Distributed optimal control problems for phase field systems with singular potential

[1] F.D. Araruna, J.L. Boldrini, B.M.R. Calsavara, Optimal control and controllability of a phase field system with one control force. Appl. Math. Optim. 70 (2014), 539-563.10.1007/s00245-014-9249-1Search in Google Scholar

[2] L'. Baňas, M. Klein, A. Prohl, Control of interface evolution in multiphase uid ows. SIAM J. Control Optim. 52 (2014), 2284-2318.10.1137/120896530Search in Google Scholar

[3] V. Barbu, M.L. Bernardi, P. Colli, G. Gilardi, Optimal control problems of phase relaxation models, J. Optim. Theory Appl. 109 (2001), 557-585.10.1023/A:1017563604922Search in Google Scholar

[4] V. Barbu, P. Colli, G. Gilardi, G. Marinoschi, Feedback stabilization of the Cahn-Hilliard type system for phase separation, J. Differential Equations 262 (2017), 2286-233410.1016/j.jde.2016.10.047Search in Google Scholar

[5] V. Barbu, P. Colli, G. Gilardi, G. Marinoschi, E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim, to appear in 2017 (see also preprint arXiv:1506.01665 [math.AP] (2015), pp. 1-28).Search in Google Scholar

[6] T. Benincasa, L.D. Donado Escobar, C. Moroșanu, Distributed and boundary optimal control of the Allen-Cahn equation with regular potential and dynamic boundary conditions. Internat. J. Control 89 (2016), 1523-1532.10.1080/00207179.2015.1137634Search in Google Scholar

[7] J.F. Blowey, C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. I. Mathematical Analysis, European J. Appl. Math. 2 (1991), 233-280.10.1017/S095679250000053XSearch in Google Scholar

[8] J.F. Blowey, C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical Analysis, European J. Appl. Math. 3 (1992), 147-179.10.1017/S0956792500000759Search in Google Scholar

[9] J.L. Boldrini, B.M.C. Caretta, E. Fernández-Cara, Some optimal control problems for a two-phase field model of solidification, Rev. Mat. Complut. 23 (2010), 49-75.10.1007/s13163-009-0012-0Search in Google Scholar

[10] H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert", North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.Search in Google Scholar

[11] M. Brokate, J. Sprekels, "Hysteresis and Phase Transitions", Springer, New York, 1996.10.1007/978-1-4612-4048-8Search in Google Scholar

[12] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal 92 (1986), 205-245.10.1007/BF00254827Search in Google Scholar

[13] L. Calatroni, P. Colli, Global solution to the Allen{Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal. 79 (2013), 12-27.10.1016/j.na.2012.11.010Search in Google Scholar

[14] L. Cherfils, A. Miranville, S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math. 79 (2011), 561-596.10.1007/s00032-011-0165-4Search in Google Scholar

[15] P. Colli, M.H. Farshbaf-Shaker, J. Sprekels, A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim. 71 (2015), 1-24.10.1007/s00245-014-9250-8Search in Google Scholar

[16] P. Colli, G. Gilardi, G. Marinoschi, A boundary control problem for a possibly singular phase field system with dynamic boundary conditions, J. Math. Anal. Appl. 434 (2016), 432-463.10.1016/j.jmaa.2015.09.011Search in Google Scholar

[17] P. Colli, G. Gilardi, G. Marinoschi, E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields 6 (2016), 95-112.10.3934/mcrf.2016.6.95Search in Google Scholar

[18] P. Colli, , G. Gilardi, P. Podio-Guidugli, J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Contin. Mech. Thermodyn. 24 (2012), 437-459.10.1007/s00161-011-0215-8Search in Google Scholar

[19] P. Colli, G. Gilardi, J. Sprekels, Analysis and optimal boundary control of a nonstandard system of phase field equations, Milan J. Math. 80 (2012), 119-149.10.1007/s00032-012-0181-zSearch in Google Scholar

[20] P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal. 4 (2015), 311-325.10.1515/anona-2015-0035Search in Google Scholar

[21] P. Colli, G. Marinoschi, E. Rocca, Sharp interface control in a Penrose- Fife model, ESAIM Control Optim. Calc. Var. 22 (2016), 473-499.10.1051/cocv/2015014Search in Google Scholar

[22] P. Colli, J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim. 53 (2015), 213-234.10.1137/120902422Search in Google Scholar

[23] A. Damlamian, N. Kenmochi, N. Sato, Subdi erential operator approach to a class of nonlinear systems for Stefan problems with phase relaxation, Nonlinear Anal. 23 (1994), 115-142.10.1016/0362-546X(94)90255-0Search in Google Scholar

[24] K. Deckelnick, C.M. Elliott, V. Styles, Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient. Inverse Problems 32 (2016), 045008, 26 pp.10.1088/0266-5611/32/4/045008Search in Google Scholar

[25] C.M. Elliott, S. Zheng, Global existence and stability of solutions to the phase-field equations, in "Free boundary problems", Internat. Ser. Numer. Math., 95, 46-58, Birkhäuser Verlag, Basel, (1990).10.1007/978-3-0348-7301-7_4Search in Google Scholar

[26] M.H. Farshbaf-Shaker, A penalty approach to optimal control of Allen- Cahn variational inequalities: MPEC-view, Numer. Funct. Anal. Optim. 33 (2012), 1321-1349.10.1080/01630563.2012.672354Search in Google Scholar

[27] M.H. Farshbaf-Shaker, C. Hecht, Optimal control of elastic vector-valued Allen-Cahn variational inequalities, SIAM J. Control Optim. 54 (2016), 129-152.10.1137/130937354Search in Google Scholar

[28] M.H. Farshbaf-Shaker, C. Heinemann, A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25 (2015), 2749-2793.10.1142/S0218202515500608Search in Google Scholar

[29] G. Gilardi, A. Miranville, G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal. 8 (2009), 881-912.10.3934/cpaa.2009.8.881Search in Google Scholar

[30] G. Gilardi, A. Miranville, G. Schimperna, Long-time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chin. Ann. Math. Ser. B 31 (2010), 679-712.10.1007/s11401-010-0602-7Search in Google Scholar

[31] M. Grasselli, A. Miranville, G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst. 28 (2010), 67-98.10.3934/dcds.2010.28.67Search in Google Scholar

[32] M. Grasselli, H. Petzeltová, G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend. 25 (2006), 51-72.10.4171/ZAA/1277Search in Google Scholar

[33] K.-H. Hoffmann, L.S. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optim. 13 (1992), 11-27.10.1080/01630569208816458Search in Google Scholar

[34] K.-H. Hoffmann, N. Kenmochi, M. Kubo, N. Yamazaki, Optimal control problems for models of phase-field type with hysteresis of play operator, Adv. Math. Sci. Appl. 17 (2007), 305-336.Search in Google Scholar

[35] N. Kenmochi, M. Niezgόdka, Evolution systems of nonlinear variational inequalities arising phase change problems, Nonlinear Anal. 22 (1994), 1163-1180.10.1016/0362-546X(94)90235-6Search in Google Scholar

[36] Ph. Laurençot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 167-185.10.1017/S0308210500030663Search in Google Scholar

[37] C. Lefter, J. Sprekels, Optimal boundary control of a phase field system modeling nonisothermal phase transitions, Adv. Math. Sci. Appl. 17 (2007), 181-194.Search in Google Scholar

[38] A. Miranville, S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci. 27 (2004), 545-582.10.1002/mma.464Search in Google Scholar

[39] G. Schimperna, Abstract approach to evolution equations of phase field type and applications, J. Differential Equations 164 (2000), 395-430.10.1006/jdeq.1999.3753Search in Google Scholar

[40] K. Shirakawa, N. Yamazaki, Optimal control problems of phase field system with total variation functional as the interfacial energy, Adv. Differential Equations 18 (2013), 309-350.10.57262/ade/1360073019Search in Google Scholar

[41] J. Sprekels, S. Zheng, Optimal control problems for a thermodynamically consistent model of phase-field type for phase transitions, Adv. Math. Sci. Appl. 1 (1992), 113-125.Search in Google Scholar