Optimal forced vibration control of a smart plate

•The relation between Kelvin–Voigt damping and external excitation is studied and stimulated for the smart plate.•Existence and uniqueness of control function and solution of state system are given by lemma and theorem.•In order to obtain the optimal control function, maximum principle is employed.•Smart material is used to control the structure.•The results about the dynamic behavior of the plate are presented in graphical and table forms.

In this paper, optimal forced vibration control problem is considered for the rectangular plate with Kelvin–Voigt damping subject to an external excitation with control exercised by piezoelectric patch actuators bonded on both sides of the plate. By deriving the maximum principle, the control problem is reduced to solving a system of partial differential equations for the state variable and the adjoint variable subjected to boundary, initial and terminal conditions. Two cases with respect to forcing function are examined and results are presented.