Optimal control of vibrations of a dynamic Gao beam in contact with a reactive foundation

In this paper, we investigate the optimal control of vibrations of a nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a reactive foundation underneath it. By the Dubovitskii and Milyutin functional analytical approach, we derive the Pontryagin maximum principle of the system governed by the Gao beam equation. And the first-order necessary optimality condition is presented for the optimal control problem in fixed final horizon case. Finally, we also sketch the numerical solution based on the obtained theoretical results.