Distributed control of spatially invariant systems over Sobolev spaces

We consider spatially invariant systems where the underlying state space is an inner-product Sobolev space. Such systems arise when considering certain state-space representations of partial differential equations of higher temporal order. We show how standard results on exponential stability, stabilizability and optimal control with quadratic criteria can be generalized to those systems. These generalizations require some bookkeeping of spatial frequency weighting functions related to the Sobolev inner products, and simple recipes for doing so are given. The results are illustrated with examples of distributed control of wave and beam equations.