Feedback Stabilization of Steady-State and Hopf Bifurcations: the Multi-input Case

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Yong Wang and Richard M. Murray
1999 Conference on Decision and Control

Classification of stabilizability is obtained for multi-input nonlinear systems possessing a simple steady-state or Hopf bifurcation with the critical mode being linearly uncontrollable. Stabilizability is defined as the existence of a sufficiently smooth state feedback such that the bifurcation for the closed loop system is supercritical, and in the meantime, the linearly controllable modes are locally asymptotically stable. Necessary and sufficient conditions of stabilizability are derived under certain nondegeneracy conditions. Explicit construction of stabilizing feedbacks is obtained for the cases when the system is stabilizable.