On controllability of fractional-order impulsive and switching systems with time delay

This article targets at addressing the controllability problem of a new introduced fractional-order impulsive and switching systems with time delay (FOISSTD). Toward this end, the algebraic method is adopted to establish the relevant controllability conditions. First, we obtain the solution representation of FOISSTD over every subinterval by resorting to the successive iterations and Laplace transform. Next, by introducing the Gramian matrices over every subinterval, we establish the controllability conditions without requiring the constraint that all impulse-dependent matrices are nonsingular. Furthermore, by introducing a row matrix composed by several Gramian matrices, we further construct a controllability test that is proved to be sufficient and necessary. In the last, a calculative example with three subsystems is worked out to confirm the theoretical controllability tests.