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  June 2021 Early access  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 and 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.

[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.

[3]

H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[11]

Y. GigaY. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187–1220. doi: 10.1023/A: 1004570921372.

[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.

[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.

[21]

S. Lang, Analysis I, Addison-Wesley Publishing Company, 1968.

[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.

[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.

[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.

[25]

R. Nakayashiki, Quasilinear type Kobayaski-Warren-Carter system including dynamic boundary condition, Adv. Math. Sci. Appl., 27 (2018), 403–437.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

H. Watanabe and K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system, Math. Bohem., 139 (2014), 381–389.

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.

[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.

[3]

H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[11]

Y. GigaY. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187–1220. doi: 10.1023/A: 1004570921372.

[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.

[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.

[21]

S. Lang, Analysis I, Addison-Wesley Publishing Company, 1968.

[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.

[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.

[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.

[25]

R. Nakayashiki, Quasilinear type Kobayaski-Warren-Carter system including dynamic boundary condition, Adv. Math. Sci. Appl., 27 (2018), 403–437.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

H. Watanabe and K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system, Math. Bohem., 139 (2014), 381–389.

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