A Note on Controllability of Time Invariant Linear Fractional h-Difference Equations

In this paper, we establish and prove two main results: (i) a Kalman-like controllability criterion, and (ii) a rank condition on the controllability matrix, defined via the discrete Mittag–Leffler function, for time-invariant linear fractional-order h-discrete systems. Using some properties of the Mittag–Leffler-type function within the framework of fractional h-discrete calculus, we state and prove the variation of constants formula for an initial value problem. Then we use this formula to prove the equivalence between two notions of controllability: complete controllability and controllability to the origin.