By Dr. Tsuyoshi Matsuo (eds.)
ISBN-10: 3540106820
ISBN-13: 9783540106821
ISBN-10: 3540386238
ISBN-13: 9783540386230
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Extra info for Realization Theory of Continuous-Time Dynamical Systems
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A : (n,-l)+ and ~ ~. , the input/output map in sophlsticated form. To complete the relation between ~-modules with readout map and n-modules with observation map, we present the following lemma. 10) LEMMA. 6), respectlvely. Then a m@p f ; X I ÷ X 2 is an n-morphism with observation map f : ((Xl,~l),Hl) ÷ only if it is an R-m0rphismwlth ((X2,~2),H2)if and readout map f : ((Xl,~l),hl) + ((X2,~2),h2). PROOF. Assume that f : (XI,#I) + (X2,~2) is an ~-morphism. If H2f = HI, then, for any ~6 ~, h2#2(~)f = hl~l(~), hence h2f = h I.
D(S(t)~)h ; 0 ~ t<__l~I} is total on X, where (XA,~ A) and (XA,(~-I) A) are the set theoretical dual a-modules of (X,~) and (X,~-l), respectively. See Sections 5. 17) LEMMA. ((X,~),h) be ~n a-module and let X ~ X be a finite set. , x~(x) = y E Y ~. ~} does not for all x ~ X ) and CC(h') s is total on X for all h'~ CC(h). Then, for any ~ ~ there exists an s ~ such that (~(~))-I(Xs) is constructlble b7 ~ ' l w ~ . PROOF. Let h ° = (~-l)~(~)h6 X ~. s Choose an ~i~ ~ such that the equivalence relation defined by the map h I = (~-l)d(~l)h ° is different from the equivalence relation defined by the map h .
U(~)x ° = x If $ (~)x = x on some interval (0,t], then u o +o holds for all T ~ R by Proposition (4 16), hence it is a contradiction. Therefore, there exist a very small t l ~ R ++ and Xl~ n such that Su(tl)x ° ~ Xl, x I ~ x O and x I ~ x u. interval (0,t], then ~u(Z)x I = x I for all z 6 ~ + hence we have a contradiction. 16), Therefore, there exist a very small x2~=n such that Su(t2)x I - x 2, x 2 ~ x I, and x 2 ~ x u. We can assume that x 2 ~ Xo, since x 2 = x ° implies that ~u ( kt2)x ° = x ° for all k ~ ~ and it contradict to the assumptions.
Realization Theory of Continuous-Time Dynamical Systems by Dr. Tsuyoshi Matsuo (eds.)
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