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

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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

Konstantinos Chrysafinos ^1, and Efthimios N. Karatzas ^1,

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

Abstract
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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.

Keywords: distributed control, discontinuous Galerkin, Symmetric error estimates, semi-linear parabolic PDE's..

Mathematics Subject Classification: Primary: 65M60, 49J2.

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 and Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473

References:

[1]	G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations, M2AN Math. Model. and Numer. Anal., 38 (2004), 261-289. doi: 10.1051/m2an:2004013.
[2]	G. Allaire and O. Pantz, Structural optimization with FreeFem++, Struct. Multidiscip. Optim., 32 (2006), 173-181. doi: 10.1007/s00158-006-0017-y.
[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}., ().
[4]	A. Borzi and R. Griesse, Distributed optimal control for lambda-omega systems, J. Numer. Math., 14 (2006), 17-40. doi: 10.1515/156939506776382120.
[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), 1586-1611. doi: 10.1137/050626600.
[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), 193-219.
[7]	K. Chrysafinos, Discontinous Galerkin approximations for distributed optimal control problems constrained by parabolic PDE's, Int. J. Numer. Anal. and Mod., 4 (2007), 690-712.
[8]	K. Chrysafinos, Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations, J. Comput. Appl. Math., 231 (2009), 327-348. doi: 10.1016/j.cam.2009.02.092.
[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), 189-206.
[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}., ().
[11]	K. Chrysafinos and N. J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 44 (2006), 349-366. doi: 10.1137/030602289.
[12]	K. Chrysafinos and N. J. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations, Math. Comp., 79 (2010), 2135-2167. doi: 10.1090/S0025-5718-10-02348-3.
[13]	K. Chrysafinos and N. J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation, M2AN Math. Model. Numer. Anal., 42 (2008), 25-55.
[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), 891-912. doi: 10.1016/j.jmaa.2005.10.053.
[15]	P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," Reprint of the 1978 original, Classics in Applied Math., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.
[16]	K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations, Numer. Math., 97 (2004), 297-320. doi: 10.1007/s00211-003-0507-4.
[17]	K. Deckelnick and M. Hinze, Variational discretization of parabolic control problems in the presence of pointwise state constraints, J. Comput. Math., 29 (2011), 1-15.
[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), 914-927. doi: 10.1137/S0036142900380431.
[19]	D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems, RAIRO Modél. Math. Anal. Numér., 27 (1993), 35-54.
[20]	K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal., 28 (1991), 43-77. doi: 10.1137/0728003.
[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), 706-740. doi: 10.1137/0732033.
[22]	K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV, Nonlinear problems, SIAM J. Numer. Anal., 32 (1995), 1729-1749. doi: 10.1137/0732078.
[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), 611-643.
[24]	L. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, AMS, Providence, RI, 1998.
[25]	R. Falk, Approximation of a class of otimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28-47. doi: 10.1016/0022-247X(73)90022-X.
[26]	A. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications," Translations of Mathematical Monographs, 187, AMS, Providence, RI, 2000.
[27]	V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms," Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986.
[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.
[29]	M. D. Gunzburger, "Perspectives in Flow Control and Optimization," Advances in Design and Control, 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.
[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), 711-748.
[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), 1481-1512. doi: 10.1137/S0036142997329414.
[32]	F. Hecht, FreeFem++, Third edition, Version 3.13, 2011. Available from: http://www.freefem.org/ff++.
[33]	M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61. doi: 10.1007/s10589-005-4559-5.
[34]	M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow, SIAM J. Control and Optim., 40 (2001), 925-946. doi: 10.1137/S0363012999361810.
[35]	K. Ito and K. Kunisch, "Lagrange Multiplier Approach to Variational Problems and Applications," Advances in Design and Control, 15, SIAM, Philadelphia, PA, 2008.
[36]	G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control and Optim., 20 (1982), 414-427. doi: 10.1137/0320032.
[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), 477-500.
[38]	I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations," Cambridge University Press, Cambridge, 2000.
[39]	J.-L. Lions, "Some Aspects of the Control of Distributed Parameter Systems," Conference Board of the Mathematical Sciences, SIAM, 1972.
[40]	W.-B. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 93 (2003), 497-521. doi: 10.1007/s002110100380.
[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), 1032-1061. doi: 10.1137/S0036142902397090.
[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), 2270-2291. doi: 10.1137/S003614290038073X.
[43]	K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems, Appl. Math. Optim., 8 (1982), 69-95. doi: 10.1007/BF01447752.
[44]	D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems, SIAM J. Control and Optim., 46 (2007), 116-142. doi: 10.1137/060648994.
[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), 1150-1177. doi: 10.1137/070694016.
[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), 2183-2211. doi: 10.1137/100809611.
[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, Marcel Dekker, Inc., New York, 1994.
[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), 345-386. doi: 10.1007/s00211-011-0409-9.
[49]	A. Rösch, Error estimates for parabolic optimal control problems with control constraints, Zeitschrift für Analysis und ihre Anwendunges, 23 (2004), 353-376.
[50]	V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer Series in Computational Mathematics, 25, Spinger-Verlag, Berlin, 1997.
[51]	F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems, in "Optimal Control" (Freiburg, 1991), International Series of Numerical Mathematics, 111, Birkhäuser, Basel, (1993), 57-68.
[52]	F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls, Appl. Math. Optim., 29 (1994), 309-329.
[53]	F. Tröltzsch, "Optimal Control of Partial Differential Equations: Theory, Methods and Applications," Graduate Studies in Mathematics, 112, American Mathematical Society, Providence, RI, 2010.
[54]	N. J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations, SIAM J. Numer. Anal, 47 (2010), 4680-4710. doi: 10.1137/080728378.
[55]	R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem, Ann. Math. Pura Appl. (4), 117 (1978), 173-206. doi: 10.1007/BF02417890.
[56]	R. Winther, Initial value methods for parabolic control problems, Math. Comp., 34 (1980), 115-125. doi: 10.1090/S0025-5718-1980-0551293-7.
[57]	E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators," Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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), 261-289. doi: 10.1051/m2an:2004013.
[2]	G. Allaire and O. Pantz, Structural optimization with FreeFem++, Struct. Multidiscip. Optim., 32 (2006), 173-181. doi: 10.1007/s00158-006-0017-y.
[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}., ().
[4]	A. Borzi and R. Griesse, Distributed optimal control for lambda-omega systems, J. Numer. Math., 14 (2006), 17-40. doi: 10.1515/156939506776382120.
[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), 1586-1611. doi: 10.1137/050626600.
[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), 193-219.
[7]	K. Chrysafinos, Discontinous Galerkin approximations for distributed optimal control problems constrained by parabolic PDE's, Int. J. Numer. Anal. and Mod., 4 (2007), 690-712.
[8]	K. Chrysafinos, Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations, J. Comput. Appl. Math., 231 (2009), 327-348. doi: 10.1016/j.cam.2009.02.092.
[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), 189-206.
[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}., ().
[11]	K. Chrysafinos and N. J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 44 (2006), 349-366. doi: 10.1137/030602289.
[12]	K. Chrysafinos and N. J. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations, Math. Comp., 79 (2010), 2135-2167. doi: 10.1090/S0025-5718-10-02348-3.
[13]	K. Chrysafinos and N. J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation, M2AN Math. Model. Numer. Anal., 42 (2008), 25-55.
[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), 891-912. doi: 10.1016/j.jmaa.2005.10.053.
[15]	P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," Reprint of the 1978 original, Classics in Applied Math., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.
[16]	K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations, Numer. Math., 97 (2004), 297-320. doi: 10.1007/s00211-003-0507-4.
[17]	K. Deckelnick and M. Hinze, Variational discretization of parabolic control problems in the presence of pointwise state constraints, J. Comput. Math., 29 (2011), 1-15.
[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), 914-927. doi: 10.1137/S0036142900380431.
[19]	D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems, RAIRO Modél. Math. Anal. Numér., 27 (1993), 35-54.
[20]	K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal., 28 (1991), 43-77. doi: 10.1137/0728003.
[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), 706-740. doi: 10.1137/0732033.
[22]	K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV, Nonlinear problems, SIAM J. Numer. Anal., 32 (1995), 1729-1749. doi: 10.1137/0732078.
[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), 611-643.
[24]	L. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, AMS, Providence, RI, 1998.
[25]	R. Falk, Approximation of a class of otimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28-47. doi: 10.1016/0022-247X(73)90022-X.
[26]	A. Fursikov, "Optimal Control of Distributed Systems. Theory and Applications," Translations of Mathematical Monographs, 187, AMS, Providence, RI, 2000.
[27]	V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms," Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986.
[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.
[29]	M. D. Gunzburger, "Perspectives in Flow Control and Optimization," Advances in Design and Control, 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.
[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), 711-748.
[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), 1481-1512. doi: 10.1137/S0036142997329414.
[32]	F. Hecht, FreeFem++, Third edition, Version 3.13, 2011. Available from: http://www.freefem.org/ff++.
[33]	M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61. doi: 10.1007/s10589-005-4559-5.
[34]	M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow, SIAM J. Control and Optim., 40 (2001), 925-946. doi: 10.1137/S0363012999361810.
[35]	K. Ito and K. Kunisch, "Lagrange Multiplier Approach to Variational Problems and Applications," Advances in Design and Control, 15, SIAM, Philadelphia, PA, 2008.
[36]	G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control and Optim., 20 (1982), 414-427. doi: 10.1137/0320032.
[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), 477-500.
[38]	I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations," Cambridge University Press, Cambridge, 2000.
[39]	J.-L. Lions, "Some Aspects of the Control of Distributed Parameter Systems," Conference Board of the Mathematical Sciences, SIAM, 1972.
[40]	W.-B. Liu and N. Yan, A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 93 (2003), 497-521. doi: 10.1007/s002110100380.
[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), 1032-1061. doi: 10.1137/S0036142902397090.
[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), 2270-2291. doi: 10.1137/S003614290038073X.
[43]	K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems, Appl. Math. Optim., 8 (1982), 69-95. doi: 10.1007/BF01447752.
[44]	D. Meidner and B. Vexler, Adaptive space-time finite element methods for parabolic optimization problems, SIAM J. Control and Optim., 46 (2007), 116-142. doi: 10.1137/060648994.
[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), 1150-1177. doi: 10.1137/070694016.
[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), 2183-2211. doi: 10.1137/100809611.
[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, Marcel Dekker, Inc., New York, 1994.
[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), 345-386. doi: 10.1007/s00211-011-0409-9.
[49]	A. Rösch, Error estimates for parabolic optimal control problems with control constraints, Zeitschrift für Analysis und ihre Anwendunges, 23 (2004), 353-376.
[50]	V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer Series in Computational Mathematics, 25, Spinger-Verlag, Berlin, 1997.
[51]	F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems, in "Optimal Control" (Freiburg, 1991), International Series of Numerical Mathematics, 111, Birkhäuser, Basel, (1993), 57-68.
[52]	F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls, Appl. Math. Optim., 29 (1994), 309-329.
[53]	F. Tröltzsch, "Optimal Control of Partial Differential Equations: Theory, Methods and Applications," Graduate Studies in Mathematics, 112, American Mathematical Society, Providence, RI, 2010.
[54]	N. J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations, SIAM J. Numer. Anal, 47 (2010), 4680-4710. doi: 10.1137/080728378.
[55]	R. Winther, Error estimates for a Galerkin approximation of a parabolic control problem, Ann. Math. Pura Appl. (4), 117 (1978), 173-206. doi: 10.1007/BF02417890.
[56]	R. Winther, Initial value methods for parabolic control problems, Math. Comp., 34 (1980), 115-125. doi: 10.1090/S0025-5718-1980-0551293-7.
[57]	E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators," Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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