Stability and controllability for fractional damped differential equations with ψ-Hilfer derivatives

Junjie Xie
School of Mathematics, Yangzhou University, Yangzhou 225000, China
Xiaoyue Han
School of Mathematics, Yangzhou University, Yangzhou 225000, China
Zhenbin Fan
School of Mathematics, Yangzhou University, Yangzhou 225000, China

Article ID: 4591

Keywords: ψ-Hilfer fractional derivative; finite-time stability; controllability; generalized Laplace transform; fixed-point theorem

Abstract

This study investigates the existence, uniqueness, finite-time stability and controllability for a class of nonlinear ψ-Hilfer fractional differential equations with damping terms, and the main mathematical contributions are achieved through several techniques including parameter selection, reasonable application of weighting methods and transformations. Specifically, by leveraging the generalized Laplace transform method and weighted norm technique, the explicit representation of solutions is derived. The existence and uniqueness of solutions are established via the generalized Banach contraction principle and Schauder fixed point theorem. Then finite-time stability criteria are obtained via inequality estimates and fractional Gronwall inequalities, each providing complementary insights into the behavior of systems within a bounded time interval. Furthermore, we establish controllability criteria for linear and nonlinear systems, the latter relies on the successive approximation method. Finally, several numerical examples are presented to validate our findings. These examples illustrate the necessity of the two existence theorems, since their respective assumptions are not always simultaneously satisfied in practical applications. In particular, we examined the magnitude of the error generated when the controllability conditions for the nonlinear system are not satisfied. These results contribute to the broader understanding of fractional damped systems governed by ψ-Hilfer derivatives and offer practical tools for analyzing stability and controllability in complex damped dynamical systems.

Published

2026-07-27

How to Cite

Xie, J., Han, X., & Fan, Z. (2026). Stability and controllability for fractional damped differential equations with ψ-Hilfer derivatives. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4591

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Vol. 33 No. 3 (2026): In Progress

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Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

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