Brockett's necessary conditions and the stabilization of nonlinear control systems

Brockett's theorem states the three necessary conditions for the existence of a continuously differentiable closed loop control that asymptotically stabilizes the nonlinear control system to an equilibrium point. Kinematic systems are shown to fail to meet Brockett's third necessary condition. A normal form is introduced so that nonholonomic control systems are defined directly over a reduced constraint distribution. In normal form, nonholonomic control systems can then easily be shown to fail to be stabilizable to a point via a C1 control. The conditions for the smooth stabilization of the nonholonomic systems to an equilibrium submanifold are then presented. For a particular case of the reduced form of mechanical control systems (Chaplygin systems), stabilization to a point can be achieved by applying the concept of geometric phase and using piecewise differentiable state controls.