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A posteriori error estimation for semilinearparabolic optimal control problems with application to model reduction by POD∗

Published online by Cambridge University Press:  21 January 2013

Eileen Kammann ,
Fredi Tröltzsch  and
Stefan Volkwein
Eileen Kammann
Affiliation:
Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany.. [email protected]; [email protected]
Fredi Tröltzsch
Affiliation:
Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany.. [email protected]; [email protected]
Stefan Volkwein
Affiliation:
Institut für Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany.; [email protected]
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Abstract

We consider the following problem of error estimation for the optimal control ofnonlinear parabolic partial differential equations: let an arbitrary admissible controlfunction be given. How far is it from the next locally optimal control? Under naturalassumptions including a second-order sufficient optimality condition for the (unknown)locally optimal control, we estimate the distance between the two controls. To do this, weneed some information on the lowest eigenvalue of the reduced Hessian. We apply thistechnique to a model reduced optimal control problem obtained by proper orthogonaldecomposition (POD). The distance between a local solution of the reduced problem to alocal solution of the original problem is estimated.

Keywords

Optimal controlsemilinear partial differential equationserror estimationproper orthogonal decomposition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 555 - 581
Copyright
© EDP Sciences, SMAI, 2013

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References

Références

A.C. Antoulas, Approximation of large-scale dynamical systems, Advances in Design and Control. Society for Industrial and Applied Mathematics SIAM, Philadelphia, PA (2005). With a foreword by Jan C. Willems.
Arada, N., Casas, E. and Tröltzsch, F., Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201–229. Google Scholar
Barrault, M., Maday, Y., Nguyen, N.C. and Patera, A.T., An ‘empirical interpolation’ method : application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004) 667–672. Google Scholar
P. Benner and E.S. Quintana-Ortí, Model reduction based on spectral projection methods, in Reduction of Large-Scale Systems, edited by P. Benner, V. Mehrmann, D.C. Sorensen, Lect. Notes Comput. Sci. Eng. 45 (2005) 5–48. CrossRef
Casas, E., De los Reyes, J.C. and Tröltzsch, F., Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM : J. Optim. 19 (2008) 616–643. Google Scholar
Casas, E. and Tröltzsch, F., First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations. SIAM : J. Control Optim. 48 (2009) 688–718. Google Scholar
Chaturantabut, S. and Sorensen, D.C., Nonlinear model reduction via discrete empirical interpolation. SIAM : J. Sci. Comput. 32 (2010) 2737–2764. Google Scholar
Dontchev, A.L., Hager, W.W., Poore, A.B. and Yang, B., Optimality, stability, and convergence in nonlinear control. Appl. Math. Optim. 31 (1995) 297–326. Google Scholar
Grepl, M.A. and Kärcher, M., Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C. R. Acad. Sci. Paris, Ser. I 349 (2011) 873–877. Google Scholar
Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S., Optimization with PDE Constraints. Springer-Verlag, Berlin. Math. Model. Theory Appl. 23 (2009). Google Scholar
Hinze, M. and Volkwein, S., Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319–345. Google Scholar
P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, coherent structures, dynamical systems and symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1996).
Ioffe, A.D., Necessary and sufficient conditions for a local minimum 3 : Second order conditions and augmented duality. SIAM : J. Control Optim. 17 (1979) 266–288. Google Scholar
K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008).
Kahlbacher, M. and Volkwein, S., POD aposteriori error based inexact SQP method for bilinear elliptic optimal control problems. ESAIM : M2AN 46 (2012) 491–511. Google Scholar
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117–148. Google Scholar
O. Lass and S. Volkwein, POD Galerkin schemes for nonlinear elliptic-parabolic systems (2011). Submitted.
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971).
Malanowski, K., Convergence of approximations vs. regularity of solutions for convex, control–constrained optimal control problems. Appl. Math. Optim. 8 (1981) 69–95. Google Scholar
Malanowski, K., Two-norm approach in stability and sensitivity analysis of optimization and optimal control problems. Adv. Math. Sci. Appl. 2 (1993) 397–443. Google Scholar
K. Malanowski, Ch. Büskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems, in Mathematical Programming with Data Perturbations, edited by Marcel-Dekker, Inc. Lect. Notes Pure Appl. Math. 195 (1997) 253–284.
Neitzel, I. and Vexler, B., A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120 (2012) 345–386. Google Scholar
Rösch, A. and Wachsmuth, D., Numerical verification of optimality conditions. SIAM J. Control Optim. 47 (2008) 2557–2581. Google Scholar
Rösch, A. and Wachsmuth, D., How to check numerically the sufficient optimality conditions for infinite-dimensional optimization problems, in Optimal control of coupled systems of partial differential equations, Int. Ser. Numer. Math. 158 (2009) 297–317. Google Scholar
Sachs, E.W. and Schu, M., A priori error estimates for reduced order models in finance. ESAIM : M2AN 47 (2013) 449–469. Google Scholar
Schittkowski, K., Numerical solution of a time–optimal parabolic boundary-value control problem. JOTA 27 (1979) 271–290. Google Scholar
A. Studinger and S. Volkwein, Numerical analysis of POD a posteriori error estimation for optimal control (2012).
Tonn, T., Urban, K. and Volkwein, S., Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear quadratic optimal control problem. Mathematical and Computer Modelling of Dynamical Systems, Math. Comput. Modell. Dyn. Syst. 17 (2011) 355–369. Google Scholar
Tröltzsch, F., Optimal Control of Partial Differential Equations. American Math. Society, Providence, Theor. Methods Appl. 112 (2010). Google Scholar
Tröltzsch, F. and Volkwein, S., POD a posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44 (2009) 83–115. Google Scholar
Volkwein, S., Optimal control of a phase-field model using proper orthogonal decomposition. ZAMM Z. Angew. Math. Mech. 81 (2001) 83–97. Google Scholar
S. Volkwein, Model Reduction using Proper Orthogonal Decomposition. Lecture notes, Institute of Mathematics and Statistics, University of Konstanz (2011).