Near-idempotents, near-nilpotents and stability of critical points for Riccati equations

Abstract

The paper introduces two algebraic concepts, near-idempotents and near-nilpotents associated to subspaces N of critical points, which can be used to re-phrase a theorem due to Boujemaa, El Qotbi and Rouiouih on stability for the Ricatti equation, ẋ = x(t)2, associated to algebra A ≈ R. Using this concepts their result corresponds to the case dim N = 1. Our main results are a generalization of the above mentioned theorem to N of arbitrary dimension and a counter-example which shows, even in the general setting, that the essential condition that critical points must be eigenvectors of a suitable multiplication operator cannot be omitted from the formulation due to Boujemaa et al.