Optimal control problem governed by a kind of Kirchhoff-type equation

In this paper, we consider an optimal control problem governed by a kind of Kirchhoff-type equation, which plays an important role in the phenomenon of beam vibration. Firstly, the existence of solution to the state equation is proved by the variational method. Secondly, for the given cost functional, we get that there exists at least an optimal state-control pair via the Sobolev's embedding theorem under the constraint of the state equation. Next, the necessary optimality condition for the optimal solution is derived by using the cone method. Finally, we give the pointwise variational inequality, minimum principles and an equivalent necessary condition for the optimal control problem according to the discussion of the variational inequality.