Results on Approximate Controllability of $$\Psi $$-Hilfer Fractional Control Semilinear Systems via Gronwall’s Inequality

• Published in: Complex analysis and operator theory Vol. 19; no. 6
• Main Authors: Chauhan, Raman, Singh, Karunesh, Shukla, Anurag
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.09.2025
• Subjects: Banach spaces; Controllability; Fixed points (mathematics); Nonlinear control; Operators (mathematics)
• ISSN: 1661-8254; 1661-8262
• DOI: 10.1007/s11785-025-01766-8

This study investigates the approximate controllability of semilinear systems governed by the Ψ-Hilfer fractional derivative, which generalizes various fractional operators. The analysis employs semigroup theory, Ψ-fractional Gronwall inequalities, and convergence of Cauchy sequences in Banach spaces. Without relying on classical fixed point theorems, controllability is established via operator-theoretic techniques based on the dense range of the controllability map. Nonlinear terms satisfy suitable growth conditions ensuring the existence of mild solutions. The results contribute to the theory of nonlocal fractional systems with memory effects. Illustrative examples validate the applicability of the proposed approach.