×
Yılmaz, Fikriye; Karasözen, Bülent

Solving optimal control problems for the unsteady Burgers equation in COMSOL multiphysics. (English) Zbl 1220.65080

J. Comput. Appl. Math. 235, No. 16, 4839-4850 (2011).
Summary: The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.

Cited in 11 Documents

MSC:

65K10 Numerical optimization and variational techniques
49K20 Optimality conditions for problems involving partial differential equations
49M37 Numerical methods based on nonlinear programming
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Keywords:

optimal control; Burgers equation; Cole-Hopf transformation; semi-smooth Newton method; finite elements; COMSOL multiphysics; gradient method; numerical results

Software:

COMSOL
Cite Review PDF
Full Text: DOI

References:

[1] Volkwein, S., Distributed control problems for the Burgers equation, Comput. Optim. Appl., 18, 115-140 (2001) · Zbl 0976.49001
[2] de los Reyes, J. C.; Kunisch, K., A comparison of algorithms for control constrained optimal control of the Burgers equation, Calcolo, 41, 203-225 (2004) · Zbl 1168.49309
[3] Hinze, M.; Volkwein, S., Analysis of instantaneous control for the Burgers equation, Nonlinear Anal., 50, 1-26 (2002) · Zbl 1022.49001
[4] Tröltzsch, F.; Volkwein, S., The SQP method for control constrained optimal control of the Burgers equation, ESAIM Control Optim. Calc. Var., 6, 649-674 (2001) · Zbl 1001.49035
[5] Volkwein, S., Second order conditions for boundary control problems of the Burgers equation, Control Cybern., 30, 249-278 (2001) · Zbl 1001.93033
[6] Noack, A.; Walther, A., Adjoint concepts for the optimal control of Burgers equation, Comput. Optim. Appl., 36, 109-133 (2007) · Zbl 1278.49028
[7] I. Neitzel, U. Prüfert, T. Slawig, Strategies for time-dependent pde control using an integrated modeling and simulation environment. part one: problems without inequality constraints, Technical Report, Matheon, Berlin, 2007.; I. Neitzel, U. Prüfert, T. Slawig, Strategies for time-dependent pde control using an integrated modeling and simulation environment. part one: problems without inequality constraints, Technical Report, Matheon, Berlin, 2007.
[8] I. Neitzel, U. Prüfert, T. Slawig, Solving time-dependent optimal control problems in COMSOL multiphysics, in: Proceedings of the COMSOL Conference Hannover, 2008.; I. Neitzel, U. Prüfert, T. Slawig, Solving time-dependent optimal control problems in COMSOL multiphysics, in: Proceedings of the COMSOL Conference Hannover, 2008.
[9] Neitzel, I.; Prüfert, U.; Slawig, T., Strategies for time-dependent pde control with inequality constraints using an integrated modeling and simulation environment, Numer. Algorithms., 50, 241-269 (2009) · Zbl 1167.65381
[10] S. Volkwein, Mesh-independence of an augmented Lagrangian-SQP method in Hilbert spaces and control problems for the Burgers equation, Dissertation at the University of Technology Berlin, 1997.; S. Volkwein, Mesh-independence of an augmented Lagrangian-SQP method in Hilbert spaces and control problems for the Burgers equation, Dissertation at the University of Technology Berlin, 1997.
[11] I. Neitzel, U. Prüfert, T. Slawig, A Smooth regularization of the projection formula for constrained parabolic optimal control problems, Technical Report, Matheon, Berlin, 21, 2010.; I. Neitzel, U. Prüfert, T. Slawig, A Smooth regularization of the projection formula for constrained parabolic optimal control problems, Technical Report, Matheon, Berlin, 21, 2010.
[12] Vedantham, R., Optimal control of the viscous Burgers equation using equivalent index method, Journal of Global Optimization, 18, 255-263 (2000) · Zbl 0974.49003
[13] Hintermüller, M.; Ito, K.; Kunisch, K., The primal-dual active set strategy as a semismooth newton method, SIAM J. Optim., 13, 865-888 (2002) · Zbl 1080.90074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.