Feedback Stabilization of Steady-State and Hopf Bifurcations

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Yong Wang and Richard Murray
1998 Conference on Decision and Control

In this paper we consider nonlinear systems with steady-state or Hopf bifurcations for which the bifurcated modes are linearly uncontrollable. The goal is to find state feedbacks such that the bifurcation for the closed loop system is supercritical, and at the same time, the linearly controllable subsystem is asymptotically stable. Necessary and sufficient conditions for the non-existence of sufficiently smooth state feedbacks have been obtained under certain nondegeneracy conditions. For the cases when those conditions are not satisfied, we give explicit constructions of the feedbacks. The construction has the following separation property: for any linear state feedback such that the controllable subsystem is asymptotically stable, we could construct gains in the nonlinear state feedbacks such that the closed loop system is asymptotically stable at the bifurcation point.