted vibrations, confirming the theoretical results and highlighting the practical relevance of the method.