On Necessary and Sufficient Conditions for Local Controllability Along a Reference Trajectory

Abstract

Consider an n — dimensional control system modelled by the differential equations

$$ \dot x(t) = f(x(t) ,u(t)), (\dot x(t) = dx/dt) $$

((1))

where f is smooth and an admissible control u is a piecewise continuous function taking values in a given set U having nonempty interior in R^m . Denote by a(t,q) the set of all points attainable at time t by solutions of (1) corresponding to admissible controls and initiating from q at time 0 . Let u* be a given control which generates a reference trajectory φ with φ(0) = p . The system (1) is locally controllable along φ at p if for all ε > 0 ,φ(ε)is an interior point of a(ε,p) . Loosely speaking, this implies the ability to control the system to a full neighborhood of the reference trajectory over an arbitrarily small interval of time.

This research was supported by the National Science Foundation under grant GP27957