Abstract
Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set strategies and interior-point methods can be used in an appropriately modified way. However, the residual-type a posteriori error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators.
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Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000)
Arada, N., Raymond, J.P.: Optimal control problems with mixed control-state constraints. SIAM J. Control Optim. 39, 1391–1407 (2000)
Babuska, I., Strouboulis, T.: The Finite Element Method and Its Reliability. Clarendon Press, Oxford (2001)
Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics. ETH-Zürich. Birkhäuser, Basel (2003)
Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: Basic concepts. SIAM J. Control Optim. 39, 113–132 (2000)
Bergounioux, M., Kunisch, K.: On the structure of the Lagrange multiplier for state-constrained optimal control problems. Syst. Control Lett. 48, 169–176 (2002)
Casas, E.: Control of an elliptic problem with pointwise state controls. SIAM J. Control Optim. 24, 1309–1318 (1986)
Casas, E., Raymond, J.-P., Zidani, H.: Pontryagin’s principle for local solutions of control problems with mixed control-state constraints. SIAM J. Control Optim. 39, 1182–1203 (2000)
Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)
Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational Differential Equations. Cambridge University Press, Cambridge (1995)
Gaevskaya, A., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: A posteriori error analysis of control constrained distributed and boundary control problems. In: Fitzgibbon, W., et al. (eds.) Proceedings of Conference on Advances in Scientific Computing, Moscow, Russia, pp. 85–108. Russian Academy of Sciences, Moscow (2006)
Gaevskaya, A., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: Convergence analysis of an adaptive finite element method for distributed control problems with control constraints. In: Leugering, G., et al. (eds.) Proceedings of Conference on Optimal Control for PDEs, Oberwolfach, Germany, pp. 47–68. Birkhäuser, Basel (2007)
Günther, A., Hinze, M.: A posteriori error control of a state constrained elliptic control problem. J. Numer. Math. 16 (2008, in press)
Hintermüller, M., Hoppe, R.H.W.: Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 47, 1721–1743 (2008)
Hintermüller, M., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM COCV 14, 540–560 (2008). doi:10.1051/cocv:2007057
Hintermüller, M., Kunisch, K.: Stationary state constrained optimal control problems. Preprint IFB, Report No. 3, Karl-Franzens-University of Graz (2006)
Hoppe, R.H.W., Iliash, Y., Iyyunni, C., Sweilam, N.: A posteriori error estimates for adaptive finite element discretizations of boundary control problems. J. Numer. Anal. 14, 57–82 (2006)
Hoppe, R.H.W., Kieweg, M.: A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems. Preprint No. 16, Inst. of Math., Univ. of Augsburg (2007)
Li, R., Liu, W., Ma, H., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41, 1321–1349 (2002)
Liu, W., Yan, N.: A posteriori error estimates for distributed optimal control problems. Adv. Comput. Math. 15, 285–309 (2001)
Liu, W., Yan, N.: A posteriori error estimates for convex boundary control problems. Preprint, Institute of Mathematics and Statistics, University of Kent, Canterbury (2003)
Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Preprint, Department of Mathematics, Berlin University of Technology (2005)
Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control problems of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, 209–228 (2006)
Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2000)
Neittaanmäki, P., Repin, S.: Reliable Methods for Mathematical Modelling. Error Control and a Posteriori Estimates. Elsevier, New York (2004)
Prüfert, U., Tröltzsch, F., Weiser, M.: The convergence of an interior point method for an elliptic control problem with mixed control-state constraints. Comput. Optim. Appl. 39, 369–383 (2008)
Rösch, A., Tröltzsch, F.: Sufficient second order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim. 17, 776–794 (2006)
Tröltzsch, F.: A minimum principle and a generalized bang-bang principle for a distributed optimal control problem with constraints on the control and the state. Z. Angew. Math. Mech. 59, 737–739 (1979)
Tröltzsch, F.: Optimale Steuerung partieller Differentialgleichungen. Vieweg, Wiesbaden (2005)
Tröltzsch, F.: Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 15, 616–634 (2005)
Verfürth, R.: A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley/Teubner, New York/Stuttgart (1996)
Vexler, B., Wollner, W.: Adaptive finite element methods for elliptic optimization problems with control constraints. SIAM J. Control Optim. 47, 509–534 (2008)
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Hoppe, R.H.W., Kieweg, M. Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems. Comput Optim Appl 46, 511–533 (2010). https://doi.org/10.1007/s10589-008-9195-4
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DOI: https://doi.org/10.1007/s10589-008-9195-4