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

This paper is devoted to analysis of relative controllability of semilinear delay differential systems with linear parts defined by permutable matrices. By introducing a notion of delay Grammian matrix, we give a sufficient and necessary condition to examine that a linear delay controlled system is relatively controllable, which is a generalized criterion for the classical linear controlled systems without delay. Thereafter, we construct a suitable control function for semilinear delay controlled system, which admits us to following the framework of fixed point methods to consider the same issue. More precisely, we apply Krasnoselskii's fixed point theorem to derive a relative controllability result for semilinear delay controlled systems. Finally, two numerical 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.