Local Feedback Stabilization and Bifurcation Control, I.Hopf Bifurcation.

Abstract

Local bifurcation control problems are defined and employed in the study of the local feedback stabilization problem for nonlinear systems in critical cases. Sufficient conditions are obtained for the local stabilizability of general nonlinear systems whose linearizations have a pair of simple, nonzero imaginary eigenvalues. The conditions show, in particular, that generically these nonlinear critical systems can be stabilized locally. The analysis also yields a direct method for computing stabilizing feedback controls. Use is made of bifurcation formulae which require only a series expansion of the vector field. The results are easily applied since they do not involve preliminary state transformations, center manifold reduction, or Liapunov functions.