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Tikhonov regularization of optimal control problems governed by semilinear partial differential equations
Institut für Mathematik, Universität Würzburg, D97974 Würzburg, Germany 
In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive apriori regularization error estimates for the control under suitable conditions. These conditions comprise secondorder sufficient optimality conditions as well as regularity conditions on the control, which consists of a source condition and a condition on the active sets. In addition, we show that these conditions are necessary for convergence rates under certain conditions. We also consider sparse optimal control problems and derive regularization error estimates for them. Numerical experiments underline the theoretical findings.
References:
[1] 
E. Casas, Second order analysis for bangbang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 23552372. 
[2] 
E. Casas, J. C. de los Reyes and F. Tröltzsch, Sufficient secondorder optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616643. 
[3] 
E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $ L^1 $ cost functional, SIAM J. Optim., 22 (2012), 795820. 
[4] 
E. Casas and F. Tröltzsch, Secondorder and stability analysis for stateconstrained elliptic optimal control problems with sparse controls, SIAM J. Control Optim., 52 (2014), 10101033. 
[5] 
N. von Daniels, Bangbang Control of Parabolic Equations, PhD thesis, University of Hamburg, 2016. 
[6] 
N. von Daniels, Tikhonov regularization of controlconstrained optimal control problems, 2017. 
[7] 
K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bangbang controls, Comput. Optim. Appl., 51 (2012), 931939. 
[8] 
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, volume 375 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996. 
[9] 
P. E. Farrell, D. A. Ham, S. W. Funke and M. E. Rognes, Automated derivation of the adjoint of highlevel transient finite element programs, SIAM J. Sci. Comput., 35 (2013), C369C393. 
[10] 
S. W. Funke and P. E. Farrell, A framework for automated pdeconstrained optimisation, CoRR abs/1302. 3894,2013. 
[11] 
F. Liu and M. Z. Nashed, Regularization of nonlinear illposed variational inequalities and convergence rates, SetValued Anal., 6 (1998), 313344. 
[12] 
A. Neubauer, Tikhonov regularisation for nonlinear illposed problems: Optimal convergence rates and finitedimensional approximation, Inverse Problems, 5 (1989), 541557. 
[13] 
F. Pörner and D. Wachsmuth, An iterative Bregman regularization method for optimal control problems with inequality constraints, Optimization, 65 (2016), 21952215. 
[14] 
G. Stadler, Elliptic optimal control problems with $ L^1 $control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159181. 
[15] 
G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189258. 
[16] 
F. Tröltzsch, Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. 
[17] 
D. Wachsmuth, Adaptive regularization and discretization of bangbang optimal control problems, Electron. Trans. Numer. Anal., 40 (2013), 249267. 
[18] 
D. Wachsmuth and G. Wachsmuth, Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints, Control Cybernet., 40 (2011), 11251158. 
[19] 
D. Wachsmuth and G. Wachsmuth, Necessary conditions for convergence rates of regularizations of optimal control problems, In System modeling and optimization, volume 391 of IFIP Adv. Inf. Commun. Technol., pages 145154, Springer, Heidelberg, 2013. 
[20] 
G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var., 17 (2011), 858886. 
show all references
Dedicated to Prof. Dr. Eduardo Casas on the occasion of his 60th birthday
References:
[1] 
E. Casas, Second order analysis for bangbang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 23552372. 
[2] 
E. Casas, J. C. de los Reyes and F. Tröltzsch, Sufficient secondorder optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616643. 
[3] 
E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $ L^1 $ cost functional, SIAM J. Optim., 22 (2012), 795820. 
[4] 
E. Casas and F. Tröltzsch, Secondorder and stability analysis for stateconstrained elliptic optimal control problems with sparse controls, SIAM J. Control Optim., 52 (2014), 10101033. 
[5] 
N. von Daniels, Bangbang Control of Parabolic Equations, PhD thesis, University of Hamburg, 2016. 
[6] 
N. von Daniels, Tikhonov regularization of controlconstrained optimal control problems, 2017. 
[7] 
K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bangbang controls, Comput. Optim. Appl., 51 (2012), 931939. 
[8] 
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, volume 375 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996. 
[9] 
P. E. Farrell, D. A. Ham, S. W. Funke and M. E. Rognes, Automated derivation of the adjoint of highlevel transient finite element programs, SIAM J. Sci. Comput., 35 (2013), C369C393. 
[10] 
S. W. Funke and P. E. Farrell, A framework for automated pdeconstrained optimisation, CoRR abs/1302. 3894,2013. 
[11] 
F. Liu and M. Z. Nashed, Regularization of nonlinear illposed variational inequalities and convergence rates, SetValued Anal., 6 (1998), 313344. 
[12] 
A. Neubauer, Tikhonov regularisation for nonlinear illposed problems: Optimal convergence rates and finitedimensional approximation, Inverse Problems, 5 (1989), 541557. 
[13] 
F. Pörner and D. Wachsmuth, An iterative Bregman regularization method for optimal control problems with inequality constraints, Optimization, 65 (2016), 21952215. 
[14] 
G. Stadler, Elliptic optimal control problems with $ L^1 $control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159181. 
[15] 
G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189258. 
[16] 
F. Tröltzsch, Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. 
[17] 
D. Wachsmuth, Adaptive regularization and discretization of bangbang optimal control problems, Electron. Trans. Numer. Anal., 40 (2013), 249267. 
[18] 
D. Wachsmuth and G. Wachsmuth, Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints, Control Cybernet., 40 (2011), 11251158. 
[19] 
D. Wachsmuth and G. Wachsmuth, Necessary conditions for convergence rates of regularizations of optimal control problems, In System modeling and optimization, volume 391 of IFIP Adv. Inf. Commun. Technol., pages 145154, Springer, Heidelberg, 2013. 
[20] 
G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var., 17 (2011), 858886. 
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