TY - JOUR
T1 - Error Estimates for Finite Element Approximation to Elliptic Optimal Control Problems with Boundary Observations in $H^{-\frac{1}{2}}(\Gamma)$
AU - Tao , Xuelin
JO - Advances in Applied Mathematics and Mechanics
VL - 5
SP - 1152
EP - 1175
PY - 2024
DA - 2024/07
SN - 16
DO - http://doi.org/10.4208/aamm.OA-2022-0172
UR - https://global-sci.org/intro/article_detail/aamm/23289.html
KW - Optimal control problems, boundary observations, finite element methods, error estimate.
AB - <p style="text-align: justify;">In this paper we consider the finite element approximation to an elliptic
optimal control problem with boundary observations in $H^{-\frac{1}{2}}(\Gamma).$ This problem is motivated by certain applications in optimal control of semiconductor devices where the
discrepancy of the current density on the boundary to the target one is the objective to
be minimized. The observation in $H^{-\frac{1}{2}}(\Gamma)$ is realized through a Neumann-to-Dirichlet
mapping which facilitates the theoretical and numerical analysis. A priori error estimate for the optimal control is derived based on the variational control discretization,
whereas the state and adjoint state variables are approximated by piecewise linear and
continuous finite elements. Second order convergence rate for the optimal control is
theoretically proved and verified by numerical results.</p>