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 Ural Math. J., 2020, Volume 6, Issue 1, Pages 84–94 (Mi umj113)

Optimal control for a controlled ill-posed wave equation without requiring the Slater hypothesis

Abdelhak Hafdallah

Laboratory of Mathematics, Informatics and Systems, University of Larbi Tébessi

Abstract: In this paper, we investigate the problem of optimal control for an ill-posed wave equation without using the extra hypothesis of Slater i.e. the set of admissible controls has a non-empty interior. Firstly, by a controllability approach, we make the ill-posed wave equation a well-posed equation with some incomplete data initial condition. The missing data requires us to use the no-regret control notion introduced by Lions to control distributed systems with incomplete data. After approximating the no-regret control by a low-regret control sequence, we characterize the optimal control by a singular optimality system.

Keywords: Ill-posed wave equation, No-regret control, Incomplete data, Carleman estimates, Null-controllability.

 Funding Agency Grant Number Directorate-General for Scientific Research and Technological Development This work was supported by the Directorate-General for Scientific Research and Technological Development (DGRSDT).

DOI: https://doi.org/10.15826/umj.2020.1.007

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Citation: Abdelhak Hafdallah, “Optimal control for a controlled ill-posed wave equation without requiring the Slater hypothesis”, Ural Math. J., 6:1 (2020), 84–94

Citation in format AMSBIB
\Bibitem{Haf20} \by Abdelhak Hafdallah \paper Optimal control for a controlled ill-posed wave equation without requiring the Slater hypothesis \jour Ural Math. J. \yr 2020 \vol 6 \issue 1 \pages 84--94 \mathnet{http://mi.mathnet.ru/umj113} \crossref{https://doi.org/10.15826/umj.2020.1.007} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=MR4128762} \elib{https://elibrary.ru/item.asp?id=43793626} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85089118612}