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Optimal Control of an Evolution Problem Involving Time-Dependent Maximal Monotone Operators

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Abstract

We consider a control problem in a finite-dimensional setting, which consists in finding a minimizer for a standard functional defined by way of two continuous and bounded below functions and a convex function, where the control functions take values in a closed convex set and the state functions solve a differential system made up of a differential inclusion governed by a maximal monotone operator; and an ordinary differential equation with a Lipschitz mapping in the right-hand side. We first show the existence of a unique absolutely continuous solution of our system, by transforming it to a sole evolution differential inclusion, and then use a result from the literature. Secondly, we prove the existence of an optimal solution to our problem. The main novelties are: the presence of the time-dependent maximal monotone operators, which may depend as well as their domains on the time variable; and the discretization scheme for the approximation of the solution.

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Acknowledgements

The authors thank a referee for pointing out important remarks and suggestions that helped in the improvement of the last version of the paper. The first and second authors were supported by the DGRSDT, PRFU project number 811 C00L03UN180120180005. Manuel D.P. Monteiro Marques was partially supported by National Funding from FCT-Fundação para a Ciência e a Technologia, under the project: UID/MAT/04561/2019.

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Authors and Affiliations

  1. Laboratoire LAOTI, FSEI, Université Mohammed Seddik Benyahia, Jijel, B.P. 98, Jijel, 18000, Algérie

    Nesrine Bouhali & Dalila Azzam-Laouir

  2. Departamento de Matemática and CMAFcIO, Faculdade de Ciências da Universidade de Lisboa, 1749-016, Campo Grande, Lisboa, Portugal

    Manuel D. P. Monteiro Marques

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  1. Nesrine Bouhali
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  2. Dalila Azzam-Laouir
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  3. Manuel D. P. Monteiro Marques
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Correspondence to Dalila Azzam-Laouir.

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Communicated by Lionel Thibault.

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Bouhali, N., Azzam-Laouir, D. & Monteiro Marques, M.D.P. Optimal Control of an Evolution Problem Involving Time-Dependent Maximal Monotone Operators. J Optim Theory Appl 194, 59–91 (2022). https://doi.org/10.1007/s10957-022-02009-y

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