Aproximación por elementos finitos de una clase problemas de control óptimo no convexos gobernados por ecuaciones diferenciales parciales elíptica

Abstract

The purpose of this study is deriving an estimation of the error generated by the approximation, by the finite element method, an optimal control problem of the elliptical type, whose cost functional has a nonconvex term that regularizes locally the quasi-norm L^q (with q\in (0.1)). Our study is based on obtaining the optimality system by formulating the cost functional as a difference of convex functions (DC), which involves a L^1 penalty optimal control problem. We consider the space of piece-wise linear continuous functions for the discretization of the state equation and the space of piece-wise constant functions for the approximation of the controls. Under certain conditions established for the parameters of regularization of the problem we obtain that the convergence of our approximation can reach a linear order of convergence as the size of the mesh decreases. Finally, numerical experiments of the obtained theoretical results will be illustrated.