Graph-Memory Selective State-Space Koopman Predictive Control of Coupled Fractional-Order Rossler Networks: A Computational Study

This paper develops a graph-structured, memory-aware Koopman model predictive control framework for chaos suppression and synchronization in coupled fractional-order Rössler networks. The continuous fractional dynamics are interpreted in the Caputo sense with lower terminal t 0 = 0 and are simulated using an explicitly initialized Grünwald–Letnikov convolution. The proposed lifted representation combines the centered physical state, a fixed normalized graph-propagation operator, an initialized fractional-memory feature, a Mamba-inspired selective state-space recurrence, and nonlinear observables. The graph and selective state-space feature maps are fixed after seeded initialization; only the Koopman matrices are identified from trajectory data by ridge regression. For the principal N = 6 network, the proposed lifted representation has dimension 109, compared with 55 for the conventional memory-free Koopman reference. On the held-out test set, the physical one-step prediction RMSE decreases from 0.039373 to 0.014456, corresponding to an approximately 63.3% reduction. A separate 109-dimensional matched ablation is used to distinguish representation effects from latent-dimension effects. In nominal closed-loop operation, the proposed controller yields post-transient regulation and synchronization errors of 0.18277 and 0.057292, respectively, compared with 0.33372 and 0.084974 for standard Koopman MPC. PID achieves a smaller nominal regulation error (0.15075), whereas the proposed method achieves a smaller synchronization error and uses substantially less accumulated control energy (7.5255 versus 13.502). Across seven Monte-Carlo uncertainty scenarios, the proposed method reduces the mean post-transient regulation and synchronization errors from 0.62171 and 0.16594 for standard Koopman MPC to 0.34445 and 0.11859, respectively. PID remains the strongest pure robust fixed-point regulator. The rate-aware composite robustness score of the proposed and standard Koopman controllers is very close and changes ordering under stronger energy/rate weighting. The resulting evidence therefore supports the proposed framework as a graph-aware and memory-aware predictive-control strategy with improved predictive accuracy and favorable regulation–synchronization–smoothness trade-offs, rather than as a universally superior replacement for classical feedback control.