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

Applied Mathematics & Optimization volume 70, pages539–563(2014)Cite this article

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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References

  1. 1.

    Benincasa, T., Moroşanu, C.: Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy–Neumann boundary conditions. Numer. Funct. Anal. Optim. 30(3–4), 199–213 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Benincasa, T., Favini, A., Moroşanu, C.: A product formula approach to a nonhomogeneous boundary optimal control problem governed by nonlinear phase-field transition system. Part I: a phase-field model. J. Optim. Theory Appl. 148(1), 14–30 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Benincasa, T., Favini, A., Moroşanu, C.: A product formula approach to a nonhomogeneous boundary optimal control problem governed by nonlinear phase-field transition system. Part II: Lie–Trotter product formula. J. Optim. Theory Appl. 148(1), 14–30 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Moroşanu, C.: Boundary optimal control problem for the phase-field transition system using fractional steps method. Control Cybern. 32(1), 5–32 (2003)

    MATH  Google Scholar 

  5. 5.

    Moroşanu, C.: The phase-field transition system with non-homogeneous Cauchy–Stefan–Boltzmann and homogeneous Neumann boundary conditions and non-constant thermal conductivity. Nonlinear Anal. 87, 22–32 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cheng, M., Warren, J.A.: An efficient algorithm for solving the phase field crystal model. J. Comput. Phys. 227(12), 6241–6248 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Hamide, M., Massoni, E., Bellet, M.: Adaptive mesh technique for thermal–metallurgical numerical simulation of arc welding processes. Int. J. Numer. Methods Eng. 73(5), 624–641 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    He, Q., Kasagi, N.: Phase-field simulation of small capillary-number two-phase flow in a microtube. Fluid Dyn. Res. 40(7–8), 497–509 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Moroşanu, C., Wang, G.: State-constrained optimal control for the phase-field transition system. Numer. Funct. Anal. Optim. 28(3–4), 379–403 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Rosam, J., Jimack, P.K., Mullis, A.A.: Fully implicit, fully adaptive time and space discretization method for phase-field simulation of binary alloy solidification. J. Comput. Phys. 225(2), 1271–1287 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Sun, Y., Beckermann, C.: Phase-Field Simulation of Two-Phase Micro-flows in a Hele-Shaw Cell, Computational Methods in Multiphase Flow III, WIT Trans. Eng. Sci., vol. 50. WIT Press, Southampton (2005)

    Google Scholar 

  12. 12.

    Tan, Z., Huang, Y.: An alternating Crank–Nicolson method for the numerical solution of the phase-field equations using adaptive moving meshes. Int. J. Numer. Methods Fluids 56(9), 1673–1693 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Zhao, P., Heinrich, J.C., Poirier, D.R.: Dendritic solidification of binary alloys with free and forced convection. Int. J. Numer. Methods Fluids 49(3), 233–266 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Ahmad, N.A., Wheeler, A.A., Boettinger, W.J., Mcfadden, G.B.: Solute trapping and solute drag in a phase-field model of rapid solidification. Phys. Rev. E Stat. Phys. Plasmas Fluids 58(3B), 3436–3450 (1998)

    Article  Google Scholar 

  15. 15.

    Boldrini, J.L., Vaz, C.L.D.: Existence and regularity of solutions of a phase field model for solidification with convection of pure materials in two dimensions. Electron. J. Differ. Equ. 109, 1–25 (2003)

    MathSciNet  Google Scholar 

  16. 16.

    Caginalp, G.: An analysis of phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–245 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Caginalp, G.: Stefan and Hele-Shaw type models as assymptotic limits of the phase-field equations. Phys. Rev. A 39(11), 5887–5896 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    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)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Caginalp, G., Jones, J.: A derivation and analysis of phase field models of thermal alloys. Annal. Phys. 237, 66–107 (1995)

    Article  Google Scholar 

  20. 20.

    Cherfils, L., Gatti, S., Miranville, A.: Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials. J. Math. Anal. Appl. 343(1), 557–566 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Colli, P., Grasselli, M., Ito, A.: On a parabolic–hyperbolic Penrose–Fife phase-field system. Electron. J. Differ. Equ. 100, 1–30 (2002)

    Google Scholar 

  22. 22.

    Gilardi, G., Marson, A.: On a Penrose–Fife type system with Dirichlet boundary conditions for temperature. Math. Methods Appl. Sci. 26(15), 1303–1325 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Gilardi, G., Rocca, E.: Convergence of phase field to phase relaxation models governed by an entropy equation with memory. Math. Methods Appl. Sci. 29(18), 2149–2179 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Jiménez-Casas, A.: Invariant regions and global existence for a phase field model. Discret. Contin. Dyn. Syst. Ser. S 1(2), 273–281 (2008)

    Article  MATH  Google Scholar 

  25. 25.

    Karma, A.: Phase-field models of microstructural pattern formation. Thermodyn. Microstruct. Plast. NATO Sci. Ser. II Math. Phys. Chem. 108, 65–89 (2003)

    MathSciNet  Google Scholar 

  26. 26.

    Krejcí, P., Rocca, E., Sprekels, J.: Non-local temperature-dependent phase-field models for non-isothermal phase transitions. J. Lond. Math. Soc. 76(1), 197–210 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Krejcí, P., Sprekels, J., Stefanelli, U.: One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions. Adv. Math. Sci. Appl. 13(2), 695–712 (2003)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Laurençot, P., Schimperna, G., Stefanelli, U.: Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions. J. Math. Anal. Appl. 271(2), 426–442 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    McFadden, G.B., Wheeler, A.A., Anderson, D.M.: Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities. Phys. D 144(1—-2), 154–168 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Moroşanu, C.: Analysis and optimal control of phase-field transition system. Nonlinear Funct. Anal. Appl. 8(3), 433–460 (2003)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Nestler, B., Garcke, H., Stinner, B.: Multicomponent alloy solidification: phase-field modeling and simulations. Phys. Rev. E 71(4), 1–6 (2005)

    Article  Google Scholar 

  32. 32.

    Penrose, O., Fife, P.C.: Thermodynamically consistent models of phase field type for the kinetics of phase transitions. Phys. D 43, 44–62 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Planas, G.: Existence of solutions to a phase-field model with phase-dependent heat absorption. Electron. J. Differ. Equ. 28, 1–12 (2007)

    MathSciNet  Google Scholar 

  34. 34.

    Stinner, B.: Weak solutions to a multi-phase field system of parabolic equations related to alloy solidification. Adv. Math. Sci. Appl. 17(2), 589–638 (2007)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Aizicovici, S., Feireisl, E.: Long-time stabilization of solutions to a phase-field model with memory. J. Evol. Equ. 1, 69–84 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Aizicovici, S., Feireisl, E., Issard-Roch, F.: Long time convergence of solutions to a phase-field system. Math. Methods Appl. Sci. 24, 277–287 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Bates, P.W., Zheng, S.: Inertial manifolds and inertial sets for phase-field equations. J. Dyn. Differ. Equ. 4, 375–397 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Brochet, D., Chen, X., Hilhorst, D.: Finite dimensional exponential attractor for the phase-field model. Appl. Anal. 49, 197–212 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Jiang, J.: Convergence to equilibrium for a parabolic–hyperbolic phase field model with Cattaneo heat flux law. J. Math. Anal. Appl. 341(1), 149–169 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Kapustyan, A.V., Melnik, V.S., Valero, J.: Attractors of multivalued dynamical processes generated by phase-field equations. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13(7), 1969–1983 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Röger, M., Tonegawa, Y.: Convergence of phase-field approximations to the Gibbs–Thomson law. Calc. Var. Partial Differ. Equ. 32(1), 111–136 (2008)

    Article  MATH  Google Scholar 

  42. 42.

    Sprekels, J., Zheng, S.: Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions. J. Math. Anal. Appl. 279(1), 97–110 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Hoffman, K., Jiang, L.: Optimal control of a phase field model for solidification. Numer. Funct. Anal. Optim. 13, 11–27 (1992)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Wang, L., Wang, G.: The optimal time control of a phase-field system. SIAM J. Control Optim. 42(4), 1483–1508 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Barbu, V.: Local controllability of the phase field system. Nonlinear Anal. 50(3), 363–372 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    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)

    MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    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)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Cao, Y.: Numerical approximations of exact controllability problems by optimal control problems for parabolic differential equations. Appl. Math. Comput. 119(2–3), 127–145 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

    Adams, R.: Sobolev Spaces. Academic Press, New York (1975)

    Google Scholar 

  50. 50.

    Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)

    Google Scholar 

  51. 51.

    Mikhaylov, V.P.: Partial Differential Equations. Mir Publishers, Moscow (1978)

    Google Scholar 

  52. 52.

    Lions, J.L.: Control of Distributed Singular Systems. Gauthier-Villars, Paris (1985)

    Google Scholar 

  53. 53.

    Chae, D., Imanuvilov, O.Y., Kim, S.M.: Exact controllability for semilinear parabolic equations with Neumann boundary conditions. J. Dyn. Control Syst. 2(4), 449–483 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  54. 54.

    Fabre, C., Puel, J.-P., Zuazua, E.: Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb. 125(1), 31–61 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  55. 55.

    Doubova, A., Fernández-Cara, E., González-Burgos, M., Zuazua, E.: On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41(3), 798–819 (2002)

    MathSciNet  Article  MATH  Google Scholar 

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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).

Author information

Affiliations

  1. Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 , João Pessoa, PB, Brazil

    F. D. Araruna

  2. Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, 13083-970 , Campinas, SP, Brazil

    J. L. Boldrini

  3. Faculdade de Ciências Aplicadas, Universidade Estadual de Campinas, R. Pedro Zaccaria, 1300, CEP 13484-350 , Limeira, SP, Brazil

    B. M. R. Calsavara

Authors
  1. F. D. Araruna
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  2. J. L. Boldrini
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  3. B. M. R. Calsavara
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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