Abstract
•Fractional differential systems with multiple delays are considered.•Delayed perturbation of two parameter Mittag-Leffler type matrix function is generated for a class of multiple delay differential equations.•Comparison between Mittag-Leffler, Delayed Mittag-Leffler and Delayed Permutation Mittag-Leffler function is given by pictorial form.•Properties, markable remarks and continuity are presented.•The explicit form of solution representation is constructed.•Using Delayed Perturbation Mittag-Leffler function, the Controllability Grammian matrix is defined.•Sufficient conditions of relative controllability are derived by using krasnoselskii’s theorem.•Numerical illustrations were given for both linear and non-linear cases.•Diagrammatic representations are explored using MATLAB for both linear and non-linear cases.
This paper is concerned with the relative controllability for a class of fractional differential equations with multiple time delays. The solution representation is introduced for this system via multiple delayed perturbations of Mittag-Leffler function. Necessary and sufficient conditions for the indicated problem to be relatively controllable are established for linear and non-linear systems. For non-linear case, the existence result is proved by using Krasnoselskii’s fixed point theorem. Numerical examples are given to illustrate the theoretical results, and its diagrammatic formulations are done by MATLAB.