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1、NonsmoothAnalysisandControlTheoryF.H.Clarke,etal.Springer1.ControlTheory.2.Nonsmoothoptimization.I.Clarke,FrancisH.II.Series.Theauthorsdedicatethisbook:toGail,Julia,andDanielle;toSofia,Simeon,andIrina;toJudy,Adam,andSach;andtoMaryandAnna.ThispageintentionallyleftblankPrefacePardonmeforwr
2、itingsuchalongletter;Ihadnotthetimetowriteashortone.—LordChesterfieldNonsmoothanalysisreferstodifferentialanalysisintheabsenceofdiffer-entiability.Itcanberegardedasasubfieldofthatvastsubjectknownasnonlinearanalysis.Whilenonsmoothanalysishasclassicalroots(weclaimtohavetraceditslineagebacktoD
3、ini),itisonlyinthelastdecadesthatthesubjecthasgrownrapidly.Tothepoint,infact,thatfurtherdevel-opmenthassometimesappearedindangerofbeingstymied,duetotheplethoraofdefinitionsandunclearlyrelatedtheories.Onereasonforthegrowthofthesubjecthasbeen,withoutadoubt,therecognitionthatnondifferentiabl
4、ephenomenaaremorewidespread,andplayamoreimportantrole,thanhadbeenthought.Philosophicallyatleast,thisisinkeepingwiththecomingtotheforeofseveralothertypesofirregularandnonlinearbehavior:catastrophes,fractals,andchaos.Inrecentyears,nonsmoothanalysishascometoplayaroleinfunctionalanalysis,op
5、timization,optimaldesign,mechanicsandplasticity,differen-tialequations(asinthetheoryofviscositysolutions),controltheory,and,increasingly,inanalysisgenerally(criticalpointtheory,inequalities,fixedpointtheory,variationalmethods...).Inthelongrun,weexpectitsmeth-odsandbasicconstructstobeviewe
6、dasanaturalpartofdifferentialanalysis.viiiPrefaceWehavefoundthatitwouldberelativelyeasytowriteaverylongbookonnonsmoothanalysisanditsapplications;severaltimes,wedid.Wehavenowmanagednottodoso,andinfactourprincipalclaimforthisworkisthatitpresentstheessentialsofthesubjectclearlyandsuccinctly
7、,togetherwithsomeofitsapplicationsandageneroussupplyofinterestingexercises.Wehavealsoincorporatedinthetextanumberofnewresultswhichclarifytherelationshipsbetweenthedifferentschoolsofthoughtinthesubject.Wehopethatthiswillhelpmakenonsmoothanalysisaccessibletoawideraudience.Inthissp
