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Least-squares finite elements for distributed optimal control problems

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Abstract

We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is \(\inf \)–\(\sup \) stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented.

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Acknowledgements

This work was supported by ANID through FONDECYT Projects and 1210391 (TF), and 1210579 (MK).

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Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

    Thomas Führer

  2. Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile

    Michael Karkulik

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  1. Thomas Führer
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  2. Michael Karkulik
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Correspondence to Thomas Führer.

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Führer, T., Karkulik, M. Least-squares finite elements for distributed optimal control problems. Numer. Math. 154, 409–442 (2023). https://doi.org/10.1007/s00211-023-01367-7

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