Differentially Controllable Subspace for Linear Periodic Systems

It is well known that the concepts of controllability and differential controllability are coincident for linear time-varying systems with analytic coefficients. In this note, we discuss the difference between these concepts for linear periodic continuous-time systems with piecewise-analytic coefficients. We propose the concept of differentially controllable subspace in order to compare with the controllable subspace, and then, the formulae for computing the differentially controllable subspace are derived in both integral and differential forms. Reachability and differential reachability are also discussed in a similar way. The significance of assuming piecewise-analytic coefficients for computing the differentially controllable subspace is explained in detail by a counterexample to the previous work.