Many important plants (e.g. flexible manipulators or heat transfer processes) are governed by partial differential equations (PDEs) and are often described by models with a significant degree of uncertainty. Some PDEs may not be robust with respect to arbitrary small time-delays in the feedback. Robust finite-dimensional controller design for PDEs is a challenging problem. In this talk two constructive methods for finite-dimensional control will be presented: Spatial decomposition (or sampling in space) method, where the spatial domain is divided into N subdomains with N sensors and actuators located in each subdomain; Modal decomposition method, where the controller is designed on the basis of a finite-dimensional system that captures the dominant dynamics of the infinite-dimensional one. Sufficient conditions ensuring the stability and performance of the closed-loop system are established in terms of simple linear matrix inequalities that are always feasible for appropriate choice of controllers. We will discuss delayed and sampled-data implementations as well as application to network-based deployment of multi-agents.
