Non-Local Methods for Pendulum-Like Feedback Systems by Volker Reitmann

By Volker Reitmann

Zero are already deecribed by way of I. Newton (116]. but it used to be 250 years later that F. Tricorni (147] conducted the 1st non-local qualitative research of equation (0.1) with arbitrary o ~ zero and "'{ ~ zero. It was once proved by way of F. Tricorni that any resolution of (0.1) with o > zero corresponds both to a rotatory movement or to a damped oscillatory movement. in addition, he confirmed that during the non-trivial case "'! :::; 1 there exists a bifurcation worth ocr("'!) resembling a separatrix-loop, i.e. to a double-asymptotic to a saddle-point trajectory. For o < ocr("'!) equation (0.1) admits damped oscillations as weil as rotatory motions. For o > ocr("'') worldwide asymptotic balance happens, i.e. each movement is a damped oscillation. The papers of F. Tricorni turned widely used instantly.

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5). 3 go back to [49]. 5) has n -1 poles with negative real part. 7). Let us choose {) = sgnRe[iwx(iw)]. 12) are fulfilled. 2, p. :; -e I z(t) 1'. 3. 3, p. 4. 2). Let us consider condition (ii). Iffor some r > 0 W(z( r), <7( r)) = W(z(O), <7(0)) it follows that W = 0 on (0, r) and consequently z(t) = 0 on [0, r]. · So (A,b) is controllable and consequently b f 0. 7) we have cp(<7(t)) = 0 on [0, r]. 7)) it follows that <7(t) = const on [0, r]. I In the next theorem we provide sufficient conditions which guarantee monostability.

4) where p > 0, u 1 and u 2 are certain numbers. 1. ,';;, sx(s) tions are true: > 0. Suppose there exists a >. ) 12 < 0, wER. 4) is Lagrange stable. 1. 1 for the case p 1 = -oo and p. 2 = p # +oo. So the matrix H is constructed here in the same way as in the previous section. 1. ij = {x : W(x- jd) ::; 0, r*(x- X2- jd) ::; 0} where j is an arbitrary integer.

50) we have y( u3) :",; y( u2)e [ ~(0"3 - u2)] . 51) Let us consider now the interval [u3 , u 1 ]. flirm that Yo(u) + y,(u) ~ 2yo(u) + ~. I u E [u3,u,]. Note that for any u E (u3,u1 ) there exists a 7f E (u,u,) suchthat Yo(u) = y~(7t)(u- u 1 ). Note that y~(7f) is negative by the requirement (b). Let c 2 besuchthat y~(u) :",; c2 < 0 for u E [u3 , u 1 ]. 'P'(u) du. , 1~ or __ 1_~ < _c_In 2h I . T < 1. 48) is fulfilled and since y 0 (u1 ) = y0(u1 - ß) = 0 it follows that Y1(ut) only Y1(u1- ß) issmall enough.

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