Generalized linear dynamical systems and the classification problem

Abstract

We consider the classification of generalized linear controllable systems over the field F = C or F = R under transformations defined by the action of the group GLn(F)×GLn(F). We review the recent results of Cobb, Helmke, Shayman, Zhou, Hinrichsen, O'Halloran, and others on the geometric structure of the set of orbits Cn,m(F) of generalized linear controllable systems, which in particular prove smoothness, compactness, and projectivity of Cn,m(F) and evaluate its dimension. We show that Cn,m(F) is a natural compactification of the set of orbits of ordinary linear controllable systems Σn,m(F), and the boundary Cn,m(F) − Σn,m(F) consists of the orbits of singular generalized systems.