Abstract
In this paper we study an optimal control problem in a physical system governed by a solidification model. The solidification system is given by a nonlinear parabolic PDE system of two equations for the unknowns the (reduced) temperature and a phase field function, with a temperature source term. The optimal control problem is defined via the source term as the control function and the objective functional given by the comparison in L 2n-norms of the real state with a given target state and the cost of the control. The main results of the paper are the existence of a global optimal solution via a minimizing sequence, and the first-order necessary conditions for local optimal solutions, by means of the application of the Dubovitskii and Milyutin formalism.
Dedicated to Prof. Enrique Fernández-Cara on the occasion of his 60th birthday.
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Acknowledgements
A. Coronel thanks the support of research project DIUBB GI 172409/C at Universidad del Bío-Bío, Chile. F.Guillen-Gonzalez has been partially supported by project MTM2015-69875-P, Spain. M.A. Rojas-Medar has been partially supported by project MTM2012-32325, Spain, and Grant 1120260, Fondecyt-Chile.
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Coronel, A., Guillén-González, F., Marques-Lopes, F., Rojas-Medar, M. (2018). The Dubovitskii and Milyutin Formalism Applied to an Optimal Control Problem in a Solidification Model. In: Doubova, A., González-Burgos, M., Guillén-González, F., Marín Beltrán, M. (eds) Recent Advances in PDEs: Analysis, Numerics and Control. SEMA SIMAI Springer Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-97613-6_11
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