Bifurcation stabilization with local output feedback

Local output feedback stabilization with smooth nonlinear controllers is studied for parametrized nonlinear systems for which the linearized system possesses either a simple zero eigenvalues, 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 eigenvalues of the linearized system is not linearly controllable. Two results are established for bifurcation stabilization: 1) stabilizability conditions for the case where the critical mode is not linearly observable through output measurement; and 2) 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 can be used to synthesize stabilizing controllers, if they exist