no. 1
Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's
ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 189-206

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).

DOI : 10.1051/m2an/2009046
Classification : 65M60, 49J20
Keywords: discontinuous Galerkin approximations, distributed controls, stability estimates, semi-linear parabolic PDE's 
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     author = {Chrysafinos, Konstantinos},
     title = {Convergence of discontinuous {Galerkin} approximations of an optimal control problem associated to semilinear parabolic {PDE's}},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {189--206},
     year = {2010},
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Chrysafinos, Konstantinos. Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 189-206. doi: 10.1051/m2an/2009046

[1] G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261-289. | Zbl | Numdam

[2] A. Borzi and R. Griesse, Distributed optimal control for lambda-omega systems. J. Numer. Math. 14 (2006) 17-40. | Zbl

[3] H. Brezis, Analyse fonctionnelle - Theorie et applications. Masson, Paris, France (1983). | Zbl

[4] K. Chrysafinos, Discontinous Galerkin approximations for distributed optimal control problems constrained to linear parabolic PDE's. Int. J. Numer. Anal. Mod. 4 (2007) 690-712. | Zbl

[5] K. Chrysafinos and N.J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006) 349-366. | Zbl

[6] K. Chrysafinos and N.J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation. ESAIM: M2AN 42 (2008) 25-55. | Zbl | Numdam

[7] 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.

[8] K. Chrysafinos, M.D. Gunzburger and L.S. Hou, Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE. J. Math. Anal. Appl. 323 (2006) 891-912.

[9] P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics 40. SIAM (2002). | Zbl

[10] K. Dechelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations. Numer. Math. 97 (2004) 297-320. | Zbl

[11] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28 (1991) 43-77. | Zbl

[12] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems 32 (1995) 706-740. | Zbl

[13] K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729-1749. | Zbl

[14] K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Modél. Math. Anal. Numér. 29 (1985) 611-643. | Numdam | Zbl | EuDML

[15] D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic equations. RAIRO Modél. Math. Anal. Numér. 27 (1993) 35-54. | Numdam | Zbl | EuDML

[16] L. Evans, Partial Differential Equations. AMS, Providence, USA (1998). | Zbl

[17] R. Falk, Approximation of a class of otimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44 (1973) 28-47. | Zbl

[18] A. Fursikov, Optimal control of distributed systems - Theory and applications. AMS, Providence, USA (2000). | Zbl

[19] M. Garvie and C. Trenchea, Optimal control of a nutrient-phytoplankton-zooplankton-fish system. SIAM J. Control Optim. 46 (2007) 775-791. | Zbl

[20] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes. Springer-Verlag, New York, USA (1986).

[21] M.D. Gunzburger, Perspectives in flow control and optimization, Advances in Design and Control. SIAM, Philadelphia, USA (2003). | Zbl

[22] M.D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481-1512. | Zbl

[23] M.D. Gunzburger, L.S. Hou and T. Svobodny, 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. | Numdam | Zbl | EuDML

[24] M.D. Gunzburger, S.-D. Yang, and W. Zhu, Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete Contin. Dyn. Syst. Ser. B 8 (2007) 569-587. | Zbl

[25] M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40 (2001) 925-946. | Zbl

[26] L.S. Hou, and H.-D. Kwon, Analysis and approximations of a terminal-state optimal control problem constrained by semilinear parabolic PDEs. Int. J. Numer. Anal. Model. 4 (2007) 713-728. | Zbl

[27] G. Knowles, Finite element approximation of parabolic time optimal control problems. SIAM J. Control Optim. 20 (1982) 414-427. | Zbl

[28] I. Lasiecka, 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. | Zbl

[29] I. Lasiecka and R. Triggiani, Control theory for partial differential equations. Cambridge University Press, Cambridge, UK (2000). | Zbl

[30] J.-L. Lions, Some aspects of the control of distributed parameter systems. Conference Board of the Mathematical Sciences, SIAM (1972).

[31] W.-B. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 93 (2003) 497-521. | Zbl

[32] W.-B. Liu, H.-P. Ma, T. Tang and N. Yan, 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. | Zbl

[33] K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optim. 8 (1981) 69-95. | Zbl

[34] D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46 (2007) 116-142. | Zbl

[35] D. Meidner and B. Vexler, 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. | Zbl

[36] P. Neittaanmaki and D. Tiba, Optimal control of nonlinear parabolic systems - Theory, algorithms and applications. M. Dekker, New York, USA (1994). | Zbl

[37] A. Rösch, Error estimates for parabolic optimal control problems with control constraints. Zeitschrift Anal. Anwendungen 23 (2004) 353-376. | Zbl

[38] R. Temam, Navier-Stokes equations. North Holland (1977). | Zbl

[39] V. Thomée, Galerkin finite element methods for parabolic problems. Spinger-Verlag, Berlin, Germany (1997). | Zbl

[40] F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems. International Series of Numerical Math. 111 (1993) 57-68. | Zbl

[41] F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems - Strong convergence of optimal controls. Appl. Math. Optim. 29 (1994) 309-329. | Zbl

[42] 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. | Zbl

[43] R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem. Ann. Math. Pura Appl. 117 (1978) 173-206. | Zbl

[44] R. Winther, Initial value methods for parabolic control problems. Math. Comp. 34 (1980) 115-125. | Zbl

[45] E. Zeidler, Nonlinear functional analysis and its applications, II/B Nonlinear monotone operators. Springer-Verlag, New York, USA (1990). | Zbl

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