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A Distributed Control Approach for the Boundary Optimal Control of the Steady MHD Equations

Published online by Cambridge University Press:  03 June 2015

G. Bornia ,
M. Gunzburger  and
S. Manservisi
G. Bornia*
Affiliation:
Laboratory of Nuclear Engineering of Montecuccolino, DIENCA, University of Bologna, Italy
M. Gunzburger*
Affiliation:
Department of Scientific Computing, Florida State University, USA
S. Manservisi*
Affiliation:
Laboratory of Nuclear Engineering of Montecuccolino, DIENCA, University of Bologna, Italy
*
Corresponding author.Email:gunzburg@fsu.edu
Email:giorgio.bornia@unibo.it
Email:sandro.manservisi@unibo.it
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Abstract

A new approach is presented for the boundary optimal control of the MHD equations in which the boundary control problem is transformed into an extended distributed control problem. This can be achieved by considering boundary controls in the form of lifting functions which extend from the boundary into the interior. The optimal solution is then sought by exploring all possible extended functions. This approach gives robustness to the boundary control algorithm which can be solved by standard distributed control techniques over the interior of the domain.

Keywords

49J2093C2075W0565N30Boundary controloptimal controlmagnetohydrodynamicsfinite element methods
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 3 , September 2013 , pp. 722 - 752
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Adams, R., Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
[2]Alekseev, G. V. and Smishliaev, A. B., Solvability of the boundary-value problems for the Boussinesq equations with inhomogeneous boundary conditions, Journal of Mathematical Fluid Mechanics, 3, pp. 18–39.Google Scholar
[3]Brizitskii, R. and Tereshko, D., On the solvability of boundary value problems for the stationary magnetohydrodynamic equations with inhomogeneous mixed boundary conditions, Differential Equations, 43, pp. 246–258.CrossRefGoogle Scholar
[4]Ciarlet, P., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[5]Davidson, P. A., An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, UK, 2001.Google Scholar
[6]Ferraro, V. C. A. and Plumpton, C., An introduction to magneto-fluid mechanics, Oxford University Press, Oxford, 1966.Google Scholar
[7]Girault, V. and Raviart, P., The Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, 1979.Google Scholar
[8]Girault, V. and Raviart, P., The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, New York, 1986.Google Scholar
[9]Griesse, R. and Kunisch, K., Optimal control for a stationary MHD system in velocity-current formulation, Siam J. Control Optim., 45 (2006), pp. 1822–1845.Google Scholar
[10]Gunzburger, M. and Manservisi, S., Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control, SIAM J. Numer. Anal., 37 (2000), pp. 1481–1512.Google Scholar
[11]Gunzburger, M. and Manservisi, S., The velocity tracking problem for Navier-Stokes flow with boundary control, SIAM Journal on Control and Optimization, 39 (2000), pp. 594–634.Google Scholar
[12]Gunzburger, M. and Trenchea, C., Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations, J. Math. Anal. Appl., 333 (2007), pp. 295–310.Google Scholar
[13]Gunzburger, M. D., Perspectives in flow control and optimization, SIAM, Philadelphia, 2003.Google Scholar
[14]Gunzburger, M.D. and Hou, L. S., Finite-dimensional approximationof a classof constrained nonlinear optimal control problems, Siam J. Control and Optimization, 34 (1996), pp. 1001–1043.Google Scholar
[15]Gunzburger, M. D., Meir, A. J., and Peterson, J. S., On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magneto-hydrodynamics, Mathematics of Computations, 56 (1991), pp. 523–563.Google Scholar
[16]Hou, L. and Peterson, J. S., Boundary optimal control for an electrically conducting fluid using boundary electrical potential controls, Nonlinear Analysis, Theory, Methods and Applications, 24 (1995), pp. 857–874.Google Scholar
[17]Hou, L. S. and Meir, A., Boundary optimal control of MHD flows, Appl. Math. Optim, 32 (1995), pp. 143–162.Google Scholar
[18]Jackson, J., Classical electrodynamics, Wiley, New York, 1975.Google Scholar
[19]Landau, L. and Lifchitz, E., Electrodynamique des milieux continus, MIR, Moscow, 1969.Google Scholar
[20]Meir, A. and Schmidt, P., Variational methods for stationary MHD flow under natural interface conditions, Nonlinear Analysis, Theory, Methods and Applications, 26 (1996), pp. 659–689.Google Scholar
[21]Ravindran, S., Real-time computational algorithm for optimal control of an MHD flow system, SIAM J. Sci. Comput., 26 (2005), pp. 1369–1388.Google Scholar
[22]Roberts, P. H., An introduction to magnetohydrodynamics, Longmans, London, 1967.Google Scholar
[23]Schotzau, D., Mixed finite element methods for stationary incompressible magneto-hydrodynamics, Numer. Math., 96 (2004), pp. 771–800.Google Scholar
[24]Temam, R., Navier-Stokes Equations, North-Holland, Amsterdam, 1979.Google Scholar
[25]Tikhomirov, V. M., Fundamental Principles of the Theory of Extremal Problems, Whiley, Chichester, 1982.Google Scholar
[26]Wiedmer, M., Finite element approximation for equations of magnetohydrodynamics, Mathematics of Computation, 69 (1999), pp. 83–101.Google Scholar
[27]Yosida, K., Functional analysis, Springer-Verlag, Berlin, 1980.Google Scholar