A counter-example to the characterization of the discontinuous value function of control problems with reflection

Ley, Olivier

Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 469-473.

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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