Optimal Control and Controllability of a Phase Field System with One Control Force

Abstract

We investigate the relation between optimal control and controllability for a phase field system modeling the solidification process of pure materials in the case that only one control force is used. Such system is constituted of one energy balance equation, with a localized control associated to the density of heat sources and sinks to be determined, coupled with a phase field equation with the classical nonlinearity derived from the two-well potential. We prove that this system has a local controllability property and we establish that a sequence of solutions of certain optimal control problems converges to a solution of such controllability problem.

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Acknowledgments

Part of this research was done while the first author was visiting Universidade Estadual de Campinas, and he expresses his thanks for their kind hospitality. The first author’s study was partially supported by INCTMat, CAPES, CNPq (Brazil). The second author’s study was partially supported by FAPESP and CNPq (Brazil). The third author’s study was partially supported by FAPESP (Brazil).

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Correspondence to B. M. R. Calsavara.

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Araruna, F.D., Boldrini, J.L. & Calsavara, B.M.R. Optimal Control and Controllability of a Phase Field System with One Control Force. Appl Math Optim 70, 539–563 (2014). https://doi.org/10.1007/s00245-014-9249-1

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Keywords

  • Phase field models
  • Solidification models
  • Optimal control
  • Controllability

Mathematics Subject Classification

  • 82B26
  • 49J20
  • 93B05