Boundary optimal control problem of semi-linear Kirchhoff plate equation

This paper examines a nonlinear Kirchhoff plate equation, where the control acts in bilinear form within the boundary of the mentioned equation. The objective is to construct a distributed control to guide such a system from the initial state to the desired state in the final time, while minimizing a quadratic functional cost defined as the sum of the norm difference between the aforementioned state and a desired equation with an energy term. We show how to approximate the solution of the nonlinear Kirchhoff plate equation to a desired objective, indicating the existence of optimal control in specific cases. and deriving the optimally conditions for a closed convex set. Moreover, it is shown that sufficient conditions ensures the uniqueness of control optimal. Furthermore, we provide a concise numerical methodology that involves the integration of finite element and finite difference discretization methods. The approach incorporates Newton's linearization method to assess the computational performance of the controlled problem, using the Freefem++ software.