Year: 2022
Author: Dongdong Liang, Wei Gong, Xiaoping Xie
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 1 : pp. 165–199
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2021-0123
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 1 : pp. 165–199
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 35
Keywords: Parabolic optimal control problem pointwise observation space-time finite element method parabolic PDE with Dirac measure error estimate.