Approximate Controllability of Abstract Discrete Fractional Systems of Order 1<α<2 via Resolvent Sequences.

We study the approximate controllability of the discrete fractional systems of order 1 < α < 2 (∗) C ∇ α u n = A u n + B v n + f (n , u n) , n ≥ 2 , subject to the initial states u 0 = x 0 , u 1 = x 1 , where A is a closed linear operator defined in a Hilbert space X, B is a bounded linear operator from a Hilbert space U into X , f : N 0 × X → X is a given sequence and C ∇ α u n is an approximation of the Caputo fractional derivative ∂ t α of u at t n : = τ n , where τ > 0 is a given step size. To do this, we first study resolvent sequences { 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 α , β n } n ∈ N 0 for some α , β > 0. Finally, we provide an example to illustrate our results. [ABSTRACT FROM AUTHOR]