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 individuals 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 overall topic of the convention used to be method idea. the most subject matters have been optimum keep an eye on, constitution and regulate of nonlinear structures, stabilization and observers, differential algebra and structures conception, nonlinear facets of Hoc conception, inflexible and versatile mechanical platforms, nonlinear research of signs. we're indebted to the medical committee John BAILLIEUL, Michel FLIESS, Bronislaw JAKUBCZYCK, Hector SUSSMANN, Jan WILLEMS. We gratefully recognize the time and notion they gave to this job. we'd additionally prefer to thank Chris BYRNES for arranging for the book of those court cases in the course of the sequence "Progress in structures and keep watch over Theory"; BIRKHAUSER. eventually, we're very thankful to the next associations who via their monetary aid contributed primarily to the good fortune of this convention : CNRS, unique 12 months " Systemes Dynamiques", DRET, MEN-DAGIC, GRECO-AUTOMATIQUE, Claude Bernard Lyon I college, 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**

**Example text**

Proof: (i) Averaging v(t) over one period yields w, and averaging v(t)2 over one period yields w 2 + ~K2. If these averaged values for v and v 2 are substituted into (15)-(16), one obtains (17)-(18). A straightforward calculation shows that (17)-(18) are Hamilton's equations relative to fl. SUPER-ARTICULATED MECHANISMS, PERIODIC FORCING 43 8 " - -- I Figure 1: Phase portmit of (9) superimposed on constant avemged energy curve. v(t) = w + K sin(at) with pammeter values w = 0, K = 1, a = 9. (q'PiV) ) - a -~~(q'PiV) T = at: .

We tacitly assume in writing this single second order equation that an arbitrary motion y(t) can be imparted to the base. Since there is no direct control over the joint configuration 0, we have here another example of a super-articulated mechanism. If we further simplify this system by linearizing about 0 = 0, and restrict our attention to sinusoidal input y(t) = sin(at), we obtain a Hill's equation .. 0+ m 2 sin(at)-g £ 0 = o. A remarkable property of this equation is that although it has a right half-plane eigenvalue when E = 0, it is nevertheless possible, by proper choice of parameters a and E, to render the equilibrium (0,8) = (0,0) stable and produce phase plane trajectories which are almost periodic.

The latter happens for Hamiltonian systems; see for instance [NV] for many examples of such Hamiltonian control systems, and the last section of [So] for conditions under which transitivity is preserved under sampling. 5 Accessibility Almost Everywhere We say here that a property holds for "almost all" x E X if it holds on an open dense subset of X. 1 Let 1: be an n-dimensional, discrete-time, invertible and analytic system. Then the following are equivalent: 1. 1: is transitive from almost all x E X.

### 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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