Qualitative analysis on approximate controllability of impulsive Hilfer fractional hemivariational inequalities

This manuscript explores the approximate controllability of Hilfer fractional hemivariational inequalities with non‐instantaneous impulses. The study is motivated by the need to establish a realistic and mathematically rigorous framework for fractional control systems by incorporating the non‐instantaneous nature of impulses, where impulsive effects act over finite time intervals rather than instantaneously, thereby providing a more accurate representation of real‐world phenomena. It captures systems exhibiting non‐smooth and non‐convex behavior, which significantly enhances the modeling effectiveness. Within the setting of fractional calculus, the model extends classical control frameworks by incorporating memory effects together with non‐smooth dynamics. To effectively describe these characteristics, a Banach space of piecewise continuous functions is introduced, which enables a precise and rigorous formulation of such systems. To prove the existence of a mild solution for the proposed system, sufficient conditions are established using a fixed‐point theorem for the multivalued maps. The approximate controllability is then proven through a piecewise‐defined control function under the assumption that the linear system corresponding to the proposed control problem is approximately controllable. The analysis relies on generalized Clarke directional derivative, generalized Clarke subdifferential, fractional calculus, and semigroup theory. An example is presented in the end as an application to illustrate the effectiveness of the main findings.