Necessary and sufficient conditions for controllability and observability of switched impulsive control systems

Many practical systems in physics, biology, engineering, and information science exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. This note first introduces the impulsive behavior into switched linear systems and studies the controllability and observability of such systems. Necessary and sufficient criteria for reachability and controllability are established. It is proved that reachability is equivalent to controllability for such systems. Then, necessary and sufficient criteria for observability and determinability of such systems are established by duality. It is also proved that observability is equivalent to determinability. Our criteria are of geometric type, they can be transformed into algebraic form conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.