Uniting local and global controllers with robustness to vanishing noise

We consider control systems for which we know two stabilizing controllers. One is globally asymptotically stabilizing, the other one is only locally a symptotically stabilizing but for some reason we insist on using it in a ne ighborhood of the origin. We look for a uniting control law being equal to the local feedback on a neighborhood of the origin, equal to the global one outside of a larger neighborhood and being a globally stabilizing controll er. We study several solutions based on continuous, discontinuous, hybrid, time-varying controllers. One criterion of the selection of a controller is the robustness of the stability to vanishing noise. This leads us in parti cular to consider a kind of generalization of Krasovskii trajectories for h ybrid systems.