Nonlinear Synthesis: Proceedings of a IIASA Workshop held in by A. A. Agrachev, S. A. Vakhrameev (auth.), Christopher I.

By A. A. Agrachev, S. A. Vakhrameev (auth.), Christopher I. Byrnes, Alexander B. Kurzhansky (eds.)

In its broadest experience, nonlinear synthesis consists of in reality the synthesis of occasionally so­ phisticated or complicated keep watch over options with the purpose of prescribing, or at the very least influencing, the evolution of complicated nonlinear structures. Nonlinear synthesis calls for the advance of methodologies for modeling advanced platforms, for the research of nonlinear versions, and for the systematic layout of keep an eye on schemes or suggestions legislation that may in attaining a wide selection of prescribed targets. The modeling, research and keep watch over of advanced platforms within the face of uncertainty shape on of the most important parts of the present examine application within the division of platforms and choice Sciences (SDS) on the foreign Institute for utilized platforms research (IIASA). In June 1989, a IIASA workshop on Nonlinear Synthesis, backed through SDS, used to be held in Sopron, Hungary. we're proud to provide this quantity because the lawsuits of this workshop, a workshop attened by way of well known researchers in nonlinear structures from either the East and the West. because the advertising and encouragement of medical cooperation among researchers within the East and within the West is likely one of the targets at IIASA, we think the Sopron convention on Nonlinear Synthesis was once very winning. furthermore, we have been specially happy through the amazing new advances provided on the workshop which, during this quantity, are actually a part of the convention record.

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Proceedings of the 28th IEEE Conference on Decision and Control, Tampa (1989),2247-2251. 8. E. Marsden and M. McCracken, ''The Hopf Bifurcation and Its Applications," Springer-Verlag, New York, Heidelberg, Berlin, 1976. 9. M. Hautus, Linear Matrix Equation. witA Application to tAe Regulator Problem, Outils and Modeles Mathematique pour l'Automatique ... S. (1983), 399-412. 10. I. Byrnes and A. tem. witA Application. to Stabilization and Adaptive Control, Proc. of 23rd IEEE Conf. on Dec. and Control, Las Vegas, NV (1984).

Unless specified otherwise we will assume that f is C1. The following theortem is due to Andreini, Bacciotti and Stefani [ABS]. 5) U where (Xl, X2) E ~P x ~m, U E ~m, F is homogeneous of some odd degree p. The system is asymptotically stabilizable by homogeneous feedback of degree p if i1 F(xl. 0) is asymptotically stable. = The following example captures the spirit of this theorem. 6) where p is an odd integer. We show that this system is asymptotically stabilizable. This is done by using an induction argument.

G. 1) initialized at x(O) = Xo is periodic, x(t) = x(t + T), of period T = 27r/w? (2) Is {x(t)} an asymptotically stable periodic orbit? 3) whereS= [~ ~], 60 = 6 [~] x E lR,n = [~~] E lR,2 and r(6) =6 2 . 1) as an exponentially stable system driven by the output of a harmonic oscillator. 2). 3); (2) by the center manifold theorem there exists a complementary invariant surface E defined by x = 7r( 6). e. for (x(O), 6(0)) E E (x(t), 6(t)) =(7r(6(t)), 6(t)) =(7r(6(t +T)), 6(t +T))= (x(t +T), 6(t +T)) where T = 27r/w.

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