# American Institute of Mathematical Sciences

June  2021, 11(2): 253-289. doi: 10.3934/mcrf.2020036

## Optimal control problems governed by 1-D Kobayashi–Warren–Carter type systems

 1 Department of Mathematical Sciences and, the Center for Mathematics and Artificial Intelligence, (CMAI), George Mason University, Fairfax, VA 22030, USA 2 Department of Mathematics and Informatics, Graduate School of Science and Engineering, Chiba University, 1-33, Yayoi-cho, Inage-ku, 263-8522, Chiba, Japan 3 Department of Mathematics, Faculty of Education, Chiba University, 1-33, Yayoi-cho, Inage-ku, 263-8522, Chiba, Japan 4 Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1, Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686, Japan

* Corresponding author: Shodai Kubota

Received  March 2020 Revised  July 2020 Published  August 2020

Fund Project: The work of the third author supported by Grant-in-Aid for Scientific Research (C) No. 16K05224 and No. 20K03672, JSPS. The work of the forth author supported by Grant-in-Aid for Scientific Research (C) No. 20K03665, JSPS. In addition, the work of the first and the third authors is partially supported by the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-19-1-0036 and NSF grants DMS-1818772 and DMS-1913004

This paper is devoted to the study of a class of optimal control problems governed by 1–D Kobayashi–Warren–Carter type systems, which are based on a phase-field model of grain boundary motion, proposed by [Kobayashi et al, Physica D, 140, 141–150, 2000]. The class consists of an optimal control problem for a physically realistic state-system of Kobayashi–Warren–Carter type, and its regularized approximating problems. The results of this paper are stated in three Main Theorems 1–3. The first Main Theorem 1 is concerned with the solvability and continuous dependence for the state-systems. Meanwhile, the second Main Theorem 2 is concerned with the solvability of optimal control problems, and some semi-continuous association in the class of our optimal control problems. Finally, in the third Main Theorem 3, we derive the first order necessary optimality conditions for optimal controls of the regularized approximating problems. By taking the approximating limit, we also derive the optimality conditions for the optimal controls for the physically realistic problem.

Citation: Harbir Antil, Shodai Kubota, Ken Shirakawa, Noriaki Yamazaki. Optimal control problems governed by 1-D Kobayashi–Warren–Carter type systems. Mathematical Control & Related Fields, 2021, 11 (2) : 253-289. doi: 10.3934/mcrf.2020036
##### References:
 [1] F. Andreu-Vaillo, V. Caselles and J. M Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Vol. 223 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7928-6.  Google Scholar [2] H. Antil, K. Shirakawa and N. Yamazaki, A class of parabolic systems associated with optimal controls of grain boundary motions, Adv. Math. Sci. Appl., 27 (2018), 299–336.  Google Scholar [3] H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar [4] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics. Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar [5] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).  Google Scholar [6] V. Caselles, A. Chambolle, S. Moll and M. Novaga, A characterization of convex calibrable sets in $\Bbb R^N$ with respect to anisotropic norms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 803–832. doi: 10.1016/j.anihpc.2008.04.003.  Google Scholar [7] P. Colli, G. Gilardi, R. Nakayashiki and K. Shirakawa, A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions, Nonlinear Anal., 158 (2017), 32–59. doi: 10.1016/j.na.2017.03.020.  Google Scholar [8] E. Emmrich, Discrete versions of gronwall's lemma and their application to the numerical analysis of parabolic problems, Technical Report 637, Institute of Mathematics, Technische Universität Berlin, "http://www3.math.tu-berlin.de/preprints/files/Preprint-637-1999.pdf", 1999. Google Scholar [9] M.-H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Jpn. J. Ind. Appl. Math., 27 (2010), 323–345. doi: 10.1007/s13160-010-0020-y.  Google Scholar [10] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, In Taniguchi Conference on Mathematics Nara '98, Vol. 31 of Adv. Stud. Pure Math., pp. 93–125. Math. Soc. Japan, Tokyo, 2001. doi: 10.2969/aspm/03110093.  Google Scholar [11] Y. Giga, Y. Kashima and N. Yamazaki, Local solvability of a constrained gradient system of total variation, Abstr. Appl. Anal., 8 (2004), 651-682.  doi: 10.1155/S1085337504311048.  Google Scholar [12] R. H. W. Hoppe and J. J. Winkle, A splitting scheme for the numerical solution of the KWC system, Numer. Math. Theory Methods Appl., 12 (2019), 661–680. doi: 10.4208/nmtma.OA-2018-0050.  Google Scholar [13] A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion, Appl. Math., 53 (2008), 433–454. doi: 10.1007/s10492-008-0035-8.  Google Scholar [14] A. Ito, N. Kenmochi and N. Yamazaki, Weak solutions of grain boundary motion model with singularity, Rend. Mat. Appl. (7), 29 (2009), 51–63.  Google Scholar [15] A. Ito, N. Kenmochi and N. Yamazaki, Global solvability of a model for grain boundary motion with constraint, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 127–146. doi: 10.3934/dcdss.2012.5.127.  Google Scholar [16] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., (https://opac.ll.chiba-u.jp/da/curator/900025033/KJ00004299282.pdf), 30 (1981), 1–87. Google Scholar [17] N. Kenmochi and N. Yamazaki, Large-time behavior of solutions to a phase-field model of grain boundary motion with constraint, In Current advances in nonlinear analysis and related topics, Vol. 32 of GAKUTO Internat. Ser. Math. Sci. Appl., pp. 389–403. Gakkōtosho, Tokyo, 2010.  Google Scholar [18] R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187–1220. doi: 10.1023/A: 1004570921372.  Google Scholar [19] R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundaries, Phys. D, 140 (2000), 141–150. doi: 10.1016/S0167-2789(00)00023-3.  Google Scholar [20] R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity, In Free Boundary Problems: Theory and Applications, II (Chiba, 1999), Vol. 14 of GAKUTO Internat. Ser. Math. Sci. Appl., pp. 283–294. Gakkōtosho, Tokyo, 2000.  Google Scholar [21] S. Lang, Analysis I, Addison-Wesley Publishing Company, 1968. Google Scholar [22] S. Moll and K. Shirakawa, Existence of solutions to the Kobayashi–Warren–Carter system, Calc. Var. Partial Differential Equations, 51 (2014), 621–656. doi: 10.1007/s00526-013-0689-2.  Google Scholar [23] S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the Kobayashi–Warren–Carter system, Nonlinearity, 30 (2017), 2752–2784. doi: 10.1088/1361-6544/aa6eb4.  Google Scholar [24] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510–585. doi: 10.1016/0001-8708(69)90009-7.  Google Scholar [25] R. Nakayashiki, Quasilinear type Kobayaski-Warren-Carter system including dynamic boundary condition, Adv. Math. Sci. Appl., 27 (2018), 403–437.  Google Scholar [26] T. Ohtsuka, K. Shirakawa and N. Yamazaki, Optimal control problem for Allen-Cahn type equation associated with total variation energy, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 159–181. doi: 10.3934/dcdss.2012.5.159.  Google Scholar [27] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Pure and Applied Mathematics, Vol. 57.  Google Scholar [28] K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, In Dissipative Phase Transitions, Vol. 71 of Ser. Adv. Math. Appl. Sci., pp. 269–288. World Sci. Publ., Hackensack, NJ, 2006. doi: 10.1142/9789812774293_0014.  Google Scholar [29] K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257–282. doi: 10.1016/j.na.2004.08.030.  Google Scholar [30] K. Shirakawa and H. Watanabe, Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 139–159. doi: 10.3934/dcdss.2014.7.139.  Google Scholar [31] K. Shirakawa and H. Watanabe, Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint, Discrete Contin. Dyn. Syst., (Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl.), 2015, 1009–1018. doi: 10.3934/proc.2015.1009.  Google Scholar [32] K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of one-dimensional phase field systems associated with grain boundary motion, Math. Ann., 356 (2013), 301–330. doi: 10.1007/s00208-012-0849-2.  Google Scholar [33] K. Shirakawa, H. Watanabe and N. Yamazaki, Phase-field systems for grain boundary motions under isothermal solidifications, Adv. Math. Sci. Appl., 24 (2014), 353–400.  Google Scholar [34] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65–96. doi: 10.1007/BF01762360.  Google Scholar [35] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Vol. 32 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. With a foreword by A. Pazy.  Google Scholar [36] H. Watanabe and K. Shirakawa, Qualitative properties of a one-dimensional phase-field system associated with grain boundary, In Nonlinear Analysis in Interdisciplinary Sciences–-Modellings, Theory and Simulations, Vol. 36 of GAKUTO Internat. Ser. Math. Sci. Appl., pp. 301–328. Gakkōtosho, Tokyo, 2013.  Google Scholar [37] H. Watanabe and K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system, Math. Bohem., 139 (2014), 381–389.  Google Scholar

show all references

##### References:
 [1] F. Andreu-Vaillo, V. Caselles and J. M Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Vol. 223 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7928-6.  Google Scholar [2] H. Antil, K. Shirakawa and N. Yamazaki, A class of parabolic systems associated with optimal controls of grain boundary motions, Adv. Math. Sci. Appl., 27 (2018), 299–336.  Google Scholar [3] H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar [4] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics. Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar [5] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).  Google Scholar [6] V. Caselles, A. Chambolle, S. Moll and M. Novaga, A characterization of convex calibrable sets in $\Bbb R^N$ with respect to anisotropic norms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 803–832. doi: 10.1016/j.anihpc.2008.04.003.  Google Scholar [7] P. Colli, G. Gilardi, R. Nakayashiki and K. Shirakawa, A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions, Nonlinear Anal., 158 (2017), 32–59. doi: 10.1016/j.na.2017.03.020.  Google Scholar [8] E. Emmrich, Discrete versions of gronwall's lemma and their application to the numerical analysis of parabolic problems, Technical Report 637, Institute of Mathematics, Technische Universität Berlin, "http://www3.math.tu-berlin.de/preprints/files/Preprint-637-1999.pdf", 1999. Google Scholar [9] M.-H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Jpn. J. Ind. Appl. Math., 27 (2010), 323–345. doi: 10.1007/s13160-010-0020-y.  Google Scholar [10] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, In Taniguchi Conference on Mathematics Nara '98, Vol. 31 of Adv. Stud. Pure Math., pp. 93–125. Math. Soc. Japan, Tokyo, 2001. doi: 10.2969/aspm/03110093.  Google Scholar [11] Y. Giga, Y. Kashima and N. Yamazaki, Local solvability of a constrained gradient system of total variation, Abstr. Appl. Anal., 8 (2004), 651-682.  doi: 10.1155/S1085337504311048.  Google Scholar [12] R. H. W. Hoppe and J. J. Winkle, A splitting scheme for the numerical solution of the KWC system, Numer. Math. Theory Methods Appl., 12 (2019), 661–680. doi: 10.4208/nmtma.OA-2018-0050.  Google Scholar [13] A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion, Appl. Math., 53 (2008), 433–454. doi: 10.1007/s10492-008-0035-8.  Google Scholar [14] A. Ito, N. Kenmochi and N. Yamazaki, Weak solutions of grain boundary motion model with singularity, Rend. Mat. Appl. (7), 29 (2009), 51–63.  Google Scholar [15] A. Ito, N. Kenmochi and N. Yamazaki, Global solvability of a model for grain boundary motion with constraint, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 127–146. doi: 10.3934/dcdss.2012.5.127.  Google Scholar [16] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., (https://opac.ll.chiba-u.jp/da/curator/900025033/KJ00004299282.pdf), 30 (1981), 1–87. Google Scholar [17] N. Kenmochi and N. Yamazaki, Large-time behavior of solutions to a phase-field model of grain boundary motion with constraint, In Current advances in nonlinear analysis and related topics, Vol. 32 of GAKUTO Internat. Ser. Math. Sci. Appl., pp. 389–403. Gakkōtosho, Tokyo, 2010.  Google Scholar [18] R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187–1220. doi: 10.1023/A: 1004570921372.  Google Scholar [19] R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundaries, Phys. D, 140 (2000), 141–150. doi: 10.1016/S0167-2789(00)00023-3.  Google Scholar [20] R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity, In Free Boundary Problems: Theory and Applications, II (Chiba, 1999), Vol. 14 of GAKUTO Internat. Ser. Math. Sci. Appl., pp. 283–294. Gakkōtosho, Tokyo, 2000.  Google Scholar [21] S. Lang, Analysis I, Addison-Wesley Publishing Company, 1968. Google Scholar [22] S. Moll and K. Shirakawa, Existence of solutions to the Kobayashi–Warren–Carter system, Calc. Var. Partial Differential Equations, 51 (2014), 621–656. doi: 10.1007/s00526-013-0689-2.  Google Scholar [23] S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the Kobayashi–Warren–Carter system, Nonlinearity, 30 (2017), 2752–2784. doi: 10.1088/1361-6544/aa6eb4.  Google Scholar [24] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510–585. doi: 10.1016/0001-8708(69)90009-7.  Google Scholar [25] R. Nakayashiki, Quasilinear type Kobayaski-Warren-Carter system including dynamic boundary condition, Adv. Math. Sci. Appl., 27 (2018), 403–437.  Google Scholar [26] T. Ohtsuka, K. Shirakawa and N. Yamazaki, Optimal control problem for Allen-Cahn type equation associated with total variation energy, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 159–181. doi: 10.3934/dcdss.2012.5.159.  Google Scholar [27] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Pure and Applied Mathematics, Vol. 57.  Google Scholar [28] K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, In Dissipative Phase Transitions, Vol. 71 of Ser. Adv. Math. Appl. Sci., pp. 269–288. World Sci. Publ., Hackensack, NJ, 2006. doi: 10.1142/9789812774293_0014.  Google Scholar [29] K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257–282. doi: 10.1016/j.na.2004.08.030.  Google Scholar [30] K. Shirakawa and H. Watanabe, Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 139–159. doi: 10.3934/dcdss.2014.7.139.  Google Scholar [31] K. Shirakawa and H. Watanabe, Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint, Discrete Contin. Dyn. Syst., (Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl.), 2015, 1009–1018. doi: 10.3934/proc.2015.1009.  Google Scholar [32] K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of one-dimensional phase field systems associated with grain boundary motion, Math. Ann., 356 (2013), 301–330. doi: 10.1007/s00208-012-0849-2.  Google Scholar [33] K. Shirakawa, H. Watanabe and N. Yamazaki, Phase-field systems for grain boundary motions under isothermal solidifications, Adv. Math. Sci. Appl., 24 (2014), 353–400.  Google Scholar [34] J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65–96. doi: 10.1007/BF01762360.  Google Scholar [35] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Vol. 32 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. With a foreword by A. Pazy.  Google Scholar [36] H. Watanabe and K. Shirakawa, Qualitative properties of a one-dimensional phase-field system associated with grain boundary, In Nonlinear Analysis in Interdisciplinary Sciences–-Modellings, Theory and Simulations, Vol. 36 of GAKUTO Internat. Ser. Math. Sci. Appl., pp. 301–328. Gakkōtosho, Tokyo, 2013.  Google Scholar [37] H. Watanabe and K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system, Math. Bohem., 139 (2014), 381–389.  Google Scholar
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