Stabilizability of linear impulsive systems

This paper establishes the equivalence of three stabilizability-related properties for a class of linear impulsive systems. The first involves a gramian-based condition inspired by results for time-varying, discrete-time linear systems introduced decades ago. The second is the ability to achieve closed-loop exponential stability via state feedback. Finally, the third property is exponential stability of an ´unreachable´ subsystem identified from a decomposition of the original system derived from an invariant subspace that characterizes the set of reachable states. A consequence of this analysis is that full state reachability of a linear impulsive system is not necessary for state feedback stabilization, a well-known fact for linear time-invariant systems. The main ideas of the paper are applied to the problem of synchronizing two Lorenz oscillators using underactuated impulsive control.