Fully Distributed Adaptive Asymptotic Consensus Control for a Network of Parabolic PDEs

This article revisits the consensus control problem of networked parabolic partial differential equation (PDE) systems perturbed to nonlinear terms and uncertain disturbances. The presence of spatial variables and reaction–diffusion terms in the system model makes designing the adaptive distributed protocol for PDE systems more challenging than for ordinary differential dynamics. A novel adaptive distributed controller is designed by including a novel compensating term in the form of the hyperbolic tangent function and a positive integral function. The asymptotic consensus can be achieved by using the Lyapunov method and PDE theory. It should be emphasized that all three types of cases-those without a leader, those with a leader, and those with multiple leaders-are looked into. In addition, in comparison with the related works, the designed control scheme does not require the global information of the graph, which is a fully distributed paradigm. Finally, three examples are utilized to show how efficient the proposed distributed algorithm is.