Linear switching systems with slow growth of trajectories

We prove the existence of positive linear switching systems (continuous time), whose trajectories grow to infinity, but slower than a given increasing function. This implies that, unlike the situation with linear ODE, the maximal growth of trajectories of linear systems may be arbitrarily slow. For systems generated by a finite set of matrices, this phenomenon is proved to be impossible in dimension 22, while in all bigger dimensions the sublinear growth may occur. The corresponding examples are provided and several open problems are formulated.