The Substratum of Impulse and Hybrid Control Systems

The behavior of the run of an impulse differential inclusion, and, in particular, of a hybrid control system, is “summarized” by the “ initialization map” associating with each initial condition the set of new initialized conditions and more generally, by its “substratum”, that is a set-valued map associating with a cadence and a state the next reinitialized state. These maps are characterized in several ways, and in particular, as “set-valued” solutions of a system of Hamilton-Jacobi partial differential inclusions, that play the same role than usual Hamilton-Jacobi-Bellman equations in optimal control.