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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 |
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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