A new exploration on the approximate controllability results for Hilfer fractional differential inclusions of order 1<μ<2$$ 1&lt;\mu &lt;2 $$ with Clarke's subdifferential type

Abstract

The main innovation of this article is to establish the existence and approximate controllability of Hilfer fractional differential inclusions of order using Clarke's subdifferential type in Hilbert spaces. By applying the fixed‐point theorem for multivalued maps, fractional calculus, generalized Clarke's subdifferential, and cosine families, we derive a novel existence result for the mild solutions of the Hilfer fractional differential inclusion system. Furthermore, we develop and demonstrate a new set of sufficient conditions for the approximate controllability of Hilfer fractional differential systems. These criteria are based on the assumption that the associated linear part of the system is approximately controllable. Finally, an example is provided to show the theoretical findings.