Observer-Based Adaptive Stabilization of a Class of Uncertain Fractional-Order Nonlinear Systems in the Caputo-Hadamard Sense

This paper addresses the observer-based adaptive stabilization problem for a class of uncertain nonlinear systems described in the Caputo-Hadamard fractional-order sense. By applying fractional Lyapunov theory, we develop an output-feedback observer and an adaptive controller that compensate for an unknown nonlinear growth bound. The proposed scheme ensures practical generalized Mittag-Leffler stability (PGMLS) of all closed-loop signals. For numerical validation, the closed-loop dynamics are discretized with the L1 scheme on the logarithmic time grid (Caputo-Hadamard L1 scheme). The simulation results support the theoretical findings, showing accurate state estimation, a Mittag-Leffler-type decay of the estimation error, and boundedness of both the adaptive parameter and the control input.