This paper states necessary conditions for the existence of ''universal stabilizers'' for smooth control systems. Roughly speaking, given a control system and a set U of reference input functions, by ''universal stabilizer'' we mean a continuous feedback law that stabilizes the state of the system asymptotically to any of the reference trajectories produced by (arbitrary) inputs in U. For an example, consider Brockett's nonholonomic integrator, with U representing a set of uniformly bounded, piecewise continuous functions of time. This system's state can be asymptotically stabilized to any reference trajectory provided the latter is persistently exciting (PE). By contrast, for constant trajectories (i.e., equilibria), which are not PE, asymptotic stabilization is impossible by means of continuous pure-state feedback, in view of Brockett's obstruction. However, since this obstruction can be circumvented by the use of time-varying state feedback, one might reasonably expect to be able to design a (time-varying) continuous control law capable of asymptotically stabilizing the state to arbitrary reference trajectories, be they PE or not. Surprisingly, a consequence of the results in this paper is that, for systems with nonholonomic constraints frequently found in control applications, if U contains reference functions that are not PE, then the ''universal stabilization'' problem cannot be solved, even if time-varying feedback is used.

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The transverse function approach to control provides a unified setting to deal with practical stabilization and tracking of arbitrary trajectories for controllable driftless systems. Controllers derived from that approach offer advantages over those based on more classical techniques for control of nonholonomic systems. Nevertheless, its extension to more general classes, such as critical underactuated mechanical systems, is not immediate. The present paper explores a possible extension by developing a framework that allows one to cast point stabilization problems for (leftinvariant) second-order systems on Lie groups, including simple mechanical systems. The approach is based on "vertical transversality," a property exhibited by derivatives of transverse functions. In this paper, we lay out the theoretical foundations of our approach and present an example to illustrate some of its features.

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This paper addresses the stabilization of postures (or con®gurations) for the general N-trailer system, also known as the N-trailer system with`o -axle' or`kingpin' hitching. In general, the kinematic model of this system cannot be transformed into the chained form, so the wealth of tools developed for the latter are not directly applicable to that system. Nevertheless, it is shown here that there exists a class of systems that may be locally approximated by a chained form, and su cient conditions for a system to be in this class are pointed out. Such a`chained form approximation' may then be used to build a local stabilizer for the original (approximated) system. After deriving its kinematic model, it is shown in this paper that the general N-trailer system has a chained form approximation at some postures, and that these may be locally »-exponentially stabilized by simple feedback controllers available in the literature. Simulation examples are given to illustrate the approach.