On duality in some problems of geometric control

The paper focuses on the analysis of duality theory in the functional, or module theoretic, approach to geometric control. Various results, previously obtained, on the characterization of controlled and conditioned invariant subspaces are related by duality. The duality is not a simple using adjoint maps. The difficulties stem from the fact that we want all characterizations to be based on left matrix fractions. Such characterizations are close to autoregressive representations of behaviors. To obtain all characterizations to be based on left matrix fractions we have to recourse to a two step process involving isomorphisms of polynomial and rational models as well as the use of dual spaces. Doubly coprime factorizations play a significant role and help to illuminate the role of behaviors in this duality theory.