Abstract
The paper provides strong convergence of solutions to a sequence of linear-quadratic (LQ) optimization problems defined in an abstract functional framework. Each problem is accompanied by the constraint of reaching a given target within a prescribed precision. We show that the problems are well-posed and characterize their solutions. The main result provides the conditions under which these solutions converge to the minimizer of the limit problem. The generality of the result allows its application to a wide range of problems: elliptic, parabolic, control ones, etc. The examples presented in the paper consider optimal approximate controls of the heat equation and optimal approximate solutions to the Poisson equation.
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Acknowledgements
This research was done while the second author was visiting Chair of Dynamics, Control and Numerics (Alexander von Humboldt Professorship) at Friedrich-Alexander-Universität Erlangen-Nürnberg, with the support of the DAAD (Research Stays for University Academics and Scientists, 2021 programme) and Alexander von Humboldt-Professorship.
The author acknowledges E. Zuazua for fruitful discussions on the subject.
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Communicated by Enrique Zuazua.
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Lazar, M. Constrained Linear-Quadratic Optimization Problems with Parameter-Dependent Entries. J Optim Theory Appl 198, 781–804 (2023). https://doi.org/10.1007/s10957-023-02257-6
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DOI: https://doi.org/10.1007/s10957-023-02257-6
Keywords
- LQ optimization
- Parametrized PDEs
- Optimal controls
- Optimal approximate solutions
- Convergence of minimizers