American Institute of Mathematical Sciences

March 2018, 8(1): 315-335. doi: 10.3934/mcrf.2018013

Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations

 Institut für Mathematik, Universität Würzburg, D-97974 Würzburg, Germany

* Corresponding author: Daniel Wachsmuth

Dedicated to Prof. Dr. Eduardo Casas on the occasion of his 60th birthday

Received  April 2017 Revised  August 2017 Published  January 2018

Fund Project: This work was supported by the German Research Foundation DFG under project grant Wa 3626/1-1

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 a-priori regularization error estimates for the control under suitable conditions. These conditions comprise second-order 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.

Citation: Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013
References:
 [1] E. Casas, Second order analysis for bang-bang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 2355-2372. [2] E. Casas, J. C. de los Reyes and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643. [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), 795-820. [4] E. Casas and F. Tröltzsch, Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls, SIAM J. Control Optim., 52 (2014), 1010-1033. [5] N. von Daniels, Bang-bang Control of Parabolic Equations, PhD thesis, University of Hamburg, 2016. [6] N. von Daniels, Tikhonov regularization of control-constrained optimal control problems, 2017. [7] K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls, Comput. Optim. Appl., 51 (2012), 931-939. [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 high-level transient finite element programs, SIAM J. Sci. Comput., 35 (2013), C369-C393. [10] S. W. Funke and P. E. Farrell, A framework for automated pde-constrained optimisation, CoRR abs/1302. 3894,2013. [11] F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6 (1998), 313-344. [12] A. Neubauer, Tikhonov regularisation for nonlinear ill-posed problems: Optimal convergence rates and finite-dimensional approximation, Inverse Problems, 5 (1989), 541-557. [13] F. Pörner and D. Wachsmuth, An iterative Bregman regularization method for optimal control problems with inequality constraints, Optimization, 65 (2016), 2195-2215. [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), 159-181. [15] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. [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 bang-bang optimal control problems, Electron. Trans. Numer. Anal., 40 (2013), 249-267. [18] D. Wachsmuth and G. Wachsmuth, Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints, Control Cybernet., 40 (2011), 1125-1158. [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 145-154, 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), 858-886.

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References:
 [1] E. Casas, Second order analysis for bang-bang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 2355-2372. [2] E. Casas, J. C. de los Reyes and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643. [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), 795-820. [4] E. Casas and F. Tröltzsch, Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls, SIAM J. Control Optim., 52 (2014), 1010-1033. [5] N. von Daniels, Bang-bang Control of Parabolic Equations, PhD thesis, University of Hamburg, 2016. [6] N. von Daniels, Tikhonov regularization of control-constrained optimal control problems, 2017. [7] K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls, Comput. Optim. Appl., 51 (2012), 931-939. [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 high-level transient finite element programs, SIAM J. Sci. Comput., 35 (2013), C369-C393. [10] S. W. Funke and P. E. Farrell, A framework for automated pde-constrained optimisation, CoRR abs/1302. 3894,2013. [11] F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6 (1998), 313-344. [12] A. Neubauer, Tikhonov regularisation for nonlinear ill-posed problems: Optimal convergence rates and finite-dimensional approximation, Inverse Problems, 5 (1989), 541-557. [13] F. Pörner and D. Wachsmuth, An iterative Bregman regularization method for optimal control problems with inequality constraints, Optimization, 65 (2016), 2195-2215. [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), 159-181. [15] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. [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 bang-bang optimal control problems, Electron. Trans. Numer. Anal., 40 (2013), 249-267. [18] D. Wachsmuth and G. Wachsmuth, Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints, Control Cybernet., 40 (2011), 1125-1158. [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 145-154, 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), 858-886.
Error $\|u_\alpha - \bar u\|_{L^2(\Omega)}$ for $f(y) = \text{exp}(y)$ in a double logarithmic plot for different values for $h$ and $\alpha$. For comparison we plotted a triangle with slope $\frac{1}{2}$.
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