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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.Thispageintentional
2、lyleftblankPrefacePardonmeforwritingsuchalongletter;Ihadnotthetimetowriteashortone.—LordChesterfieldNonsmoothanalysisreferstodifferentialanalysisintheabsenceofdiffer-entiability.Itcanberegardedasasubfieldofthatvastsubjectknownasnonlinearanalysis.Whilenonsmooth
3、analysishasclassicalroots(weclaimtohavetraceditslineagebacktoDini),itisonlyinthelastdecadesthatthesubjecthasgrownrapidly.Tothepoint,infact,thatfurtherdevel-opmenthassometimesappearedindangerofbeingstymied,duetotheplethoraofdefinitionsandunclearlyrelatedtheo
4、ries.Onereasonforthegrowthofthesubjecthasbeen,withoutadoubt,therecognitionthatnondifferentiablephenomenaaremorewidespread,andplayamoreimportantrole,thanhadbeenthought.Philosophicallyatleast,thisisinkeepingwiththecomingtotheforeofseveralothertypesofirregular
5、andnonlinearbehavior:catastrophes,fractals,andchaos.Inrecentyears,nonsmoothanalysishascometoplayaroleinfunctionalanalysis,optimization,optimaldesign,mechanicsandplasticity,differen-tialequations(asinthetheoryofviscositysolutions),controltheory,and,increasin
6、gly,inanalysisgenerally(criticalpointtheory,inequalities,fixedpointtheory,variationalmethods...).Inthelongrun,weexpectitsmeth-odsandbasicconstructstobeviewedasanaturalpartofdifferentialanalysis.viiiPrefaceWehavefoundthatitwouldberelativelyeasytowriteaverylon
7、gbookonnonsmoothanalysisanditsapplications;severaltimes,wedid.Wehavenowmanagednottodoso,andinfactourprincipalclaimforthisworkisthatitpresentstheessentialsofthesubjectclearlyandsuccinctly,togetherwithsomeofitsapplicationsandageneroussupplyofinterestingexerc
8、ises.Wehavealsoincorporatedinthetextanumberofnewresultswhichclarifytherelationshipsbetweenthedifferentschoolsofthoughtinthesubject.Wehopethatthiswillhelpmakenonsmoothanalysisaccessibletoawideraudience.Inthissp