Switched linear systems on time scales with relaxed commutativity constraints

A fundamental stability result for arbitrarily switched linear systems in continuous time assumes that the set of system coefficient matrices are commutative with one another. This result was recently generalized to include arbitrarily switched linear system on arbitrary time scales T, with additional constraints imposed upon the graininess of the time scales. In the following analysis we explore the case when pairs of switched systems are non-commutative by visualizing the space of common Lyapunov solutions graphically. We deduce that there are cases in which a common Lyapunov solution exists for a non-commutative switched system if the time scale graininess is limited to some upper bound.