On the optimality of localised distributed controllers

Design of optimal distributed controllers with a priori assigned localisation constraints is a difficult problem. Alternatively, one can ask the following question: given a localised distributed exponentially stabilising controller, is it inversely optimal with respect to some cost functional? We study this problem for linear spatially invariant systems and establish a frequency domain criterion for inverse optimality (in the LQR sense). We utilise this criterion to separate localised controllers that are never optimal from localised controllers that are optimal. For the latter, we provide examples to demonstrate optimality with respect to physically appealing cost functionals. These are characterised by state penalties that are not fully decentralised and they provide insight about spatial extent of the LQR weights that lead to localised controllers.