Approximate Controllability of Abstract Discrete Fractional Systems of Order $$1

We study the approximate controllability of the discrete fractional systems of order $$1 0$$ τ > 0 is a given step size. To do this, we first study resolvent sequences $$\{S_{\alpha ,\beta }^n\}_{n\in {\mathbb {N}}_0}$$ { S α , β n } n ∈ N 0 generated by closed linear operators to obtain some subordination results. In addition, we discuss the existence of solutions to $$(*)$$ ( ∗ ) and next, we study the existence of optimal controls to obtain the approximate controllability of the discrete fractional system $$(*)$$ ( ∗ ) in terms of the resolvent sequence $$\{S_{\alpha ,\beta }^n\}_{n\in {\mathbb {N}}_0}$$ { S α , β n } n ∈ N 0 for some $$\alpha ,\beta >0.$$ α , β > 0 . Finally, we provide an example to illustrate our results.