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

2022/09/23 by Pedro Merino, Merino, Pedro, Alexander Nenjer +1 · 1 citation
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Elasticity and Material Modeling #FOS: Mathematics #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2209.11384

openalex publication_date 2022/09/23 · openalex created_date 2022/09/27 · openalex updated_date 2026/07/28

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∈(0,1)) in the cost function. Our approach is based on the 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 L¹-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

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