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1、NonsmoothAnalysisandControlTheoryF.H.Clarke,etal.Springer1.ControlTheory.2.Nonsmoothoptimization.I.Clarke,FrancisH.II.Series.Theauthorsdedicatethisbook:toGail,Julia,andDanielle;toSo?a,Simeon,andIrina;toJudy,Adam,andSach;andtoMaryandAnna.ThispageintentionallyleftblankPrefacePardonmeforwr
2、itingsuchalongletter;Ihadnotthetimetowriteashortone.—LordChester?eldNonsmoothanalysisreferstodi?erentialanalysisintheabsenceofdi?er-entiability.Itcanberegardedasasub?eldofthatvastsubjectknownasnonlinearanalysis.Whilenonsmoothanalysishasclassicalroots(weclaimtohavetraceditslineagebacktoD
3、ini),itisonlyinthelastdecadesthatthesubjecthasgrownrapidly.Tothepoint,infact,thatfurtherdevel-opmenthassometimesappearedindangerofbeingstymied,duetotheplethoraofde?nitionsandunclearlyrelatedtheories.Onereasonforthegrowthofthesubjecthasbeen,withoutadoubt,therecognitionthatnondi?erentiabl
4、ephenomenaaremorewidespread,andplayamoreimportantrole,thanhadbeenthought.Philosophicallyatleast,thisisinkeepingwiththecomingtotheforeofseveralothertypesofirregularandnonlinearbehavior:catastrophes,fractals,andchaos.Inrecentyears,nonsmoothanalysishascometoplayaroleinfunctionalanalysis,op
5、timization,optimaldesign,mechanicsandplasticity,di?eren-tialequations(asinthetheoryofviscositysolutions),controltheory,and,increasingly,inanalysisgenerally(criticalpointtheory,inequalities,?xedpointtheory,variationalmethods...).Inthelongrun,weexpectitsmeth-odsandbasicconstructstobeviewe
6、dasanaturalpartofdi?erentialanalysis.viiiPrefaceWehavefoundthatitwouldberelativelyeasytowriteaverylongbookonnonsmoothanalysisanditsapplications;severaltimes,wedid.Wehavenowmanagednottodoso,andinfactourprincipalclaimforthisworkisthatitpresentstheessentialsofthesubjectclearlyandsuccinctly
7、,togetherwithsomeofitsapplicationsandageneroussupplyofinterestingexercises.Wehavealsoincorporatedinthetextanumberofnewresultswhichclarifytherelationshipsbetweenthedi?erentschoolsofthoughtinthesubject.Wehopethatthiswillhelpmakenonsmoothanalysisaccessibletoawideraudience.Inthissp
