Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
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)
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)
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)
Peskin, C.S.: The immersed boundary method. Acta Numer., 419–517 (2002)
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)
Haslinger, J., Neittaanmäki, P.: Finite element approximation for optimal shape, material and topology design, 2nd edn. Wiley, Chichester (1996)
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)
Douglas Jr., J., Gunn, J.E.: A general formulation of alternating direction methods. Numer. Math. 6(1), 428–453 (1964)
Yanenko, N.: The Method of Fractional Steps. Springer (1971)
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)
Samarsky, A.A.: Theory of Difference Schemes. Marcel Dekker (2001)
Agoshkov V. I., Dubovski, P. B., Shutyaev, V. P.: Methods for Solving Mathematical Physics Problems. Cambridge International Science Publishing (2006)
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)
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)
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer (1997)
Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971)
Agoshkov, V.I.: Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics, 2nd edn. INM RAS, Moscow (2016) (in Russian)
Ekeland, I., Temam, R.: Convex analysis and variational problems. North-Holland, Amsterdam (1976)
Lapin, A.: Preconditioned Uzawa type methods for finite-dimensional constrained saddle point problems. Lobachevskii J. Math. 31(4), 309–322 (2010)
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)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Lapin, A., Laitinen, E. (2019). Solution of a Parabolic Optimal Control Problem Using Fictitious Domain Method. In: Petrov, I., Favorskaya, A., Favorskaya, M., Simakov, S., Jain, L. (eds) Smart Modeling for Engineering Systems. GCM50 2018. Smart Innovation, Systems and Technologies, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-030-06228-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-06228-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-06227-9
Online ISBN: 978-3-030-06228-6
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)