Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 1, p. 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 : https://doi.org/10.1051/m2an/2009046
Classification:  65M60,  49J20
Keywords: discontinuous Galerkin approximations, distributed controls, stability estimates, semi-linear parabolic PDE's
@article{M2AN_2010__44_1_189_0,
     author = {Chrysafinos, Konstantinos},
     title = {Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {1},
     year = {2010},
     pages = {189-206},
     doi = {10.1051/m2an/2009046},
     zbl = {1191.65074},
     mrnumber = {2647758},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_1_189_0}
}
Chrysafinos, Konstantinos. Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 1, pp. 189-206. doi : 10.1051/m2an/2009046. http://www.numdam.org/item/M2AN_2010__44_1_189_0/

[1] G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261-289. | Numdam | Zbl 1085.65094

[2] A. Borzi and R. Griesse, Distributed optimal control for lambda-omega systems. J. Numer. Math. 14 (2006) 17-40. | Zbl 1104.65063

[3] H. Brezis, Analyse fonctionnelle - Theorie et applications. Masson, Paris, France (1983). | Zbl 1147.46300

[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 1149.65046

[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 1112.65086

[6] K. Chrysafinos and N.J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation. ESAIM: M2AN 42 (2008) 25-55. | Numdam | Zbl 1136.65089

[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. | Zbl pre05797904

[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. | Zbl pre05077876

[9] P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics 40. SIAM (2002). | Zbl 0999.65129

[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 1055.76027

[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 0732.65093

[12] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems 32 (1995) 706-740. | Zbl 0830.65094

[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 0835.65116

[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 0589.65070

[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 0768.65065

[16] L. Evans, Partial Differential Equations. AMS, Providence, USA (1998). | Zbl 0902.35002

[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 0268.49036

[18] A. Fursikov, Optimal control of distributed systems - Theory and applications. AMS, Providence, USA (2000). | Zbl 1027.93500

[19] M. Garvie and C. Trenchea, Optimal control of a nutrient-phytoplankton-zooplankton-fish system. SIAM J. Control Optim. 46 (2007) 775-791. | Zbl 1357.49090 | Zbl pre05288504

[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 1088.93001

[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 0963.35150

[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 0737.76045

[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 1260.49006 | Zbl pre05232627

[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 1012.49026

[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 1141.49005

[27] G. Knowles, Finite element approximation of parabolic time optimal control problems. SIAM J. Control Optim. 20 (1982) 414-427. | Zbl 0481.49026

[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 0549.49024

[29] I. Lasiecka and R. Triggiani, Control theory for partial differential equations. Cambridge University Press, Cambridge, UK (2000). | Zbl 0942.93001

[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 1049.65057

[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 1085.65054

[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 0479.49017

[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 1149.65051

[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 1161.49026

[36] P. Neittaanmaki and D. Tiba, Optimal control of nonlinear parabolic systems - Theory, algorithms and applications. M. Dekker, New York, USA (1994). | Zbl 0812.49001

[37] A. Rösch, Error estimates for parabolic optimal control problems with control constraints. Zeitschrift Anal. Anwendungen 23 (2004) 353-376. | Zbl 1052.49031

[38] R. Temam, Navier-Stokes equations. North Holland (1977). | Zbl 0335.35077 | Zbl 0383.35057

[39] V. Thomée, Galerkin finite element methods for parabolic problems. Spinger-Verlag, Berlin, Germany (1997). | Zbl 0884.65097 | Zbl 0528.65052

[40] F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems. International Series of Numerical Math. 111 (1993) 57-68. | Zbl 0790.49025

[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 0802.49017

[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 1252.65169 | Zbl pre05840696

[43] R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem. Ann. Math. Pura Appl. 117 (1978) 173-206. | Zbl 0434.65092

[44] R. Winther, Initial value methods for parabolic control problems. Math. Comp. 34 (1980) 115-125. | Zbl 0428.35043

[45] E. Zeidler, Nonlinear functional analysis and its applications, II/B Nonlinear monotone operators. Springer-Verlag, New York, USA (1990). | Zbl 0684.47029