Total approximate controllability of non-instantaneous impulsive fractional differential systems involving Riemann-Liouville derivatives of order <inline-formula><tex-math id="M1">$ \beta \in (1,2) $</tex-math></inline-formula>

Abstract

This artifact introduces the concept of total approximate controllability in order to seek approximate controllability of the impulsive systems at break points in addition to the final point. The chosen system of inspection is a nonlinear fractional differential system affected by non-instantaneous impulses. The system is governed by Riemann-Liouville derivatives of higher-order with fixed lower limits. The appropriate integral-type initial conditions are determined differently depending on the impulsive functions. Firstly, mild solution of the concerned system is constructed, followed by a sufficient set of assumptions required to manifest the existence, uniqueness, and controllability results. Next, the definition of total approximate controllability is interposed. Further, the total approximate controllability of the concerned fractional system is established using Riemann-Liouville fractional resolvent, appropriately defined interval-wise Nemytskii operators, and an iterative technique. Lastly, an illustration is put forward to validate the proposed methodology.