Stability of a set of matrices: a control theoretic approach

We define and investigate asymptotic stability and stabilizability of a set of matrices. Asymptotic stability of a set of matrices requires that all infinite products of matrices from that set tend to zero. Asymptotic stabilizability of a set of matrices, however, requires that there is at least one such sequence. We define the upper and lower spectral radius of a set as the two possible generalizations of the regular spectral radius. We provide necessary and sufficient conditions for asymptotic stability and stabilizability. We give some methods using Lyapunov theory and linear matrix inequalities to check asymptotic stability of a set of matrices. Finally we consider some problems from different areas of control and show that the concept of stability of matrix sets may be helpful in analysis and control design.