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Convergence of discontinuous Galerkinapproximations of an optimal control problem associated tosemilinear parabolic PDE's

Published online by Cambridge University Press:  16 December 2009

Konstantinos Chrysafinos
Konstantinos Chrysafinos*
Affiliation:
National Technical University of Athens, Department of Mathematics, Zografou Campus, Athens 15780, Greece. chrysafinos@math.ntua.gr
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Abstract

A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at arbitrary time points under minimal regularity assumptions, and a discrete compactness argument for discontinuous Galerkin schemes (see Walkington [SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4).

Keywords

Discontinuous Galerkin approximationsdistributed controlsstability estimatessemi-linear parabolic PDE's.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 1 , January 2010 , pp. 189 - 206
Copyright
© EDP Sciences, SMAI, 2009

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References

Akrivis, G. and Makridakis, C., Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261–289. CrossRef
Borzi, A. and Griesse, R., Distributed optimal control for lambda-omega systems. J. Numer. Math. 14 (2006) 17–40. CrossRef
H. Brezis, Analyse fonctionnelle – Theorie et applications. Masson, Paris, France (1983).
Chrysafinos, K., Discontinous Galerkin approximations for distributed optimal control problems constrained to linear parabolic PDE's. Int. J. Numer. Anal. Mod. 4 (2007) 690–712.
Chrysafinos, K. and Walkington, N.J., Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006) 349–366. CrossRef
Chrysafinos, K. and Walkington, N.J., Lagrangian and moving mesh methods for the convection diffusion equation. ESAIM: M2AN 42 (2008) 25–55. CrossRef
K. Chrysafinos and N.J. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations. Math. Comp. (to appear), available at http://www.math.cmu.edu/ noelw.
Chrysafinos, K., Gunzburger, M.D. and Hou, L.S., Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE. J. Math. Anal. Appl. 323 (2006) 891–912. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics 40. SIAM (2002).
Dechelnick, K. and Hinze, M., Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations. Numer. Math. 97 (2004) 297–320.
Eriksson, K. and Johnson, C., Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28 (1991) 43–77. CrossRef
Eriksson, K. and Johnson, C., Adaptive finite element methods for parabolic problems. II. Optimal error estimates in $L_{\infty}(L^2)$ and $L_{\infty}(L_{\infty})$ . SIAM J. Numer. Anal. 32 (1995) 706–740. CrossRef
Ericksson, K. and Johnson, C., Adaptive finite element methods for parabolic problems IV: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729–1749. CrossRef
Eriksson, K., Johnson, C. and Thomée, V., Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Modél. Math. Anal. Numér. 29 (1985) 611–643. CrossRef
Estep, D. and Larsson, S., The discontinuous Galerkin method for semilinear parabolic equations. RAIRO Modél. Math. Anal. Numér. 27 (1993) 35–54. CrossRef
L. Evans, Partial Differential Equations. AMS, Providence, USA (1998).
Falk, R., Approximation of a class of otimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44 (1973) 28–47. CrossRef
A. Fursikov, Optimal control of distributed systems – Theory and applications. AMS, Providence, USA (2000).
Garvie, M. and Trenchea, C., Optimal control of a nutrient-phytoplankton-zooplankton-fish system. SIAM J. Control Optim. 46 (2007) 775–791. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes. Springer-Verlag, New York, USA (1986).
M.D. Gunzburger, Perspectives in flow control and optimization, Advances in Design and Control. SIAM, Philadelphia, USA (2003).
Gunzburger, M.D. and Manservisi, S., Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481–1512. CrossRef
Gunzburger, M.D., Hou, L.S. and Svobodny, T., Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. RAIRO Modél. Math. Anal. Numer. 25 (1991) 711–748. CrossRef
Gunzburger, M.D., Yang, S.-D., and Zhu, W., Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete Contin. Dyn. Syst. Ser. B 8 (2007) 569–587.
Hinze, M. and Kunisch, K., Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40 (2001) 925–946. CrossRef
Hou, L.S., and Kwon, H.-D., Analysis and approximations of a terminal-state optimal control problem constrained by semilinear parabolic PDEs. Int. J. Numer. Anal. Model. 4 (2007) 713–728.
Knowles, G., Finite element approximation of parabolic time optimal control problems. SIAM J. Control Optim. 20 (1982) 414–427. CrossRef
Lasiecka, I., Rietz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions. SIAM J. Control Optim. 22 (1984) 477–500. CrossRef
I. Lasiecka and R. Triggiani, Control theory for partial differential equations. Cambridge University Press, Cambridge, UK (2000).
J.-L. Lions, Some aspects of the control of distributed parameter systems. Conference Board of the Mathematical Sciences, SIAM (1972).
Liu, W.-B. and Yan, N., A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 93 (2003) 497–521. CrossRef
Liu, W.-B., Ma, H.-P., Tang, T. and Yan, N., A posteriori error estimates for DG time-stepping method for optimal control problems governed by parabolic equations. SIAM J. Numer. Anal. 42 (2004) 1032–1061. CrossRef
Malanowski, K., Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optim. 8 (1981) 69–95. CrossRef
Meidner, D. and Vexler, B., Adaptive space-time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46 (2007) 116–142. CrossRef
Meidner, D. and Vexler, B., A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part I: Problems without control constraints. SIAM J. Control Optim. 47 (2008) 1150–1177. CrossRef
P. Neittaanmaki and D. Tiba, Optimal control of nonlinear parabolic systems – Theory, algorithms and applications. M. Dekker, New York, USA (1994).
Rösch, A., Error estimates for parabolic optimal control problems with control constraints. Zeitschrift Anal. Anwendungen 23 (2004) 353–376. CrossRef
R. Temam, Navier-Stokes equations. North Holland (1977).
V. Thomée, Galerkin finite element methods for parabolic problems. Spinger-Verlag, Berlin, Germany (1997).
Tröltzsch, F., Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems. International Series of Numerical Math. 111 (1993) 57–68.
Tröltzsch, F., Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems – Strong convergence of optimal controls. Appl. Math. Optim. 29 (1994) 309–329. CrossRef
N.J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations. SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw.
Winther, R., Error estimates for a Galerkin approximation of a parabolic control problem. Ann. Math. Pura Appl. 117 (1978) 173–206. CrossRef
Winther, R., Initial value methods for parabolic control problems. Math. Comp. 34 (1980) 115–125. CrossRef
E. Zeidler, Nonlinear functional analysis and its applications, II/B Nonlinear monotone operators. Springer-Verlag, New York, USA (1990).