Approximation of inverse problems for fractional differential equations in abstract spaces

This article focuses on approximating a fractional-order inverse problem (IP) for an abstract differential equation in a Hilbert space. The main tools to find out the results are fixed-point methods, the $\alpha$-resolvent family, and optimal control (OC) theory. We have defined an optimal control problem corresponding to the original inverse problem, and then by using an approximate optimal control problem, we have proved that the sequence of solutions to this approximate optimal control problem will converge to the solution of our original inverse problem. Furthermore, the fractional Crank–Nicolson scheme and a matrix optimization algorithm are utilized to derive approximation results, ensuring the convergence of the proposed numerical method. Finally, an example is presented to illustrate and validate the theoretical findings.