The Dubovitskii and Milyutin Formalism Applied to an Optimal Control Problem in a Solidification Model

  • Aníbal Coronel
  • Francisco Guillén-González
  • Francisco Marques-Lopes
  • Marko Rojas-Medar
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)


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 L2n-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.


Optimal control Parabolic systems Diffuse-interface phase field Solidification Optimality conditions 



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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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Aníbal Coronel
    • 1
  • Francisco Guillén-González
    • 2
  • Francisco Marques-Lopes
    • 3
  • Marko Rojas-Medar
    • 4
  1. 1.GMA, Dpto. de Ciencias BásicasFac. de Ciencias, Universidad del Bío-BíoChillánChile
  2. 2.Dpto. EDAN and IMUS, Fac. de MatemáticasUniversidad de SevillaSevillaSpain
  3. 3.Dpto. de MatemáticaUFPABelémBrazil
  4. 4.Instituto de Alta InvestigaciónUniversidad de TarapacáAricaChile

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