| Abstract: | We consider the classification of generalized linear controllable systems over the field  = ℂ or  = ℝ under transformations defined by the action of the group GL n (  ) × GL n (  ). We review the recent results of Cobb, Helmke, Shayman, Zhou, Hinrichsen, O’Halloran, and others on the geometric structure of the set of orbits C n,m (  ) of generalized linear controllable systems, which in particular prove smoothness, compactness, and projectivity of C n,m (  ) and evaluate its dimension. We show that C n,m (  ) is a natural compactification of the set of orbits of ordinary linear controllable systems ∑ n,m (  ) and the boundary C n,m (  ) − ∑ n,m (  ) consists of the orbits of singular generalized systems. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 153–166, 2004. |
