Maximum principle for optimal control of vibrations of a dynamic Gao beam in contact with a rigid foundation

In our preceding paper, we studied an optimal control problem of vibrations of a dynamic Gao beam in contact with a reactive foundation and derived the Pontryagin maximum principle for the controlled system in fixed final horizon case. As a follow-up, in this paper, we focus on the investigation of the Gao beam that may come in contact with a rigid foundation underneath it. In this case, the nonlinear viscoelastic beam equation is equipped with the Signorini condition. By the Dubovitskii and Milyutin functional analytical approach, we investigate the new optimal control problem with multiple inequality constraints and present further original results of current interests.