Duality of the optimal distributed control for spatially invariant systems

We consider the problem of optimal distributed control of spatially invariant systems. We develop an input-output framework for problems of this class. Spatially invariant systems are viewed as multiplication operators from a particular Hilbert function space into itself in the Fourier domain. Optimal distributed performance is then posed as a distance minimization in a general L-infinity space from a vector function to a subspace with a mixed L^∞ and H^∞ space structure. In this framework, a generalized version of the Youla parametrization plays a central role. The duality structure of the problem is characterized by computing various dual and pre-dual spaces. The annihilator and pre-annihilator subspaces are also calculated for the dual and pre-dual problems. Furthermore, the latter is used to show the existence of optimal distributed controllers and dual extremal functions under certain conditions. The dual and pre-dual formulations lead to finite dimensional convex optimizations which approximate the optimal solution within desired accuracy. These optimizations can be solved using convex programming methods. Our approach is purely input-output and does not use any state space realization.