Reciprocals of Integrated Nodes.

Abstract

We survey recent results solving the classical problems of linear optimal control and normalized coprime factorizations for a wide class of infinite-dimensional linear systems called integrated nodes. These include well-posed linear systems, but form a subclass of operator nodes that were studied in Staffans [16, Section 4.7]. These results were obtained using the reciprocal approach introduced in Curtain [4], [5] and illustrate its usefulness in solving system theoretic problems for systems with unbounded input and output operators. As is customary, a system has a state space Z, an input space U and an output space Y ; we assume that all three spaces are separable Hilbert spaces. An operator node is specified by three generating operators A,B,C and a characteristic function G. These are assumed to satisfy: • A is a closed densely defined operator on Z with nonempty resolvent set. • C ∈ L(D(A), Y ) is bounded where D(A) is equiped with the graph norm. • B∗ ∈ L(D(A∗), U) is bounded where D(A∗) is equiped with the graph norm. • G : ρ(A) → L(U, Y ) satisfies the following for α, s ∈ ρ(A)