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:
[1]

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

[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. doi: 10.1006/jfan.2000.3698.

[3]

H. Attouch, "Variational Convergence for Functions and Operators,", Pitman Advanced Publishing Program, (1984).

[4]

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

[5]

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

[7]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).

[8]

L. A. Fernández, Integral state-constrained optimal control problems for some quasilinear parabolic equations,, Nonlinear Anal., 39 (2000), 977. 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., 31 (2001), 93.

[10]

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

[11]

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

[12]

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

[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 (1967).

[14]

U. Mosco, Convergence of convex sets and of solutions variational inequalities,, Advances Math., 3 (1969), 510. 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, 29 (2008), 329.

[16]

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

[17]

T. Ohtsuka, K. Shirakawa and N. Yamazaki, Convergence of numerical algorithm for optimal control problem of Allen-Cahn type equation with constraint,, in, 29 (2008), 441.

[18]

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

[19]

K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials,, in, 11 (1998), 287.

[20]

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

[21]

N. Yamazaki, Optimal control of nonlinear evolution equations associated with time-dependent subdifferentials and applications,, in, 86 (2009), 313.

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.

[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. doi: 10.1006/jfan.2000.3698.

[3]

H. Attouch, "Variational Convergence for Functions and Operators,", Pitman Advanced Publishing Program, (1984).

[4]

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

[5]

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

[7]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).

[8]

L. A. Fernández, Integral state-constrained optimal control problems for some quasilinear parabolic equations,, Nonlinear Anal., 39 (2000), 977. 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., 31 (2001), 93.

[10]

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

[11]

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

[12]

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

[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 (1967).

[14]

U. Mosco, Convergence of convex sets and of solutions variational inequalities,, Advances Math., 3 (1969), 510. 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, 29 (2008), 329.

[16]

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

[17]

T. Ohtsuka, K. Shirakawa and N. Yamazaki, Convergence of numerical algorithm for optimal control problem of Allen-Cahn type equation with constraint,, in, 29 (2008), 441.

[18]

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

[19]

K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials,, in, 11 (1998), 287.

[20]

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

[21]

N. Yamazaki, Optimal control of nonlinear evolution equations associated with time-dependent subdifferentials and applications,, in, 86 (2009), 313.

[1]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703

[2]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669

[3]

Haydi Israel. Well-posedness and long time behavior of an Allen-Cahn type equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2811-2827. doi: 10.3934/cpaa.2013.12.2811

[4]

Jean-Paul Chehab, Alejandro A. Franco, Youcef Mammeri. Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 87-100. doi: 10.3934/dcdss.2017005

[5]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

[6]

Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947

[7]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015

[8]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679

[9]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577

[10]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319

[11]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009

[12]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024

[13]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407

[14]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019077

[15]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

[16]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517

[17]

Ken Shirakawa. Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies. Conference Publications, 2009, 2009 (Special) : 697-707. doi: 10.3934/proc.2009.2009.697

[18]

Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391

[19]

Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823

[20]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (3)

[Back to Top]