Periodic descriptor systems: solvability and conditionability

The authors consider discrete-time linear periodic descriptor systems and study the concepts of solvability and conditionability, introduced by Luenberger. They prove that solvability is equivalent to conditionability, just as in the time-invariant case. We give a characterization of solvability/conditionability in terms of a cyclic matrix pencil and, furthermore, propose a simple test via the periodic Schur decomposition to check for either property. This could lead to further systematic study of these systems.