# American Institute of Mathematical Sciences

September  2015, 35(9): 4477-4501. doi: 10.3934/dcds.2015.35.4477

## Uniform convergence of the POD method and applications to optimal control

 1 Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria, Austria

Received  May 2014 Revised  September 2014 Published  April 2015

We consider proper orthogonal decomposition (POD) based Galerkin approximations to parabolic systems and establish uniform convergence with respect to forcing functions. The result is used to prove convergence of POD approximations to optimal control problems that automatically update the POD basis in order to avoid problems due to unmodeled dynamics in the POD reduced order system. A numerical example illustrates the results.
Citation: Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477
##### References:
 [1] D. Chapelle, A. Gariah and J. Sainte-Marie, Galerkin approximation with proper orthogonal decomposition: New error estimates and illustrative examples, ESAIM: Mathematical Modelling and Numerical Analysis, 46 (2012), 731-757. doi: 10.1051/m2an/2011053. [2] S. Chaturantabut and D. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM Journal Scientific Computing, 32 (2010), 2737-2764. doi: 10.1137/090766498. [3] T. Henri, Réduction de Modèles par des Méthodes de Décomposition Orthogonale Propre, PhD thesis, Rennes, 2003. [4] T. Henri and J. Yvon, Convergence estimates of POD-Galerkin methods for parabolic problems, System modeling and optimization, IFIP Int. Fed. Inf. Process., Kluwer Acad. Publ., Boston, MA, 166 (2005), 295-306. doi: 10.1007/0-387-23467-5_21. [5] D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using POD, Mathematical and Computer Modelling, 38 (2003), 1003-1028. doi: 10.1016/S0895-7177(03)90102-6. [6] K. Kunisch and S. Volkwein, Galerkin POD methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148. doi: 10.1007/s002110100282. [7] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems, ESAIM: M2AN, 42 (2008), 1-23. doi: 10.1051/m2an:2007054. [8] J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin [u.a], 1971. [9] G. Lube, Theorie und Numerik Instationärer Probleme, 2007,, Lecture Notes, (). [10] R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford graduate texts in mathematics, Clarendon Press, 1997. [11] M. Müller, Uniform Convergence of the POD Method and Applications to Optimal Control, PhD thesis, University of Graz, 2011. [12] L. Schwartz, Analyse Hilbertienne, Hermann Paris, 1979. [13] H. Triebel, Higher Analysis, Johann Ambrosius Barth, 1992. [14] H. Triebel, Theory of Function Spaces III, Birkhäuser Verlag, 2006. [15] 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. [16] S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, Control and Cybernetics, 40 (2011), 1109-1124. [17] K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition, Balanced Model Reduction via the Proper Orthogonal Decomposition, AIAA Journal, 40 (2002), 2323-2330. doi: 10.2514/2.1570. [18] J. Wloka, Partielle Differentialgleichungen, B.G. Teubner, Stuttgart, 1982.

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##### References:
 [1] D. Chapelle, A. Gariah and J. Sainte-Marie, Galerkin approximation with proper orthogonal decomposition: New error estimates and illustrative examples, ESAIM: Mathematical Modelling and Numerical Analysis, 46 (2012), 731-757. doi: 10.1051/m2an/2011053. [2] S. Chaturantabut and D. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM Journal Scientific Computing, 32 (2010), 2737-2764. doi: 10.1137/090766498. [3] T. Henri, Réduction de Modèles par des Méthodes de Décomposition Orthogonale Propre, PhD thesis, Rennes, 2003. [4] T. Henri and J. Yvon, Convergence estimates of POD-Galerkin methods for parabolic problems, System modeling and optimization, IFIP Int. Fed. Inf. Process., Kluwer Acad. Publ., Boston, MA, 166 (2005), 295-306. doi: 10.1007/0-387-23467-5_21. [5] D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using POD, Mathematical and Computer Modelling, 38 (2003), 1003-1028. doi: 10.1016/S0895-7177(03)90102-6. [6] K. Kunisch and S. Volkwein, Galerkin POD methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148. doi: 10.1007/s002110100282. [7] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems, ESAIM: M2AN, 42 (2008), 1-23. doi: 10.1051/m2an:2007054. [8] J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin [u.a], 1971. [9] G. Lube, Theorie und Numerik Instationärer Probleme, 2007,, Lecture Notes, (). [10] R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford graduate texts in mathematics, Clarendon Press, 1997. [11] M. Müller, Uniform Convergence of the POD Method and Applications to Optimal Control, PhD thesis, University of Graz, 2011. [12] L. Schwartz, Analyse Hilbertienne, Hermann Paris, 1979. [13] H. Triebel, Higher Analysis, Johann Ambrosius Barth, 1992. [14] H. Triebel, Theory of Function Spaces III, Birkhäuser Verlag, 2006. [15] 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. [16] S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, Control and Cybernetics, 40 (2011), 1109-1124. [17] K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition, Balanced Model Reduction via the Proper Orthogonal Decomposition, AIAA Journal, 40 (2002), 2323-2330. doi: 10.2514/2.1570. [18] J. Wloka, Partielle Differentialgleichungen, B.G. Teubner, Stuttgart, 1982.
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