Invariant dual cones for hybrid systems

The concepts of invariance and duality have been analyzed and exploited in systems science and related fields, both in continuous state domain and in graphs theory but we are not aware of a theory developed for hybrid systems, i.e., dynamic systems resulting from the interaction of continuous and discrete state systems. In this paper we introduce invariance concepts for hybrid cones: controlled reset invariance and conditioned reset invariance. We show that a reset conditioned invariant cone and a reset controlled invariant cone are related each other by a duality relation, obtained by associating an appropriate hybrid system to the given one. A discussion on system theoretic interpretation of these cones is provided.