Grobner bases approach to stability and infinitesimal V-stability of maps arising in robust control

We will investigate stability and infinitesimal V-stability properties of maps that arise in Robust Control. Commutative algebra methods for the study of the infinitesimal V-stability of the critical points of general maps are first reviewed and then applied to different maps. Grobner Bases are used to simplify the calculations allowing a practical approach to infinitesimal V-stability. These ideas are then applied to the Riccati map which depends on a solution to the Algebraic Riccati Equation. This allows us to further classify the structurally unstable solutions to the Algebraic Riccati Equation. An interesting connection between this new classification, the quadratic differential, and the eigenstructure of the Hamiltonian is proved in a dimension by dimension basis. The conjectures are supported by much evidence and are proved for the case of the 2 x 2 Algebraic Riccati Equation.