2021/06/01 by Matthew D. Kvalheim, Kvalheim, Matthew D., Daniel E. Koditschek +1 · 1 citation
Computer Science · Engineering · #34H05 #70G40 #93B24 #93B52 #93C35 #93D20 #Control and Dynamics of Mobile Robots #Control and Stability of Dynamical Systems #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Optimization and Control (math.OC) #Robotic Path Planning Algorithms #Robotics (cs.RO)
paper · pdf · doi:10.48550/arxiv.2106.00215
openalex publication_date 2021/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Brockett's necessary condition yields a test to determine whether a system can be made to stabilize about some operating point via continuous, purely state-dependent feedback. For many real-world systems, however, one wants to stabilize sets which are more general than a single point. One also wants to control such systems to operate safely by making obstacles and other "dangerous" sets repelling. We generalize Brockett's necessary condition to the case of stabilizing general compact subsets having a nonzero Euler characteristic in general ambient state spaces (smooth manifolds). Using this generalization, we also formulate a necessary condition for the existence of "safe" control laws. We illustrate the theory in concrete examples and for some general classes of systems including a broad class of nonholonomically constrained Lagrangian systems. We also show that, for the special case of stabilizing a point, the specialization of our general stabilizability test is stronger than Brockett's.