Controllability and stabilization of one-dimensional systems near bifurcation points

Abstract Bifurcation theory deals with the change of qualitative behavior in a parameter dependent family of differential equations. For one-dimensional equations the possible bifurcation scenarios are well understood. If the family of differential equations can be controlled by admissible controls with compact range, the question arises, whether the systems are controllable near a bifurcation point and whether stabilization around unstable bifurcation branches via bounded feedback is possible. In this paper we show that controllability for parametrized families of one-dimensional control systems can be characterized in terms of two parameters, the original bifurcation parameter and the size of the control range. These results are used to construct (nonsmooth) stabilizing feedbacks and to describe the set of initial values, from which stabilization is possible. Furthermore, robustness properties of the stabilizing feedback are discussed.