Résumé en anglais

In this paper we explain that various (possibly discontinuous) value functions for optimal control problem under state-constraints can be approached by a sequence of value functions for suitable discretized systems. The key- point of this approach is the characterization of epigraphs of the value functions as suitable viability kernels. We provide new results for estimation of the convergence rate of numerical schemes and discuss conditions for the convergence of discrete optimal controls to the optimal control for the initial problem.