Local L<inf>2</inf> gain of Hopf bifurcation stabilization
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Abstract
Local L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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 L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain are derived by solving the Hamilton-Jacobi-Bellman (HJB) inequality. Local robust analysis can then be conducted through computing the local L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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.
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