Applied Mathematics & Optimization

, Volume 78, Issue 2, pp 267–296 | Cite as

Local Exact Controllability of Two-Phase Field Solidification Systems with Few Controls

  • F. D. ArarunaEmail author
  • B. M. R. Calsavara
  • E. Fernández-Cara


We analyze a control problem for a phase field system modeling the solidification process of materials that allow two different types of crystallization coupled to a Navier–Stokes system and a nonlinear heat equation, with a reduced number of controls. We prove that this system is locally exactly controllable to suitable trajectories, with controls acting only on the motion and heat equations.


Phase field models Solidification models Controllability Observability 

Mathematics Subject Classification

82B26 35K55 93B05 93B07 



This paper was written in part during several visits of the first and second authors to the Institute of Mathematics of the University of Sevilla (IMUS). The authors are indebted to this Institute for its assistance. The authors are also indebted to the anonymous referee for his/her comments and suggestions, that led to essential improvements of the paper. F.D. Araruna was partially supported by INCTMat, CAPES, CNPq (Brazil) and MathAmSud COSIP. B.M.R. Calsavara was partially supported by FAPESP, Grant Nos. 2012/15379-2 and 2014/16802-1, and CNPq (Brazil), Grant No 211473/2013-8. E. Fernández-Cara was partially supported by Grant No MTM2013-41286-P, DGI-MICINN (Spain) and CAPES (Brazil).


  1. 1.
    Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control, Translated from the Russian by V. M. Volosov, Contemporary Soviet Mathematics. Consultants Bureau, New York (1987)CrossRefGoogle Scholar
  2. 2.
    Ammar Khodja, F., Benabdallah, A., Dupaix, C., Kostin, I.: Controllability to the trajectories of phase-field models by one control force. SIAM J. Control Optim. 42(5), 1661–1680 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ammar Khodja, F., Benabdallah, A., González-Burgos, M., De Teresa, L.: Recent results on the controllability of linear coupled parabolic problems: a survey. Math. Control Relat. Fields 1(3), 267–306 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barbu, V.: Local controllability of the phase field system. Nonlinear Anal. 50(3), 363–372 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benabdallah, A., Cristofol, M., Gaitan, P., de Teresa, L.: Controllability to trajectories for some parabolic systems of three and two equations by one control force. Math. Control Relat. Fields 4(1), 17–44 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bennon, W.D., Incropera, F.P., Continuum, A.: Model for momentum, heat and species transport in binary solid–liquid phase change systems: I. Model formulation. Int. J. Heat Mass Transf. 30, 2161–2170 (1987)CrossRefGoogle Scholar
  8. 8.
    Bennon, W.D., Incropera, F.P., Continuum, A.: Model for momentum, heat and species transport in binary solid liquid phase change systems: II. Application to solidification in a rectangular cavity. Int. J. Heat Mass Transf. 30, 2171–2187 (1987)CrossRefGoogle Scholar
  9. 9.
    Caginalp, G.: An analysis of phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–245 (1986)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Caginalp, G.: Stefan and Hele–Shaw type models as asymptotic limits of the phase-field equations. Phys. Rev. A 39(11), 5887–5896 (1989)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Caginalp, G.: Phase field computations of single-needle crystals, crystal growth and motion by mean curvature. SIAM J. Sci. Comput. 15(1), 106–126 (1994)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Caginalp, G., Jones, J.: A derivation and analysis of phase field models of thermal alloys. Ann. Phys. 237, 66–107 (1995)CrossRefGoogle Scholar
  13. 13.
    Calsavara, B.M.R., Boldrini, J.L.: On a system coupling two-crystallization Allen–Cahn equations and a singular Navier–Stokes system. Commun. Math. Sci. 12(2), 66–107 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Carreño, N.: Local controllability of the \(N\)-dimensional Boussinesq system with \(N-1\) scalar controls in an arbitrary control domain. Math. Control Relat. Fields 2(4), 361–382 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Carreño, N., Guerrero, S.: Local null controllability of the \(N\)-dimensional Navier–Stokes system with \(N-1\) scalar controls in an arbitrary control domain. J. Math. Fluid Mech. 15(1), 139–153 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chae, D., Imanuvilov, OYu., Kim, S.M.: Exact controllability for semilinear parabolic equations with Neumann boundary conditions. J. Dynam. Control Syst. 2(4), 449–483 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Coron, J.-M.: Control and nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence (2007)Google Scholar
  18. 18.
    Coron, J.-M., Lissy, P.: Local null controllability of the three-dimensional Navier–Stokes system with a distributed control having two vanishing components. Invent. Math. 198(3), 833–880 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fernández-Cara, E., González-Burgos, M., Guerrero, S., Puel, J.-P.: Null controllability of the heat equation with boundary Fourier conditions: the linear case. ESAIM Control Optim. Calc. Var. 12, 442–465 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fernández-Cara, E., Guerrero, S.: Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45(4), 1395–1446 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Fernández-Cara, E., Guerrero, S., Imanuvilov, OYu., Puel, J.-P.: Local exact controllability of the Navier–Stokes system. J. Math. Pures Appl. 83, 1501–1542 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Y., Puel, J.-P.: Some controllability results for the \(N\)-dimensional Navier–Stokes and Boussinesq systems with \(N-1\) scalar controls. SIAM J. Control Optim. 45(1), 146–173 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fursikov, A.V., Imanuvilov, O.Y.: Controllability of evolutions equations. Lectures Notes Series, vol. 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996)Google Scholar
  24. 24.
    González-Burgos, M., Pérez-García, R.: Controllability results for some nonlinear coupled parabolic systems by one control force. Asymptot. Anal. 46(2), 123–162 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Guerrero, S.: Local exact controllability to the trajectories of the Boussinesq system. Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 29–61 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hoffman, K., Jiang, L.: Optimal control of a phase field model for solidification. Numer. Funct. Anal. Optim. 13, 11–27 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Imanuvilov, OYu.: Remarks on exact controllability for the Navier–Stokes equations. ESAIM Control Optim. Cal. Var. 6, 39–72 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Imanuvilov, OYu., Puel, J.-P.: Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. C. R. Math. Acad. Sci. Paris 335, 33–38 (2002)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ni, J., Beckermann, C.: A volume-averaged two-phase model for solidication transport phenomena. Metallurgical Transactions B 22, 349–361 (1991)CrossRefGoogle Scholar
  30. 30.
    Penrose, O., Fife, P.C.: Thermodynamically consistent models of phase field type for the kinetics of phase transitions. Physica D 43, 44–62 (1990)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (2001)CrossRefGoogle Scholar
  32. 32.
    Silverman, L.M., Meadows, H.E.: Controllability and observability in time-variable linear systems. SIAM J Control 5(1), 64–73 (1967)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • F. D. Araruna
    • 1
    Email author
  • B. M. R. Calsavara
    • 2
  • E. Fernández-Cara
    • 3
  1. 1.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil
  2. 2.Instituto de Matemática, Estatística e Computação CientíficaUniversidade Estadual de CampinasCampinasBrazil
  3. 3.Dpto. EDAN and IMUSUniversity of SevillaSevillaSpain

Personalised recommendations