Numerical procedure to approximate a singular optimal control problem
Di Marco, Silvia C. ; González, Roberto L. V.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007), p. 461-484 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

In this work we deal with the numerical solution of a Hamilton-Jacobi-Bellman (HJB) equation with infinitely many solutions. To compute the maximal solution - the optimal cost of the original optimal control problem - we present a complete discrete method based on the use of some finite elements and penalization techniques.

Publié le : 2007-01-01
DOI : https://doi.org/10.1051/m2an:2007028
Classification:  49L20,  49L99,  93C15,  65L70 
@article{M2AN_2007__41_3_461_0,
     author = {Di Marco, Silvia C. and Gonz\'alez, Roberto L. V.},
     title = {Numerical procedure to approximate a singular optimal control problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {41},
     year = {2007},
     pages = {461-484},
     doi = {10.1051/m2an:2007028},
     mrnumber = {2355708},
     zbl = {pre05289381},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2007__41_3_461_0}
}

Di Marco, Silvia C.; González, Roberto L. V. Numerical procedure to approximate a singular optimal control problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) pp. 461-484. doi : 10.1051/m2an:2007028. http://gdmltest.u-ga.fr/item/M2AN_2007__41_3_461_0/

[1] G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271-283. | Zbl 0729.65077

[2] M.J. Brooks and K.P. Horn, Shape from shading. MIT Press, Cambridge, MA (1989). | MR 1062877

[3] F. Camilli and L. Grüne, Numerical approximation of the maximal solutions for a class of degenerate Hamilton-Jacobi equations. SIAM J. Num. Anal. 38 (2000) 1540-1560. | Zbl 0988.65077

[4] F. Camilli and A. Siconolfi, Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems. Indiana Univ. Math. J. 48 (1999) 271-283. | Zbl 0939.49019

[5] E. Cristiani and M. Falcone, Fast semi-Lagrangian schemes for the eikonal equation and applications. http://cpde.iac.rm.cnr.it/file_ uploaded/EFX30053.pdf.

[6] S.C. Di Marco and R.L.V. González, Minimax optimal control problems. Numerical analysis of the finite horizon case. ESAIM: M2AN 33 (1999) 23-54. | Numdam | Zbl 0918.65049

[7] S.C. Di Marco and R.L.V. González, Numerical approximation of a singular optimal control problem2003).

[8] S.C. Di Marco and R.L.V. González, Penalization methods in the numerical solution of the eikonal equation. Mecánica Computacional, Vol XXII, ISSN 1666-6070 (2003).

[9] H. Ishii and M. Ramaswamy, Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients. Comm. Partial Diff. Eq. 20 (1995) 2187-2213. | Zbl 0842.35019

[10] P.-L. Lions, E. Rouy and A. Tourin, Shape from shading, viscosity solutions and edges. Numer. Math. 64 (1993) 323-353. | Zbl 0804.68160

[11] J. Sethian, Fast marching methods. SIAM Rev. 41 (1999) 199-235. | Zbl 0926.65106

[12] H. Whitney, A function not constant on a connected set of critical points. Duke Math. J. 1 (1935) 514-517. | JFM 61.1117.01