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 \(1<\alpha <2\) $1<\alpha <2$ \(\begin{aligned} (*)\quad \,_C\nabla ^{\alpha } u^n=Au^n+Bv^n+f(n,u^n), \quad n\ge 2, \end{aligned}\) $$\begin{array}{c}\hfill (\ast )_C^{}{\mathrm{\nabla }}^{\alpha }{u}^n=A{u}^n+B{v}^n+f(n,u^n),n\ge 2,\end{array}$$ subject to the initial states \(u^0=x_0,u^1=x_1,\) $u^0=x_0,u^1=x_1,$ 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 \(X, f:{\mathbb {N}}_0\times X\rightarrow X\) $X,f:{\mathbb{N}}_0\times X\to X$ is a given sequence and \(\,_C\nabla ^{\alpha } u^n\) $_C^{}{\mathrm{\nabla }}^{\alpha }{u}^n$ is an approximation of the Caputo fractional derivative \(\partial ^\alpha _t\) ${\partial }_t^{\alpha }$ of u at \(t_n:=\tau n,\) $t_n:=\tau n,$ where \(\tau >0\) $\tau >0$ is a given step size. To do this, we first study resolvent sequences \(\{S_{\alpha ,\beta }^n\}_{n\in {\mathbb {N}}_0}\) ${\{S_{\alpha ,\beta }^n\}}_{n\in {\mathbb{N}}_0}$ generated by closed linear operators to obtain some subordination results. In addition, we discuss the existence of solutions to \((*)\) $(\ast )$ and next, we study the existence of optimal controls to obtain the approximate controllability of the discrete fractional system \((*)\) $(\ast )$ in terms of the resolvent sequence \(\{S_{\alpha ,\beta }^n\}_{n\in {\mathbb {N}}_0}\) ${\{S_{\alpha ,\beta }^n\}}_{n\in {\mathbb{N}}_0}$ for some \(\alpha ,\beta >0.\) $\alpha ,\beta >0.$ Finally, we provide an example to illustrate our results.