Rosa Maria Bianchini

Controllability along a trajectory: a variational approach

This paper unifies and improves most sufficient conditions of local controllability both along a trajectory and at a point. This is accomplished by defining high-order variations that can be continuously summed. The peculiar property of these variations is that they may be generated by thin conditions as relations in the Lie algebra associated with a control system. This property leads to a variational interpretation of Sussmann-type sufficient conditions of local controllability (neutralization of obstructions), and it allows the use of different weights for neutralization.

Normal local controllability of order one

This paper gives a sufficient condition of normal local controllability at a point x0 for a family of analytical vector fields. The families considered are of the following

Needle Variations that Cannot be Summed

where Ω is a bounded subset of Rκ, The condition is of the first order, that is it depends only on the values of the fields and their first derivatives at x0. It is proved that this condition is the most general condition of this type. In fact, if this condition is not satisfied, there are families of vector fields which at the first order coincide with the given family, and which are not normally locally controllable at x0

Sufficient conditions of local controllability

This article analyzes sets of higher order tangent vectors to reachable sets of analytic control systems (affine in the control). Both small-time local output controllability and small-time local controllability about a nonstationary reference trajectory are considered. In a series of purposefully constructed examples it is shown that the cones generated by needle variations may fail to be convex. The examples demonstrate that the usual technical condition that needle variations must be movable is essential to guarantee desirable convexity properties. Moreover, new doubts are cast on the structural stability of controllability properties, as apparently higher order perturbations can reverse the (lack of) controllability of lower order nilpotent approximating systems, thereby providing new insights about the ultimate question of whether controllability is finitely determined.

Local Controllability, Rest States and Cyclic Points

A C¿ control system of type x = fo(x) + ¿i=1 m ui fi(x) |ui| ¿ 1 is considered. Under the assumption fo (xo)=0, a sufficient condition of local controllability at xo is given. The condition generalizes the one given by Sussmann [10].

Graded approximations and controllability along a trajectory

The problem of steering a neighbourhood of the state x of a linear time-invariant control system, to the point itself is considered. The set of points for which this property holds when the control is subject to magnitude constraints, is characterised. Moreover it is proved that this set is strictly related to the set of points that can be reached from themselves by an admissible control map.

Lack of convexity for tangent cones of needle variations

A class of approximating nilpotent systems of an affine control system is defined. The systems of this class are used to study the local controllability along a trajectory.<<ETX>>

A Note on the Reachable Sets of Control Systems

The article provides a carefully constructed sequence of simple systems which show that even for very benign systems, the usual conditions that needle variations do not collide (or that they are moveable by sufficiently large amounts) are essential for guaranteeing convexity of the tangent objects. This, in turn, is essential for practical applicability to decide optimality. These examples also further raise deep questions about the structural stability of nonlinear controllability properties: they demonstrate that the controllability (or the lack thereof) of nilpotent approximating systems need not reflect the controllability (or the lack thereof) of the original systems. These suggest limitations to extending the usual arguments using nilpotent approximations have played a critical role in obtaining many classical controllability and optimality results.

Local Controllabilifty Along a Reference Trajectory

Let M be a finite dimensional differential manifold and let (∑u) be a timeindependent control system on M, that is a family of differential equations on M