Finite element error estimates for a bilinear optimal control problem with pointwise tracking governed by a semilinear elliptic PDE
Abstract
We study finite element approximations for an optimal control problem with pointwise tracking governed by a semilinear elliptic PDE. The control variable enters the state equation as a reaction coefficient, resulting in a bilinear control, and the cost functional includes point evaluations of the state variable. We propose two discretization schemes: a semidiscrete scheme, where the control variable is not discretized, and a fully discrete scheme, where the control variable is discretized using piecewise constant functions. In two and three dimensions, for both schemes, we establish convergence of discrete solutions and prove a priori error bounds that behave as 𝒪(h|log h|) in the L2(Ω)-norm for approximating a locally optimal control. In two dimensions, these bounds are improved to 𝒪(h) for the fully discrete scheme and 𝒪(h2|log h|2) for the semidiscrete scheme. We conclude with numerical experiments that demonstrate the performance of the proposed schemes.
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BibTeX
@misc{otarola2026finite,
title = {{Finite element error estimates for a bilinear optimal control problem with pointwise tracking governed by a semilinear elliptic PDE}},
author = {Enrique Otarola and Daniel Quero and Matias Sasso},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31987},
eprint = {2609.31987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31987},
}
RIS
TY - UNPB
TI - Finite element error estimates for a bilinear optimal control problem with pointwise tracking governed by a semilinear elliptic PDE
AU - Enrique Otarola
AU - Daniel Quero
AU - Matias Sasso
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31987
UR - https://arxiv.org/abs/2609.31987
AB - We study finite element approximations for an optimal control problem with pointwise tracking governed by a semilinear elliptic PDE. The control variable enters the state equation as a reaction coefficient, resulting in a bilinear control, and the cost functional includes point evaluations of the state variable. We propose two discretization schemes: a semidiscrete scheme, where the control variable is not discretized, and a fully discrete scheme, where the control variable is discretized using piecewise constant functions. In two and three dimensions, for both schemes, we establish convergence of discrete solutions and prove a priori error bounds that behave as 𝒪(h|log h|) in the L^2(Ω)-norm for approximating a locally optimal control. In two dimensions, these bounds are improved to 𝒪(h) for the fully discrete scheme and 𝒪(h^2|log h|^2) for the semidiscrete scheme. We conclude with numerical experiments that demonstrate the performance of the proposed schemes.
ER -