Adaptive Neural Control for a Network of Parabolic PDEs With Event-Triggered Mechanism

This paper investigates the finite-time consensus problem for nonlinear parabolic networks by designing a new tracking controller. For undirected topology, the newly designed controller allows to optimize the consensus time by adjusting the parameter beta(0 10). These stages correspond to the transition of flow regimes from the viscous Darcy regime to the weak inertia regime, and further developing into the strong inertia regime. Among them, Re = 1 can be considered as the critical point for the onset of the non-Darcy flow. Furthermore, As Re increases, the evolution of b(ih) exhibits four stages influenced by fluid inertia effects and main flow width in the fracture: stability, slight increase, slight decrease, and rapid increase. Then, based on 892 sets of simulation results (Re >= 1), the expression of b(ih) was obtained using Gene Expression Programming. Compared to the four existing empirical models of b(h), the present b(ih) exhibits the highest accuracy and the lowest errors (R-2 = 0.994, MAE = 0.008, RMSE = 0.013). Finally, the proposed b(ih) is further employed to modify the Forchheimer equation. This study enhances the understanding of hydraulic conductivity in 3-D rough single fractures.