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《differential-geometric and group-theoretical methods in optimal control theory》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、DIFFERENTIAL-GEOMETRICANDGROUP-THEORETICALMETHODSINOPTIMALCONTROLTHEORYA.A.Agrachev,S.A.Vakhrameev,UDC517.977.1+514.7andR.V.GamkrelidzeProblemsarisinginthegeometryofattainablesetsofnonlinearcontrolsystemsareconsidered.Oldandnewresultsarediscussed.INTRODUCTIONThepresentworkisdevotedtocertainas
2、pectsofcontemporarycontroltheorywhichhavecometotheattentionofspecialistsonlyrecently.Wediscussbothknownandnewre-suits.Someoftheresultsaregivenwithfullproofs.Weusetheapparatusofchronologicalcalculus,basedontheexponentialrepresentationofflows[2,3].Someelementsofthiscal-culusaregivenintheAppendi
3、x.Weshallnotgivehereageneralsurveyofthecontentofourpaper,becauseitissuf-ficientlyreflectedbytheheadingsofthesections.Instead,weshallbrieflyexplainthenotationusedbelow.Asusual,weletRndenotethen-dimensionalarithmeticspacewhosepointsaretreatedascolumn-vectorsandarealwaysdenotedbyLatinletters;for
4、example,ylx'lRowvectorsaredenotedbyGreekletter,forexample,andmatrixmultiplicationisusedtowritethescalarproductofarowvectorandacolumnvector:BythemodulusofannxmmatrixA=(a~),a=1,...,n,B=1,...,m,wemeanthequan-titynlIAl=maxlast.p--Il<=5、en-vectorX~Rnandthatoftherowm-vector~=(~i,...,Em).TheJacobimatrixofanm-dimensionalvector-functionx§g(x)withrespecttothecoor-dinatesofthevectorkERnwillbedenotedbyTranslatedfromItogiNaukiiTekhniki,SeriyaProblemyGeometrii,Vol.14,pp.3-56,1983.0090-4104/85/2802-0145509.50@1985PlenumPublishingCorpo
6、ration145og(.~){o~~~,Bysmoothweshallalwaysmeaninfinitelydifferentiable.LetMbeasmoothmanifold,smoothlyembeddedinRd,andletEdenotetherestrictionoftheidentitymapofRdtoM.Further,letr=C~(M)denotethealgebraofallsmoothfunctionsonM.+AvectorfieldfonMisanyderivationofthealgebrari.e.,anylinearmap/:~-+~,w
7、hichobeystheruleof"differentiationofaproduct":/(~.,)=(/~).,+~(/,)vg,~.WedenotethesetofallderivationsofalgebrarbyDer(rItisknownthateveryderiva-tionofrisafirst-orderdifferentialoperatorandactsaccordingtotherule(/~)(x)=VxEM,~.Inpart
