Adaptive Synchronization of Diffusively Coupled Systems

We present an adaptive algorithm that guarantees synchronization in diffusively coupled systems. We first consider compartmental systems of ODEs where variables in each compartment are interconnected through diffusion terms with like variables in other compartments. Each set of variables may have its own weighted undirected graph describing the topology of the interconnection between compartments. The link weights are updated adaptively according to the magnitude of the difference between neighboring agents connected by each link. We show that an incremental passivity property is fundamental in guaranteeing output synchronization. We next consider reaction-diffusion PDEs with Neumann boundary conditions and derive an analogous algorithm guaranteeing spatial homogenization of the solutions. We provide several numerical examples demonstrating the results.