May 3, 2004

Dmitry Yakubovich

Title: Linearly similar Nagy--Foias model in a domain and control theory

Abstract:

In this talk we discuss a Nagy-Foias type model of a linear operator in a domain $ \Omega$ in the complex plane. A model of a (possibly unbounded) operator $ A$ is constructed with the aid of an auxiliary operator $ B$. This model is up to the similarity; different operators $ B$ give rise to different models. The model is formulated in terms of a generalized characteristic function of $ A$, which is an $ H^\infty$ operator-valued function in $ \Omega$.

This construction is closely related to the control theory. An operator $ B$ can be used as an auxiliary operator in modelling operator $ A$ in the left half-plane if and only if the system $ \dot x(t)=Ax(t) +B u(t)$ is infinite time exactly controllable.

Examples include all generators of $ C_0$ 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 $ A$ and $ A^*$. 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.