Regularity and Stability Sets for Families of Sequences of Matrices

We consider the notions of Lyapunov regularity and of Lyapunov stability and asymptotic stability for a dynamics defined by a continuous 1-parameter family of sequences of matrices. In particular, we identify all classes of sets that can be the regularity set, the stability set and the asymptotic stability set of any such family. Moreover, we construct explicitly families of sequences of matrices whose regularity set, stability set or asymptotic stability set is a given set in those classes.