On the Lie Algebra of a Linear Impulsive System

Recent investigations have focused on characterizing uniform exponential stability of linear impulsive systems in terms of properties of an associated Lie algebra together with familiar stability criteria. Of interest in this regard is determining whether this Lie algebra is solvable. Necessary and sufficient conditions have previously been derived that explicitly involve the system data but are nonconstructive and thus potentially difficult to verify. In this paper, necessary and sufficient conditions for solvability are once again cast directly in terms of the system description, except these new conditions are constructive. Our approach is motivated by a decades old result that addressed the problem of finding a common eigenvector of a pair of matrices.