    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 & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159
##### References:
  F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential and Integral Equations, 14 (2001), 321-360. Google Scholar  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.  Google Scholar  H. Attouch, "Variational Convergence for Functions and Operators," Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984. Google Scholar  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.  Google Scholar  H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973. Google Scholar  E. Casas, L. A. Fernández and J. Yong, Optimal control of quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 545-565. Google Scholar  L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. Google Scholar  N. Kenmochi and K. Shirakawa, Stability for a parabolic variational inequality associated with total variation functional, Funkcial. Ekvac., 44 (2001), 119-137. Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  T. Ohtsuka, K. Shirakawa and N. Yamazaki, Optimal control problems of singular diffusion equation with constraint, Adv. Math. Sci. Appl., 18 (2008), 1-28. Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257-282. Google Scholar  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. Google Scholar

show all references

##### References:
  F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential and Integral Equations, 14 (2001), 321-360. Google Scholar  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.  Google Scholar  H. Attouch, "Variational Convergence for Functions and Operators," Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984. Google Scholar  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.  Google Scholar  H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973. Google Scholar  E. Casas, L. A. Fernández and J. Yong, Optimal control of quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 545-565. Google Scholar  L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. Google Scholar  N. Kenmochi and K. Shirakawa, Stability for a parabolic variational inequality associated with total variation functional, Funkcial. Ekvac., 44 (2001), 119-137. Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  T. Ohtsuka, K. Shirakawa and N. Yamazaki, Optimal control problems of singular diffusion equation with constraint, Adv. Math. Sci. Appl., 18 (2008), 1-28. Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257-282. Google Scholar  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. Google Scholar
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