Approximate Controllability of Abstract Discrete Fractional Systems of Order $$1<\alpha <2$$ via Resolvent Sequences

We study the approximate controllability of the discrete fractional systems of order \(1<\alpha <2\)

$$\begin{aligned} (*)\quad \,_C\nabla ^{\alpha } u^n=Au^n+Bv^n+f(n,u^n), \quad n\ge 2, \end{aligned}$$

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 \(X, f:{\mathbb {N}}_0\times X\rightarrow X\) is a given sequence and \(\,_C\nabla ^{\alpha } u^n\) is an approximation of the Caputo fractional derivative \(\partial ^\alpha _t\) of u at \(t_n:=\tau n,\) where \(\tau >0\) is a given step size. To do this, we first study resolvent sequences \(\{S_{\alpha ,\beta }^n\}_{n\in {\mathbb {N}}_0}\) 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 \(\{S_{\alpha ,\beta }^n\}_{n\in {\mathbb {N}}_0}\) for some \(\alpha ,\beta >0.\) Finally, we provide an example to illustrate our results.