A sequential approach for approximate controllability of Ψ-Hilfer semilinear fractional control systems

Abstract This paper aims to study the approximate controllability of Ψ-Hilfer-type semilinear fractional control systems through a sequential approach under novel conditions. Our concept is to find a sequence of control functions in the control space (Hilbert space). That sequence would be a Cauchy sequence. The Cauchy sequence of control functions converges to a limiting control function that leads to the solution of the given system in the ɛ neighborhood of the desired state by solution mapping. The proof is obtained by using a Cauchy sequence in a Hilbert space. The nonlinear function satisfies the Lipschitz condition. The paper also provides some examples to support the proof.