[Un contre-exemple à la caractérisation de la fonction-valeur discontinue d'un problème de contrôle réfléchi]
Nous nous intéressons à un problème de contrôle optimal déterministe réfléchi en horizon fini, avec un coût final discontinu. Dans le but d'étudier la fonction-valeur, qui est alors elle-même discontinue, nous utilisons l'approche discontinue de Barles et Perthame. Nous obtenons que la fonction-valeur est une solution de viscosité d'une équation de Hamilton–Jacobi avec condition de Neumann ; de plus, les solutions de viscosité discontinues maximale et minimale de cette équation peuvent être exprimées comme des fonctions-valeurs de problèmes de contrôle réfléchis, éventuellement relaxés. Néanmoins, par un contre-exemple, nous montrons que la fonction-valeur du problème n'est pas l'unique solution de l'équation.
We consider a finite horizon deterministic optimal control problem with reflection. The final cost is assumed to be merely a locally bounded function which leads to a discontinuous value function. We address the question of the characterization of the value function as the unique solution of an Hamilton–Jacobi equation with Neumann boundary conditions. We follow the discontinuous approach developed by Barles and Perthame for problems set in the whole space. We prove that the minimal and maximal discontinuous viscosity solutions of the associated Hamilton–Jacobi can be written in terms of value functions of control problems with reflection. Nethertheless, we construct a counter-example showing that the value function is not the unique solution of the equation.
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@article{CRMATH_2002__335_5_469_0,
author = {Ley, Olivier},
title = {A counter-example to the characterization of the discontinuous value function of control problems with reflection},
journal = {Comptes Rendus. Math\'ematique},
pages = {469--473},
publisher = {Elsevier},
volume = {335},
number = {5},
year = {2002},
doi = {10.1016/S1631-073X(02)02505-0},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02505-0/}
}
TY - JOUR AU - Ley, Olivier TI - A counter-example to the characterization of the discontinuous value function of control problems with reflection JO - Comptes Rendus. Mathématique PY - 2002 SP - 469 EP - 473 VL - 335 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02505-0/ DO - 10.1016/S1631-073X(02)02505-0 LA - en ID - CRMATH_2002__335_5_469_0 ER -
%0 Journal Article %A Ley, Olivier %T A counter-example to the characterization of the discontinuous value function of control problems with reflection %J Comptes Rendus. Mathématique %D 2002 %P 469-473 %V 335 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02505-0/ %R 10.1016/S1631-073X(02)02505-0 %G en %F CRMATH_2002__335_5_469_0
Ley, Olivier. A counter-example to the characterization of the discontinuous value function of control problems with reflection. Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 469-473. doi : 10.1016/S1631-073X(02)02505-0. http://www.numdam.org/articles/10.1016/S1631-073X(02)02505-0/
[1] Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations: a guided visit, Nonlinear Anal., Volume 20 (1993) no. 9, pp. 1123-1134
[2] Solutions de viscosité des équations de Hamilton–Jacobi, Springer-Verlag, Paris, 1994
[3] Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér., Volume 21 (1987) no. 4, pp. 557-579
[4] Semicontinuous viscosity solutions of Hamilton–Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, Volume 15 (1990) no. 12, pp. 1713-1740
[5] Optimal control and semicontinuous viscosity solutions, Proc. Amer. Math. Soc., Volume 113 (1991) no. 2, pp. 397-402
[6] Deterministic exit time control problems with discontinuous exit costs, SIAM J. Control Optim., Volume 35 (1997) no. 2, pp. 399-434
[7] Lower semicontinuous solutions of Hamilton–Jacobi–Bellman equations, SIAM J. Control Optim., Volume 31 (1993) no. 1, pp. 257-272
[8] Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ., Volume 28 (1985), pp. 33-77
[9] H. Ishii, Lectures at Brown University, 1988
[10] O. Ley, Thèse de doctorat, Université de Tours, 2001
[11] Neumann type boundary conditions for Hamilton–Jacobi equations, Duke Math. J., Volume 52 (1985), pp. 793-820
[12] Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., Volume 37 (1984), pp. 511-537
[13] O.S. Serea, Prépublication, Université de Bretagne Occidentale, 2001
[14] Discontinuous viscosity solutions to Dirichlet problems for Hamilton–Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, Volume 18 (1993) no. 9–10, pp. 1493-1514
[15] Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972
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