Relative controllability of conformable delay differential systems with linear parts defined by permutable matrices

We study relative controllability of linear and nonlinear conformable delay differential systems with linear parts defined by permutable matrices. By using a notion of delay Grammian matrix, we give a sufficient and necessary condition to examine that a linear delay controlled systems is relatively controllable. Thereafter, we construct a suitable control function for nonlinear delay controlled system, which admits us to adopt the framework of fixed point methods to investigate the same issue. More precisely, we apply Krassnoselskii?s fixed point theorem to derive a relative controllability result. Finally, two examples are presented to illustrate our theoretical results with the help of computing the desired control functions and inverse of delay Grammian matrix as well.