Transition rule of differentially controllable subspace for linear periodic systems

In our previous works, we have proposed the concepts of differentially controllable subspace and the differentially reachable subspace for linear periodic continuous-time systems with piecewise-analytic coefficients. Those subspaces are determined from pointwise information of coefficients matrices, and therefore, the transition rules of those subspaces are not trivial. In this note, we discuss the transition rules of those subspaces. It is shown that those subspaces are interval-wise invariant with respect to the state transition matrix.