Necessary conditions for feedback stabilization and safety

Brockett's necessary condition yields a test to determine whether system can be made stabilize about some operating point via continuous, purely state-dependent feedback. For many real-world systems, however, one wants sets which are more general than single point. One also control such systems operate safely by making obstacles and other "dangerous" repelling. style='text-indent:20px;'>We generalize Brockett's the case of stabilizing compact subsets having nonzero Euler characteristic in ambient state spaces (smooth manifolds). Using this generalization, we formulate for existence "safe" laws. We illustrate theory concrete examples classes including broad class nonholonomically constrained Lagrangian systems. show that, special point, specialization our stabilizability is stronger Brockett's.