Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T13:05:53.513Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical procedure to approximate a singular optimal control problem

Published online by Cambridge University Press:  02 August 2007

Silvia C. Di Marco  and
Roberto L.V. González
Silvia C. Di Marco
Affiliation:
CONICET – Dpto. Matemática, FCEIA, Universidad Nacional de Rosario, Rosario, Argentine. dimarco@fceia.unr.edu.ar
Roberto L.V. González
Affiliation:
CONICET – Dpto. Matemática, FCEIA, Universidad Nacional de Rosario, Rosario, Argentine. dimarco@fceia.unr.edu.ar
Get access

Abstract

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.

Keywords

Multiple solutionseikonal equationsingular optimal control problemspenalization methodsnumerical approximation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 3 , May 2007 , pp. 461 - 484
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barles, G. and Souganidis, P.E., Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271–283.
M.J. Brooks and K.P. Horn, Shape from shading. MIT Press, Cambridge, MA (1989).
Camilli, F. and Grüne, L., Numerical approximation of the maximal solutions for a class of degenerate Hamilton-Jacobi equations. SIAM J. Num. Anal. 38 (2000) 1540–1560. CrossRef
Camilli, F. and Siconolfi, A., Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems. Indiana Univ. Math. J. 48 (1999) 271–283. CrossRef
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.
Di Marco, S.C. and González, R.L.V., Minimax optimal control problems. Numerical analysis of the finite horizon case. ESAIM: M2AN 33 (1999) 23–54. CrossRef
S.C. Di Marco and R.L.V. González, Numerical approximation of a singular optimal control problem. Anales de Simposio Argentino en Investigación Operativa, 32° Jornadas Argentinas de Informática e Investigación Operativa, Sociedad Argentina de Informática e Investigación Operativa (2003).
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).
Ishii, H. and Ramaswamy, M., Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients. Comm. Partial Diff. Eq. 20 (1995) 2187–2213. CrossRef
Lions, P.-L., Rouy, E. and Tourin, A., Shape from shading, viscosity solutions and edges. Numer. Math. 64 (1993) 323–353. CrossRef
Sethian, J., Fast marching methods. SIAM Rev. 41 (1999) 199–235. CrossRef
Whitney, H., A function not constant on a connected set of critical points. Duke Math. J. 1 (1935) 514–517. CrossRef