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Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems

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  • The main focus of this paper is on an a-posteriori analysis for different model-order strategies applied to optimal control problems governed by linear parabolic partial differential equations. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the reduced-order model, is from the (unknown) exact one. For the model-order reduction, $\mathcal H_{2,\alpha}$-norm optimal model reduction (H2), balanced truncation (BT), and proper orthogonal decomposition (POD) are studied. The proposed approach is based on semi-discretization of the underlying dynamics for the state and the adjoint equations as a large scale linear time-invariant (LTI) system. This system is reduced to a lower-dimensional one using Galerkin (POD) or Petrov-Galerkin (H2, BT) projection. The size of the reduced-order system is iteratively increased until the error in the optimal control, computed with the a-posteriori error estimator, satisfies a given accuracy. The method is illustrated with numerical tests.
    Mathematics Subject Classification: Primary: 49K20, 90C20; Secondary: 35K10.


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  • [1]

    K. Afanasiev and M. Hinze, Adaptive control of a wake flow using proper orthogonal decomposition, Lect. Notes Pure Appl. Math., 216 (2001), 317-332.


    A. C. Antoulas, "Approximation of Large-Scale Dynamical Systems," SIAM, Philadelphia, (2005).doi: 10.1137/1.9780898718713.


    N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, 23 (2002), 201-219.doi: 10.1023/A:1020576801966.


    P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems, SIAM Journal on Control and Optimization , 49 (2011), 686-711.doi: 10.1137/09075041X.


    P. Benner and J. SaakA Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations, 2010. Available from: http://www.am.uni-erlangen.de/home/spp1253/wiki/index.php/Preprints.


    P. Benner and E. S. Quintana-Ortí, Model reduction based on spectral projection methods, In "Reduction of Large-Scale Systems" (eds. P. Benner, V. Mehrmann and D. C. Sorensen), Lecture Notes in Computational Science and Engineering, 45 (2005), 5-48.


    A. Bunse-Gerstner, D. Kubalinska, G. Vossen and D. Wilczek, $h_2$-norm optimal model reduction for large-scale discrete dynamical MIMO systems, Journal of Computational and Applied Mathematics, 233 (2010), 1202-1216.doi: 10.1016/j.cam.2008.12.029.


    A. L. Dontchev, W. W. Hager, A. B. Poore and B. Yang, Optimality, stability, and convergence in nonlinear control, Appl. Math. and Optim., 31 (1995), 297-326.doi: 10.1007/BF01215994.


    K. Glover, All optimal Hankel-norm approximations of linear multi-variable systems and their $L_\infty$ error bounds, International Journal of Control, 39 (1984), 1115-1193.doi: 10.1080/00207178408933239.


    M. A. Grepl and M. Kärcher, 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.


    S. Gugercin, A. C. Antoulas and C. A. Beattie, $H_2$ model reduction for large-scale linear dynamical systems, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 609-638.doi: 10.1137/060666123.


    M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition, Comput. Optim. and Appl., 39 (2008), 319-345.doi: 10.1007/s10589-007-9058-4.


    P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry," Cambridge Univ. Press, New York, 1996.doi: 10.1017/CBO9780511622700.


    M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semi-smooth Newton method, SIAM J. Optimization, 13 (2003), 865-888.


    C. Joerres, G. Vossen and M. Herty, On an inexact gradient method using POD for a parabolic optimal control problem, submitted, 2011.


    E. A. Jonckheere and L. M. Silverman, A new set of invariants for linear systems - Application to reduced order compensator design, IEEE Trans. Automat. Control, 28 (1983), 953-964.doi: 10.1109/TAC.1983.1103159.


    E. Kammann, F. Tröltzsch and S. Volkwein, A method of a-posteriori error estimation with application to proper orthogonal decomposition, submitted, 2011.


    D. Kubalinska, "Optimal Interpolation-Based Model Reduction," PhD thesis, University of Bremen, 2008.


    K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148.doi: 10.1007/s002110100282.


    K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems, ESAIM: Mathematical Modelling and Numerical Analysis, 42 (2008), 1-23.doi: 10.1051/m2an:2007054.


    E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction, Statistical Forecasting Scientific Rep. 1, Department of Meteorology, Massachusetts Institute of Technology, Cambridge, MA, 1956.


    L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera and D. V. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, CR Acad Sci Paris Series I, 331 (2000), 1531-1548.


    K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems, in "Mathematical Programming with Data Perturbation" (eds. A. V. Fiacco and Marcel Dekker), Inc., New York, (1997), 253-284.


    H. Maurer and J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Mathematical Programming, 16 (1979), 98-110.


    L. Meier and D. Luenberger, Approximation of linear constant systems, IEEE Transactions on Automatic Control, 12 (1967), 585-588.doi: 10.1109/TAC.1967.1098680.


    B. C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Automatic Control, 26 (1981), 17-32.doi: 10.1109/TAC.1981.1102568.


    A. T. Patera and G. Rozza, "Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations," MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2006.


    S. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, SIAM J. Sci. Comput., 15 (2000), 457-478.


    J. C. De Los Reyes and T. Stykel, A balanced truncation based strategy for optimal control of evolution problems, Optim. Methods Software, 26 (2011), 673-694.doi: 10.1080/10556788.2010.526756.


    J. Saak, "Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction," PhD thesis, TU Chemnitz, 2009.


    E. W. Sachs and M. Schu, A priori error estimates for reduced order models in finance, submitted, 2011.


    T. Stykel, Gramian-based model reduction for descriptor systems, Math. Control Signals Systems, 16 (2004), 297-319.doi: 10.1007/s00498-004-0141-4.


    T. Tonn, K. Urban and S. Volkwein, 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, Special Issue: Model order reduction of parameterized problems, 17 (2011), 355-369.


    F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems, Computational Optimization and Applications, 44 (2009), 83-115.doi: 10.1007/s10589-008-9224-3.


    F. Tröltzsch., "Optimal Control of Partial Differential Equations. Theory, Methods and Applications," American Math. Society, Providence, 112, 2010.


    R. Usmani, Inversion of a tridiagonal Jacobi matrix, Linear Algebra Appl. , 212/213 (1994), 413-414.doi: 10.1016/0024-3795(94)90414-6.


    S. Volkwein, Model reduction using proper orthogonal decomposition, Lecture Notes, Institute of Mathematics and Statistics, University of Constance, 2011.


    S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, to appear in Control and Cybernetics, 2012.


    G. Vossen, $\mathcal H_{2,\alpha}$-norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations, submitted, 2011.

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