Bifurcation stabilization with local output feedback

Local output feedback stabilization with smooth nonlinear controllers is studied for parameterized nonlinear systems for which the linearized system possesses either a simple zero eigenvalue, or a pair of imaginary eigenvalues, and the bifurcated solution is unstable at the critical value of the parameter. It is assumed that the unstable mode corresponding to the critical eigenvalue of the linearized system is not linearly controllable. Two results are established for bifurcation stabilization. The first one is stabilizability conditions for the case where the critical mode is not linearly observable through output measurement. It is shown that nonlinear controllers do not offer any advantage over the linear ones for bifurcation stabilization. The second one is stabilizability conditions for the case when the critical mode is linearly observable through output measurement. It is shown that linear controllers are adequate for stabilization of transcritical bifurcation, and quadratic controllers are adequate for stabilization of pitchfork and Hopf bifurcations, respectively. The results in this paper can be used to synthesize stabilizing controllers, if they exist