Approximate Controllability of Fractional Differential Systems with Nonlocal Conditions of Order $q\in ]1,2[$

TL;DR

This paper investigates the approximate controllability of fractional nonlinear differential systems with nonlocal conditions of order q in (1,2), providing new sufficient conditions and an illustrative example.

Contribution

It introduces new sufficient conditions for approximate controllability of fractional systems with nonlocal conditions, extending existing results in the field.

Findings

Established approximate controllability under new conditions

Improved upon previous results in fractional control theory

Provided an example demonstrating practical application

Abstract

This manuscript is concerned with the approximate controllability of fractional nonlinear differential equations with nonlocal conditions of order $1<q<2$ in Banach spaces. As far as we know, few articles have investigated this issue. The idea is to see under which sufficient conditions the proposed control problem is approximately controllable. The discussion is based on the theory of resolvent operator, fractional calculus techniques and Krasnoselskii's fixed point theorem under the assumption that the associated linear system is approximately controllable. The obtained results improve some existing analogous ones on this topic. Finally, an example is provided to illustrate the applications of the obtained results.