# American Institute of Mathematical Sciences

October  2014, 34(10): 3985-4017. doi: 10.3934/dcds.2014.34.3985

## Analysis and optimal control of some solidification processes

 1 Departamento de Ciencias Básicas, Universidad del Bío-Bío, Campus Fernando May, Casilla 447, Chillán, Chile 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain

Received  October 2012 Revised  January 2013 Published  April 2014

In this paper we consider a mathematical model that describes the solidification of a binary alloy. We prove some existence and uniqueness results for a regularized problem, depending on a small parameter $\epsilon$. We also analyze the behavior of the regularized solutions as $\epsilon \to 0$. Then, we consider some associated optimal control problems. We prove existence and optimality results and we present and discuss some iterative methods.
Citation: Roberto C. Cabrales, Gema Camacho, Enrique Fernández-Cara. Analysis and optimal control of some solidification processes. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3985-4017. doi: 10.3934/dcds.2014.34.3985
##### References:
 [1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. [2] W. J. Boettinger, J. A. Warren, C. Beckermann and A. Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002), 163-194. doi: 10.1007/BF02648953. [3] J. Ni and C. Beckermann, A volume-averaged two-phase model for solidication transportphenomena, Metallurgical Transactions B, 22 (1991), 349-361. [4] W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid-Liquid Phase Change Systems: I. Model Formulation, Int. J. Heat Mass Transfer, 30 (1987), 2161-2170. doi: 10.1016/0017-9310(87)90094-9. [5] W. D. Bennon and F. P. Incropera, A Continuum 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 Transfer, 30 (1987), 2171-2187. doi: 10.1016/0017-9310(87)90095-0. [6] Ph. Blanc, L. Gasser and J. Rappaz, Existence for a stationary model of binary alloy solidification, RAIRO Modél. Math. Anal. Numér., 29 (1995), 687-699. [7] J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Contin. Dyn. Syst., 13 (2005), 429-450. doi: 10.3934/dcds.2005.13.429. [8] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. [9] M. Durán, E. Ortega-Torres and J. Rappaz, Weak solution of a stationary convection-diffusion model describing binary alloy solidification processes, Math. Comput. Modelling, 42 (2005), 1269-1286. doi: 10.1016/j.mcm.2005.01.035. [10] M. Durán and E. Ortega-Torres, Existence of weak solutions for a non-homogeneous solidification model, Electron. J. Differential Equations, 2006 17 pp. [11] I. Ekeland and R. Temam, Convex analysis and Variational Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1999. doi: 10.1137/1.9781611971088. [12] J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595. [13] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967. [14] G. Planas and J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. Appl., 303 (2005), 669-687. doi: 10.1016/j.jmaa.2004.08.068. [15] J. C. Ramirez and C. Beckermann, Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Materialia, 53 (2005), 1721-1736. doi: 10.1016/j.actamat.2004.12.021. [16] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differ. Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X. [17] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 64-96. doi: 10.1007/BF01762360. [18] J. Simon, A constructive proof of a theorem of G. de Rham, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1167-1172. [19] R. Temam, Navier Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. [2] W. J. Boettinger, J. A. Warren, C. Beckermann and A. Karma, Phase-field simulation of solidification, Annual Review of Materials Research, 32 (2002), 163-194. doi: 10.1007/BF02648953. [3] J. Ni and C. Beckermann, A volume-averaged two-phase model for solidication transportphenomena, Metallurgical Transactions B, 22 (1991), 349-361. [4] W. D. Bennon and F. P. Incropera, A Continuum Model for Momentum, Heat and Species Transport in Binary Solid-Liquid Phase Change Systems: I. Model Formulation, Int. J. Heat Mass Transfer, 30 (1987), 2161-2170. doi: 10.1016/0017-9310(87)90094-9. [5] W. D. Bennon and F. P. Incropera, A Continuum 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 Transfer, 30 (1987), 2171-2187. doi: 10.1016/0017-9310(87)90095-0. [6] Ph. Blanc, L. Gasser and J. Rappaz, Existence for a stationary model of binary alloy solidification, RAIRO Modél. Math. Anal. Numér., 29 (1995), 687-699. [7] J. L. Boldrini and G. Planas, A tridimensional phase-field model with convection for phase change of an alloy, Discrete Contin. Dyn. Syst., 13 (2005), 429-450. doi: 10.3934/dcds.2005.13.429. [8] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. [9] M. Durán, E. Ortega-Torres and J. Rappaz, Weak solution of a stationary convection-diffusion model describing binary alloy solidification processes, Math. Comput. Modelling, 42 (2005), 1269-1286. doi: 10.1016/j.mcm.2005.01.035. [10] M. Durán and E. Ortega-Torres, Existence of weak solutions for a non-homogeneous solidification model, Electron. J. Differential Equations, 2006 17 pp. [11] I. Ekeland and R. Temam, Convex analysis and Variational Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1999. doi: 10.1137/1.9781611971088. [12] J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595. [13] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967. [14] G. Planas and J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. Appl., 303 (2005), 669-687. doi: 10.1016/j.jmaa.2004.08.068. [15] J. C. Ramirez and C. Beckermann, Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Materialia, 53 (2005), 1721-1736. doi: 10.1016/j.actamat.2004.12.021. [16] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differ. Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X. [17] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 64-96. doi: 10.1007/BF01762360. [18] J. Simon, A constructive proof of a theorem of G. de Rham, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1167-1172. [19] R. Temam, Navier Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.
 [1] Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305 [2] Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control and Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183 [3] N. U. Ahmed. Weak solutions of stochastic reaction diffusion equations and their optimal control. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1011-1029. doi: 10.3934/dcdss.2018059 [4] Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 383-392. doi: 10.3934/dcdss.2016002 [5] Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control and Related Fields, 2020, 10 (1) : 113-140. doi: 10.3934/mcrf.2019032 [6] Harbir Antil, Ciprian G. Gal, Mahamadi Warma. A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1883-1918. doi: 10.3934/dcdss.2022012 [7] Lorena Bociu, Barbara Kaltenbacher, Petronela Radu. Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications. Evolution Equations and Control Theory, 2013, 2 (2) : i-ii. doi: 10.3934/eect.2013.2.2i [8] Roberto Castelli, Susanna Terracini. On the regularization of the collision solutions of the one-center problem with weak forces. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1197-1218. doi: 10.3934/dcds.2011.31.1197 [9] Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777 [10] Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 [11] Rainer Buckdahn, Christian Keller, Jin Ma, Jianfeng Zhang. Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 7-. doi: 10.1186/s41546-020-00049-8 [12] Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems and Imaging, 2021, 15 (4) : 599-618. doi: 10.3934/ipi.2021006 [13] Claus Kirchner, Michael Herty, Simone Göttlich, Axel Klar. Optimal control for continuous supply network models. Networks and Heterogeneous Media, 2006, 1 (4) : 675-688. doi: 10.3934/nhm.2006.1.675 [14] Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 [15] Alexander Komech. Attractors of Hamilton nonlinear PDEs. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6201-6256. doi: 10.3934/dcds.2016071 [16] Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial and Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275 [17] Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 [18] Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control and Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016 [19] Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control and Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013 [20] Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897

2020 Impact Factor: 1.392