Numerical procedure to approximate a singular optimal control problem

Silvia C. Di Marco; Roberto L.V. González

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 3, page 461-484
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Di Marco, Silvia C., and González, Roberto L.V.. "Numerical procedure to approximate a singular optimal control problem." ESAIM: Mathematical Modelling and Numerical Analysis 41.3 (2007): 461-484. <http://eudml.org/doc/250069>.

@article{DiMarco2007,
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. },
author = {Di Marco, Silvia C., González, Roberto L.V.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Multiple solutions; eikonal equation; singular optimal control problems; penalization methods; numerical approximation.; Hamilton-Jacobi-Bellman equation; maximal solution},
language = {eng},
month = {8},
number = {3},
pages = {461-484},
publisher = {EDP Sciences},
title = {Numerical procedure to approximate a singular optimal control problem},
url = {http://eudml.org/doc/250069},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Di Marco, Silvia C.
AU - González, Roberto L.V.
TI - Numerical procedure to approximate a singular optimal control problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/8//
PB - EDP Sciences
VL - 41
IS - 3
SP - 461
EP - 484
AB - 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.
LA - eng
KW - Multiple solutions; eikonal equation; singular optimal control problems; penalization methods; numerical approximation.; Hamilton-Jacobi-Bellman equation; maximal solution
UR - http://eudml.org/doc/250069
ER -

References

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  1. G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal.4 (1991) 271–283.  
  2. M.J. Brooks and K.P. Horn, Shape from shading. MIT Press, Cambridge, MA (1989).  
  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.  
  4. F. Camilli and A. Siconolfi, Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems. Indiana Univ. Math. J.48 (1999) 271–283.  
  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: M2AN33 (1999) 23–54.  
  7. 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).  
  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.  
  10. P.-L. Lions, E. Rouy and A. Tourin, Shape from shading, viscosity solutions and edges. Numer. Math.64 (1993) 323–353.  
  11. J. Sethian, Fast marching methods. SIAM Rev.41 (1999) 199–235.  
  12. H. Whitney, A function not constant on a connected set of critical points. Duke Math. J.1 (1935) 514–517.  

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