Networked control systems and distributed parameter systems can be viewed as instances of dynamical systems distributed in discrete and continuum space, respectively. This unified perspective provides insightful connections, and motivates new questions in both areas. Owing to the large number of degrees of freedom, these systems often display complex dynamical responses and phenomena.
Understanding these responses is an important challenge for analysis; mitigating these responses and quantifying fundamental performance limitations in the presence of architectural constraints on distributed controllers is the challenge for synthesis.
I will summarize some new directions in distributed systems research by outlining fascinating connections between distributed systems theory on the one hand, and canonical problems in turbulence and statistical mechanics on the other. In one class of problems, spatio-temporal dynamical analysis clarifies old and vexing questions in the theory of shear flow turbulence. In another class of problems, structured, distributed control design exhibits dimensionality-dependence and phase transition phenomena similar to those in statistical mechanics. It appears that such structured design problems, while difficult and non-convex for finite size systems, have sharp answers in the large system limit. I will argue that such results can be used to build a theory of fundamental performance limitations that are induced by network/spatial topological constraints.
These new directions provide exciting research opportunities and suggest that contact with other disciplines enriches both applications and theory of networked and distributed parameter systems. The study of systems with special structure provides informative answers to difficult analysis and synthesis problems. The systems theory and applications for such classes of problems are arguably still in their infancy, and many challenges with significant intellectual and societal impact remain wide open.