Local L2 gain of Hopf bifurcation stabilization

Local L₂ gain analysis of a class of stabilizing controllers for nonlinear systems with Hopf bifurcations is studied. In particular, a family of Lyapunov functions is first constructed for the corresponding critical system, and simplified sufficient conditions to compute the L₂ gain are derived by solving the Hamilton-Jacobi-Bellman (HJB) inequality. Local robust analysis can then be conducted through computing the local L₂ gain achieved by the stabilizing controllers at the critical situation. The theoretical results obtained in this paper provide useful guidance for selecting a robust controller from a given class of stabilizing controllers under Hopf bifurcation. As an example, application to a modified Van der Pol oscillator is presented.