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MS 1 MINISYMPOSIUM: ADVANCED IMAGING FOR INDUSTRIAL APPLICATION Read chapter Read first chapter

2016 | Supplement | Chapter

HJB-POD Feedback Control for Navier-Stokes Equations

Authors : Alessandro Alla, Michael Hinze

Published in: Progress in Industrial Mathematics at ECMI 2014

Publisher: Springer International Publishing

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Abstract

In this report we present the approximation of an infinite horizon optimal control problem for the evolutive Navier-Stokes system. The method is based on a model reduction technique, using a POD approximation, coupled with a Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function of the corresponding control problem for the reduced system. Although the approximation schemes available for the HJB are shown to be convergent for any dimension, in practice we need to restrict the dimension to rather small numbers and this limitation affects the accuracy of the POD approximation. We will present numerical tests for the control of the time-dependent Navier-Stokes system in two-dimensional spatial domains to illustrate our approach and to show the effectiveness of the method.

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previous chapter Analysis of the Contraction Condition in the Co-simulation of a Specific Electric Circuit
next chapter MS 35 MINISYMPOSIUM: PARTICLE METHODS AND THEIR APPLICATIONS
Literature
1.
Alla, A., Falcone, M.: An adaptive POD approximation method for the control of advection-diffusion equations. In: Control and Optimization with PDE Constraints. International Series of Numerical Mathematics. Birkhauser, Basel (2013)CrossRefMATH
2.
Alla, A., Falcone, M.: A time-adaptive POD method for optimal control problems. In: Proceedings of the 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations (2013)
3.
Atwell, J.A., King, B.B.: Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33, 1–19 (2001)MathSciNetCrossRefMATH
4.
Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Basel (1997)CrossRefMATH
5.
Chaturantabut, S., Sorensen, D.C.: Discrete empirical interpolation for nonlinear model reduction. SIAM J. Sci. Comput. 32, 2737–2764 (2010)MathSciNetCrossRefMATH
6.
Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, Philadelphia, PA (2013)CrossRefMATH
7.
Kunisch, K., Volkwein, S.: Control of Burgers’ equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102, 345–371 (1999)MathSciNetCrossRefMATH
8.
Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117–148 (2001)MathSciNetCrossRefMATH
9.
Kunisch, K., Xie, L.: POD-based feedback control of Burgers equation by solving the evolutionary HJB equation. Comput. Math. Appl. 49, 1113–1126 (2005)MathSciNetCrossRefMATH
10.
Kunisch, K., Volkwein, S., Xie, L.: HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 4, 701–722 (2004)MathSciNetCrossRefMATH
11.
12.
Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I–II. Q. Appl. Math. XVL, 561–590 (1987)
13.
Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. American Mathematical Society, Philadelphia, PA (2001)CrossRefMATH
14.
Metadata
Title
HJB-POD Feedback Control for Navier-Stokes Equations
Authors
Alessandro Alla
Michael Hinze
Copyright Year
2016
Book
Progress in Industrial Mathematics at ECMI 2014

Print ISBN: 978-3-319-23412-0

Electronic ISBN: 978-3-319-23413-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-23413-7_120

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