Stabilization and detection of time varying linear systems

We introduce the weaker notion of the P-stabilizability and P-detectability of the linear time varying systems. Then, by a naive differential algebraic setting, using earlier results on the characterization of the controllability and observability of those systems by Kalmans type rank conditions, it is proven that a time varying system is p-stabilizable if and only if the non-controllable part of the system is asymptotically stable. Analogously, those are P-detectable by a higher order Luenberger observer if and only if the non-observable part of the systems are asymptotically stable.