Controllability and Ulam stabilities for a fractional damped evolution equation using the ρ -Caputo derivative

This paper investigates the existence, uniqueness, stability, and controllability of mild solutions for a class of nonlinear fractional differential equations involving the ρ-Caputo derivative. By employing resolvent operator theory within Hilbert spaces, we first establish explicit representation formulas for the mild solutions. Sufficient conditions for their existence and uniqueness are then derived via the Banach fixed-point theorem. Furthermore, we analyse Ulam-Hyers, Ulam-Hyers-Rassias stability, and semi Ulam-Hyers-Rassias stability providing quantitative estimates on the robustness of solutions to perturbations. Controllability is addressed for both linear and nonlinear cases using controllability operators and Gramian techniques, establishing criteria for exact and approximate controllability. The applicability of the theoretical framework is demonstrated through an illustrative example. These findings extend existing results in fractional evolution equations and underscore the efficacy of the ρ-Caputo operator for modelling systems with memory and hereditary properties.