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In a recent article, we presented a framework to control nonlinear partial differential equations (PDEs) by means of Koopman operator based reduced models and concepts from switched systems.
Hamiltonian Systems and Transformations in Hilbert Space
B. O. Koopman · 1931
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Turbulence and the dynamics of coherent structures part I: coherent structures
L. Sirovich · 1987
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Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition
K. Kunisch and S. Volkwein · 1999
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A dynamic programming approach for optimal control of switched systems
X. Xu and P. Antsaklis · 2000
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Computational Methods for Fluid Dynamics
J. H. Ferziger and M. Peric · 2002
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Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition
C. W. Rowley · 2005
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Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control
M. Hinze and S. Volkwein · 2005
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Spectral Properties of Dynamical Systems, Model Reduction and Decompositions
I. Mezić · 2005
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OpenFOAM : A C++ Library for Complex Physics Simulations
H. Jasak, A. Jemcov, and Z. Tukovic · 2007
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Optimization and Control of Bilinear Systems
P. M. Pardalos and V. Yatsenko · 2008
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Spectral analysis of nonlinear flows
C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson · 2009
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Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control
S. Sager · 2009
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Bilinear Control Systems - Matrices in Action
D. L. Elliott · 2009
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Dynamic mode decomposition of numerical and experimental data
P. J. Schmid · 2010
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Applied Koopmanism
M. Budišić, R. Mohr, and I. Mezić · 2012
Cited alongside, same era.
The localized reduced basis multiscale method
F. Albrecht, B. Haasdonk, S. Kaulmann, and M. Ohlberger · 2012
Cited alongside, same era.
Analysis of Fluid Flows via Spectral Properties of the Koopman Operator
I. Mezić · 2013
Cited alongside, same era.
Model order reduction in fluid dynamics : challenges and perspectives
T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza · 2014
Cited alongside, same era.
On Dynamic Mode Decomposition: Theory and Applications
J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz · 2014
Cited alongside, same era.
Closed-Loop Turbulence Control: Progress and Challenges
S. L. Brunton and B. R. Noack · 2015
Cited alongside, same era.
Machine Learning Control – Taming Nonlinear Dynamics and Turbulence
T. Duriez, S. L. Brunton, and B. R. Noack · 2017
Later among the works it cites.
Data-driven discovery of Koopman eigenfunctions for control
E. Kaiser, J. N. Kutz, and S. L. Brunton · 2017
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Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator
H. Arbabi and I. Mezić · 2017
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Chaos as an intermittently forced linear system
S. L. Brunton, B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz · 2017
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Q. Li, F. Dietrich, E. M. Bollt, and I. G. Kevrekidis · 2017
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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
P. Benner, S. Gugercin, and K. Willcox · 2015
Cited alongside, same era.
A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
M. O. Williams, I. G. Kevrekidis, and C. W. Rowley · 2015
Cited alongside, same era.
Dynamic mode decomposition with control
J. L. Proctor, S. L. Brunton, and J. N. Kutz · 2015
Cited alongside, same era.
Applied dynamic programming
R. E. Bellmann and E. D. Stuart · 2015
Cited alongside, same era.
On the numerical approximation of the Perron-Frobenius and Koopman operator
S. Klus, P. Koltai, and C. Schütte · 2016
Cited alongside, same era.
Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
S. L. Brunton, B. W. Brunton, J. L. Proctor, and J. N. Kutz · 2016
Cited alongside, same era.
Set-Oriented Multiobjective Optimal Control of PDEs using Proper Orthogonal Decomposition
D. Beermann, M. Dellnitz, S. Peitz, and S. Volkwein · 2018
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Tensor-based dynamic mode decomposition
S. Klus, P. Gelß, S. Peitz, and C. Schütte · 2018
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Generalizing Koopman Theory to allow for inputs and control
J. L. Proctor, S. L. Brunton, and J. N. Kutz · 2018
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Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
M. Korda and I. Mezić · 2018
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On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator
M. Korda and I. Mezić · 2018
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Route to chaos in the fluidic pinball
N. Deng, L. Pastur, M. Morzyński, and B. R. Noack · 2018
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Data-driven model reduction and transfer operator approximation
S. Klus, F. Nüske, P. Koltai, H. Wu, I. Kevrekidis, C. Schütte, and F. Noé · 2018
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Koopman operator-based model reduction for switched-system control of PDEs
S. Peitz and S. Klus · 2019
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