Smooth stabilizing feedback for a class of mechanical systems with a pitchfork bifurcation (Q1329774)

The paper deals with the local stabilization of a class of mechanical systems depending on a physical parameter that takes a critical value for which the open-loop system has a pitchfork bifurcation and its tangent linearization contains two zero uncontrollable modes. The approach, relying on center manifold theory, proves that a nonlinear cubic feedback law ensures local asymptotic stabilization. The results have been applied to the stabilization of a double inverted pendulum when the flexibility coefficient of the spring (joining the two sticks) takes the critical value.