Integral contractor approach on controllability of Hilfer–Langevin system

The primary goal of this study is to investigate the necessary conditions for the Hilfer–Langevin dynamical system with initial circumstances to be controllable. Using a sequencing technique, we have employed the concept of a bounded integral contractor to reach our results. Unlike previous studies, we do not need to specify the induced inverse of the controllability operator, and the relevant nonlinear function does not have to satisfy a Lipschitz condition to prove our accurate controllability conclusions. We have obtained a theoretical result that is shown to be critically important through an example.