Unbounded Hamilton-Jacobi-Bellman equations with one co-dimensional discontinuities

Abstract

The aim of this thesis is to deal, of the point of view of viscosity solutions, with a discontinuous Hamilton-Jacobi equation in the whole euclidian N-dimensional space where the discontinuity is located on an hyperplane. The typical questions that arise this directions are concern the existence and uniqueness of solutions, and of course the definition itself of solution. Here we consider viscosity solutions in the sense of Ishii. Since we consider convex Hamiltonians, we can also associate the problem to a control problem with specific cost and dynamics given on each side of the hyperplane. We assume that those are Lipshichitz continuous but potentially unbounded, as well as the control spaces. Using Bellman’s approach we construct two value functions which turn out to be the minimal and maximal solutions in the sense of Ishii. Moreover, we also build a whole family of value functions, which are still solutions in the sense of Ishii and connect continuously the minimal solution to the maximal one.