May 3, 2004
Dmitry Yakubovich
Title: Linearly similar Nagy--Foias model in a domain and control theory
In this talk we discuss a Nagy-Foias type model of a linear operator in a domain in the complex plane. A model of a (possibly unbounded) operator is constructed with the aid of an auxiliary operator . This model is up to the similarity; different operators give rise to different models. The model is formulated in terms of a generalized characteristic function of , which is an operator-valued function in .
This construction is closely related to the control theory. An operator can be used as an auxiliary operator in modelling operator in the left half-plane if and only if the system is infinite time exactly controllable.
Examples include all generators of groups, unbounded perturbations of unbounded self-adjoint operators and generators of neutral linear systems with delays.
We will pay special attention to a Cauchy duality between the models of and . We explain how in concrete cases, this duality helps one to prove the similarity result.
Upper and lower estimates of the least time of exact controllability in terms of certain ``mean winding numbers'' and the dimension of the control will also be discussed.