@Article{NMTMA-15-1,
author = {Liang, Dongdong and Wei, Gong and Xie, Xiaoping},
title = {Finite Element Error Estimation for Parabolic Optimal Control Problems with Pointwise Observations},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2022},
volume = {15},
number = {1},
pages = {165--199},
abstract = {<p style="text-align: justify;">In this paper, we consider parabolic distributed control problems with
cost functional of pointwise observation type either in space or in time. First, we
show the well-posedness of the optimization problems and derive the first order optimality systems, where the adjoint state can be expressed as the linear combination
of solutions to two backward parabolic equations that involve the Dirac delta distribution as source either in space or in time. Second, we use a space-time finite
element method to discretize the control problems, where the state variable is approximated by piecewise constant functions in time and continuous piecewise linear
polynomials in space, and the control variable is discretized by following the variational discretization concept. We obtain a priori error estimates for the control and
state variables with order $\mathcal{O}(k^{\frac{1}{2}}+h)$ up to a logarithmic factor under the $L^2$-norm.
Finally, we perform several numerical experiments to support our theoretical results.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2021-0123},
url = {https://global-sci.com/article/90248/finite-element-error-estimation-for-parabolic-optimal-control-problems-with-pointwise-observations}
}