September  2015, 35(9): 4439-4453. doi: 10.3934/dcds.2015.35.4439

Dynamic programming using radial basis functions

1. 

Center for Mathematics, Technische Universität München, 85747 Garching bei München, Germany, Germany

Received  May 2014 Revised  October 2014 Published  April 2015

We propose a discretization of the optimality principle in dynamic programming based on radial basis functions and Shepard's moving least squares approximation method. We prove convergence of the value iteration scheme, derive a statement about the stability region of the closed loop system using the corresponding approximate optimal feedback law and present several numerical experiments.
Citation: Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439
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show all references

References:
[1]

http://www-m3.ma.tum.de/Allgemeines/JuSch_DP_RBF, ., ().   Google Scholar

[2]

J. Glob. Opt., 52 (2012), 305-322. doi: 10.1007/s10898-011-9667-4.  Google Scholar

[3]

Birkhäuser, Boston, Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[4]

Princeton University Press, Princeton, NJ, 1957.  Google Scholar

[5]

Prentice Hall Inc., Englewood Cliffs, NJ, 1987.  Google Scholar

[6]

Appl. Math. Optim., 10 (1983), 367-377. doi: 10.1007/BF01448394.  Google Scholar

[7]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 161-183.  Google Scholar

[8]

Comput. Vis. Sci., 7 (2004), 15-29. doi: 10.1007/s00791-004-0124-5.  Google Scholar

[9]

J. Comp. Phys., 196 (2004), 327-347. doi: 10.1016/j.jcp.2003.11.010.  Google Scholar

[10]

Appl. Math. Optim., 15 (1987), 1-13. doi: 10.1007/BF01442644.  Google Scholar

[11]

Numer. Math., 67 (1994), 315-344. doi: 10.1007/s002110050031.  Google Scholar

[12]

in Nonlinear variational problems and partial differential equations (Isola d'Elba, 1990), vol. 320 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, (1995), 197-209.  Google Scholar

[13]

World Scientific, 2007. doi: 10.1142/6437.  Google Scholar

[14]

J. Diff. Eq. Appl., 13 (2007), 523-546. doi: 10.1080/10236190601135209.  Google Scholar

[15]

Journal of Approximation Theory, 153 (2008), 184-211. doi: 10.1016/j.jat.2008.01.007.  Google Scholar

[16]

E. Gottzein, R. Meisinger and L. Miller, Anwendung des "Magnetischen Rades" in Hochgeschwindigkeitsmagnetschwebebahnen,, ZEV-Glasers Annalen, 103 ().   Google Scholar

[17]

Numer. Math., 75 (1997), 319-337. doi: 10.1007/s002110050241.  Google Scholar

[18]

Numer. Math., 99 (2004), 85-112. doi: 10.1007/s00211-004-0555-4.  Google Scholar

[19]

Syst. Cont. Lett., 54 (2005), 169-180. doi: 10.1016/j.sysconle.2004.08.005.  Google Scholar

[20]

Automatica, 42 (2006), 2201-2207. doi: 10.1016/j.automatica.2006.07.013.  Google Scholar

[21]

ESAIM Control Optim. Calc. Var., 10 (2004), 259-270. doi: 10.1051/cocv:2004006.  Google Scholar

[22]

SIAM J. Numer. Anal., 42 (2005), 2612-2632. doi: 10.1137/S0036142902419600.  Google Scholar

[23]

Mat. Sb. (N.S.), 98 (1975), 450-493, 496.  Google Scholar

[24]

IEEE Trans. Auto. Ctrl., 51 (2006), 1249-1260. doi: 10.1109/TAC.2006.878720.  Google Scholar

[25]

Birkhäuser, Boston, 2006.  Google Scholar

[26]

Numerical Algorithms, 24 (2000), 239-254. doi: 10.1023/A:1019105612985.  Google Scholar

[27]

Technical report, DTIC Document, 2000. Google Scholar

[28]

SIAM J. Numer. Anal., 41 (2003), 325-363. doi: 10.1137/S0036142901392742.  Google Scholar

[29]

in Proc. 23rd ACM, ACM, (1968), 517-524. doi: 10.1145/800186.810616.  Google Scholar

[30]

Adv. Comp. Math., 4 (1995), 389-396. doi: 10.1007/BF02123482.  Google Scholar

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