Inverse problem for an abstract neutral differential equation of Sobolev-type

In this manuscript, we consider an inverse problem for the neutral differential equation of Sobolev type with a deviated argument and identify the parameter using an overdetermined condition on a mild solution. A direct approach using Volterra integral equations for sufficiently regular data and an optimal control approach for less regular data are the main techniques to find the result. Under certain hypotheses, the characterization of the limit of the sequence of approximate solutions demonstrates that it is a solution to the original inverse problem under specific hypotheses. At last, an example is provided in the support of our results.