Synchronization under space and time-dependent heterogeneities

We study output synchronization of nonlinear systems subject to a class of spatially and temporally-varying heterogeneities. We begin by addressing reaction–diffusion PDEs, and show that an incremental passivity property in the nominal reaction dynamics is fundamental in guaranteeing spatial homogenization of trajectories. Our distributed control achieves homogenization by defining an internal model subsystem for each output corresponding to the respective heterogeneity. We also design a similar distributed internal model control that achieves output synchronization of a network of incrementally passive nonlinear ODE systems in the presence of heterogeneities. We prove that our control laws effectively compensate for desynchronizing effects due to the differences between the heterogeneous inputs. We illustrate our algorithms on ring oscillators and bistable systems. The main contribution of the paper is a constructive and unified approach to guaranteeing homogenization under heterogeneities for both reaction–diffusion PDE and diffusively-coupled ODE systems.