February  2012, 5(1): 159-181. doi: 10.3934/dcdss.2012.5.159

Optimal control problem for Allen-Cahn type equation associated with total variation energy

1. 

Division of Mathematical Sciences, Graduate School of Engineering, Gunma University, 4-2 Aramaki-cho, Maebashi, 371-8510, Japan

2. 

Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

3. 

Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, 221-8686

Received  March 2009 Revised  December 2009 Published  February 2011

In this paper we study an optimal control problem for a singular diffusion equation associated with total variation energy. The singular diffusion equation is derived as an Allen-Cahn type equation, and then the observing optimal control problem corresponds to a temperature control problem in the solid-liquid phase transition. We show the existence of an optimal control for our singular diffusion equation by applying the abstract theory. Next we consider our optimal control problem from the view-point of numerical analysis. In fact we consider the approximating problem of our equation, and we show the relationship between the original control problem and its approximating one. Moreover we show the necessary condition of an approximating optimal pair, and give a numerical experiment of our approximating control problem.
Citation: Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159
References:
[1]

F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential and Integral Equations, 14 (2001), 321-360.

[2]

F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The Dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), 347-403. doi: 10.1006/jfan.2000.3698.

[3]

H. Attouch, "Variational Convergence for Functions and Operators," Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984.

[4]

G. Bellettini, V. Caselles and M. Novaga, The total variation flow in RN J. Differential Equations, 184 (2002), 475-525. doi: 10.1006/jdeq.2001.4150.

[5]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973.

[6]

E. Casas, L. A. Fernández and J. Yong, Optimal control of quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 545-565.

[7]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[8]

L. A. Fernández, Integral state-constrained optimal control problems for some quasilinear parabolic equations, Nonlinear Anal., 39 (2000), 977-996. doi: 10.1016/S0362-546X(98)00264-8.

[9]

M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, Proc. Taniguchi Conf. on Math., Advanced Studies in Pure Math., 31 (2001), 93-125.

[10]

Y. Giga, Y. Kashima and N. Yamazaki, Local solvability of a constrained gradient system of total variation, Abstr. Appl. Anal., 2004 (2004), 651-682. doi: 10.1155/S1085337504311048.

[11]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.

[12]

N. Kenmochi and K. Shirakawa, Stability for a parabolic variational inequality associated with total variation functional, Funkcial. Ekvac., 44 (2001), 119-137.

[13]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967.

[14]

U. Mosco, Convergence of convex sets and of solutions variational inequalities, Advances Math., 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.

[15]

T. Ohtsuka, Numerical simulations for optimal controls of an Allen-Cahn type equation with constraint, in "Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation-Mathematical Analysis, Modelling and Simulation," GAKUTO Intern. Ser. Math. Appl., vol. 29, Gakkotosho, Tokyo, (2008), 329-339.

[16]

T. Ohtsuka, K. Shirakawa and N. Yamazaki, Optimal control problems of singular diffusion equation with constraint, Adv. Math. Sci. Appl., 18 (2008), 1-28.

[17]

T. Ohtsuka, K. Shirakawa and N. Yamazaki, Convergence of numerical algorithm for optimal control problem of Allen-Cahn type equation with constraint, in "Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation-Mathematical Analysis, Modelling and Simulation," GAKUTO Intern. Ser. Math. Appl., vol 29, Gakkotosho, Tokyo, (2008), 441-462.

[18]

K. Shirakawa, Asymptotic convergence of $p$-Laplace equations with constraint as $p$ tends to 1, Math. Methods Appl. Sci., 25 (2002), 771-793. doi: 10.1002/mma.314.

[19]

K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials, in "Recent Developments in Domain Decomposition Methods and Flow Problems," GAKUTO Internat. Ser. Math. Sci. Appl., vol 11, Gakkōtosho, Tokyo, (1998), 287-310.

[20]

K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257-282.

[21]

N. Yamazaki, Optimal control of nonlinear evolution equations associated with time-dependent subdifferentials and applications, in "Nonlocal and Abstract Parabolic Equations and Their Applications," Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 313-327.

show all references

References:
[1]

F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential and Integral Equations, 14 (2001), 321-360.

[2]

F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The Dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), 347-403. doi: 10.1006/jfan.2000.3698.

[3]

H. Attouch, "Variational Convergence for Functions and Operators," Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984.

[4]

G. Bellettini, V. Caselles and M. Novaga, The total variation flow in RN J. Differential Equations, 184 (2002), 475-525. doi: 10.1006/jdeq.2001.4150.

[5]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973.

[6]

E. Casas, L. A. Fernández and J. Yong, Optimal control of quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 545-565.

[7]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[8]

L. A. Fernández, Integral state-constrained optimal control problems for some quasilinear parabolic equations, Nonlinear Anal., 39 (2000), 977-996. doi: 10.1016/S0362-546X(98)00264-8.

[9]

M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, Proc. Taniguchi Conf. on Math., Advanced Studies in Pure Math., 31 (2001), 93-125.

[10]

Y. Giga, Y. Kashima and N. Yamazaki, Local solvability of a constrained gradient system of total variation, Abstr. Appl. Anal., 2004 (2004), 651-682. doi: 10.1155/S1085337504311048.

[11]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.

[12]

N. Kenmochi and K. Shirakawa, Stability for a parabolic variational inequality associated with total variation functional, Funkcial. Ekvac., 44 (2001), 119-137.

[13]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1967.

[14]

U. Mosco, Convergence of convex sets and of solutions variational inequalities, Advances Math., 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.

[15]

T. Ohtsuka, Numerical simulations for optimal controls of an Allen-Cahn type equation with constraint, in "Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation-Mathematical Analysis, Modelling and Simulation," GAKUTO Intern. Ser. Math. Appl., vol. 29, Gakkotosho, Tokyo, (2008), 329-339.

[16]

T. Ohtsuka, K. Shirakawa and N. Yamazaki, Optimal control problems of singular diffusion equation with constraint, Adv. Math. Sci. Appl., 18 (2008), 1-28.

[17]

T. Ohtsuka, K. Shirakawa and N. Yamazaki, Convergence of numerical algorithm for optimal control problem of Allen-Cahn type equation with constraint, in "Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation-Mathematical Analysis, Modelling and Simulation," GAKUTO Intern. Ser. Math. Appl., vol 29, Gakkotosho, Tokyo, (2008), 441-462.

[18]

K. Shirakawa, Asymptotic convergence of $p$-Laplace equations with constraint as $p$ tends to 1, Math. Methods Appl. Sci., 25 (2002), 771-793. doi: 10.1002/mma.314.

[19]

K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials, in "Recent Developments in Domain Decomposition Methods and Flow Problems," GAKUTO Internat. Ser. Math. Sci. Appl., vol 11, Gakkōtosho, Tokyo, (1998), 287-310.

[20]

K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257-282.

[21]

N. Yamazaki, Optimal control of nonlinear evolution equations associated with time-dependent subdifferentials and applications, in "Nonlocal and Abstract Parabolic Equations and Their Applications," Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 313-327.

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