Numerical procedure to approximate a singular optimal control problem
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 3, pp. 461-484.

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.

DOI : 10.1051/m2an:2007028
Classification : 49L20, 49L99, 93C15, 65L70
Mots clés : multiple solutions, eikonal equation, singular optimal control problems, penalization methods, numerical approximation
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Di Marco, Silvia C.; González, Roberto L. V. Numerical procedure to approximate a singular optimal control problem. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 3, pp. 461-484. doi : 10.1051/m2an:2007028. http://www.numdam.org/articles/10.1051/m2an:2007028/

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

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

[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

[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

[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

[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

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

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

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

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