Relative controllability of damped fractional differential system with distributed delays and impulses

In this paper, we deal with the relative controllability of linear and nonlinear damped fractional differential system with distributed delays and impulses. The sufficient and necessary conditions for the relative controllability of the linear system under consideration are given by using the controllability Gramian matrix. We also prove a sufficient condition on the nonlinear term to ensure that the above system is relative controllable. Two instances are provided to verify that our theoretical results are accurate.