Solution of a Parabolic Optimal Control Problem Using Fictitious Domain Method

  • Alexander Lapin
  • Erkki Laitinen
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 133)


A linear-quadratic parabolic optimal control problem in a cylinder with spatial curvilinear domain is solved numerically by means of embedding fictitious domain method and finite difference approximation on a regular orthogonal mesh. A regularized mesh problem in a parallelepiped containing initial space-time domain is constructed. This problem is loaded by additional constraints for control and state functions in the fictitious subdomain. The restriction of its solution to the initial domain tends to the solution of the initial problem as regularization parameter tends to zero. Efficiently implementable finite difference methods can be used for the state and co-state problems of the new optimal control problem. The optimal control problem is solved by an iterative method, the rate of its convergence is proved. Numerical tests demonstrate the efficiency of the proposed approach for solving formulated problem.


Parabolic optimal control Fictitious domain method Finite-difference approximation 



The work of the first author was supported by Russian Foundation of Basic Researches, project 16-01-00408, and by Academy of Finland, project 318175. The work of the second author was supported by Academy of Finland, project 318303.


  1. 1.
    Saulev, V.: On solution of some boundary value problems on high performance computers by fictitious domain method. Siberian Math. J. 4, 912–925 (1963). (in Russian)Google Scholar
  2. 2.
    Astrakhantsev, G.: Method of fictitious domains for a second-order elliptic equation with natural boundary conditions. USSR Comput. Math. Math. Phys. 18, 114–121 (1978)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Glowinski, R., Pan, T.W., Periaux, J.: A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 11193(4), 283–303 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Glowinski, R., Kuznetsov, Y.: On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrange multiplier method. C. R. l’Academie Sci. – Ser. I – Math. 327(7), 693–698 (1998)CrossRefGoogle Scholar
  5. 5.
    Glowinski, R., Kuznetsov, Y.: Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems. Comput. Methods Appl. Mech. Eng. 196(8), 1498–1506 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Peskin, C.S.: The immersed boundary method. Acta Numer., 419–517 (2002)Google Scholar
  7. 7.
    Del Pino, S., Pironneau, O.: A fictitious domain based general PDE solver. In: Heikkola, E., Kuznetsov, Y., Neittaanmäki, P., Pironneau, O. (eds.) Numerical Methods for Scientific Computing. Variational Problems and Applications. CIMNE, Barcelona (2003)Google Scholar
  8. 8.
    Haslinger, J., Neittaanmäki, P.: Finite element approximation for optimal shape, material and topology design, 2nd edn. Wiley, Chichester (1996)zbMATHGoogle Scholar
  9. 9.
    D’Yakonov, E.G.: The method of alternating directions in the solution of finite difference equations. Dokl. Akad. Nauk. SSSR 138, 271–274 (1961). (in Russian)MathSciNetGoogle Scholar
  10. 10.
    Douglas Jr., J., Gunn, J.E.: A general formulation of alternating direction methods. Numer. Math. 6(1), 428–453 (1964)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Yanenko, N.: The Method of Fractional Steps. Springer (1971)Google Scholar
  12. 12.
    Marchuk, G.: Splitting and alternating direction methods. In: Handbook of Numerical Mathematics, V.1: Finite Difference Methods. Elsevier Science Publisher B.V., North-Holland (1990)CrossRefGoogle Scholar
  13. 13.
    Samarsky, A.A.: Theory of Difference Schemes. Marcel Dekker (2001)Google Scholar
  14. 14.
    Agoshkov V. I., Dubovski, P. B., Shutyaev, V. P.: Methods for Solving Mathematical Physics Problems. Cambridge International Science Publishing (2006)Google Scholar
  15. 15.
    Ladyženskaja, O.A., Solonnikov V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23, AMS, Providence, RI (1968)Google Scholar
  16. 16.
    Grisvard, P.: Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. In: Numerical Solution of Partial Differential Equations, III. Academic Press, New York (1976)CrossRefGoogle Scholar
  17. 17.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer (1997)Google Scholar
  18. 18.
    Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971)Google Scholar
  19. 19.
    Agoshkov, V.I.: Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics, 2nd edn. INM RAS, Moscow (2016) (in Russian)Google Scholar
  20. 20.
    Ekeland, I., Temam, R.: Convex analysis and variational problems. North-Holland, Amsterdam (1976)zbMATHGoogle Scholar
  21. 21.
    Lapin, A.: Preconditioned Uzawa type methods for finite-dimensional constrained saddle point problems. Lobachevskii J. Math. 31(4), 309–322 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Laitinen, E., Lapin, A., Lapin, S.: On the iterative solution of finite-dimensional inclusions with applications to optimal control problems. Comp. Methods Appl. Math. 10(3), 283–301 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Information TechnologiesKazan Federal UniversityKazanRussian Federation
  2. 2.Coordinated Innovation Center for Computable Modeling in Management ScienceTianjin University of Finance and EconomicsTianjinPeople’s Republic of China
  3. 3.Faculty of Science, Research Unit of Mathematical SciencesUniversity of OuluOuluFinland

Personalised recommendations