By Dirk Aeyels (auth.), Bernard Bonnard, Bernard Bride, Jean-Paul Gauthier, Ivan Kupka (eds.)

ISBN-10: 1461232147

ISBN-13: 9781461232148

ISBN-10: 146127835X

ISBN-13: 9781461278351

The convention "Analysis of managed Dynamical platforms" was once held in July 1990 on the college of LYON FRANCE. approximately hundred contributors 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 method idea. the most themes have been optimum keep an eye on, constitution and keep an eye on of nonlinear structures, stabilization and observers, differential algebra and platforms conception, nonlinear features of Hoc idea, inflexible and versatile mechanical platforms, nonlinear research of indications. we're indebted to the clinical committee John BAILLIEUL, Michel FLIESS, Bronislaw JAKUBCZYCK, Hector SUSSMANN, Jan WILLEMS. We gratefully recognize the time and idea they gave to this job. we might additionally prefer to thank Chris BYRNES for arranging for the booklet of those complaints during the sequence "Progress in platforms and regulate Theory"; BIRKHAUSER. eventually, we're very thankful to the subsequent associations who via their monetary aid contributed basically to the good fortune of this convention : CNRS, particular yr " Systemes Dynamiques", DRET, MEN-DAGIC, GRECO-AUTOMATIQUE, Claude Bernard Lyon I collage, Entreprise Rhone-Alpes overseas, Conseil normal du RhOne, the towns of LYON and VILLEURBANNE.

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**Additional info for Analysis of Controlled Dynamical Systems: Proceedings of a Conference held in Lyon, France, July 1990**

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The following controllability result is an analogue for discrete-time systems of the result in [Lo]. The proof is very similar, but it uses the facts just established. 5 Assume that the state space X is a compact Riemannian analytic manifold, and that for all u E U the map fu is a measure preserving transformation (for the natural measure in X). Then ~ is transitive if and only if ~ is controllable. Proof. We need only to prove that transitivity implies controllability. For each u, since lu is a measure preserving map, by the Poincare Recurrence Theorem the set of positively Poisson stable points for lu is known to be dense in X.

4) below we have: B 0 {(k,y)lk2:1, kE'Ji" -k~y~k} { (k,y) I k 2: 0, k E 'Ji" -k ~ y ~ k } 6 Bt U {(O,O)}. Low-Dimensional Cases In this section we make some remarks about one- and two-dimensional systems. 1 Dimension One There we consider systems for which the state space X is of dimension one. The pointwise versions of [JS], Theorem 3, hold for these systems as follows. 1 Let 1. if ~ 2. if ~ ~ ~ be as above, and pick x E X. Then: is smooth then is forward accessible from x if and only if dim r+ ( x) ~ = 1, is analytic and U is connected then is forward accessible from x if and only if dim L +(x) = l.

We shall treat similar examples in detail below. The remainder of the paper is organized as follows. In Section 2 we define what we mean by a Lagrangian control system. For certain types of systems in which a symmetry group acts, there is a natural reduction in the order of the model. It is shown that typically (unless special structure is present in the model) the reduced order Lagrangian will contain terms which are linear in the velocities-so-called gyroscopic terms. In Section 3, we rewrite the reduced order Lagrangian control systems in Hamiltonian form.

### Analysis of Controlled Dynamical Systems: Proceedings of a Conference held in Lyon, France, July 1990 by Dirk Aeyels (auth.), Bernard Bonnard, Bernard Bride, Jean-Paul Gauthier, Ivan Kupka (eds.)

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