P-type iterative learning control for impulsive pantograph equations with Hilfer fractional derivative

This paper investigates the application of $ P $-type iterative learning control (ILC) to pantograph equations with impulsive effects, incorporating the Hilfer fractional derivative. The focus is on achieving accurate trajectory tracking with an initial state offset constraint over a finite time interval. A bounded learning law is employed to ensure stability and convergence of the tracking error, even under varying reference trajectories. The theoretical results are validated using detailed numerical examples, which demonstrate the efficiency of the proposed approach in reducing the tracking error across iterations.