An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain

•A non-linear Hilfer system has been studied for mild solution with the aid of Banach contraction principle.•Mönch fixed point theorem ensures the exact controllability of the system concerned.•Existence of optimal pair verified for the Hilfer nonlinear system.•Some numerical approaches have been done to exemplify the criteria of obtained results using the software ‘mathematica’.•Comparison analysis was done with the major classifications of fractional systems. In this article, the controllability results of the non-dense Hilfer neutral fractional derivative (HNFD) are presented. The results are acknowledged using semigroup theory, fractional calculus, Banach contraction principle, and Mönch technique. Moreover, a numerical analysis is given to enhance our model.