Rank condition of relative controllability for linear quaternion differential equations with delay

We focus on the relative controllability of linear quaternion differential equations (LQDEs) with delay. By delayed quaternion matrix function, the rank condition of relative controllability is obtained without the restriction of permutation matrices. Then, we get the relationship of controllability between LQDEs and LQDEs with delay. Moreover, the control function is obtained. Finally, two examples are shown to verify the results.