July  2012, 17(5): 1473-1506. doi: 10.3934/dcdsb.2012.17.1473

Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's

1. 

National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Department of Mathematics, Zografou Campus, 15780, Athens, Greece, Greece

Received  September 2011 Revised  December 2011 Published  March 2012

A discontinuous Galerkin finite element method for an optimal control problem having states constrained to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. It is shown that under suitable assumptions, the error estimates of the corresponding optimality system are of the same order to the standard linear (uncontrolled) parabolic problem. These estimates have symmetric structure and are also applicable for higher order elements.
Citation: Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473
References:
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G. Allaire and O. Pantz, Structural optimization with FreeFem++,, Struct. Multidiscip. Optim., 32 (2006), 173. doi: 10.1007/s00158-006-0017-y. Google Scholar

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T. Apel and T. Flaig, Crank-Nicolson schemes for optimal control problems with evolution equations,, submitted. Available from: \url{http://www.unibw.de/bauv1/personen/apel/papers}., (). Google Scholar

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A. Borzi and R. Griesse, Distributed optimal control for lambda-omega systems,, J. Numer. Math., 14 (2006), 17. doi: 10.1515/156939506776382120. Google Scholar

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E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equation,, SIAM J. Control and Optim., 45 (2006), 1586. doi: 10.1137/050626600. Google Scholar

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E. Casas, M. Mateos and F. Tröltzsch, Error estimates for the numerical approximation of boundary semilinear elliptic control problem,, Comput. Optim. and Appl., 31 (2005), 193. Google Scholar

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K. Chrysafinos, Discontinous Galerkin approximations for distributed optimal control problems constrained by parabolic PDE's,, Int. J. Numer. Anal. and Mod., 4 (2007), 690. Google Scholar

[8]

K. Chrysafinos, Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations,, J. Comput. Appl. Math., 231 (2009), 327. doi: 10.1016/j.cam.2009.02.092. Google Scholar

[9]

K. Chrysafinos, Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's,, M2AN Math. Model. Numer. Anal., 44 (2010), 189. Google Scholar

[10]

K. Chrysafinos, Convergence of discontinuous time-stepping schemes for a Robin boundary control problem under minimal regularity assumptions,, submitted. Available from: \url{http://www.math.ntua.gr/~chrysafinos}., (). Google Scholar

[11]

K. Chrysafinos and N. J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations,, SIAM J. Numer. Anal., 44 (2006), 349. doi: 10.1137/030602289. Google Scholar

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K. Chrysafinos and N. J. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations,, Math. Comp., 79 (2010), 2135. doi: 10.1090/S0025-5718-10-02348-3. Google Scholar

[13]

K. Chrysafinos and N. J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation,, M2AN Math. Model. Numer. Anal., 42 (2008), 25. Google Scholar

[14]

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. doi: 10.1016/j.jmaa.2005.10.053. Google Scholar

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K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations,, Numer. Math., 97 (2004), 297. doi: 10.1007/s00211-003-0507-4. Google Scholar

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K. Deckelnick and M. Hinze, Variational discretization of parabolic control problems in the presence of pointwise state constraints,, J. Comput. Math., 29 (2011), 1. Google Scholar

[18]

T. F. Dupont and Y. Liu, Symmetric error estimates for moving mesh Galerkin methods for advection-diffusion equations,, SIAM J. Numer. Anal., 40 (2002), 914. doi: 10.1137/S0036142900380431. Google Scholar

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D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems,, RAIRO Modél. Math. Anal. Numér., 27 (1993), 35. Google Scholar

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K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem,, SIAM J. Numer. Anal., 28 (1991), 43. doi: 10.1137/0728003. Google Scholar

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K. Eriksson and C. Johnson, 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. doi: 10.1137/0732033. Google Scholar

[22]

K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV, Nonlinear problems,, SIAM J. Numer. Anal., 32 (1995), 1729. doi: 10.1137/0732078. Google Scholar

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K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method,, RAIRO Modél. Math. Anal. Numér., 19 (1985), 611. Google Scholar

[24]

L. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar

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R. Falk, Approximation of a class of otimal control problems with order of convergence estimates,, J. Math. Anal. Appl., 44 (1973), 28. doi: 10.1016/0022-247X(73)90022-X. Google Scholar

[26]

A. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications,", Translations of Mathematical Monographs, 187 (2000). Google Scholar

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V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms,", Springer Series in Computational Mathematics, 5 (1986). Google Scholar

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W. Gong, M. Hinze and Z. Zhou, A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control,, submitted. Available from: \url{http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2011-07.pdf}., (): 2011. Google Scholar

[29]

M. D. Gunzburger, "Perspectives in Flow Control and Optimization,", Advances in Design and Control, 5 (2003). Google Scholar

[30]

M. D. Gunzburger, L. S. Hou and Th. P. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls,, RAIRO Modél. Math. Anal. Numér., 25 (1991), 711. Google Scholar

[31]

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. doi: 10.1137/S0036142997329414. Google Scholar

[32]

F. Hecht, FreeFem++,, Third edition, (2011). Google Scholar

[33]

M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comput. Optim. Appl., 30 (2005), 45. doi: 10.1007/s10589-005-4559-5. Google Scholar

[34]

M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow,, SIAM J. Control and Optim., 40 (2001), 925. doi: 10.1137/S0363012999361810. Google Scholar

[35]

K. Ito and K. Kunisch, "Lagrange Multiplier Approach to Variational Problems and Applications,", Advances in Design and Control, 15 (2008). Google Scholar

[36]

G. Knowles, Finite element approximation of parabolic time optimal control problems,, SIAM J. Control and Optim., 20 (1982), 414. doi: 10.1137/0320032. Google Scholar

[37]

I. Lasiecka, Rietz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions,, SIAM J. Control and Optim., 22 (1984), 477. Google Scholar

[38]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations,", Cambridge University Press, (2000). Google Scholar

[39]

J.-L. Lions, "Some Aspects of the Control of Distributed Parameter Systems,", Conference Board of the Mathematical Sciences, (1972). Google Scholar

[40]

W.-B. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by parabolic equations,, Numer. Math., 93 (2003), 497. doi: 10.1007/s002110100380. Google Scholar

[41]

W.-B. Liu, H.-P. Ma, T. Tang and N. Yan, A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations,, SIAM J. Numer. Anal., 42 (2004), 1032. doi: 10.1137/S0036142902397090. Google Scholar

[42]

Y. Liu, R. E. Bank, T. F. Dupont, S. Garcia and R. F. Santos, Symmetric error estimates for moving mesh mixed methods for advection-diffusion equations,, SIAM J. Numer. Anal., 40 (2002), 2270. doi: 10.1137/S003614290038073X. Google Scholar

[43]

K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems,, Appl. Math. Optim., 8 (1982), 69. doi: 10.1007/BF01447752. Google Scholar

[44]

D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems,, SIAM J. Control and Optim., 46 (2007), 116. doi: 10.1137/060648994. Google Scholar

[45]

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 and Optim., 47 (2008), 1150. doi: 10.1137/070694016. Google Scholar

[46]

D. Meidner and B. Vexler, A-priori error analysis of the Petrov-Galerkin Crank-Nicolson scheme for parabolic optimal control problems,, SIAM J. Control and Optim., 49 (2011), 2183. doi: 10.1137/100809611. Google Scholar

[47]

P. Neittaanmäki and D. Tiba, "Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications,", Monographs and Textbooks in Pure and Applied Mathematics, 179 (1994). Google Scholar

[48]

I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems,, Numer. Math., 120 (2011), 345. doi: 10.1007/s00211-011-0409-9. Google Scholar

[49]

A. Rösch, Error estimates for parabolic optimal control problems with control constraints,, Zeitschrift für Analysis und ihre Anwendunges, 23 (2004), 353. Google Scholar

[50]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Springer Series in Computational Mathematics, 25 (1997). Google Scholar

[51]

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems,, in, 111 (1993), 57. Google Scholar

[52]

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls,, Appl. Math. Optim., 29 (1994), 309. Google Scholar

[53]

F. Tröltzsch, "Optimal Control of Partial Differential Equations: Theory, Methods and Applications,", Graduate Studies in Mathematics, 112 (2010). Google Scholar

[54]

N. J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations,, SIAM J. Numer. Anal, 47 (2010), 4680. doi: 10.1137/080728378. Google Scholar

[55]

R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem,, Ann. Math. Pura Appl. (4), 117 (1978), 173. doi: 10.1007/BF02417890. Google Scholar

[56]

R. Winther, Initial value methods for parabolic control problems,, Math. Comp., 34 (1980), 115. doi: 10.1090/S0025-5718-1980-0551293-7. Google Scholar

[57]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators,", Springer-Verlag, (1990). doi: 10.1007/978-1-4612-0985-0. Google Scholar

show all references

References:
[1]

G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations,, M2AN Math. Model. and Numer. Anal., 38 (2004), 261. doi: 10.1051/m2an:2004013. Google Scholar

[2]

G. Allaire and O. Pantz, Structural optimization with FreeFem++,, Struct. Multidiscip. Optim., 32 (2006), 173. doi: 10.1007/s00158-006-0017-y. Google Scholar

[3]

T. Apel and T. Flaig, Crank-Nicolson schemes for optimal control problems with evolution equations,, submitted. Available from: \url{http://www.unibw.de/bauv1/personen/apel/papers}., (). Google Scholar

[4]

A. Borzi and R. Griesse, Distributed optimal control for lambda-omega systems,, J. Numer. Math., 14 (2006), 17. doi: 10.1515/156939506776382120. Google Scholar

[5]

E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equation,, SIAM J. Control and Optim., 45 (2006), 1586. doi: 10.1137/050626600. Google Scholar

[6]

E. Casas, M. Mateos and F. Tröltzsch, Error estimates for the numerical approximation of boundary semilinear elliptic control problem,, Comput. Optim. and Appl., 31 (2005), 193. Google Scholar

[7]

K. Chrysafinos, Discontinous Galerkin approximations for distributed optimal control problems constrained by parabolic PDE's,, Int. J. Numer. Anal. and Mod., 4 (2007), 690. Google Scholar

[8]

K. Chrysafinos, Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations,, J. Comput. Appl. Math., 231 (2009), 327. doi: 10.1016/j.cam.2009.02.092. Google Scholar

[9]

K. Chrysafinos, Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's,, M2AN Math. Model. Numer. Anal., 44 (2010), 189. Google Scholar

[10]

K. Chrysafinos, Convergence of discontinuous time-stepping schemes for a Robin boundary control problem under minimal regularity assumptions,, submitted. Available from: \url{http://www.math.ntua.gr/~chrysafinos}., (). Google Scholar

[11]

K. Chrysafinos and N. J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations,, SIAM J. Numer. Anal., 44 (2006), 349. doi: 10.1137/030602289. Google Scholar

[12]

K. Chrysafinos and N. J. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations,, Math. Comp., 79 (2010), 2135. doi: 10.1090/S0025-5718-10-02348-3. Google Scholar

[13]

K. Chrysafinos and N. J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation,, M2AN Math. Model. Numer. Anal., 42 (2008), 25. Google Scholar

[14]

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. doi: 10.1016/j.jmaa.2005.10.053. Google Scholar

[15]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Reprint of the 1978 original, (1978). Google Scholar

[16]

K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations,, Numer. Math., 97 (2004), 297. doi: 10.1007/s00211-003-0507-4. Google Scholar

[17]

K. Deckelnick and M. Hinze, Variational discretization of parabolic control problems in the presence of pointwise state constraints,, J. Comput. Math., 29 (2011), 1. Google Scholar

[18]

T. F. Dupont and Y. Liu, Symmetric error estimates for moving mesh Galerkin methods for advection-diffusion equations,, SIAM J. Numer. Anal., 40 (2002), 914. doi: 10.1137/S0036142900380431. Google Scholar

[19]

D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems,, RAIRO Modél. Math. Anal. Numér., 27 (1993), 35. Google Scholar

[20]

K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem,, SIAM J. Numer. Anal., 28 (1991), 43. doi: 10.1137/0728003. Google Scholar

[21]

K. Eriksson and C. Johnson, 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. doi: 10.1137/0732033. Google Scholar

[22]

K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV, Nonlinear problems,, SIAM J. Numer. Anal., 32 (1995), 1729. doi: 10.1137/0732078. Google Scholar

[23]

K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method,, RAIRO Modél. Math. Anal. Numér., 19 (1985), 611. Google Scholar

[24]

L. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar

[25]

R. Falk, Approximation of a class of otimal control problems with order of convergence estimates,, J. Math. Anal. Appl., 44 (1973), 28. doi: 10.1016/0022-247X(73)90022-X. Google Scholar

[26]

A. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications,", Translations of Mathematical Monographs, 187 (2000). Google Scholar

[27]

V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms,", Springer Series in Computational Mathematics, 5 (1986). Google Scholar

[28]

W. Gong, M. Hinze and Z. Zhou, A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control,, submitted. Available from: \url{http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2011-07.pdf}., (): 2011. Google Scholar

[29]

M. D. Gunzburger, "Perspectives in Flow Control and Optimization,", Advances in Design and Control, 5 (2003). Google Scholar

[30]

M. D. Gunzburger, L. S. Hou and Th. P. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls,, RAIRO Modél. Math. Anal. Numér., 25 (1991), 711. Google Scholar

[31]

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. doi: 10.1137/S0036142997329414. Google Scholar

[32]

F. Hecht, FreeFem++,, Third edition, (2011). Google Scholar

[33]

M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comput. Optim. Appl., 30 (2005), 45. doi: 10.1007/s10589-005-4559-5. Google Scholar

[34]

M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow,, SIAM J. Control and Optim., 40 (2001), 925. doi: 10.1137/S0363012999361810. Google Scholar

[35]

K. Ito and K. Kunisch, "Lagrange Multiplier Approach to Variational Problems and Applications,", Advances in Design and Control, 15 (2008). Google Scholar

[36]

G. Knowles, Finite element approximation of parabolic time optimal control problems,, SIAM J. Control and Optim., 20 (1982), 414. doi: 10.1137/0320032. Google Scholar

[37]

I. Lasiecka, Rietz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions,, SIAM J. Control and Optim., 22 (1984), 477. Google Scholar

[38]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations,", Cambridge University Press, (2000). Google Scholar

[39]

J.-L. Lions, "Some Aspects of the Control of Distributed Parameter Systems,", Conference Board of the Mathematical Sciences, (1972). Google Scholar

[40]

W.-B. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by parabolic equations,, Numer. Math., 93 (2003), 497. doi: 10.1007/s002110100380. Google Scholar

[41]

W.-B. Liu, H.-P. Ma, T. Tang and N. Yan, A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations,, SIAM J. Numer. Anal., 42 (2004), 1032. doi: 10.1137/S0036142902397090. Google Scholar

[42]

Y. Liu, R. E. Bank, T. F. Dupont, S. Garcia and R. F. Santos, Symmetric error estimates for moving mesh mixed methods for advection-diffusion equations,, SIAM J. Numer. Anal., 40 (2002), 2270. doi: 10.1137/S003614290038073X. Google Scholar

[43]

K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems,, Appl. Math. Optim., 8 (1982), 69. doi: 10.1007/BF01447752. Google Scholar

[44]

D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems,, SIAM J. Control and Optim., 46 (2007), 116. doi: 10.1137/060648994. Google Scholar

[45]

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 and Optim., 47 (2008), 1150. doi: 10.1137/070694016. Google Scholar

[46]

D. Meidner and B. Vexler, A-priori error analysis of the Petrov-Galerkin Crank-Nicolson scheme for parabolic optimal control problems,, SIAM J. Control and Optim., 49 (2011), 2183. doi: 10.1137/100809611. Google Scholar

[47]

P. Neittaanmäki and D. Tiba, "Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications,", Monographs and Textbooks in Pure and Applied Mathematics, 179 (1994). Google Scholar

[48]

I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems,, Numer. Math., 120 (2011), 345. doi: 10.1007/s00211-011-0409-9. Google Scholar

[49]

A. Rösch, Error estimates for parabolic optimal control problems with control constraints,, Zeitschrift für Analysis und ihre Anwendunges, 23 (2004), 353. Google Scholar

[50]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Springer Series in Computational Mathematics, 25 (1997). Google Scholar

[51]

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems,, in, 111 (1993), 57. Google Scholar

[52]

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls,, Appl. Math. Optim., 29 (1994), 309. Google Scholar

[53]

F. Tröltzsch, "Optimal Control of Partial Differential Equations: Theory, Methods and Applications,", Graduate Studies in Mathematics, 112 (2010). Google Scholar

[54]

N. J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations,, SIAM J. Numer. Anal, 47 (2010), 4680. doi: 10.1137/080728378. Google Scholar

[55]

R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem,, Ann. Math. Pura Appl. (4), 117 (1978), 173. doi: 10.1007/BF02417890. Google Scholar

[56]

R. Winther, Initial value methods for parabolic control problems,, Math. Comp., 34 (1980), 115. doi: 10.1090/S0025-5718-1980-0551293-7. Google Scholar

[57]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators,", Springer-Verlag, (1990). doi: 10.1007/978-1-4612-0985-0. Google Scholar

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