Second-order linear switching systems with arbitrary control sets: stability and invariant norms

We show that the stability problem and the problem of constructing Barabanov norms can be resolved for planar linear switching systems in an explicit form. This can be done for every compact control set of $2 \times 2$ matrices. If the control set does not contain a dominant matrix with a real spectrum, then the invariant norm is always unique (up to a multiplier) and belongs to~$C^1$. Otherwise, there may be infinitely many such norms, including non-smooth ones. All of them can be found and classified. In particular, every symmetric convex body is a unit ball of the Barabanov norm of a suitable linear switching system. Several examples of control sets such as matrix Frobenius balls and matrix polyhedra are analysed.