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Space-time finite element discretization of parabolic optimal control problems with energy regularization
TLDR
In this paper, the authors analyzed space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space and proved unique solvability in the continuous case.Abstract:
We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuska's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.read more
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References
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MonographDOI
Linear and Quasi-linear Equations of Parabolic Type
TL;DR: In this article, the authors introduce a system of linear and quasi-linear equations with principal part in divergence (PCI) in the form of systems of linear, quasilinear and general systems.
Book
Theory and practice of finite elements
Alexandre Ern,Jean-Luc Guermond +1 more
TL;DR: Theoretical Foundations for Finite Element Interpolation and Banach Spaces by Galerkin Methods are given in this article, along with a discussion of the application of the Banach and Hilbert spaces in data-structuring and mesh generation.
Book
Galerkin Finite Element Methods for Parabolic Problems
TL;DR: The standard Galerkin method is based on more general approximations of the elliptic problem as discussed by the authors, and is used to solve problems in algebraic systems at the time level.
BookDOI
The boundary value problems of mathematical physics
TL;DR: In this paper, the method of finite differences is used to compare Equations of Elliptic Type, Parabolic Type, Hyperbolic Type, and Equation of Parabolical Type.
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