Necessary conditions for structural and strong structural controllability of linear time-varying systems

In this paper, the notion of structural controllability of linear time-invariant systems is extended to the time-varying case ẋ(t) = A(t) · x(t) + B(t) · u(t). We provide two Examples which show that neither the conditions for structural controllability of time-invariant systems are necessary, nor the conditions for strong structural controllability of time-invariant systems are sufficient for the controllability of time-varying systems. We present a necessary condition for structural controllability of linear time-varying systems and in our main result a necessary condition for strong structural controllability of linear time-varying systems is given. In a previous work, this necessary condition for strong structural controllability of linear time-invariant systems was shown to be sufficient, so that the strong structural controllability of linear time-invariant systems is now characterized. We want to emphasize, that our results cover the single and the multi-input case.