Practical stabilization by Robin boundary controls for fractional‐order nonlinear dynamical systems of partial differential equations (PDEs) with time‐delay

Abstract Stabilization of classical nonlinear dynamical systems (with integer‐order derivation) through boundary control is frequently studied in the literature. However, there are few works that have focused on this type of problems in the framework of fractional‐order derivation and with time‐delay. In this context, this work studies and presents a method for stabilizing a nonlinear dynamical system of PDE with time‐delay via the Caputo–Hadamard fractional derivative (NSTDCH). In fact, the NSTDCH is a generalization of the classical nonlinear dynamical system of PDE. The method consists in designing robust Robin‐type boundary controller for the practical stabilization of NSTDCH. Using some technical inequality, the properties of the Mittag–Leffler function and the Lyapunov method, some sufficient conditions for the practical stabilization of the NSTDCH by both right and left Robin boundary controls have been established. Also, by the same tools, we obtained sufficient conditions for the practical stabilization of the NSTDCH by distributed control. Note that the practical stabilization for NSTDCH through Robin boundary controls is novel. Finally, some numerical examples have been presented to show the effectiveness and robustness of Robin boundary controls for the practical stabilization of the NSTDCH.