is a research project funded by the ANR call ARPEGE and specifically within the third thematic axis dedicated to control, measure and associated tools. Recall the objective of this call: "The program supports projects aiming at the development of specific technological bricks by targeting domains for which it is necessary to maintain or to develop a strong capacity of innovation, a leading role: embedded systems, large scale systems, etc". Needless to say that hybrid systems framework is the formalism suited to analysis and control design for embedded systems. The proposed research plan aims at maintaining the top high level of the involved teams in the framework of hybrid systems. This project brings together control scientists and mathematicians to investigate fundamental questions in hybrid systems theory.