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

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)

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)

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)

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

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)

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

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)

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)

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

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)

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)

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)

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)

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)

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)

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

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

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)

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

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)

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

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)

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)

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

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

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)

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)

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)

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)

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

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

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

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

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)

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

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)

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

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

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)

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)

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

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)

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

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

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

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)

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)

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)

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

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)

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

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

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)

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

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)