| Limits Of Highly Oscillatory Controls And The Approximation Of General Paths By Admissible Trajectories (1991) | |||||||||||||||||
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| We describe sufficient conditions, extending earlier work by Kurzweil and Jarnik, for a sequence of inputs u j = (u j 1 ; : : : ; u j m ) 2 L 1 ([0; T ]; IR m ) to be such that, for every m-tuple (f 1 ; : : : ; f m ) of smooth vector fields, the trajectories of x(t) = P m k=1 u j k (t)f k (x(t)) converge to those of an "extended system" x(t) = P r k=1 v k (t)f k (x(t)), where the new vector fields fm+1 ; : : : ; f r are Lie brackets of the original f k 's. Using these conditions, we can solve the inverse problem: given a trajectory fl of the extended system, find trajectories of the original system that converge to fl. This is done by means of a universal construction that only involves a knowledge of the v's. These results can be applied to solve the problem of approximate tracking for a controllable system without drift. 1. Introduction In this paper we report a number of results on the relation between the solutions of an equation x(t) = m X k=1 u k (t)f k (x(... | |||||||||||||||||
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