Stability and invariant norms of arbitrary sets of 2x2 matrices

Invariant norms, also called Barabanov norms, are defined in Rᵈ for any compact family A of d × d matrices. They correspond to the linear switching system, which is a differential equation x(t) = A(t)x(t), where A(t) ∈ A for each t. The invariant norm identifies the trajectories $x(t)$ of the fastest asymptotic growth as t→ +∞. It also solves the stability problem. This norm is difficult to construct even for a pair of matrices. We show that in case $d=2$ the invariant norm can be found explicitly for every compact matrix family A. If A does not contain a dominant matrix with a real spectrum, then this norm is always unique (up to a multiplier) and is C¹, otherwise, there may be infinitely many norms. All of them can be found and classified.