Operator theoretic approach to the optimal distributed control problem for spatially invariant systems

This paper considers the problem of optimal distributed control of spatially invariant systems. The Banach space duality structure of the problem is characterized in terms of tensor product spaces. This complements the prior study undertaken by the authors, where the dual and pre-dual formulations were in terms of abstract spaces. Here, we show that these spaces together with the pre-annihilator and annihilator subspaces can be realized explicitly as specific tensor spaces and subspaces, respectively. The tensor space formulation leads to a solution in terms of an operator given by a tensor product. Specifically, the optimal distributed control performance for spatially invariant systems is equal to the operator induced norm of this operator. The results obtained in this paper bridge the gap between control theory and the metric theory of tensor product spaces.