Fractional Interconnected Systems with Boundary Feedback: A GNN-Based Computational Approach

In this paper, we present an applied numerical study inspired by recent theoretical advances on boundary feedback control for fractional coupled PDE–ODE systems. While earlier works have mainly focused on proving the existence, uniqueness, and stability of solutions within the fractional Lyapunov framework, our contribution lies in translating these theoretical results into a practical setting for graph neural networks (GNNs). In this model, the partial differential equation describes the diffusion of information signals across the network topology, while the fractional-order ordinary differential equation captures the nonlinear and memory-dependent update rules of the node states. By employing the backstepping algorithm, we design a boundary controller that guarantees Mittag–Leffler stability of the coupled system. Numerical simulations demonstrate that, in the absence of control, the network dynamics exhibit instability and divergence, whereas the proposed boundary control gradually stabilizes the information propagation process. These results underline the effectiveness of the method and its potential relevance for the fractional modeling and regulation of graph-based neural architectures. An example is given with a consensus problem that shows that the boundary controller stabilizes the network dynamics.