Viability kernels and capture basins of sets under differential inclusions

This paper provides a characterization of viability kernels and capture bas ins of a target viable in a constrained subset as a unique closed subset be tween the target and the constrained subset satisfying tangential condition s or, by duality, normal conditions. It is based on a method devised by Hel ene Frankowska for characterizing the value function of an optimal control problem as generalized (contingent or viscosity) solutions to Hamilton Jaco bi equations. These abstract results, interesting by themselves, can be app lied to epigraphs of functions or graphs of maps and happen to be very effi cient for solving other problems, such as stopping time problems, dynamical games, boundary-value problems for systems of partial differential equatio ns, and impulse and hybrid control systems, which are the topics of other c ompanion papers.