By Dirk Aeyels (auth.), Bernard Bonnard, Bernard Bride, Jean-Paul Gauthier, Ivan Kupka (eds.)
The convention "Analysis of managed Dynamical structures" was once held in July 1990 on the collage of LYON FRANCE. approximately hundred members attended this convention which lasted 4 days : there have been 50 audio system from departments of Engineering and arithmetic in east and west Europe, united states and USSR. the final topic of the convention used to be procedure concept. the most subject matters have been optimum regulate, constitution and keep watch over of nonlinear structures, stabilization and observers, differential algebra and platforms idea, nonlinear elements of Hoc idea, inflexible and versatile mechanical platforms, nonlinear research of signs. we're indebted to the clinical committee John BAILLIEUL, Michel FLIESS, Bronislaw JAKUBCZYCK, Hector SUSSMANN, Jan WILLEMS. We gratefully recognize the time and proposal they gave to this job. we'd additionally prefer to thank Chris BYRNES for arranging for the e-book of those lawsuits in the course of the sequence "Progress in structures and keep an eye on Theory"; BIRKHAUSER. ultimately, we're very thankful to the next associations who via their monetary help contributed basically to the good fortune of this convention : CNRS, designated 12 months " Systemes Dynamiques", DRET, MEN-DAGIC, GRECO-AUTOMATIQUE, Claude Bernard Lyon I collage, Entreprise Rhone-Alpes overseas, Conseil basic du RhOne, the towns of LYON and VILLEURBANNE.
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Extra info for Analysis of Controlled Dynamical Systems: Proceedings of a Conference held in Lyon, France, July 1990
By continuity, Zz -t z, which proves (4). 4 Assume that the system ~ is analytic, and that there exists an Xo E X and a ko ~ 0 for which R,ko (xo) =I- 0. 1) imply that: for all k ~ ko. Moreover, since 'IfIk,,,,[U] is analytic also with respect to the x-variable, this particular ko works also for an open dense set of states x EX. 1 ~ ko and for almost all x EX. Regular Points We call x a regular point if p", is constant in a neighbourhood of x. The following fact will be useful later; it is of course a well-known general fact about smooth mappings.
Take first k = O. 1 ofuo+v (x) = 0 since f(x, 0) is independent of u (~ is not forward accessible from x). eIf(j=-1: X;;' (x) since = :vlv=o f- 1 (x, 0) is independent of u (~ f"oof;;'~v(x) = 0 is not backward accessible from x). Take now any k > 0 and note that: From the first part of the proof, we have that ~ is also neither forward nor backward accessible from f~: (x), so, by inductive assumption, this last vector is zero. 3 A consequence of the two previous Lemmas is that, for each x: 1. dimL(x) = 1 if and only if dimL+(x) = 1 or dimL-(x) = 1.
Qn), (Ul, ... , urn) variables. The system dynamics are given by the to modelling the dy& VAN DER SCHAFT function L(q,qiU) of and the control u = Euler-Lagrange equa- tions (1) For many of the mechanical systems we wish to study, we may write the Lagrangian function as L(q,qiU) = Lo(q,q) + qTpu (2) where Lo( q, q) is the sum of a kinetic energy term ~qT M( q)q which is quadratic in the velocities and a potential energy term - V( q), and P is an orthogonal projection operator (on the cotangent bundle of the configuration space).