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. 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.

翻译：布洛基特的必备条件产生一个测试,以确定一个系统是否可以通过连续的、纯粹依赖国家的反馈来稳定某些运行点。 但是,对于许多现实世界的系统来说,人们想要稳定比一个点更普通的系统。 我们还想通过设置障碍和其他“危险”的反射装置来控制这些系统的安全运行。 我们把布罗基特的必备条件推广到稳定具有非零电荷特性的一般常规子集的情况中。 使用这种概括化, 我们还为存在“ 安全” 控制法制定了一个必要条件。 我们用具体的例子和一些普通的系统类别来说明理论, 包括广泛的非光学性约束拉格朗吉亚系统。 我们还表明,在稳定某个点的特殊情况下,我们一般可稳定性测试的专业化比布罗基特的要强。