Controllability Analysis of Hilfer Fractional Systems with State-Dependent Delays and Neural Network Approximation of Nonlinear Trajectories

This research investigates the controllability of Hilfer fractional dynamical systems with state-dependent delays in Rn, including systems with a linear control component and nonlinear delayed dynamics. Using Mittag–Leffler matrix functions, a mild solution representation is established. For the system with a state-dependent delay and a linear control component, controllability is studied through the associated controllability Gramian and its invertibility. The Gramian’s invertibility alone does not establish controllability of the full system; the sufficient conditions also account for the state-dependent delayed term. For the nonlinear system, a sufficient condition for relative controllability is established using the Schauder fixed-point theorem, provided that the associated operator is well-defined and continuous; maps a suitable nonempty, closed, bounded, and convex set into itself; and has a relatively compact image, as assumed in Theorem 2. The theoretical results are illustrated through a nonlinear numerical example involving fractional dynamics, nonlinear delayed interactions, and a state-dependent delay. Furthermore, an artificial neural network framework is developed to approximate the resulting numerical state trajectories. Its performance is evaluated using independent test data, trajectory comparisons, absolute-error analysis, and regression metrics. The root mean squared error on the test data is 0.009669, while the coefficients of determination are 0.999961 for the first state component and 0.999121 for the second state component. These results demonstrate the effectiveness of the proposed analytical framework for investigating controllability and the artificial neural network approach for approximating computed trajectories of nonlinear Hilfer fractional systems with memory- and state-dependent delay effects.