Results on approximate controllability sobolev-type differential system of order σ ∈ (0, 1) in a hilbert space

This paper is devoted to the study of the approximate controllability for a class of Sobolev-type fractional differential systems of order σ ∈ (0, 1) in a separable Hilbert space. The proposed control system is governed by the Hilfer fractional derivative, which provides a unified framework that interpolates between the Riemann.Liouville and Caputo fractional derivatives. By employing resolvent operator theory and semigroup techniques combined with suitable fixed-point arguments, sufficient conditions for the existence of mild solutions and approximate controllability are established under nonlocal initial conditions. Unlike many existing results, the present analysis does not rely on the compactness assumption of the associated semigroup, thereby extending the applicability of the controllability criteria to a broader class of systems with implicit dynamics. The obtained results are further extended to Sobolev-type fractional integro-differential systems. An illustrative example is presented to demonstrate the effectiveness and applicability of the theoretical findings.