By A. A. Milyutin and N. P. Osmolovskii
The idea of a Pontryagin minimal is built for difficulties within the calculus of diversifications. the applying of the inspiration of a Pontryagin minimal to the calculus of adaptations is a particular characteristic of this ebook. a brand new thought of quadratic stipulations for a Pontryagin minimal, which covers damaged extremals, is built, and corresponding enough stipulations for a robust minimal are bought. a few classical theorems of the calculus of diversifications are generalized
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Extra info for Calculus of variations and optimal control
Example text
2 fur (x, y)# (0,0), /(0,0):= O. Zeige: Die iterierten Limites x y +(x-y lim lim/(x, y) und lim lim/(x, y) sind vorhanden, lim /(x, y) existiert jedoeh nicht. / ist . 7. y_O also im Nullpunkt unstetig. 114 Lineare Abbildungen von RP nach Rq Wir wenden uns nun ganz speziell den linearen Funktionen von RP nach Rq zu. Abweichend von unseren bisher eingehaltenen Benennungs- und Bezeichnungskonventionen in Nr. 111 nennt man solche Funktionen lieber wieder (lineare) Abbildungen, bezeichnet sie statt mit/.
X2, ... ). 2) k~CX) Strebt Xn:= (x~"), xi"), .. )~x:= (Xt. X2, ... +co ist, kurz: Aus x,,~X k_oo folgt stets AXn~Ax. 3. 6 formulierten Voraussetzungen. 3) k(s, t)f(t)dt erklart wird, so ist A eine Selbstabbildung des Banachraumes Aus fn~f folgt stets Afn~Af (s. 6b und d). qa, b], und es gilt: 4. Auf dem Intervall [a, b] seien drei reellwertige Funktionen Xl (t), X2 (t), X3 (t) erklart. R3 werde mit irgendeiner Norm ausgestattet. 4) eine Abbildung A der Teilmenge [a, b] des Banachraumes R in den Banachraum R3 definiert.
1) f(t)dt1). 4 gilt: Aus f,,~ffolgt stets Af,,~Af2). 2. Nun definieren wir eine Abbildung A des Banachraumes (c) aller konvergenten Zahlenfolgen in den Banachraum R durch Ax:= lim Xk flir x:=(xt. X2, ... ). 2) k~CX) Strebt Xn:= (x~"), xi"), .. )~x:= (Xt. X2, ... +co ist, kurz: Aus x,,~X k_oo folgt stets AXn~Ax. 3. 6 formulierten Voraussetzungen. 3) k(s, t)f(t)dt erklart wird, so ist A eine Selbstabbildung des Banachraumes Aus fn~f folgt stets Afn~Af (s. 6b und d). qa, b], und es gilt: 4. Auf dem Intervall [a, b] seien drei reellwertige Funktionen Xl (t), X2 (t), X3 (t) erklart.