Abstract
We consider an optimal control problem where the dynamics is given by the propagation of a one-dimensional graph controlled by its normal speed. A target corresponding to the final configuration of the front is given and we want to minimize the cost to reach the target. We want to solve this optimal control problem via the dynamic programming approach but it is well known that these methods suffer from the “curse of dimensionality” so that we can not apply the method to the semi-discrete version of the dynamical system. However, this is made possible by a reduced-order model for the level set equation which is based on Proper Orthogonal Decomposition. This results in a new low-dimensional dynamical system which is sufficient to track the dynamics. By the numerical solution of the Hamilton-Jacobi-Bellman equation related to the POD approximation we can compute the feedback law and the corresponding optimal trajectory for the nonlinear front propagation problem. We discuss some numerical issues of this approach and present a couple of numerical examples.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alla, A., Falcone, M.: An adaptive POD approximation method for the control of advection-diffusion equations. In: Kunisch, K., Bredies, K., Clason, C., von Winckel, G. (eds.) Control and Optimization with PDE Constraints. International Series of Numerical Mathematics, vol. 164, pp. 1–17. Birkhäuser, Basel (2013)
Alla, A., Falcone, M., Kalise, D.: An efficient policy iteration algorithm for dynamic programming equations. SIAM J. Sci. Comput. 37, 181–200 (2015)
Alla, A., Falcone, M., Volkwein, S.: Error Analysis for POD approximations of infinite horizon problems via the dynamic programming principle. SIAM J. Control. Optim. (submitted, 2015)
Alla, A., Schmidt, A., Haasdonk, B.: Model order reduction approaches for infinite horizon optimal control problems via the HJB equation. In: Benner, P., et al. (eds.) Model Reduction of Parametrized Systems. MS&A, vol. 17. Springer International Publishing, Cham (2017). doi:10.1007/978-3-319-58786-8_21
Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Basel (1997)
Barles, G.: Solutions de Visocité des Equations de Hamilton-Jacobi. Springer, Berlin (1994)
Deckelnick, K., Elliott, C.M.: Propagation of graphs in two-dimensional inhomogeneous media. Appl. Numer. Math. 56, 3, 1163–1178 (2006)
Deckelnick, K., Elliott, C.M., Styles, V.: Optimal control of the propagation of a graph in inhomogeneous media. SIAM J. Control. Optim. 48, 1335–1352 (2009)
Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, Philadelphia (2014)
Grepl, M., Veroy, K.: A level set reduced basis approach to parameter estimation. C. R. Math. 349, 1229–1232 (2011)
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, Berlin, (2009)
Kröner, A., Kunisch, K., Zidani, H.: Optimal feedback control of undamped wave equations by solving a HJB equation. ESAIM: Control Optim. Calc. Var. 21, 442–464 (2014)
Kunisch, K., Xie, L.: POD-based feedback control of Burgers equation by solving the evolutionary HJB equation. Comput. Math. Appl. 49, 1113–1126 (2005)
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)
Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories. In: Abstract Parabolic Systems. Encyclopedia of Mathematics and Its Applications 74, vol. I. Cambridge University Press, Cambridge (2000)
Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories. In: Abstract Hyperbolic-Like Systems Over a Finite Time Horizon. Encyclopedia of Mathematics and Its Applications 74, vol. II. Cambridge University Press, Cambridge (2000)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations, Band 170. Springer, New York/Berlin (1971)
Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003)
Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)
Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I-II. Q. Appl. Math. XVL, 561–590 (1987)
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Application. American Mathematical Society, Providence (2010)
Volkwein, S.: Model reduction using proper orthogonal decomposition. Lecture Notes, University of Konstanz (2013). http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/scripts.php
Acknowledgements
The first author is supported by US Department of Energy grant number DE-SC0009324.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Alla, A., Fabrini, G., Falcone, M. (2017). A HJB-POD Approach to the Control of the Level Set Equation. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-58786-8_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58785-1
Online ISBN: 978-3-319-58786-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)