Optimality conditions for a Timoshenko beam equation with periodic constraint

This article is concerned with an optimization problem governed by a Timoshenko beam equation with periodic constraints. Firstly, a detail spectral analysis is conducted for the Timoshenko operator related to the homogeneous system. Then, we derive a closed range property of the state equation. Finally, we present the main results: the optimality conditions and the maximum principle for our problem via the closed range theorem.