Classifications of linear controlled systems-a mathematical analysis approach

This paper is devoted to providing a solution to the linear, differential and topological classification problem for linear controlled systems governed by ordinary differential equations in a uniform way. The main difficulty lies in the topological classification. The topological classification for linear controlled systems governed by ordinary differential equations was first resolved by J. C. Willems mainly by an algebra approach. Here, we show an alternative approach. By means of an elementary analysis of the special form of the topological transformation function, we determine the principle topological invariants for the linear controlled system, i.e., the numbers (counting the algebraic multiplicity) of eigenvalues with negative and positive real parts for the completely uncontrollable subsystem and the Jordan block matrices corresponding to the eigenvalues with vanishing real parts for the same subsystem, and the structure of the completely controllable subsystem.