Relative controllability of linear state-delay fractional systems

In this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new \(\alpha \) $\alpha$ -Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples.