We consider the Marguerre–von Kármán equations that model the deformation of a thin, nonlinearly elastic, shallow shell, subjected to a specific class of boundary conditions of von Kármán’s type. Next, we reduce these equations to a single equation with a cubic operator following Berger’s classical method, whose second member depends on the function defining the middle surface of the shallow shell and the resultant of the vertical forces acting on the shallow shell. We also prove the existence and uniqueness of a weak solution to the reduced equation. Then, we prove the existence theorem for the optimal control problem governed by Marguerre–von Kármán equations, with a control variable on the resultant of the vertical forces. Using the Fréchet differentiability of the state function with respect to the control variable, we prove the uniqueness of the optimal control and derive the necessary optimality condition. As a result, this work addresses the more general geometry of Marguerre–von Kármán shallow shells to study the quadratic cost optimal control problems governed by these equations.
