Numerical approximation of regularized non-convex elliptic optimal control problems by the finite element method

TL;DR

This paper develops a finite element method for approximating solutions to non-convex elliptic optimal control problems involving $L^q$-quasinorm regularization, providing convergence analysis and numerical validation.

Contribution

It introduces a difference-of-convex formulation and finite element discretization for non-convex regularized control problems, with proven convergence rates.

Findings

Convergence rate of order $h^{1/2}$ under certain conditions.

Numerical experiments confirm theoretical convergence behavior.

Method effectively approximates non-convex regularized optimal controls.

Abstract

We investigate the numerical approximation of an elliptic optimal control problem which involves a nonconvex local regularization of the $L^q$-quasinorm penalization (with $q\in(0,1)$) in the cost function. Our approach is based on the \emph{difference-of-convex} function formulation, which leads to first-order necessary optimality conditions, which can be regarded as the optimality system of an auxiliar convex $L^1$-penalized optimal control problem. We consider piecewise-constant finite element approximation for the controls, whereas the state equation is approximated using piecewise-linear basis functions. Then, convergence results are obtained for the proposed approximation. Under certain conditions on the support's boundary of the optimal control, we deduce an order of $h^\frac12$ approximation rate of convergence where $h$ is the associated discretization parameter. We illustrate our theoretical findings with numerical experiments that show the convergence behavior of the numerical approximation