By J. Mawhin, M. Willem
ISBN-10: 354096908X
ISBN-13: 9783540969082
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Extra resources for Critical Point Theory and Hamiltonian Systems
Example text
L I I I ¢ ~() Im I L I I I I FTI m H e r e e ( ' ) is the nonlinear characteristic of the servo device, while r is a scalar parameter which we can choose. Clearly u(t) : t / ~(y - ru(T))dT, 0 or G = ~(y - ru). 2) 0 = ¢(y - ru).
1 . 3 ) . Let £ be a neighbourhood of R, V: ~ ÷~ a continuously d i f f e r e n t i a b l e function, and ~I c_ ~ an open s e t such t h a t (a) V(x) > 0 and V(x) > 0 in ~i; (b) V(x) = 0 for x E R q a~ I (c) ~ e a~ I Under t h e s e conditions ~ i s an unstable r e s t point. We close this subsection with the remark that i t has been established that, in principle, the question of the s t a b i l i t y , asymptotic s t a b i l i t y or i n s t a b i l i t y of a rest point can always be decided using these methods.
Last l I 1 o I I 9--0_! 0 Thus the A-matrix is almost block-companion - this is spoiled only by the non-zero entries in the nl-th, (n1+n2)-th . . rows. 7). The following theorem is due to Luenberger, and its proof is a clever extension of that sketched above for single-input systems. THEOREM 2 . 1 . 2 state exists dlme~ion Given any c o m p l e t e l y c o n t r o l l a b l e n a~d i n p u t dimension r ~ n w i t h r a n k B = r , t h e r e a state tra~ fo~ation ( 2 . 1 . 2 ) and a n o n - s i n g u l a r t~ans f o r m a t i o n T i n c o n t r o l s p a c e R r , s u c h t h a t w i t h n~w s t a t e variable~ = Sx, •the.
Critical Point Theory and Hamiltonian Systems by J. Mawhin, M. Willem
by William
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