Zur Lösung gewisser linear optimaler Prozesse mit dem Maximumprinzip

Dietze, S.

doi:10.1002/zamm.19680480108

Cited by 4 publications

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“…Some statements of these theorems are known. For a concrete and the general problem (3) with U and 8 according to (13), resp., results concerning the differentiability and subdifferentiability of the function Fl, resp., are contained in [7], 191, resp. The Prbchet derivative PI, = fa in the case of the abstract problem (4) has been calculated in 121.…”

Section: Problems 1n the Spacementioning

confidence: 99%

“…Dual problems: The problem 8. DIETZE : On Control-Approximation Problems -cquation (7) c ix dual to (4) (see Proposition 2), wherc g* denotes t,hc conjugate function of g I U 3 XI, g(u) = -z lluiillb + + IQ(u). A simple calculation gives C q*(,U*) = sup ( ( U * , U ) U ---l ( u -.iill;:.u E a} = 2 generalized equation (9) Yrohlemx which arive from o p t i m a l i t y conditions: Applying well-known optimality conditiom for convex oytiniizatioii probleins to (3), (4), (a), ( G ) , respectively, we will obtain in thie paper the respective equivalent equations for u E U (t > 0 fixed);…”

Section: Introductionmentioning

confidence: 99%

“…for u E U (t > 0 fixed); dual problem (6) for X * E X. In (7) and (8) the number t > 0 can be chosen arbitrarily, but fixed. lines indicate the connections between the problems.…”

Section: Introductionmentioning

confidence: 99%

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On Control‐Approximation Problems in Hilbert Spaces and Related Dual Problems: Subdifferentiability Properties and Estimates

Dietze¹

1983

Z Angew Math Mech

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Oedioated to the memory of the 100t.h birthday of ~I E D R I C H ADOLP W I~E B S I n cler Arbeit toerden detrakte 8teuer-Appvxim&mapoblerne in Hilbert-Riunaen, zuge-e dude Aufgaben und iiquivdente Aufgden, welche aua O@kdWsbedingungelc resultkren, bctrachtet. Ee werden die Beziehungen zwisehen diesen Aujgaben angegeben und verdiedena EigCneohaften der Aujguben untcreuclJ, mangig Pigenachuften, w&he die 8ubd~ferenzierbmrkeit der Zidfunktionen betreffen. A u J e r h wenlen AbecWungen fiir die Liisungen und die optinaden Zieijunktimwerte hergekitet. Die Arbed kt Bedeutung DOT dlem fiir die fimng cwn r 9 t e z c e r -A~p~x~v n~i~q~r o b l e n t e n niit Rive dualer Aujpben, und z t~a r direkt oder im Zuscmrmenibang mi+? Regularisierungmrjahren.In thie paper we consider detract eontrd-qqroximtion problem h Hilbert spcscea, related dual probleiiu and equivalent problem which h e from optimdity conditions. We 've the connections betwccrs these problem and study several properties of them, particularly pperties eOnCenting the subdi&entiability of the wst junctbrpds. Furtheme, rue derive estinwtes for the solutbiur and the optimal funetbud vduea. The results of this paper are h e dl uaejul for solvinq the wnfrolwpprozimnlion problcrns, by the aid of the dud problem, directly or in wnnedbn with regularizQtion niethwlr. pa(i0l.e 1)aCCMaTpnBaloTCH 3aUdYH THlla yllpaBJIelin~-alIllpOKCnMa~H B lMdlb6epTOBHX II~OCTiiHCTtlaX, COOT-UCTCTBJ'IOIIIH ABOfiCTBeHHbIe 3aAaYH H 3KBEIBaSIeATIIble 3Wa71E11 KOTOPhIe pfBYJIbTElpyIOT E 1 3 YCJIOSEIfi 3KCTpe-IIOCTEI, CBOi&CTBP, KOTOpHe KaCaIOTCII Cy6AH@@epeHLV ¶pyeMOCTH IWIeBhIX @YHKqHfit. DaJIee BbIBOgflTCH 04eH~11 6emo Ansi PeIIIeHHFI 3aAas TElIIa J'npaBJIeHUH-aIlIIpOKCHM~H C nOMOIl@dO ABOhTBeHHMX aaAa=l, na6o Mynta. Y~a a~a a m~c s i C B H~H Memy ~T W M H 3 a~a s a~u n nccnenymicn HeKoTopwe cBoticTsa aazaq, B oco6eii-JIJIH PemeHHi EI AJIR OIITEMNIbHblX 3HaseHHZt UWeBblX @ Y € I € q d . PfByJIbTaTbI pa6o~b1 HMeI OT 3~a~e~n e OCO-HenocpeRcTBemo nnGo B c~n3a c MeToRaMH perynflpaaaaa.Ii,((i) := +m for a Q IR. Then the conjugate function I 2 of I,, is given by 1 : 1 U + R1, G(u*) = max {(tt*, u )~: u E $2) ,

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Section: Problems 1n the Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%