Local $ell_2$ 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.