Н.Л. Григоренко [25] получил необходимые и достаточные условия r–поимки одного убегающего группой преследователей при условии


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ÔÅÄÅÐÀËÜÍÎÅÃÎÑÓÄÀÐÑÒÂÅÍÍÎÅÁÞÄÆÅÒÍÎÅ ÎÁÐÀÇÎÂÀÒÅËÜÍÎÅÓ×ÐÅÆÄÅÍÈÅÂÛÑØÅÃÎÎÁÐÀÇÎÂÀÍÈß ¾ÓÄÌÓÐÒÑÊÈÉÃÎÑÓÄÀÐÑÒÂÅÍÍÛÉÓÍÈÂÅÐÑÈÒÅÒ¿ Íàïðàâàõðóêîïèñè ÑÎËÎÂÜÅÂÀÍÀÄÅÆÄÀÀËÅÊÑÀÍÄÐÎÂÍÀ ÃÐÓÏÏÎÂÎÅÏÐÅÑËÅÄÎÂÀÍÈÅÂÐÅÊÓÐÐÅÍÒÍÛÕ ÄÈÔÔÅÐÅÍÖÈÀËÜÍÛÕÈÃÐÀÕ 01.01.02äèôôåðåíöèàëüíûåóðàâíåíèÿ, äèíàìè÷åñêîåóïðàâëåíèåèîïòèìàëüíîåóïðàâëåíèå Äèññåðòàöèÿíàñîèñêàíèåó÷¼íîéñòåïåíè êàíäèäàòàôèçèêî-ìàòåìàòè÷åñêèõíàóê Íàó÷íûéðóêîâîäèòåëü: äîêòîðôèçèêî-ìàòåìàòè÷åñêèõíàóê,ïðîôåññîð ÏåòðîâÍ.Í. Èæåâñê2016 Îãëàâëåíèå Ââåäåíèå.....................................3 Ãëàâà1.Ëèíåéíûåðåêóððåíòíûåäèôôåðåíöèàëüíûåèãðû........26 1.1Ãðóïïîâîåïðåñëåäîâàíèåîäíîãîóáåãàþùåãîâëèíåéíûõðåêóð- ðåíòíûõäèôôåðåíöèàëüíûõèãðàõ.................26 1.2Ïîèìêàãðóïïûñêîîðäèíèðîâàííûõóáåãàþùèõâëèíåéíûõðåêóð- ðåíòíûõäèôôåðåíöèàëüíûõèãðàõ.................35 1.3Ïîèìêàçàäàííîãî÷èñëàóáåãàþùèõ................44 Ãëàâà2.ÏðèìåðË.Ñ.Ïîíòðÿãèíàñîìíîãèìèó÷àñòíèêàìè........49 2.1ÏîèìêàîäíîãîóáåãàþùåãîâðåêóððåíòíîìïðèìåðåË.Ñ.Ïîíòðÿ- ãèíà...................................49 2.2Ãðóïïîâîåïðåñëåäîâàíèåñôàçîâûìèîãðàíè÷åíèÿìèâðåêóððåíò- íîìïðèìåðåË.Ñ.Ïîíòðÿãèíà...................54 2.3ÌíîãîêðàòíàÿïîèìêàâðåêóððåíòíîìïðèìåðåË.Ñ.Ïîíòðÿãèíà ñôàçîâûìèîãðàíè÷åíèÿìè.....................68 Çàêëþ÷åíèå....................................82 Ñïèñîêîáîçíà÷åíèé...............................83 Ñïèñîêëèòåðàòóðû...............................84 3 Ââåäåíèå Àêòóàëüíîñòüèñòåïåíüðàçðàáîòàííîñòèòåìûèññëåäîâàíèÿ. Òåîðèÿ äèôôåðåíöèàëüíûõèãðèçó÷àåòçàäà÷èêîíôëèêòíîãîóïðàâëåíèÿïðèíàëè÷èè äâóõèëèáîëååñòîðîí,äâèæåíèÿêîòîðûõîïèñûâàþòñÿäèôôåðåíöèàëüíûìè óðàâíåíèÿìè.Ïðàêòè÷åñêèåçàäà÷èèçîáëàñòèýêîíîìèêè,ýêîëîãèè,áèîëîãèè, óïðàâëåíèÿìåõàíè÷åñêèìèñèñòåìàìè,àòàêæåâîåííîãîäåëàÿâëÿþòñÿëèøü íåêîòîðûìèïðèëîæåíèÿìèòåîðèèäèôôåðåíöèàëüíûõèãð. Îäíîéèçïåðâûõðàáîòñëåäóåòñ÷èòàòüðàáîòóÃ.Øòåéíãàóçà[132],îïóá- ëèêîâàííóþâ1925ãîäó,âêîòîðîéîíñôîðìóëèðîâàëçàäà÷óïðåñëåäîâàíèÿ. Òåîðèÿäèôôåðåíöèàëüíûõèãðíà÷àëàðàçâèâàòüñÿâíà÷àëå50õãîäîâ. Îäíèìèèçïåðâûõñåðüåçíûõèññëåäîâàíèéÿâëÿþòñÿðàáîòûàìåðèêàíñêîãî ìàòåìàòèêàÐ.Àéçåêñà,êîòîðûéèââåëòåðìèí¾äèôôåðåíöèàëüíàÿèãðà¿.Îí âñâîåéìîíîãðàôèè[3]ðàçâèëîðèãèíàëüíûéìåòîäðåøåíèÿâåñüìàîáùèõäèô- ôåðåíöèàëüíûõèãð,ðàññìîòðåëöåëûéðÿäïðèêëàäíûõçàäà÷èïîëó÷èëèíòå- ðåñíûåðåçóëüòàòû. Âíàøåéñòðàíåäèíàìè÷åñêèåçàäà÷èêîíôëèêòíîãîóïðàâëåíèÿðàññìàò- ðèâàþòñÿñíà÷àëà60õãîäîâïðîøëîãîâåêàèñâÿçàíûñèìåíàìèñîâåòñêèõ ìàòåìàòèêîâÍ.Í.Êðàñîâñêîãî[4345],Ë.Ñ.Ïîíòðÿãèíà[7984],Ë.À.Ïåòðîñÿ- íà[71,72],Á.Í.Ïøåíè÷íîãî[8588]. Ñðåäèðàáîòçàðóáåæíûõàâòîðîâêîíöà60-õíà÷àëà70-õãîäîâïðîøëîãî âåêàîòìåòèìðàáîòûL.D.Berkovitz,A.Blaqui`ere,J.V.Breakwell,W.H.Fleming, A.Friedman,G.Leitmann,A.W.Merz(ñì.[120,121,125,128]èáèáëèîãðàôèþê íèì).Âíèõðàññìàòðèâàëèñüòåîðåìûñóùåñòâîâàíèÿôóíêöèèöåíûâïîäõî- äÿùåìêëàññåñòðàòåãèé,èðàçâèâàëñÿìåòîäÐ.Àéçåêñàðåøåíèÿäèôôåðåíöè- àëüíûõèãðïðèïîìîùèïîñòðîåíèÿñèíãóëÿðíûõïîâåðõíîñòåé. ÊðóïíûéâêëàäâðàçâèòèèòåîðèèäèôôåðåíöèàëüíûõèãðâíåñëèÀ.À.Àçà- ìîâ,Ý.Ã.Àëüáðåõò,Â.Ä.Áàòóõòèí,Ì.Ñ.Ãàáðèýëÿí,Ð.Â.Ãàìêðåëèäçå, Í.Ë.Ãðèãîðåíêî,Ï.Á.Ãóñÿòíèêîâ,Â.È.Æóêîâñêèé,Ä.Çîííåâåíä,Ð.Ï.Èâà- 4 íîâ,À.Ô.Êëåéìåíîâ,À.Í.Êðàñîâñêèé,À.Â.Êðÿæèìñêèé,À.Á.Êóðæàí- ñêèé,Â.Í.Ëàãóíîâ,Þ.Ñ.Ëåäÿåâ,Äæ.Ëåéòìàí,Í.Þ.Ëóêîÿíîâ,À.Â.Ìå- çåíöåâ,À.À.Ìåëèêÿí,Å.Ô.Ìèùåíêî,Ì.Ñ.Íèêîëüñêèé,Þ.Ñ.Îñèïîâ, Â.Â.Îñòàïåíêî,À.Ã.Ïàøêîâ,Â.Ñ.Ïàöêî,Í.Í.Ïåòðîâ,Í.Íèêàíäð.Ïåò- ðîâ,Ã.Ê.Ïîæàðèöêèé,Å.Ñ.Ïîëîâèíêèí,È.Ñ.Ðàïïîïîðò,Á.Á.Ðèõ- ñèåâ,Í.Þ.Ñàòèìîâ,À.È.Ñóááîòèí,Í.Í.Ñóááîòèíà,Â.Å.Òðåòüÿêîâ, Â.Í.Óøàêîâ,Â.È.Óõîáîòîâ,À.Ã.×åíöîâ,Ô.Ë.×åðíîóñüêî,À.À.×è- êðèé,Ñ.Â.×èñòÿêîâ,Ð.Ýëëèîò,Ë.Ï.Þãàéèìíîãèåäðóãèåìàòåìàòèêè (ñì.[1,2,2430,3840,42,49,50,5259,62,92,96,99,103,107109,111,128]èáèáëèî- ãðàôèþêíèì). Â1974ãîäóáûëàîïóáëèêîâàíàêíèãàÍ.Í.ÊðàñîâñêîãîèÀ.È.Ñóááîòèíà ¾Ïîçèöèîííûåäèôôåðåíöèàëüíûåèãðû¿[45].Âíåé,â÷àñòíîñòè,ïðåäëîæåíà ïîçèöèîííàÿôîðìàëèçàöèÿäèôôåðåíöèàëüíûõèãðèäîêàçàíàòåîðåìàîáàëü- òåðíàòèâå,ðîäñòâåííàÿòåîðåìåñóùåñòâîâàíèÿôóíêöèèöåíû.Ðàññìàòðèâàåò- ñÿóïðàâëÿåìàÿñèñòåìà,òåêóùèåñîñòîÿíèÿêîòîðîéîïèñûâàþòñÿååôàçîâûì âåêòîðîì x = x ( t ) ,èçìåíÿþùèìñÿâîâðåìåíè t âñîîòâåòñòâèèñäèôôåðåíöè- àëüíûìóðàâíåíèåìäâèæåíèÿ _ x = f ( t;x;u;v ) ; ãäå x 2 R n ;u 2 P;v 2 Q;P  R m è Q  R k çàìêíóòûåìíîæåñòâà, f  íåïðåðûâíàÿôóíêöèÿ.Âïðîñòðàíñòâå R n çàäàíûçàìêíóòûåìíîæåñòâà M c è N .Ôîðìóëèðóåòñÿèãðàñáëèæåíèÿóêëîíåíèÿ,êîòîðàÿñêëàäûâàåòñÿèçäâóõ çàäà÷.Ïåðâàÿçàäà÷à(ñòîÿùàÿïåðåäïåðâûìèãðîêîì)çàäà÷àîñáëèæåíèè ñöåëåâûììíîæåñòâîì M c âíóòðèçàäàííûõîãðàíè÷åíèé N ;âòîðàÿçàäà÷à (ñòîÿùàÿïåðåäâòîðûìèãðîêîì)çàäà÷àîáóêëîíåíèèâåêòîðà x îò M c . ¾Öåíòðàëüíûéðåçóëüòàòñîñòàâëÿåòòåîðåìàîáàëüòåðíàòèâå,êîòîðàÿ óòâåðæäàåò,÷òîïðèâûïîëíåíèèëîêàëüíîãîóñëîâèÿñåäëîâîéòî÷êèìàëåíüêîé èãðûâñòàíäàðòíîéèãðåñáëèæåíèÿóêëîíåíèÿäëÿâñÿêîéíà÷àëüíîéïîçèöèè 5 f t 0 ;x 0 g ñïðàâåäëèâîîäíîèçäâóõóòâåðæäåíèé:ëèáîñóùåñòâóåòïîçèöèîííàÿ ñòðàòåãèÿïåðâîãîèãðîêàñîþçíèêà,êîòîðàÿîáåñïå÷èâàåòâñòðå÷óäâèæåíèÿ x [ t ] ñíàçíà÷åííîéöåëüþ M c ,êàêáûíèäåéñòâîâàëâòîðîéèãðîêïðîòèâíèê,ëè- áîñóùåñòâóåòïîçèöèîííàÿñòðàòåãèÿâòîðîãîèãðîêàñîþçíèêà,êîòîðàÿîáåñ- ïå÷èâàåòóêëîíåíèåäâèæåíèÿ x [ t ] îòóêàçàííîéöåëè M c ,êàêáûíèäåéñòâîâàë ïåðâûéèãðîêïðîòèâíèê.¿[45,ñ.14]. Èäåþðàññìàòðèâàòüäèôôåðåíöèàëüíóþèãðóñäâóõòî÷åêçðåíèÿïðåäëî- æèëèðàçâèëË.Ñ.Ïîíòðÿãèí[79].Âðàáîòå[80]Ë.Ñ.Ïîíòðÿãèíûìïîëó÷åíû äîñòàòî÷íûåóñëîâèÿäëÿâîçìîæíîñòèçàâåðøåíèÿïðåñëåäîâàíèÿâëèíåéíûõ äèôôåðåíöèàëüíûõèãðàõ.Âñâîåéðàáîòå[81]îíèñïîëüçîâàëôîðìàëèçìïðèí- öèïàìàêñèìóìàîäíîãîèçöåíòðàëüíûõìåòîäîâìàòåìàòè÷åñêîéòåîðèè óïðàâëåíèÿ.Îñíîâíîéðåçóëüòàòçàêëþ÷àåòñÿâîïèñàíèèìíîæåñòâàíà÷àëü- íûõïîçèöèé,èçêîòîðûõãàðàíòèðóåòñÿâîçìîæíîñòüçàâåðøåíèÿïðåñëåäîâà- íèÿ,àòàêæåââû÷èñëåíèèâðåìåíèïðåñëåäîâàíèÿ,èñïîñîáôîðìèðîâàíèÿ óïðàâëåíèÿïðåñëåäîâàòåëÿ,ðåàëèçóþùåãîïðîöåññïðåñëåäîâàíèÿ. Á.Í.Ïøåíè÷íûì[87]áûëèðàññìîòðåíûíåëèíåéíûåäèôôåðåíöèàëüíûå èãðûîáùåãîâèäà,äëÿêîòîðûõèìïðåäëîæåíàïðîöåäóðà,îïðåäåëÿþùàÿíåîá- õîäèìûåèäîñòàòî÷íûåóñëîâèÿðàçðåøèìîñòèçàäà÷èïðåñëåäîâàíèÿ.Èíòå- ðåñíûåðåçóëüòàòûïîëó÷åíûïðèèññëåäîâàíèèëèíåéíûõäèôôåðåíöèàëüíûõ èãð[88]. Âòåîðèèäèôôåðåíöèàëüíûõèãðáîëååîáùåéÿâëÿåòñÿñèòóàöèÿ,êîãäà âèãðåïðèíèìàþòó÷àñòèåíåñêîëüêîïðåñëåäîâàòåëåéèîäèíèëèíåñêîëüêî óáåãàþùèõ.Âýòîìñëó÷àåäèôôåðåíöèàëüíàÿèãðàíàçûâàåòñÿäèôôåðåíöè- àëüíîéèãðîéìíîãèõëèö.Òàêèåèãðûîõâàòûâàþòìíîãèåçàäà÷è,íàïðèìåð, çàäà÷óóáåãàíèÿîäíîãîóïðàâëÿåìîãîîáúåêòàîòãðóïïûïðåñëåäîâàòåëåé,çà- äà÷óèçáåæàíèÿñòîëêíîâåíèÿñíåñêîëüêèìèïðåïÿòñòâèÿìèèäðóãèå.Çàäà- ÷åïðåñëåäîâàíèÿâäèôôåðåíöèàëüíûõèãðàõìíîãèõëèöïîñâÿùåíûðàáîòû Í.Ë.Ãðèãîðåíêî[24],Á.Í.Ïøåíè÷íîãî[87],À.È.×èêðèÿèÈ.Ñ.Ðàïïîïîð- òà[115],Í.ÑàòèìîâàèÌ.Ø.Ìàìàòîâà[96],Ï.Á.Ãóñÿòíèêîâà[29]. 6 Îäíîéèçïåðâûõðàáîò,ïîñâÿùåííûõçàäà÷åãðóïïîâîãîïðåñëåäîâàíèÿ,áû- ëàðàáîòàË.À.Ïåòðîñÿíà[72],ãäåáûëîââåäåíîïîíÿòèåñòðàòåãèèïàðàëëåëü- íîãîïðåñëåäîâàíèÿ. ÂðàáîòåÁ.Í.Ïøåíè÷íîãî[87]ðàññìàòðèâàëàñüçàäà÷àïðîñòîãîïðåñëåäî- âàíèÿãðóïïîéïðåñëåäîâàòåëåéîäíîãîóáåãàþùåãî,ïðèóñëîâèè,÷òîñêîðîñòè óáåãàþùåãîèïðåñëåäîâàòåëåéïîíîðìåíåïðåâîñõîäÿòåäèíèöû.Áûëèïîëó- ÷åíûíåîáõîäèìûåèäîñòàòî÷íûåóñëîâèÿïîèìêè:ïîèìêàïðîèñõîäèòòîãäàè òîëüêîòîãäà,êîãäàíà÷àëüíàÿïîçèöèÿóáåãàþùåãîïðèíàäëåæèòâíóòðåííîñòè âûïóêëîéîáîëî÷êèíà÷àëüíûõïîçèöèéïðåñëåäîâàòåëåé. Ô.Ë.×åðíîóñüêî[107]ðàññìàòðèâàëçàäà÷óóêëîíåíèÿóïðàâëÿåìîéòî÷êè, ñêîðîñòüêîòîðîéîãðàíè÷åíàïîâåëè÷èíå,îòâñòðå÷èñëþáûìêîíå÷íûì÷èñ- ëîìïðåñëåäóþùèõòî÷åê,ñêîðîñòèêîòîðûõòàêæåîãðàíè÷åíûïîâåëè÷èíå èñòðîãîìåíüøåñêîðîñòèóêëîíÿþùåéñÿòî÷êè.Áûëïîñòðîåíòàêîéñïîñîá óïðàâëåíèÿ,êîòîðûéîáåñïå÷èâàåòóêëîíåíèåîòâñåõïðåñëåäîâàòåëåéíàêî- íå÷íîåðàññòîÿíèå,ïðè÷åìäâèæåíèåóêëîíÿþùåéñÿòî÷êèîñòàåòñÿâôèêñè- ðîâàííîéîêðåñòíîñòèçàäàííîãîäâèæåíèÿ. Ð.Ï.Èâàíîâ[39]ðàññìîòðåëçàäà÷óïðîñòîãîïðåñëåäîâàíèÿãðóïïîéïðåñëå- äîâàòåëåéîäíîãîóáåãàþùåãîïðèóñëîâèè,÷òîóáåãàþùèéíåïîêèäàåòïðåäåëû âûïóêëîãîêîìïàêòàñíåïóñòîéâíóòðåííîñòüþ.Áûëîäîêàçàíî,÷òîåñëè÷èñëî ïðåñëåäîâàòåëåéìåíüøåðàçìåðíîñòèìíîæåñòâà,òîáóäåòóêëîíåíèå,èíà÷å ïîèìêàèïîëó÷åíàîöåíêàâðåìåíèïîèìêè.ÐàáîòàÍ.Í.Ïåòðîâà[58]îáîáùà- åòðåçóëüòàòÐ.Ï.Èâàíîâàíàñëó÷àé,êîãäàóáåãàþùèéíåïîêèäàåòïðåäåëû âûïóêëîãîìíîãîãðàííîãîìíîæåñòâàñíåïóñòîéâíóòðåííîñòüþ. Í.Ë.Ãðèãîðåíêî[24]ïîëó÷èëíåîáõîäèìûåèäîñòàòî÷íûåóñëîâèÿóêëîíå- íèÿîòâñòðå÷èîäíîãîóáåãàþùåãîîòíåñêîëüêèõïðåñëåäîâàòåëåéïðèóñëîâèè, ÷òîóáåãàþùèéèïðåñëåäîâàòåëèîáëàäàþòïðîñòûìäâèæåíèåì,èìíîæåñòâî óïðàâëåíèéêàæäîãîèçèãðîêîâîäèíèòîòæåâûïóêëûéêîìïàêò. Âðàáîòå[114]À.À.×èêðèåìèÏ.Â.Ïðîêîïîâè÷åìðàññìàòðèâàëàñüçàäà÷à óêëîíåíèÿãðóïïûèç m óáåãàþùèõîòãðóïïûèç n ïðåñëåäîâàòåëåé,âêîòîðîé 7 çàêîíäâèæåíèÿêàæäîãîèçó÷àñòíèêîâèìååòâèä _ z = Az + w;w 2 V;z 2 R k ; ãäå A êâàäðàòíàÿìàòðèöà, V âûïóêëûéêîìïàêò, k � 2 .Âòåðìèíàõ íà÷àëüíûõïîçèöèéèïàðàìåòðîâèãðûáûëèïîëó÷åíûäîñòàòî÷íûåóñëîâèÿ óêëîíåíèÿõîòÿáûîäíîãîóáåãàþùåãîîòãðóïïûïðåñëåäîâàòåëåéèççàäàí- íûõíà÷àëüíûõïîçèöèéèèçëþáûõíà÷àëüíûõïîçèöèé(âïîñëåäíåìñëó÷àå ïðåäïîëàãàåòñÿ,÷òîâåëè÷èíû n;m ôèêñèðîâàíû). Âðàáîòå[96]Í.ÑàòèìîâèÌ.Ø.Ìàìàòîâðàññìîòðåëèçàäà÷óïðåñëåäîâà- íèÿãðóïïîéïðåñëåäîâàòåëåéãðóïïûóáåãàþùèõïðèóñëîâèè,÷òîïðåñëåäîâà- òåëèîáëàäàþòïðîñòûìäâèæåíèåìñåäèíè÷íîéïîíîðìåìàêñèìàëüíîéñêî- ðîñòüþèóáåãàþùèå,êðîìåòîãî,èñïîëüçóþòîäíîèòîæåóïðàâëåíèå(æåñòêî ñêîîðäèíèðîâàííûåóáåãàþùèå).Öåëüãðóïïûïðåñëåäîâàòåëåéïîéìàòüõîòÿ áûîäíîãîóáåãàþùåãî.Áûëèïðèâåäåíûäîñòàòî÷íûåóñëîâèÿïîèìêè.Ðàáîòû Ä.À.ÂàãèíàèÍ.Í.Ïåòðîâà[19,59]äîïîëíÿþòïðåäûäóùóþðàáîòó. Í.Í.ÏåòðîâèÂ.À.Ïðîêîïåíêî[63]ðàññìàòðèâàëèçàäà÷óïðîñòîãîïðåñëå- äîâàíèÿãðóïïîéïðåñëåäîâàòåëåéãðóïïûóáåãàþùèõïðèóñëîâèè,÷òîñêîðî- ñòèâñåõó÷àñòíèêîâïîíîðìåíåïðåâîñõîäÿòåäèíèöå,êàæäûéïðåñëåäîâàòåëü ëîâèòíåáîëååîäíîãîóáåãàþùåãî,àóáåãàþùèåâíà÷àëüíûéìîìåíòâðåìåíè âûáèðàþòñâîåóïðàâëåíèåíàèíòåðâàëå [0; 1 ) .Áûëèïîëó÷åíûíåîáõîäèìûå èäîñòàòî÷íûåóñëîâèÿïîèìêè. Âðàáîòå[106]Á.Ê.Õàéäàðîâðàññìîòðåëçàäà÷óïîçèöèîííîé l ïîèìêèîä- íîãîóáåãàþùåãîãðóïïîéïðåñëåäîâàòåëåéïðèóñëîâèè,÷òîêàæäûéèçèãðî- êîâîáëàäàåòïðîñòûìäâèæåíèåì.Í.Ë.Ãðèãîðåíêî[25]ïîëó÷èëíåîáõîäèìûå èäîñòàòî÷íûåóñëîâèÿ r ïîèìêèîäíîãîóáåãàþùåãîãðóïïîéïðåñëåäîâàòåëåé ïðèóñëîâèè,÷òîâñåèãðîêèîáëàäàþòïðîñòûìäâèæåíèåìñìàêñèìàëüíîé ïîíîðìåñêîðîñòüþ,ðàâíîéåäèíèöå.À.À.×èêðèåì[110]áûëèïîëó÷åíûäî- ñòàòî÷íûåóñëîâèÿìíîãîêðàòíîéïîèìêèâêîíôëèêòíîóïðàâëÿåìûõïðîöåñ- 8 ñàõ.Âðàáîòàõ[11,13,14,17]À.È.Áëàãîäàòñêèõïðèâîäèòäîñòàòî÷íûåóñëîâèÿ ìíîãîêðàòíîé,íåñòðîãîéîäíîâðåìåííîéèîäíîâðåìåííîéìíîãîêðàòíîéïîè- ìîê;â÷àñòíîñòè,äëÿçàäà÷èïðîñòîãîãðóïïîâîãîïðåñëåäîâàíèÿñðàâíûìè âîçìîæíîñòÿìèïîëó÷åíûíåîáõîäèìûåèäîñòàòî÷íûåóñëîâèÿîäíîâðåìåííîé ìíîãîêðàòíîéïîèìêè.Âðàáîòå[10]ââåäåíîïîíÿòèåèïîëó÷åíûíåîáõîäèìûå èäîñòàòî÷íûåóñëîâèÿìíîãîêðàòíîéèîäíîâðåìåííîéìíîãîêðàòíîéïîèìîê âçàäà÷åïðîñòîãîãðóïïîâîãîïðåñëåäîâàíèÿñðàâíûìèâîçìîæíîñòÿìèïðè íàëè÷èèòðåòüåéãðóïïûó÷àñòíèêîâçàùèòíèêîâóáåãàþùèõ. Îáîáùåíèåìçàäà÷èïðîñòîãîïðåñëåäîâàíèÿÿâëÿåòñÿïðèìåðÏîíòðÿãè- íà[79].Âðàáîòå[56]Í.Í.Ïåòðîâðàññìîòðåëçàäà÷óïðåñëåäîâàíèÿãðóïïîé ïðåñëåäîâàòåëåéîäíîãîóáåãàþùåãîâïðèìåðåÏîíòðÿãèíàñðàâíûìèäèíàìè- ÷åñêèìèèèíåðöèîííûìèâîçìîæíîñòÿìèèãðîêîâ.Áûëèïîëó÷åíûäîñòàòî÷- íûåóñëîâèÿïîèìêè.Âðàáîòå[53]ðàññìîòðåíàçàäà÷àîìíîãîêðàòíîéïîèìêå îäíîãîóáåãàþùåãîãðóïïîéïðåñëåäîâàòåëåéâïðèìåðåÏîíòðÿãèíàñôàçîâûìè îãðàíè÷åíèÿìè.Çàäà÷àïðåñëåäîâàíèÿæåñòêîñêîîðäèíèðîâàííûõóáåãàþùèõ ãðóïïîéïðåñëåäîâàòåëåéâïðèìåðåÏîíòðÿãèíàïðèðàâíûõäèíàìè÷åñêèõè èíåðöèîííûõâîçìîæíîñòÿõó÷àñòíèêîâðàññìîòðåíàâ[20].Ïîëó÷åíûäîñòà- òî÷íûåóñëîâèÿïîèìêèõîòÿáûîäíîãîóáåãàþùåãî.Âðàáîòå[55]Í.Í.Ïåòðîâ ðàññìîòðåëçàäà÷óïðåñëåäîâàíèÿãðóïïîéïðåñëåäîâàòåëåéãðóïïûóáåãàþùèõ âïðèìåðåÏîíòðÿãèíàñðàâíûìèäèíàìè÷åñêèìèèèíåðöèîííûìèâîçìîæíî- ñòÿìèèãðîêîâïðèóñëîâèè,÷òîêàæäûéïðåñëåäîâàòåëüëîâèòíåáîëååîäíîãî óáåãàþùåãî,àóáåãàþùèåâíà÷àëüíûéìîìåíòâðåìåíèâûáèðàþòñâîåóïðàâ- ëåíèåíàèíòåðâàëå [0; 1 ) èíåïîêèäàþòïðåäåëûìíîæåñòâà D . Á.Ò.Ñàìàòîââðàáîòàõ[94,95]ðàññìîòðåëçàäà÷óïðåñëåäîâàíèÿóáåãàíèÿ äëÿñëó÷àÿ,êîãäàíàêëàññóïðàâëåíèéïðåñëåäîâàòåëÿíàëàãàåòñÿèíòåãðàëü- íîåîãðàíè÷åíèå,äîïóñêàþùååëèíåéíîåèçìåíåíèåñòå÷åíèåìâðåìåíè,êîòîðîå ÿâëÿåòñÿîáîáùåíèåìêàêèíòåãðàëüíûõ,òàêèãåîìåòðè÷åñêèõîãðàíè÷åíèé,à íàêëàññóïðàâëåíèéóáåãàþùåãîòîëüêîãåîìåòðè÷åñêîå.Ïðèýòîìçàäà÷àîï- òèìàëüíîãîïðåñëåäîâàíèÿðåøàåòñÿïîñðåäñòâîìîáîáùåííîéñòðàòåãèèïàðàë- 9 ëåëüíîãîïðåñëåäîâàíèÿ,àâçàäà÷åóáåãàíèÿóñòàíàâëèâàþòñÿíèæíèåîöåíêè äëÿðàññòîÿíèÿìåæäóïðåñëåäîâàòåëåìèóáåãàþùèì. ÂñâîèõðàáîòàõÑ.À.Ãàíåáíûé,Ñ.Ñ.Êóìêîâ,Ñ.ËåÌåíåê,Â.Ñ.Ïàöêî [23,46,47]ðàññìîòðåëèäèôôåðåíöèàëüíóþèãðóñäâóìÿäîãîíÿþùèìèèîäíèì óáåãàþùèì.Äèíàìèêàêàæäîãîèçîáúåêòîâîïèñàíàëèíåéíîéñòàöèîíàðíîé ñèñòåìîéîáùåãîâèäàñîñêàëÿðíûìóïðàâëÿþùèìâîçäåéñòâèåì.Ïëàòîéÿâëÿ- åòñÿìèíèìóìèçäâóõîäíîìåðíûõïðîìàõîâìåæäóïåðâûìïðåñëåäîâàòåëåì èóáåãàþùèìèìåæäóâòîðûìïðåñëåäîâàòåëåìèóáåãàþùèì.Ïðîìàõèïîä- ñ÷èòûâàþòñÿâôèêñèðîâàííûåçàðàíååìîìåíòûâðåìåíè.Îïèñûâàåòñÿñïîñîá ïîñòðîåíèÿìíîæåñòâóðîâíÿôóíêöèèöåíû(ìíîæåñòâðàçðåøèìîñòèèãðîâîé çàäà÷è)äëÿðàçëè÷íûõâàðèàíòîâïàðàìåòðîâçàäà÷è.Äëÿñëó÷àÿñèëüíûõ ïðåñëåäîâàòåëåéäàþòñÿñïîñîáûïîñòðîåíèÿîïòèìàëüíûõñòðàòåãèé. Öåëüèçàäà÷èèññëåäîâàíèÿ. Öåëüäàííîéðàáîòûñîñòîèòâïîëó÷åíèè óñëîâèéðàçðåøèìîñòèíîâûõêëàññîâèãðîâûõçàäà÷ãðóïïîâîãîïðåñëåäîâà- íèÿïðèäîïîëíèòåëüíûõ,òèïà¾ôàçîâûõ¿,îãðàíè÷åíèÿõíàñîñòîÿíèåóáåãà- þùåãî.Âäèññåðòàöèèèññëåäóþòñÿñëåäóþùèåçàäà÷è:çàäà÷àïðåñëåäîâàíèÿ ãðóïïîéïðåñëåäîâàòåëåéîäíîãîèëèíåñêîëüêèõóáåãàþùèõâëèíåéíûõíåñòà- öèîíàðíûõäèôôåðåíöèàëüíûõèãðàõïðèóñëîâèè,÷òîôóíäàìåíòàëüíàÿìàò- ðèöàîäíîðîäíîéñèñòåìûÿâëÿåòñÿðåêóððåíòíîéïîÇóáîâó;çàäà÷àãðóïïîâîãî ïðåñëåäîâàíèÿîäíîãîóáåãàþùåãîâíåñòàöèîíàðíîìïðèìåðåË.Ñ.Ïîíòðÿãèíà ïðèóñëîâèè,÷òîíåêîòîðûåôóíêöèè,îïðåäåëÿåìûåíà÷àëüíûìèóñëîâèÿìèè ïàðàìåòðàìèèãðû,ÿâëÿþòñÿðåêóððåíòíûìèïîÇóáîâó. Íàó÷íàÿíîâèçíà. Âñåîñíîâíûåðåçóëüòàòûäèññåðòàöèèÿâëÿþòñÿíîâû- ìè. Òåîðåòè÷åñêàÿèïðàêòè÷åñêàÿöåííîñòü. Ðàáîòàíîñèòòåîðåòè÷åñêèé õàðàêòåð.Ïîëó÷åííûåðåçóëüòàòûìîãóòáûòüèñïîëüçîâàíûäëÿäàëüíåéøåãî èññëåäîâàíèÿçàäà÷ãðóïïîâîãîïðåñëåäîâàíèÿ. Ìåòîäîëîãèÿèìåòîäûèññëåäîâàíèÿ. Âðàáîòåèñïîëüçóþòñÿìåòîäû òåîðèèäèôôåðåíöèàëüíûõèãð,îïòèìàëüíîãîóïðàâëåíèÿ,âûïóêëîãîàíàëèçà. 10 Ïîëîæåíèÿ,âûíîñèìûåíàçàùèòó. Âðàáîòåïîëó÷åíû: 1 : Äîñòàòî÷íûåóñëîâèÿïîèìêèãðóïïîéïðåñëåäîâàòåëåéîäíîãîóáåãàþ- ùåãîâëèíåéíûõíåñòàöèîíàðíûõäèôôåðåíöèàëüíûõèãðàõâïðåäïîëîæåíèè, ÷òîôóíäàìåíòàëüíàÿìàòðèöàîäíîðîäíîéñèñòåìûÿâëÿåòñÿðåêóððåíòíîéïî Çóáîâó; 2 : Äîñòàòî÷íûåóñëîâèÿïîèìêèõîòÿáûîäíîãîóáåãàþùåãîäëÿëèíåéíîé íåñòàöèîíàðíîéçàäà÷èïðåñëåäîâàíèÿãðóïïîéïðåñëåäîâàòåëåéãðóïïûóáåãà- þùèõ,ïðèóñëîâèè,÷òîôóíäàìåíòàëüíàÿìàòðèöàîäíîðîäíîéñèñòåìûÿâëÿ- åòñÿðåêóððåíòíîéïîÇóáîâóôóíêöèåé,àâñåóáåãàþùèåèñïîëüçóþòîäíîèòî æåóïðàâëåíèå; 3 : Äîñòàòî÷íûåóñëîâèÿïîèìêèçàäàííîãî÷èñëàóáåãàþùèõäëÿëèíåéíîé íåñòàöèîíàðíîéçàäà÷èïðåñëåäîâàíèÿãðóïïîéïðåñëåäîâàòåëåéãðóïïûóáåãà- þùèõ,ïðèóñëîâèè,÷òîôóíäàìåíòàëüíàÿìàòðèöàîäíîðîäíîéñèñòåìûÿâëÿåò- ñÿðåêóððåíòíîéïîÇóáîâóôóíêöèåéèêàæäûéïðåñëåäîâàòåëüìîæåòïîéìàòü íåáîëååîäíîãîóáåãàþùåãî; 4 : Äîñòàòî÷íûåóñëîâèÿðàçðåøèìîñòèçàäà÷èïðåñëåäîâàíèÿâîáîáùåííîì íåñòàöèîíàðíîìïðèìåðåÀ.Ñ.Ïîíòðÿãèíàñîìíîãèìèó÷àñòíèêàìèïðèîäè- íàêîâûõäèíàìè÷åñêèõèèíåðöèîííûõâîçìîæíîñòÿõâñåõèãðîêîââïðåäïîëî- æåíèèðåêóððåíòíîñòèïîÇóáîâóíåêîòîðûõôóíêöèéâòåðìèíàõíà÷àëüíûõ ïîçèöèéèïàðàìåòðîâèãðû. Ñòåïåíüäîñòîâåðíîñòèèàïðîáàöèÿðåçóëüòàòîâ. Îñíîâíûåìàòå- ðèàëûäèññåðòàöèèîïóáëèêîâàíûâ14ðàáîòàõ[21,22,61,6470,100102,129], èçíèõñåìüïóáëèêàöèé[22,6668,100,102,129]îïóáëèêîâàíûââåäóùèõðå- öåíçèðóåìûõíàó÷íûõæóðíàëàõèèçäàíèÿõ:ðîññèéñêèõèçÏåðå÷íÿÂÀÊ [22,6668,100,102]èçàðóáåæíûõ[129],âõîäÿùèõâìåæäóíàðîäíóþðåôåðà- òèâíóþáàçóäàííûõScopus.Âñåðåçóëüòàòûäèññåðòàöèèñòðîãîäîêàçàíû. Îñíîâíûåðåçóëüòàòûäèññåðòàöèèäîêëàäûâàëèñüíàìåæäóíàðîäíûõèâñå- ðîññèéñêèõêîíôåðåíöèÿõ:Ìåæäóíàðîäíàÿêîíôåðåíöèÿ¾Àêòóàëüíûåïðî- áëåìûïðèêëàäíîéìàòåìàòèêèèèíôîðìàöèîííûõòåõíîëîãèéàëü-Õîðåçìè 11 2012¿(Íàö.óí-òÓçáåêèñòàíàèì.ÌèðçîÓëóãáåêà,Òàøêåíò,1922äåê.2012ã.), Êîíôåðåíöèÿ¾Äèôôåðåíöèàëüíûåóðàâíåíèÿèîïòèìàëüíîåóïðàâëåíèå¿,ïî- ñâÿùåííàÿ90ëåòèþñîäíÿðîæäåíèÿàêàäåìèêàÅâãåíèÿÔðîëîâè÷àÌèùåíêî (Ìàò.èí-òèì.Â.À.ÑòåêëîâàÐÀÍ,Ìîñêâà,1617àïðåëÿ2012ã.),Ìåæäóíà- ðîäíàÿêîíôåðåíöèÿïîìàòåìàòè÷åñêîéòåîðèèóïðàâëåíèÿèìåõàíèêå(Ìàò. èí-òèì.Â.À.ÑòåêëîâàÐÀÍ,Âëàäèìèð.ãîñ.óí-òèì.ÀëåêñàíäðàÃðèãîðüåâè- ÷àèÍèêîëàÿÃðèãîðüåâè÷àÑòîëåòîâûõ,Ìîñê.ãîñ.óí-òèì.Ì.Â.Ëîìîíîñîâà, Ñóçäàëü,2013.),Ìåæäóíàðîäíàÿêîíôåðåíöèÿ¾Äèíàìèêàñèñòåìèïðîöåññû óïðàâëåíèÿ¿,ïîñâÿù.90ëåòèþñîäíÿðîæä.àêàä.Í.Í.Êðàñîâñêîãî,(Èí-ò ìàòåìàòèêèèìåõàíèêèèì.Í.Í.ÊðàñîâñêîãîÓðÎÐÀÍ,Åêàòåðèíáóðã,1520 ñåíò.2014ã.),IIÌåæäóíàðîäíûéñåìèíàð¾Òåîðèÿóïðàâëåíèÿèòåîðèÿîáîá- ùåííûõðåøåíèéóðàâíåíèéÃàìèëüòîíàßêîáè¿,ïîñâ.70ëåòèþñîäíÿðîæäå- íèÿàêàä.À.È.Ñóááîòèíà(Åêàòåðèíáóðã,13àïðåëÿ2015ã.),Âñåðîññèéñêàÿ êîíôåðåíöèÿñìåæäóíàðîäíûìó÷àñòèåì¾Òåîðèÿóïðàâëåíèÿèìàòåìàòè÷å- ñêîåìîäåëèðîâàíèå¿,ïîñâÿù.ïàìÿòèïðîô.Í.Â.Àçáåëåâàèïðîô.Å.Ë.Òîí- êîâà(ÔÃÁÎÓÂÏξÓäìóðòñêèéãîñóäàðñòâåííûéóíèâåðñèòåò¿,Èæåâñê,911 èþíÿ2015ã.).Òåçèñûäîêëàäîâîïóáëèêîâàíûâ[21,61,65,69,70,101].Ðåçóëü- òàòûîáñóæäàëèñüòàêæåíàñåìèíàðåîòäåëàäèíàìè÷åñêèõñèñòåìÈíñòèòóòà ìàòåìàòèêèèìåõàíèêèèì.Í.Í.ÊðàñîâñêîãîÓðÎÐÀÍ(ðóêîâîäèòåëè ÷ëåí-êîððåñïîíäåíòÐÀÍÂ.Í.Óøàêîâ,ïðîôåññîðÀ.Ì.Òàðàñüåâ;2016ã.)è íàñåìèíàðàõïîäèôôåðåíöèàëüíûìóðàâíåíèÿìèòåîðèèóïðàâëåíèÿêàôåä- ðûäèôôåðåíöèàëüíûõóðàâíåíèéÓäÃÓ. Âñåîñíîâíûåðåçóëüòàòûäèññåðòàöèèàâòîðïîëó÷èëëè÷íî.Âñîâìåñòíûõ ñòàòüÿõñíàó÷íûìðóêîâîäèòåëåì[22,64,6668,129]ÏåòðîâóÍ.Í.ïðèíàäëå- æàòïîñòàíîâêàçàäà÷èèîáùååðóêîâîäñòâîïðîâîäèìûìèèññëåäîâàíèÿìè.Èç ðåçóëüòàòîâðàáîòû[22]âäèññåðòàöèþâêëþ÷åíàëåììà3,ïðèíàäëåæàùàÿàâ- òîðó. Îñíîâíîåñîäåðæàíèåðàáîòû. Äèññåðòàöèÿñîñòîèòèçââåäåíèÿ,äâóõ ãëàâ,çàêëþ÷åíèÿ,ñïèñêàîáîçíà÷åíèéèñïèñêàëèòåðàòóðû.Îáúåìðàáîòû96 12 ñòðàíèö.Ñïèñîêëèòåðàòóðûâêëþ÷àåò132íàèìåíîâàíèé. Ðàáîòàïîñâÿùåíàäèôôåðåíöèàëüíûìèãðàìïðåñëåäîâàíèÿñó÷àñòèåì äâóõãðóïï(ïðåñëåäîâàòåëåéèóáåãàþùèõ).Âñåäèôôåðåíöèàëüíûåèãðûðàñ- ñìàòðèâàþòñÿâïðîñòðàíñòâå R k ( k � 2) .Âïåðâîéãëàâåäèññåðòàöèèðàññìàò- ðèâàþòñÿëèíåéíûåðåêóððåíòíûåäèôôåðåíöèàëüíûåèãðû.Ïåðâàÿãëàâàñî- ñòîèòèçòðåõïàðàãðàôîâ. Âïåðâîìïàðàãðàôåðàññìàòðèâàåòñÿëèíåéíàÿíåñòàöèîíàðíàÿçàäà÷àïðå- ñëåäîâàíèÿãðóïïîéïðåñëåäîâàòåëåé P 1 ;:::;P n îäíîãîóáåãàþùåãî E ñðàâíû- ìèäèíàìè÷åñêèìèèèíåðöèîííûìèâîçìîæíîñòÿìèâñåõó÷àñòíèêîâ. Çàêîíäâèæåíèÿêàæäîãîèçïðåñëåäîâàòåëåé P i èìååòâèä _ x i = A ( t ) x i + u i ;u i 2 V: Çàêîíäâèæåíèÿóáåãàþùåãî E èìååòâèä _ y = A ( t ) y + v;v 2 V: Çäåñüèäàëåå x i ;y;u i ;v 2 R k ;i 2 I = f 1 ; 2 ;:::;n g , A ( t ) íåïðåðûâíàÿíà [ t 0 ; 1 ) êâàäðàòíàÿìàòðèöàïîðÿäêà k , V ñòðîãîâûïóêëûéêîìïàêò R k ñ ãëàäêîéãðàíèöåé.Ïðè t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x i ( t 0 )= x 0 i ;y ( t 0 )= y 0 ; ïðè÷åì x 0 i 6 = y 0 äëÿâñåõ i: Ðàññìîòðèìñèñòåìóñíà÷àëüíûìèóñëîâèÿìè _ z i = A ( t ) z i + u i � v;u i ;v 2 V;z i ( t 0 )= z 0 i = x 0 i � y 0 : Îòìåòèì,÷òî z 0 i 6 =0 . Íàçîâåìïðåäûñòîðèåéóïðàâëåíèÿ v ( t ) óáåãàþùåãî E âìîìåíòâðåìåíè 13 t;t 2 [ t 0 ; 1 ) ìíîæåñòâî v t (  )= f v ( s ) ;s 2 [ t 0 ;t ] ;v � èçìåðèìàÿôóíêöèÿ. g Îïðåäåëåíèå1.1 Áóäåìãîâîðèòü,÷òîçàäàíàêâàçèñòðàòåãèÿ U i ïðå- ñëåäîâàòåëÿ P i ; åñëèîïðåäåëåíîîòîáðàæåíèå U i ( t;z 0 ;v t (  )) ; ñòàâÿùååâñî- îòâåòñòâèåíà÷àëüíîìóñîñòîÿíèþ z 0 =( z 0 1 ;:::;z 0 n ) ,ìîìåíòó t èïðîèç- âîëüíîéïðåäûñòîðèèóïðàâëåíèÿ v t (  ) óáåãàþùåãî E èçìåðèìóþôóíêöèþ u i ( t )= U i ( t;z 0 ;v t (  )) ñîçíà÷åíèÿìèâ V: Ïðèýòîìïðåäïîëàãàåòñÿ,÷òîäîëæíîáûòüâûïîëíåíîóñëîâèå¾ôèçè÷åñêîé îñóùåñòâèìîñòè¿,òîåñòüåñëè v 1 ;v 2 äâàäîïóñòèìûõóïðàâëåíèÿóáåãàþùåãî E; ïðè÷åì v 1 ( t )= v 2 ( t ) äëÿïî÷òèâñåõ t; òîñîîòâåòñòâóþùèåèìïðèîòîáðà- æåíèè U i ( t;z 0 ;v t (  )) ôóíêöèè u 1 ;u 2 òàêæåðàâíûïî÷òèâñþäóïðè t � 0 : Îáîçíà÷èìäàííóþèãðó÷åðåç � 1 . Îïðåäåëåíèå1.2 Âèãðå � 1 ïðîèñõîäèòïîèìêà,åñëèñóùåñòâóåòìî- ìåíò T 0 = T ( z 0 ) ,êâàçèñòðàòåãèè U 1 ;:::; U n ïðåñëåäîâàòåëåé P 1 ;:::;P n ,òà- êèå,÷òîäëÿëþáîéèçìåðèìîéôóíêöèè v (  ) , v ( t ) 2 V , t 2 [0 ;T 0 ] íàéäóòñÿ íîìåð q 2f 1 ;:::;n g èìîìåíò  6 T 0 òàêèå,÷òî z q (  )=0 . Îïðåäåëåíèå1.3 ([33]) Ôóíêöèÿ F : R 1 ! R k íàçûâàåòñÿðåêóððåíòíîé ïîÇóáîâó(äàëååðåêóððåíòíîé),åñëèäëÿëþáîãî "� 0 ñóùåñòâóåò T ( " ) � 0 òàêîå,÷òîäëÿëþáûõ t;a 2 R 1 ñóùåñòâóåò  ( t ) 2 [ a;a + T ( " )] ,äëÿêîòîðûõ âûïîëíåíîíåðàâåíñòâî k F ( t +  ( t )) � F ( t ) k ": Îïðåäåëåíèå1.4 Ôóíêöèÿ f :[ t 0 ; 1 ) ! R k íàçûâàåòñÿðåêóððåíòíîéïîÇó- áîâó(äàëååðåêóððåíòíîé)íà [ t 0 ; 1 ) ,åñëèñóùåñòâóåòðåêóððåíòíàÿôóíê- öèÿ F : R 1 ! R k òàêàÿ,÷òî f ( t )= F ( t ) äëÿâñåõ t 2 [ t 0 ; 1 ) : Îáîçíà÷èì÷åðåç ( t ) ôóíäàìåíòàëüíóþìàòðèöóñèñòåìû _ ! = A ( t ) !; 14 ãäå ( t 0 )  E , E åäèíè÷íàÿìàòðèöà. Òåîðåìà1.1 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ìàòðèöà ( t ) ðåêóððåíòíàíà [ t 0 ; 1 ) ,àååïåðâàÿïðîèçâîäíàÿðàâíî- ìåðíîîãðàíè÷åíàíà [ t 0 ; 1 ) ; 2 : 0 2 Intco f z 0 1 ;:::;z 0 n g . Òîãäàâèãðå � 1 ïðîèñõîäèòïîèìêà. Ïðèâåäåíïðèìåðñèñòåìû,âêîòîðîéôóíäàìåíòàëüíàÿìàòðèöàÿâëÿåòñÿ ðåêóððåíòíîé. Âîâòîðîìïàðàãðàôåðàññìàòðèâàåòñÿëèíåéíàÿçàäà÷àïðåñëåäîâàíèÿ ãðóïïûñêîîðäèíèðîâàííûõóáåãàþùèõ E 1 ;:::E m .Ïîëó÷åíûäîñòàòî÷íûåóñëî- âèÿïîèìêèõîòÿáûîäíîãîóáåãàþùåãîâïðåäïîëîæåíèè,÷òîóáåãàþùèåèñ- ïîëüçóþòîäíîèòîæåóïðàâëåíèåèôóíäàìåíòàëüíàÿìàòðèöàîäíîðîäíîé ñèñòåìûÿâëÿåòñÿðåêóððåíòîé. Çàêîíäâèæåíèÿêàæäîãîèçïðåñëåäîâàòåëåé P i èìååòâèä _ x i = A ( t ) x i + u i ;u i 2 V: Çàêîíäâèæåíèÿêàæäîãîèçóáåãàþùèõ E j èìååòâèä _ y j = A ( t ) y j + v;v 2 V: Çäåñüèäàëåå x i ;y j ;u i ;v 2 R k ;i 2 I = f 1 ;:::;n g ;j 2 J = f 1 ;:::;m g ;A ( t )  íåïðåðûâíàÿíà [ t 0 ; 1 ) êâàäðàòíàÿìàòðèöàïîðÿäêà k , V ñòðîãîâûïóêëûé êîìïàêò R k ñãëàäêîéãðàíèöåé.Ïðè t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x i ( t 0 )= x 0 i ;y j ( t 0 )= y 0 j ; ïðè÷åì x 0 i 6 = y 0 j äëÿâñåõ i;j: 15 Ðàññìîòðèìñèñòåìóñíà÷àëüíûìèóñëîâèÿìè _ z ij = A ( t ) z ij + u i � v;u i ;v 2 V;z ij ( t 0 )= z 0 ij = x 0 i � y 0 j : Îòìåòèì,÷òî z 0 ij 6 =0 . Îòìåòèì,÷òîäåéñòâèÿóáåãàþùèõìîæíîòðàêòîâàòüñëåäóþùèìîáðàçîì: èìååòñÿöåíòð,êîòîðûéäëÿâñåõóáåãàþùèõ E j âûáèðàþòîäíîèòîæåóïðàâ- ëåíèå v: Îïðåäåëåíèå1.5 Áóäåìãîâîðèòü,÷òîçàäàíàêâàçèñòðàòåãèÿ U i ïðå- ñëåäîâàòåëÿ P i ; åñëèîïðåäåëåíîîòîáðàæåíèå U i ( t;z 0 ;v t (  )) ; ñòàâÿùååâñîîò- âåòñòâèåíà÷àëüíîìóñîñòîÿíèþ z 0 =( z 0 11 ;:::;z 0 nm ) ,ìîìåíòó t èïðîèçâîëü- íîéïðåäûñòîðèèóïðàâëåíèÿ v t (  ) óáåãàþùèõ E 1 ;:::;E m èçìåðèìóþôóíêöèþ u i ( t )= U i ( t;z 0 ;v t (  )) ñîçíà÷åíèÿìèâ V: Îáîçíà÷èìäàííóþèãðó÷åðåç � 2 . Îïðåäåëåíèå1.6 Âèãðå � 2 ïðîèñõîäèòïîèìêà,åñëèñóùåñòâóåòìî- ìåíò T 0 = T ( z 0 ) ,êâàçèñòðàòåãèè U 1 ;:::; U n ïðåñëåäîâàòåëåé P 1 ;:::;P n ,òà- êèå,÷òîäëÿëþáîéèçìåðèìîéôóíêöèè v (  ) , v ( t ) 2 V , t 2 [0 ;T 0 ] íàéäóòñÿ íîìåðà q 2f 1 ;:::;n g ;p 2f 1 ;:::;m g èìîìåíò  6 T 0 òàêèå,÷òî z qp (  )=0 . Òåîðåìà1.2 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ìàòðèöà ( t ) ðåêóððåíòíàíà [ t 0 ; 1 ) ,àååïåðâàÿïðîèçâîäíàÿðàâíî- ìåðíîîãðàíè÷åíàíà [ t 0 ; 1 ) ; 2 : Intco f x 0 1 ;:::;x 0 n g\ co f y 0 1 ;:::;y 0 m g6 = ; : Òîãäàâèãðå � 2 ïðîèñõîäèòïîèìêà. Òðåòèéïàðàãðàôïîñâÿùåíïîèìêåçàäàííîãî÷èñëàóáåãàþùèõ.Ðàññìàò- ðèâàåòñÿäèôôåðåíöèàëüíàÿèãðà n + m ëèö: n ïðåñëåäîâàòåëåé P 1 ;:::;P n è m óáåãàþùèõ E 1 ;:::E m . Çàêîíäâèæåíèÿêàæäîãîèçïðåñëåäîâàòåëåé P i èìååòâèä _ x i = A ( t ) x i + u i ;u i 2 V: 16 Çàêîíäâèæåíèÿêàæäîãîèçóáåãàþùèõ E j èìååòâèä _ y j = A ( t ) y j + v j ;v j 2 V: Çäåñüèäàëåå x i ;y j ;u i ;v j 2 R k ;i 2 I = f 1 ;:::;n g ;j 2 J = f 1 ;:::;m g ;A ( t )  íåïðåðûâíàÿíà [ t 0 ; 1 ) êâàäðàòíàÿìàòðèöàïîðÿäêà k , V ñòðîãîâûïóêëûé êîìïàêò R k ñãëàäêîéãðàíèöåé.Ïðè t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x i ( t 0 )= x 0 i ;y j ( t 0 )= y 0 j ; ïðè÷åì x 0 i 6 = y 0 j äëÿâñåõ i;j: Öåëüãðóïïûïðåñëåäîâàòåëåéïîéìàòüíåìåíåå÷åì q (1 6 q 6 m ) óáå- ãàþùèõ,ïðèóñëîâèè,÷òîñíà÷àëàóáåãàþùèåâûáèðàþòñâîèóïðàâëåíèÿ,à çàòåìïðåñëåäîâàòåëè,çíàÿèíôîðìàöèþîâûáîðåóáåãàþùèõ,âûáèðàþòñâîè óïðàâëåíèÿ,ïðè÷åìêàæäûéïðåñëåäîâàòåëüìîæåòïîéìàòüíåáîëååîäíîãî óáåãàþùåãî. Ñ÷èòàåì,÷òî n � q . Ðàññìîòðèìñèñòåìóñíà÷àëüíûìèóñëîâèÿìè _ z ij = A ( t ) z ij + u i � v j ;u i ;v j 2 V;z ij ( t 0 )= z 0 ij = x 0 i � y 0 j : Îòìåòèì,÷òî z 0 ij 6 =0 . Îáîçíà÷èìäàííóþèãðó÷åðåç � 3 . Îïðåäåëåíèå1.8 Âèãðå � 3 ïðîèñõîäèòïîèìêà q óáåãàþùèõ,åñëèñóùå- ñòâóåòìîìåíò T 0 = T ( z 0 ) òàêîé,÷òîäëÿëþáîéñîâîêóïíîñòèäîïóñòèìûõ óïðàâëåíèé v j ( t ) óáåãàþùèõ E j , t 2 [ t 0 ;T 0 ] íàéäóòñÿäîïóñòèìûåóïðàâëåíèÿ ïðåñëåäîâàòåëåé P 1 ;:::;P n u i ( t )= u i ( t;z 0 ij ;v j ( s ) ;s 2 [ t 0 ;T 0 ]) ; 17 òàêèå,÷òîñóùåñòâóþòìíîæåñòâà N  I;M  J; j N j = j M j = q èäëÿ êàæäîãîíîìåðà 2 M íàéäóòñÿíîìåð = ( ) 2 N èìîìåíò  2 [ t 0 ;T 0 ] , äëÿêîòîðûõâûïîëíåíî z (  )=0 èïðèýòîì ( 1 ) 6 = ( 2 ) äëÿëþáûõ 1 6 = 2 . 18 Òåîðåìà1.3 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ìàòðèöà ( t ) ðåêóððåíòíàíà [ t 0 ; 1 ) ,àååïåðâàÿïðîèçâîäíàÿðàâíî- ìåðíîîãðàíè÷åíàíà [ t 0 ; 1 ) ; 2 : Äëÿêàæäîãî s 2f 0 ; 1 ;:::;q � 1 g âåðíîñëåäóþùåå:äëÿëþáîãîìíîæå- ñòâà N  I , j N j = n � s íàéäåòñÿòàêîåìíîæåñòâî M  J; j M j = q � s , ÷òîäëÿâñåõ 2 M âûïîëíåíî 0 2 Intco f z 0 ; 2 N g : Òîãäàâèãðå � 3 ïðîèñõîäèòïîèìêàíåìåíåå q óáåãàþùèõ. ÂòîðàÿãëàâàïîñâÿùåíàîáîáùåííîìóïðèìåðóË.Ñ.Ïîíòðÿãèíà.Âïåð- âîìïàðàãðàôåðàññìàòðèâàåòñÿíåñòàöèîíàðíàÿäèôôåðåíöèàëüíàÿèãðàñ n ïðåñëåäîâàòåëÿìèèîäíèìóáåãàþùèìïðèîäèíàêîâûõäèíàìè÷åñêèõèèíåð- öèîííûõâîçìîæíîñòÿõâñåõèãðîêîâ. Çàêîíäâèæåíèÿêàæäîãîèçïðåñëåäîâàòåëåé P i èìååòâèä x ( l ) i + a 1 ( t ) x ( l � 1) i +  + a l ( t ) x i = u i ;u i 2 V: Çàêîíäâèæåíèÿóáåãàþùåãî E èìååòâèä y ( l ) + a 1 ( t ) y ( l � 1) +  + a l ( t ) y = v;v 2 V: Çäåñüèäàëåå x i ;y;u i ;v 2 R k ;i 2 I = f 1 ; 2 ;:::;n g ;a 1 ( t ) ;:::;a l ( t ) íåïðå- ðûâíûåíà [ t 0 ; 1 ) ôóíêöèè, V ñòðîãîâûïóêëûéêîìïàêòâ R k ñãëàäêîé ãðàíèöåé. Ïðè t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x ( q ) i ( t 0 )= x q i ;y ( q ) ( t 0 )= y q ; ïðè÷åì x 0 i 6 = y 0 äëÿâñåõ i: Çäåñüèäàëåå q =0 ; 1 ;:::;l � 1 : Îáîçíà÷èìäàííóþèãðó÷åðåç � 4 . 19 Ðàññìîòðèìñèñòåìó z ( l ) i + a 1 ( t ) z ( l � 1) i +  + a l ( t ) z i = u i � v;u i ;v 2 V: ñíà÷àëüíûìèóñëîâèÿìè z ( q ) i ( t 0 )= z q i = x q i � y q : Ïðåñëåäîâàòåëèèñïîëüçóþòêâàçèñòðàòåãèè. Îïðåäåëåíèå2.2 Âèãðå � 4 ïðîèñõîäèòïîèìêà,åñëèñóùåñòâóåòìî- ìåíò T 0 = T ( z 0 ) èêâàçèñòðàòåãèè U 1 ( t;z 0 ;v t (  )) ;:::; U n ( t;z 0 ;v t (  )) ïðåñëåäî- âàòåëåé P 1 ;:::;P n òàêèå,÷òîäëÿëþáîéèçìåðèìîéôóíêöèè v (  ) , v ( t ) 2 V , t 2 [ t 0 ;T ( z 0 )] íàéäóòñÿíîìåð 2f 1 ;:::;n g èìîìåíò  2 [ t 0 ;T ( z 0 )] òàêèå, ÷òî z (  )=0 : Îáîçíà÷èì÷åðåç ' q ( t;s ) ;q =0 ;:::;l � 1 ; ( t � s � t 0 ) ðåøåíèÿóðàâíåíèÿ ! ( l ) + a 1 ( t ) ! ( l � 1) +  + a l ( t ) ! =0 ñíà÷àëüíûìèóñëîâèÿìè ! ( j ) ( s )=0 ;j =0 ;:::;q � 1 ;q +1 :::;l � 1 ;! ( q ) ( s )=1 : Ïóñòüäàëåå  i ( t )= ' 0 ( t;t 0 ) z 0 i + ' 1 ( t;t 0 ) z 1 i + ::: + ' l � 1 ( t;t 0 ) z l � 1 i : Îáîçíà÷èì H i = f  i ( t ) ;t 2 [ t 0 ; 1 ) g . 20 Îïðåäåëèìôóíêöèè:  ( t;s )= 8 � � : 1 ; åñëè ' l � 1 ( t;s ) � 0 ; � 1 ; åñëè ' l � 1 ( t;s ) 0 ( t 0 6 s 6 t ) ;  ( v;;h i )=sup f  :  � 0 ;v � h i 2 V g ; G ( t;h i )= Z t t 0 j ' l � 1 ( t;s ) j  ( v ( s ) ; ( t;s ) ;h i ) ds: Ïîëàãàåìäàëåå h =( h 1 ;h 2 ;:::;h n ) ;D = D " ( h 0 1 )  D " ( h 0 2 )  D " ( h 0 n ) : Âòåðìèíàõíà÷àëüíûõïîçèöèéèïàðàìåòðîâèãðûïîëó÷åíûäîñòàòî÷íûå óñëîâèÿðàçðåøèìîñòèçàäà÷èïðåñëåäîâàíèÿ. Ïóñòü T ( z 0 )=min f t � 0:inf v (  ) min h 2 D max i 2 I G ( t;h i ) � 1 g : Äîêàçàíî,÷òî T ( z 0 ) 1 : Òåîðåìà2.1 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ôóíêöèè  i ( t ) ðåêóððåíòíûíà [ t 0 ; 1 ) ; 2 : Ñóùåñòâóþò h 0 i 2 H i ;h 0 i 6 =0 òàêèå,÷òî 0 2 Intco f h 0 1 ;h 0 2 ;:::;h 0 n g ; 3 : Ñóùåñòâóþòìîìåíòû  i � T ( z 0 ) òàêèå,÷òî (a)  i (  i ) 2 D " ( h 0 i ) ; (b) inf v (  ) max i G (  i ; i (  i )) � 1 . Òîãäàâèãðå � 4 ïðîèñõîäèòïîèìêà. Ñëåäñòâèå2.1 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ôóíêöèè  i ( t ) ðåêóððåíòíûíà [ t 0 ; 1 ) ; 2 : 0 2 Intco f z 0 1 ;:::;z 0 n g : Òîãäàâèãðå � 4 ïðîèñõîäèòïîèìêà. 21 Âîâòîðîìïàðàãðàôåðàññìàòðèâàåòñÿçàäà÷àïðåñëåäîâàíèÿãðóïïîéïðå- ñëåäîâàòåëåéîäíîãîóáåãàþùåãîïðèðàâíûõäèíàìè÷åñêèõèèíåðöèîííûõâîç- ìîæíîñòÿõèãðîêîâ.Ïðåäïîëàãàåòñÿ,÷òîóáåãàþùèéâïðîöåññåèãðûíåïîêèäà- åòïðåäåëûâûïóêëîãîìíîãîãðàííîãîìíîæåñòâà,òåðìèíàëüíûåìíîæåñòâà íà÷àëîêîîðäèíàò. Äâèæåíèåêàæäîãîïðåñëåäîâàòåëÿ P i îïèñûâàåòñÿóðàâíåíèåì x ( l ) i + a 1 ( t ) x ( l � 1) i + a 2 ( t ) x ( l � 2) i + ::: + a l ( t ) x i = u i ;u i 2 V; Çàêîíäâèæåíèÿóáåãàþùåãî E èìååòâèä y ( l ) + a 1 ( t ) y ( l � 1) + a 2 y ( l � 2) + ::: + a l ( t ) y = v;v 2 V; ãäå x i ;y j ;u i ;v j 2 R k ; ôóíêöèè a 1 ( t ) ;a 2 ( t ) ;:::;a l ( t ) íåïðåðûâíûíàïðîìåæóòêå [ t 0 ; 1 ) ;V ñòðîãîâûïóêëûéêîìïàêòâ R k ñãëàäêîéãðàíèöåé. Âìîìåíò t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x ( q ) i ( t 0 )= x q i ;y ( q ) ( t 0 )= y q ; ïðè÷åì x 0 i � y 0 = 2 M i äëÿâñåõ i; ãäå M i çàäàííûåâûïóêëûåêîìïàêòû.Çäåñüèäàëåå i 2 I = f 1 ; 2 ;:::;n g ; q =0 ; 1 ;:::;l � 1 : Äîïîëíèòåëüíîïðåäïîëàãàåòñÿ,÷òîóáåãàþùèéíåïîêèäàåòïðåäåëûâû- ïóêëîãîìíîæåñòâà B = f y : y 2 R k ; ( p c ;y ) 6  c ;c =1 ; 2 ;:::;r g ; ñíåïóñòîéâíóòðåííîñòüþ,ãäå ( a;b ) ñêàëÿðíîåïðîèçâåäåíèåâåêòîðîâ a è b; p 1 ;:::;p r åäèíè÷íûåâåêòîðû R k ; 1 ;:::; r âåùåñòâåííûå÷èñëà. 22 Ðàññìîòðèìóðàâíåíèå z ( l ) i + a 1 ( t ) z ( l � 1) i + a 2 ( t ) z ( l � 2) i + ::: + a l ( t ) z i = u i � v; ñíà÷àëüíûìèóñëîâèÿìè z ( q ) i ( t 0 )= z q i = x q i � y q : ×åðåç ' q ( t;s )( t � s � t 0 ) îáîçíà÷èìðåøåíèåóðàâíåíèÿ ! ( l ) + a 1 ( t ) ! ( l � 1) + a 2 ( t ) ! ( l � 2) + ::: + a l ( t ) ! =0 ; ñíà÷àëüíûìèóñëîâèÿìè ! ( s )=0 ;:::;! ( q � 1) ( s )=0 ;! ( q ) ( s )=1 ;! ( q +1) ( s )=0 ;:::;! ( l � 1) ( s )=0 : Ïóñòüäàëåå  i ( t )= ' 0 ( t;t 0 ) z 0 i + ' 1 ( t;t 0 ) z 1 i + ::: + ' l � 1 ( t;t 0 ) z l � 1 i ;  ( t )= ' 0 ( t;t 0 ) y 0 + ' 1 ( t;t 0 ) y 1 + ::: + ' l � 1 ( t;t 0 ) y l � 1 : Ñ÷èòàåì,÷òî  i ( t ) = 2 M i äëÿâñåõ i;t � t 0 : Îïðåäåëåíèå2.4 Âèãðå �( n;B ) ïðîèñõîäèòïîèìêà,åñëèñóùåñòâó- åòìîìåíò T ( z 0 ) ; êâàçèñòðàòåãèè U 1 ( t;z 0 ;v t (  )) ;:::;U n ( t;z 0 ;v t (  )) ïðåñëåäî- âàòåëåé P 1 ;:::;P n òàêèå,÷òîäëÿëþáîéèçìåðèìîéôóíêöèè v (  ) , v ( t ) 2 V , y ( t ) 2 B;t 2 [ t 0 ;T ( z 0 )] ñóùåñòâóþòìîìåíò  2 [ t 0 ;T ( z 0 )] èíîìåð 2 I; ÷òî z (  ) 2 M : Ïóñòü  ( t;s )= 8 � � : 1 ; åñëè ' l � 1 ( t;s ) � 0 ; � 1 ; åñëè ' l � 1 ( t;s ) 0 ( t 0 6 s 6 t ) ; 23  ( v;;b i )=sup f  j�  ( b i � M i ) \ ( V � v ) 6 = ;g ; G ( t;v (  ) ;b i )= t Z t 0 j ' l � 1 ( t;s ) j  ( v ( s ) ; ( t;s ) ;b i )) ds; F ( t )= t Z t 0 j ' l � 1 ( t;s ) j ds: Ïðåäïîëîæåíèå2.1 1 : Ôóíêöèè  i ( t ) ÿâëÿþòñÿðåêóððåíòíûìèíà [ t 0 ; 1 ); 2 : Ôóíêöèÿ  ( t ) îãðàíè÷åíàíà [ t 0 ; 1 ); 3 : lim t !1 F ( t )= 1 ; Ïðåäïîëîæåíèå2.2 Ñóùåñòâóþòìîìåíòû  0 i � t 0 ; ïîëîæèòåëüíûå ÷èñëà "; òàêèå,÷òî 1 : Äëÿâñåõ i èäëÿâñåõ h i 2 D " (  i (  0 i )) âûïîëíåíî h i = 2 M i ; 2 : Äëÿâñåõ h i 2 D " (  i (  0 i )) ñïðàâåäëèâûíåðàâåíñòâà min v max  max i  ( v; +1 ;h i ) ; max j ( p j ;v ) � ; min v max  max i  ( v; � 1 ;h i ) ; max j ( � p j ;v ) � : Îïðåäåëèì÷èñëî T 0 : T 0 =min f t � t 0 :inf v (  ) min h 2 D max i 2 I G ( t;v (  ) ;h i ) � 1 g : Ïðåäïîëîæåíèå2.3 Ñóùåñòâóþòìîìåíòû  i � T 0 òàêèå,÷òî 1 : i (  i ) 2 D " (  i (  0 i )) äëÿâñåõ i ; 2 : inf v (  ) max i G (  i ;v (  ) ; i (  i )) � 1 : Òåîðåìà2.2 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 1 , 2 : 2 , 2 : 3 , r =1 : Òîãäà âèãðå �( n;B ) ïðîèñõîäèòïîèìêà. 24 Ïðåäïîëîæåíèå2.4 Ñóùåñòâóþò  0 i � t 0 òàêèå,÷òî 0 2 Intco   i (  0 i ) � M i ;i 2 I;p 1 ;:::;p r Ðàññìîòðèììíîæåñòâî B 1 = f x x 2 R k ; ( p;x ) 6  g ; ãäå  = 1  1 +  + r  r : Ïðåäïîëîæåíèå2.5 Äëÿëþáîãî h 2 D âìíîæåñòâå n S i =1 ( h i � M i ) ñóùå- ñòâóåò k ëèíåéíîíåçàâèñèìûõâåêòîðîâ. Òåîðåìà2.4 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 1 , 2 : 4 , 2 : 5 èñóùåñòâóþò  i � T 0 òàêèå,÷òî 1 : i (  i ) 2 D " (  i (  0 i )); 2 : inf v (  ) max i G (  i ;v (  ) ; i (  i )) � 1 âèãðå �( n;B 1 ) : Òîãäàâèãðå �( n;B 1 ) ïðîèñõîäèòïîèìêà. Ïðèâåäåíïðèìåð,èëëþñòðèðóþùèéïîëó÷åííûåóñëîâèÿ. Òðåòèéïàðàãðàôïîñâÿùåíìíîãîêðàòíîéïîèìêåâðåêóððåíòíîìïðèìå- ðàÏîíòðÿãèíàïðèîäèíàêîâûõäèíàìè÷åñêèõèèíåðöèîííûõâîçìîæíîñòÿõ èãðîêîâèôàçîâûìèîãðàíè÷åíèÿìèíàñîñòîÿíèÿóáåãàþùåãî. Îïðåäåëåíèå2.6 Âèãðå �( n;B ) ïðîèñõîäèò m -êðàòíàÿïîèìêà (ïðè m =1 ïîèìêà),åñëèñóùåñòâóþòìîìåíò T ( z 0 ) ; êâàçèñòðàòåãèè U 1 ( t;z 0 ;v t (  )) ;:::;U n ( t;z 0 ;v t (  )) ïðåñëåäîâàòåëåé P 1 ;:::;P n òàêèå,÷òîäëÿëþ- áîéèçìåðèìîéôóíêöèè v (  ) ;v ( t ) 2 V;y ( t ) 2 B;t 2 [ t 0 ;T ( z 0 )] ñóùåñòâóþòìî- ìåíòû  1 ;:::; m 2 [ t 0 ;T ( z 0 )] ; ïîïàðíîðàçëè÷íûåèíäåêñû i 1 ;:::;i m 2 I; ÷òî z i s (  s )=0 ;s =1 ;:::;m: Ïóñòü ( p )= f ( i 1 ;:::;i p ) j i 1 ;:::;i p 2 I èïîïàðíîðàçëè÷íû g Ïðåäïîëîæåíèå2.6 1 :n � m + k � 1; 2 : Ôóíêöèè  i ( t ) ÿâëÿþòñÿðåêóððåíòíûìèíà [ t 0 ; 1 ); 25 3 : Ôóíêöèÿ  ( t ) îãðàíè÷åíàíà [ t 0 ; 1 ); 4 : lim t !1 F ( t )= 1 ; 5 :V = D 1 (0) : Ïðåäïîëîæåíèå2.7 Ñóùåñòâóþòìîìåíòû  0 i 2 [ t 0 ; 1 ) òàêèå,÷òîäëÿ âñåõ  2 ( n � m +1) âûïîëíåíîâêëþ÷åíèå 0 2 Intco f  j (  0 j ) ;j 2  ;p 1 ;:::;p r g : Îïðåäåëèì÷èñëî T 0 =min f t � t 0 j min v (  ) min h 2 D max  2 ( m ) min j 2  G ( t;v (  ) ;h j ) � 1 g : Ïðåäïîëîæåíèå2.8 Ñóùåñòâóþòìîìåíòû  i � T 0 òàêèå,÷òî 1 : i (  i ) 2 D " (  i (  0 i )) äëÿâñåõ i ; 2 : inf v (  ) max  2 ( m ) min j 2  G (  j ;v (  ) ; j (  j )) � 1 : Òåîðåìà2.5 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 6 , 2 : 7 , 2 : 8 , r =1 : Òîãäà âèãðå �( n;B ) ïðîèñõîäèò m -êðàòíàÿïîèìêà. 26 Ãëàâà1.Ëèíåéíûåðåêóððåíòíûå äèôôåðåíöèàëüíûåèãðû 1.1.Ãðóïïîâîåïðåñëåäîâàíèåîäíîãîóáåãàþùåãîâëèíåéíûõðåêóð- ðåíòíûõäèôôåðåíöèàëüíûõèãðàõ Âïðîñòðàíñòâå R k ( k � 2) ðàññìàòðèâàåòñÿäèôôåðåíöèàëüíàÿèãðà n +1 ëèö: n ïðåñëåäîâàòåëåé P 1 ;:::;P n èóáåãàþùèé E . Çàêîíäâèæåíèÿêàæäîãîèçïðåñëåäîâàòåëåé P i èìååòâèä _ x i = A ( t ) x i + u i ;u i 2 V: (1.1) Çàêîíäâèæåíèÿóáåãàþùåãî E èìååòâèä _ y = A ( t ) y + v;v 2 V: (1.2) Çäåñüèäàëåå x i ;y;u i ;v 2 R k ;i 2 I = f 1 ; 2 ;:::;n g , A ( t ) íåïðåðûâíàÿíà [ t 0 ; 1 ) êâàäðàòíàÿìàòðèöàïîðÿäêà k , V ñòðîãîâûïóêëûéêîìïàêò R k ñ ãëàäêîéãðàíèöåé. Ïðè t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x i ( t 0 )= x 0 i ;y ( t 0 )= y 0 ; (1.3) ïðè÷åì x 0 i 6 = y 0 äëÿâñåõ i: Âìåñòîñèñòåì (1 : 1)  (1 : 3) ðàññìîòðèìñèñòåìóñíà÷àëüíûìèóñëîâèÿìè _ z i = A ( t ) z i + u i � v;u i ;v 2 V;z i ( t 0 )= z 0 i = x 0 i � y 0 : (1.4) Îòìåòèì,÷òî z 0 i 6 =0 . 27 Íàçîâåìïðåäûñòîðèåéóïðàâëåíèÿ v ( t ) óáåãàþùåãî E âìîìåíòâðåìåíè t;t 2 [ t 0 ; 1 ) ìíîæåñòâî v t (  )= f v ( s ) ;s 2 [ t 0 ;t ] ;v � èçìåðèìàÿôóíêöèÿ. g Îïðåäåëåíèå1.1 Áóäåìãîâîðèòü,÷òîçàäàíàêâàçèñòðàòåãèÿ U i ïðåñëå- äîâàòåëÿ P i ; åñëèîïðåäåëåíîîòîáðàæåíèå U i ( t;z 0 ;v t (  )) ; ñòàâÿùååâñîîò- âåòñòâèåíà÷àëüíîìóñîñòîÿíèþ z 0 =( z 0 1 ;:::;z 0 n ) ,ìîìåíòó t èïðîèç- âîëüíîéïðåäûñòîðèèóïðàâëåíèÿ v t (  ) óáåãàþùåãî E èçìåðèìóþôóíêöèþ u i ( t )= U i ( t;z 0 ;v t (  )) ñîçíà÷åíèÿìèâ V: Ïðèýòîìïðåäïîëàãàåòñÿ,÷òîäîëæíîáûòüâûïîëíåíîóñëîâèå¾ôèçè÷åñêîé îñóùåñòâèìîñòè¿,òîåñòüåñëè v 1 ;v 2 äâàäîïóñòèìûõóïðàâëåíèÿóáåãàþùåãî E; ïðè÷åì v 1 ( t )= v 2 ( t ) äëÿïî÷òèâñåõ t; òîñîîòâåòñòâóþùèåèìïðèîòîáðà- æåíèè U i ( t;z 0 ;v t (  )) ôóíêöèè u 1 ;u 2 òàêæåðàâíûïî÷òèâñþäóïðè t � 0 : Îáîçíà÷èìäàííóþèãðó÷åðåç � 1 . Îïðåäåëåíèå1.2 Âèãðå � 1 ïðîèñõîäèòïîèìêà,åñëèñóùåñòâóåòìîìåíò T 0 = T ( z 0 ) ,êâàçèñòðàòåãèè U 1 ;:::; U n ïðåñëåäîâàòåëåé P 1 ;:::;P n ,òàêèå,÷òî äëÿëþáîéèçìåðèìîéôóíêöèè v (  ) , v ( t ) 2 V , t 2 [0 ;T 0 ] íàéäóòñÿíîìåð q 2f 1 ;:::;n g èìîìåíò  6 T 0 òàêèå,÷òî z q (  )=0 . Îïðåäåëåíèå1.3 ([33]) Ôóíêöèÿ F : R 1 ! R k íàçûâàåòñÿðåêóððåíòíîéïî Çóáîâó(äàëååðåêóððåíòíîé),åñëèäëÿëþáîãî "� 0 ñóùåñòâóåò T ( " ) � 0 òàêîå,÷òîäëÿëþáûõ t;a 2 R 1 ñóùåñòâóåò  ( t ) 2 [ a;a + T ( " )] ,äëÿêîòîðûõ âûïîëíåíîíåðàâåíñòâî k F ( t +  ( t )) � F ( t ) k ": Åñëèìîæíîâûáðàòü  ( t ) íåçàâèñÿùèìîò t äëÿâñåõ t ,òîôóíêöèÿ f ( t ) íàçûâàåòñÿïî÷òèïåðèîäè÷åñêîé. 28 Îïðåäåëåíèå1.4 Ôóíêöèÿ f :[ t 0 ; 1 ) ! R k íàçûâàåòñÿðåêóððåíòíîéïîÇó- áîâó(äàëååðåêóððåíòíîé)íà [ t 0 ; 1 ) ,åñëèñóùåñòâóåòðåêóððåíòíàÿôóíê- öèÿ F : R 1 ! R k òàêàÿ,÷òî f ( t )= F ( t ) äëÿâñåõ t 2 [ t 0 ; 1 ) : Ëåììà1.1 Ïóñòü 0 2 Intco f b 0 1 ;:::;b 0 n g ,ïðè÷åì b 0 j 6 =0 äëÿâñåõ j .Òîãäàñó- ùåñòâóåò "� 0 òàêîå,÷òîâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : 0 = 2 D 2 " ( b 0 i ) äëÿâñåõ i ,ãäå D r ( a )= f z : k z � a k 6 r g ; 2 : Äëÿëþáûõ b 1 2 D 2 " ( b 0 1 ) ;:::;b n 2 D 2 " ( b 0 n ) âûïîëíåíî 0 2 Intco f b 1 ;:::;b n g : Äîêàçàòåëüñòâî.Ïóíêò1î÷åâèäíîâûïîëíåí.Äîêàæåìïóíêò2.Ìíî- æåñòâî co f b 0 1 ;:::;b 0 n g ÿâëÿåòñÿâûïóêëûììíîãîãðàííèêîìñâåðøèíàìèâòî÷- êàõ b 0 j ;j 2 K f 1 ;:::;n g Èçóñëîâèÿëåììûñëåäóåò,÷òî 0 2 Intco f b 0 j ;j 2 K g . Ìíîæåñòâî Intco f b 0 j ;j 2 K g îòêðûòî.Ñëåäîâàòåëüíî,ñóùåñòâóåò "� 0 òà- êîå,÷òîäëÿëþáûõ b j 2 D 2 " ( b 0 j ) ñïðàâåäëèâî 0 2 Intco f b j ;j 2 K g .Òàêêàê Intco f b j ;j 2 K g Intco f b 1 ;:::;b n g ,òîïîëó÷àåìóòâåðæäåíèÿëåììû. Ñëåäñòâèå1.1 Ïóñòüâûïîëíåíîóñëîâèå Intco f b 0 1 ;:::;b 0 n g\ co f c 0 1 ;:::;c 0 m g6 = ; : Òîãäàñóùåñòâóåò "� 0 òàêîå,÷òî Intco f b 1 ;:::;b n g\ co f c 1 ;:::;c m g6 = ; : äëÿëþáûõ b 1 ;:::;b n ;c 1 ;:::;c m òàêèõ,÷òî b i 2 D 2 " ( b 0 i ) ;c j 2 D 2 " ( c 0 j ) Âäàëüíåéøåìñ÷èòàåì,÷òî "� 0 âûáðàíîâñîîòâåòñòâèèñëåììîé 1 : 1 . Îáîçíà÷èì÷åðåç ( t ) ôóíäàìåíòàëüíóþìàòðèöóñèñòåìû _ ! = A ( t ) !; 29 ãäå ( t 0 )  E , E åäèíè÷íàÿìàòðèöà. Îïðåäåëèìôóíêöèè  ( v;h )=sup f  :  � 0 ; � h 2 V � v g ïðè h 6 =0 ; J ( t;b )= t Z t 0  ( v ( s ) ; ( s ) b ) ds: Ëåììà1.2 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ìàòðèöà ( t ) ðåêóððåíòíàíà [ t 0 ; 1 ) ,àååïåðâàÿïðîèçâîäíàÿðàâíî- ìåðíîîãðàíè÷åíàíà [ t 0 ; 1 ) ; 2 : 0 2 Intco f z 0 1 ;:::;z 0 n g . Òîãäàñóùåñòâóåò T�t 0 òàêîå,÷òîäëÿëþáîãîäîïóñòèìîãîóïðàâëåíèÿ v (  ) ñóùåñòâóåò 2 I òàêîå,÷òî J ( T;z 0 ) � 1 Äîêàçàòåëüñòâî.Îïðåäåëèìäâàìíîæåñòâà = f t � t 0 :( t ) z 0 i 2 D 2 " ( z 0 i ) äëÿâñåõ i g ; Q = f q 2 I :( t ) z 0 q 2 D 2 " ( z 0 q ) äëÿâñåõ t � t 0 g :  ( G ) ìåðàËåáåãàìíîæåñòâà G  R 1 : Âîçìîæíûäâàñëó÷àÿ: 1. Q = I: Òîãäà  ( )= 1 : 2. Q 6 = I: Áóäåìñ÷èòàòü,÷òî Q = ; ,òîåñòüçíà÷åíèåêàæäîéèçôóíêöèé ( t ) z 0 i âíåêîòîðûéìîìåíòíåïðèíàäëåæèòøàðó D 2 " ( z 0 i ) .Äîêàæåì,÷òî èâýòîìñëó÷àå  ( )= 1 : Òàêêàêôóíêöèè ( t ) z 0 i ÿâëÿþòñÿðåêóððåíòíûìè,òîïî " ñóùåñòâóåò T ( " ) òàêîå,÷òîäëÿëþáîãî j ñóùåñòâóåò  j ( t 0 ) 2 [ t 0 + T ( " ) j ; t 0 + T ( " ) j + T ( " )] ,äëÿ êîòîðûõâûïîëíåíîíåðàâåíñòâî k ( t 0 +  j ( t 0 )) z 0 i � ( t 0 ) z 0 i k " 30 äëÿâñåõ i: Ïóñòü j = f t : t 2 [  j ( t 0 ) ; j +1 ( t 0 )) ; ( t 0 + t ) z 0 i 2 D 2 " ( z 0 i ) äëÿâñåõ i g , dist( D 1 ;D 2 )=inf d 1 2 D 1 ;d 2 2 D 2 k d 1 � d 2 k : Ïîóñëîâèþôóíêöèè _ ( t ) z 0 i ðàâíîìåðíîîãðàíè÷åíû,òîåñòüíàéäåòñÿòàêîå ïîëîæèòåëüíîå÷èñëî M ,÷òî max t 2 [ t 0 ; 1 ) k _ ( t ) z 0 i k 6 M äëÿâñåõ i: Èçòåîðåìûîñðåäíåì([41])èìååì,÷òîäëÿëþáûõ t 2 �t 1 �t 0 k ( t 2 ) z 0 i � ( t 1 ) z 0 i ) k 6 sup t 2 [ t 1 ;t 2 ] k _ ( t ) z 0 i kj t 2 � t 1 j 6 M j t 2 � t 1 j : Ïîýòîìó,åñëè k ( t 2 ) z 0 i � ( t 1 ) z 0 i ) k � L ,òîñïðàâåäëèâîíåðàâåíñòâî t 2 � t 1 + L M : Òàêêàê dist( @D " ( z 0 i ) ;@D 2 " ( z 0 i ))= "; ( t 0 +  j ( t 0 )) z 0 i 2 Int D " ( z 0 i ) äëÿâñåõ i;j ,òî [  j ( t 0 ) ; j ( t 0 )+ " M ]  j äëÿâñåõ j .Ñëåäîâàòåëüíî,  ( ) �  ( 1 S j =0 j )= 1 . Âñèëóëåììû 1 : 1 äëÿëþáîãî h =( h 1 ;h 2 ;:::;h n ) 2 D = D 2 " ( z 0 1 )  D 2 " ( z 0 2 )  D 2 " ( z 0 n ) ñïðàâåäëèâîíåðàâåíñòâî  ( h )=min v 2 V max i 2 I  ( v;h i ) � 0 : Äîêàæåì,÷òîôóíêöèÿ  íåïðåðûâíàíà D ,òîåñòüäëÿëþáîãî "� 0 ñóùå- ñòâóåò � 0 òàêîå,÷òîäëÿâñåõ h ,óäîâëåòâîðÿþùèõíåðàâåíñòâó j h � h  j  31 âûïîëíåíî j  ( h ) �  ( h  ) j " . Ðàññìîòðèìðàçíîñòü j  ( h ) �  ( h  ) j = j min v 2 V max i 2 I  ( v;h i ) � min v 2 V max i 2 I  ( v;h  i ) j 6 6 max v 2 V j max i 2 I  ( v;h i ) � max i 2 I  ( v;h  i ) j 6 6 max v 2 V max i 2 I j  ( v;h i ) �  ( v;h  i ) j : Ïîëåììå1.3.13([110])ôóíêöèÿ  íåïðåðûâíà,ïîýòîìóäëÿëþáîãî "� 0 ñóùå- ñòâóåò � 0 òàêîå,÷òîäëÿâñåõ h i ,óäîâëåòâîðÿþùèõíåðàâåíñòâó j h i � h  i j  âûïîëíåíî j  ( v;h i ) �  ( v;h  i ) j " .Ñëåäîâàòåëüíî,ôóíêöèÿ  íåïðåðûâíàíà D . Òàêêàê D êîìïàêò,òîïîëó÷èì r =min h 2 D min v 2 V max i 2 I  ( v;h i )=min h 2 D  ( h ) � 0 : Òàêèìîáðàçîì,âåëè÷èíà  =min t 2 min v 2 V max i 2 I  ( v; ( t ) z 0 i ) � min h 2 D min v 2 V max i 2 I  ( v;h i )= r� 0 : Äàëåå max i 2 I J ( t;z 0 i )=max i 2 I t Z t 0  ( v ( s ) ; ( s ) z 0 i ) ds � max i 2 I Z [ t 0 ;t ] \  ( v ( s ) ; ( s ) z 0 i ) ds � � 1 n Z [ t 0 ;t ] \ X i 2 I  ( v ( s ) ; ( s ) z 0 i ) ds � 1 n Z [ t 0 ;t ] \ ds =  n  ([ t 0 ;t ] \ ) : Îòìåòèì,÷òî lim t !1  ([ t 0 ;t ] \ )= 1 ,òàêêàê  ( )= 1 .Òîãäàäëÿìîìåíòà T , îïðåäåëÿåìîãîèçóñëîâèÿ  n  ([ t 0 ;T ] \ ) � 1 ; 32 èíåêîòîðîãî 2 I âûïîëíåíî J ( T;z 0 ) � 1 : Ëåììàäîêàçàíà. Ïóñòü T ( z 0 )=min f t � 0:inf v (  ) min h 2 D max i 2 I J ( t;h i ) � 1 g : Âñèëóëåììû1.2 T ( z 0 ) 1 : Òåîðåìà1.1 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ìàòðèöà ( t ) ðåêóððåíòíàíà [ t 0 ; 1 ) ,àååïåðâàÿïðîèçâîäíàÿðàâíî- ìåðíîîãðàíè÷åíàíà [ t 0 ; 1 ) ; 2 : 0 2 Intco f z 0 1 ;:::;z 0 n g . Òîãäàâèãðå � 1 ïðîèñõîäèòïîèìêà. Äîêàçàòåëüñòâî.ÏîôîðìóëåÊîøèðåøåíèåçàäà÷è (1 : 4) ïðèëþ- áûõäîïóñòèìûõóïðàâëåíèÿõèìååòâèä z i ( t )=( t ) z 0 i + t R t 0  � 1 ( s )( u i ( s ) � v ( s )) ds ! äëÿâñåõ t � t 0 : Ïóñòü v (  ) ;t 0 6  6 T 0 = T ( z 0 ) ïðîèçâîëüíîåäîïóñòèìîåóïðàâëåíèå óáåãàþùåãî E è t 1 �t 0 íàèìåíüøèéêîðåíüôóíêöèèâèäà F ( t )=1 � max i 2 I t Z t 0  ( v ( s ) ; ( s ) z 0 i ) ds: Îòìåòèì,÷òî,âñèëóîïðåäåëåíèÿ T 0 ,ìîìåíò t 1 ñóùåñòâóåòè t 1 6 T 0 : Çàäàåìóïðàâëåíèåïðåñëåäîâàòåëåé P i ñëåäóþùèìîáðàçîì: u i ( t )= v ( t ) �  ( v ( t ) ; ( t ) z 0 i )( t ) z 0 i äëÿâñåõ t 2 [ t 0 ;T 0 ] : Ñ÷èòàåì,÷òî  ( v ( t ) ; ( t ) z 0 i )=0 äëÿâñåõ t 2 [ t 1 ;T 0 ] : Òîãäà,ñó÷åòîìôîðìóëû Êîøè, z i ( t 1 )=( t 1 ) z 0 i 0 @ 1 � t 1 Z t 0  ( v ( s ) ; ( s ) z 0 i ) ds 1 A : Âñèëóîïðåäåëåíèÿ t 1 ,äëÿíåêîòîðîãî 2 I âûðàæåíèåâñêîáêàõîáðàùàåòñÿ âíîëü,ïîýòîìó z ( t 1 )=0 : Òåîðåìàäîêàçàíà. 33 Çàìå÷àíèå1.1 Åñëèìàòðèöà ( t ) íåÿâëÿåòñÿðåêóððåíòíîé,òîóñëîâèå 2 íåãàðàíòèðóåòïîèìêóâèãðå � 1 . Ñîîòâåòñòâóþùèéïðèìåðïðèâåäåíâ[17,ñ.119]. Òàêêàêâñÿêàÿïî÷òèïåðèîäè÷åñêàÿôóíêöèÿÿâëÿåòñÿðåêóððåíòíîé,òî ñïðàâåäëèâî Ñëåäñòâèå1.2 ([15]) Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ìàòðèöà ( t ) ïî÷òèïåðèîäè÷åñêàÿíà [ t 0 ; 1 ) ,àååïåðâàÿïðîèçâîäíàÿ ðàâíîìåðíîîãðàíè÷åíàíà [ t 0 ; 1 ) ; 2 : 0 2 Intco f z 0 1 ;:::;z 0 n g . Òîãäàâèãðå � 1 ïðîèñõîäèòïîèìêà. Ïðèìåð1.1 Ïóñòü A ( t )= ! ( t ) E; ãäå ! ( t )= 8 : 0 ; åñëè t 2 [0 ; 2  ] sin t; åñëè t= 2 [0 ; 2  ] Äîêàæåì,÷òîôóíêöèÿ ! ( t ) ðåêóððåíòíà. Äëÿëþáîãî "� 0 âîçüìåì T ( " )=4  .Ðàññìîòðèìäâàñëó÷àÿ: 1. t= 2 [0 ; 2  ] : Òîãäàäëÿëþáîãî a 2 R 1 ñóùåñòâóåò k 2 N òàêîå,÷òî  ( t )=2 k 2 [ a;a +4  ] ,äëÿêîòîðûõâûïîëíåíî j ! ( t +  ( t )) � ! ( t ) j = j sin( t +2 k ) � sin( t ) j =0 " 2. t 2 [0 ; 2  ] : Òîãäàäëÿëþáîãî a 2 R 1 âûáåðåì k 2 N òàêîå,÷òî k 2 [ a;a +4  ] ,à k + = 2 [ a;a +4  ] èñóùåñòâóåò  ( t )= k � t ,  ( t ) 2 [ a;a +4  ] ; äëÿêîòîðûõâûïîëíåíî j ! ( t +  ( t )) � ! ( t ) j = j ! ( k )) � ! ( t ) j =0 " 34 Ïóñòü t 0 =0 : Òîãäàôóíäàìåíòàëüíàÿìàòðèöàñèñòåìû _ x = A ( t ) x; (0)= E èìååòâèä ( t )= g ( t ) E; ãäå g ( t )= 8 : 1 ; åñëè t 2 [0 ; 2  ] e � cos t +1 ; åñëè t 2 (2 ; 1 ) Ôóíêöèÿ g ( t ) ÿâëÿåòñÿðåêóððåíòíîéíà [0 ; 1 ) èïîýòîìóôóíêöèÿ ( t ) ðå- êóððåíòíà. Äîêàæåì,÷òîôóíêöèÿ g ( t ) íåÿâëÿåòñÿïî÷òèïåðèîäè÷åñêîé.Ïðåäïîëî- æèì,÷òîôóíêöèÿ g ( t ) ïî÷òèïåðèîäè÷åñêàÿ.Òîãäàïî " = 1 2 íàéäåòñÿ T� 0 ; ÷òîâëþáîìïðîìåæóòêå [ a;a + T ] ñóùåñòâóåòõîòÿáûîäíî÷èñëî ; ïðèêîòîðîì j g ( t +  ) � g ( t ) j 1 2 äëÿâñåõ t: Ïóñòü  2 [2 ; 2  + T ] : Òîãäà,â÷àñòíîñòè,äëÿâñåõ t 2 [0 ; 2  ] ñïðàâåäëèâî íåðàâåíñòâî g ( t +  ) 3 2 : Ñäðóãîéñòîðîíû, t +  2 [ ; 2  +  ] èïîýòîìóñóùåñòâóåò t 0 ; ÷òî g ( t 0 +  )= e 2 � 3 2 : Ïîëó÷èëèïðîòèâîðå÷èå.Ñëåäîâàòåëüíî,ôóíêöèÿ g íåÿâëÿåòñÿïî÷òèïåðèî- äè÷åñêîé. Óòâåðæäåíèå1.1 Ïóñòü A ( t )= ! ( t ) E è 0 2 Intco f z 0 1 ;:::;z 0 n g : Òîãäàâèãðå � 1 ïðîèñõîäèòïîèìêà. Çàìå÷àíèå1.2 Îòìåòèì,÷òîåñëè A ( t )  0 äëÿâñåõ t � t 0 ,òî ( t )= E  ðåêóððåíòíàÿôóíêöèÿ. Ñëåäñòâèå1.3 ([87]) Ïóñòü A ( t )  0 è 0 2 Intco f z 0 1 ;:::;z 0 n g : Òîãäàâèãðå � 1 ïðîèñõîäèòïîèìêà. 35 1.2.Ïîèìêàãðóïïûñêîîðäèíèðîâàííûõóáåãàþùèõâëèíåéíûõðå- êóððåíòíûõäèôôåðåíöèàëüíûõèãðàõ Âïðîñòðàíñòâå R k ( k � 2) ðàññìàòðèâàåòñÿäèôôåðåíöèàëüíàÿèãðà n + m ëèö: n ïðåñëåäîâàòåëåé P 1 ;:::;P n è m óáåãàþùèõ E 1 ;:::E m . Çàêîíäâèæåíèÿêàæäîãîèçïðåñëåäîâàòåëåé P i èìååòâèä _ x i = A ( t ) x i + u i ;u i 2 V: (1.5) Çàêîíäâèæåíèÿêàæäîãîèçóáåãàþùèõ E j èìååòâèä _ y j = A ( t ) y j + v;v 2 V: (1.6) Çäåñüèäàëåå x i ;y j ;u i ;v 2 R k , i 2 I = f 1 ;:::;n g ;j 2 J = f 1 ;:::;m g ;A ( t )  íåïðåðûâíàÿíà [ t 0 ; 1 ) êâàäðàòíàÿìàòðèöàïîðÿäêà k , V ñòðîãîâûïóêëûé êîìïàêò R k ñãëàäêîéãðàíèöåé.Ïðè t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x i ( t 0 )= x 0 i ;y j ( t 0 )= y 0 j ; (1.7) ïðè÷åì x 0 i 6 = y 0 j äëÿâñåõ i;j: Âìåñòîñèñòåì (1 : 5) , (1 : 6) , (1 : 7) ðàññìîòðèìñèñòåìóñíà÷àëüíûìèóñëîâè- ÿìè _ z ij = A ( t ) z ij + u i � v;u i ;v 2 V;z ij ( t 0 )= z 0 ij = x 0 i � y 0 j : (1.8) Îòìåòèì,÷òî z 0 ij 6 =0 . Îòìåòèì,÷òîäåéñòâèÿóáåãàþùèõìîæíîòðàêòîâàòüñëåäóþùèìîáðàçîì: èìååòñÿöåíòð,êîòîðûéäëÿâñåõóáåãàþùèõ E j âûáèðàþòîäíîèòîæåóïðàâ- ëåíèå v: Îïðåäåëåíèå1.5 Áóäåìãîâîðèòü,÷òîçàäàíàêâàçèñòðàòåãèÿ U i ïðåñëå- äîâàòåëÿ P i ; åñëèîïðåäåëåíîîòîáðàæåíèå U i ( t;z 0 ;v t (  )) ; ñòàâÿùååâñîîò- 36 âåòñòâèåíà÷àëüíîìóñîñòîÿíèþ z 0 =( z 0 11 ;:::;z 0 nm ) ,ìîìåíòó t èïðîèçâîëü- íîéïðåäûñòîðèèóïðàâëåíèÿ v t (  ) óáåãàþùèõ E 1 ;:::;E m èçìåðèìóþôóíêöèþ u i ( t )= U i ( t;z 0 ;v t (  )) ñîçíà÷åíèÿìèâ V: Îáîçíà÷èìäàííóþèãðó÷åðåç � 2 . Îïðåäåëåíèå1.6 Âèãðå � 2 ïðîèñõîäèòïîèìêà,åñëèñóùåñòâóåòìîìåíò T 0 = T ( z 0 ) ,êâàçèñòðàòåãèè U 1 ;:::; U n ïðåñëåäîâàòåëåé P 1 ;:::;P n ,òàêèå, ÷òîäëÿëþáîéèçìåðèìîéôóíêöèè v (  ) , v ( t ) 2 V , t 2 [0 ;T 0 ] íàéäóòñÿíîìåðà q 2f 1 ;:::;n g ;p 2f 1 ;:::;m g èìîìåíò  6 T 0 òàêèå,÷òî z qp (  )=0 . ×åðåç ( t ) îáîçíà÷èìôóíäàìåíòàëüíóþìàòðèöóñèñòåìû _ ! = A ( t ) !; ãäå ( t 0 )  E , E åäèíè÷íàÿìàòðèöà. Îïðåäåëåíèå1.7 ([57]) Âåêòîðû a 1 ;a 2 ;:::;a s îáðàçóþòïîëîæèòåëüíûéáà- çèñ R k ,åñëèäëÿëþáîãî x 2 R k ñóùåñòâóþòïîëîæèòåëüíûåâåùåñòâåííûå ÷èñëà 1 ; 2 ;:::; s òàêèå,÷òî x = 1 a 1 + 2 a 2 +  + s a s : Ëåììà1.3 ([110]) Ïóñòü b 1 ;:::;b n 2 R k ;V ñòðîãîâûïóêëûéêîìïàêòñ ãëàäêîéãðàíèöåé.Ñëåäóþùèåóòâåðæäåíèÿðàâíîñèëüíû. 1 : =min v 2 V max i  ( v;b i ) � 0 ; ãäå  ( v;b i )=sup f  :  � 0 ; � b i 2 V � v g 2 : Âåêòîðû b 1 ;:::;b n îáðàçóþòïîëîæèòåëüíûéáàçèñ R k ; 3 : 0 2 Intco f b 1 ;b 2 ;:::;b n g : Áóäåìïðåäïîëàãàòüâäàëüíåéøåì,÷òîíà÷àëüíûåïîçèöèè x 0 i ;y 0 j òàêîâû, ÷òî 37 1.Åñëè n�k ,òîäëÿëþáîãîíàáîðàèíäåêñîâ I f 1 ;:::;n g , j I j � k +1 ñïðàâåäëèâî Intco f x 0 i ;i 2 I g6 = ; ; 2.Ëþáûå k +1 òî÷êèèçíàáîðà f x 0 i ;y 0 j g àôôèííîíåçàâèñèìû. Òåîðåìà1.2 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ìàòðèöà ( t ) ðåêóððåíòíàíà [ t 0 ; 1 ) ,àååïåðâàÿïðîèçâîäíàÿðàâíî- ìåðíîîãðàíè÷åíàíà [ t 0 ; 1 ) ; 2 : Intco f x 0 1 ;:::;x 0 n g\ co f y 0 1 ;:::;y 0 m g6 = ; : (1.9) Òîãäàâèãðå � 2 ïðîèñõîäèòïîèìêà. Äîêàçàòåëüñòâî.Èçóñëîâèÿòåîðåìûñëåäóåò,÷òî n + m � k +2 : Èç[19,ëåììà8]ñëåäóåò,÷òîñóùåñòâóþòìíîæåñòâà I 0  I;J 0  J òàêèå,÷òî rico f x 0 i ;i 2 I 0 g\ rico f y 0 j ;j 2 J 0 g6 = ; è j I 0 j + j J 0 j = k +2 : Áóäåìñ÷èòàòü,÷òî I 0 = f 1 ;:::;q g ;J 0 = f 1 ;:::;l g ; ïðè÷åì q + l = k +2 : Ïîëåììå3ðàáîòû[19] f z 0 ij ;i 2 I 0 ;j 2 J 0 g îáðàçóþòïîëîæèòåëüíûéáàçèñ. Åñëè j J 0 j =1 ; òîïîèìêàñëåäóåòèçòåîðåìû 1 : 1 . Ñ÷èòàåì,÷òî j J 0 j � 2 : Ïóñòü c = y 0 � y 0 : Òîãäà z 0 i = z 0 i 1 + c 1 äëÿâñåõ i 2 I 0 ; 2 J 0 ; 6 =1 : Ïîýòîìó f z 0 i 1 ;i 2 I 0 ;c 1 ; 2 J 0 ; 6 =1 g îáðàçóþòïîëîæèòåëüíûé áàçèñ.Òàêêàê n � k +1 ,òî q + � 1 2f q +1 ;:::;n g äëÿâñåõ 6 =1 ; 2 J 0 : Âñèëóñëåäñòâèÿ1ðàáîòû[19]íàáîð f z 0 i; 1 ;i 2 I;z 0 q + � 1 ; 1 + c 1 ; 6 =1 ; 2 J 0 g (1.10) îáðàçóåòïîëîæèòåëüíûéáàçèñïðèíåêîòîðîì � 0 : Âîçüìåì "� 0 òàê,÷òîáûïîîòíîøåíèþêíàáîðó( 1 : 10 )áûëèñïðàâåäëèâû 38 ëåììà 1 : 1 èñëåäñòâèå 1 : 1 .Âñèëóëåììû 1 : 2 ÷èñëî T 0 = T ( z 0 )=min f t � 0:inf v (  ) min z 2 D max s J ( t;z s ) � 1 g (1.11) êîíå÷íî,ãäå D = D 2 " ( z 0 11 )  :::  D 2 " ( z 0 q 1 )  D 2 " ( z 0 q +1 ; 1 + c 2 1 )  :::  D 2 " ( z 0 q + l � 1 ; 1 + c l 1 ) : Ïóñòüäàëåå v (  ) ;t 0 6  6 T 0 ïðîèçâîëüíîåäîïóñòèìîåóïðàâëåíèåóáåãàþ- ùåãî, t 1 �t 0 íàèìåíüøèéêîðåíüôóíêöèèâèäà F ( t )=1 � max s t Z t 0  ( v ( s ) ; ( s ) w 0 s ) ds; ãäå w 0 s = 8 � � : z 0 s 1 ;s 2 I 0 ; z 0 q + s � 1 ; 1 + c s 1 ;s 2 J 0 ;s 6 =1 Îòìåòèì,÷òîâñèëóîïðåäåëåíèÿ T 0 ìîìåíò t 1 ñóùåñòâóåòè t 1 6 T 0 : Çàäàåìóïðàâëåíèÿïðåñëåäîâàòåëåéñëåäóþùèìîáðàçîì: u i ( t )= v ( t ) �  ( v ( t ) ; ( t ) z 0 i 1 )( t ) z 0 i 1 ;i 2 I 0 u q + � 1 ; 1 ( t )= v ( t ) �  ( v ( t ) ; ( t )( z 0 q + � 1 ; 1 + c 1 ))( t )( z 0 q + � 1 + c 1 ) ; 2 J 0 ; 6 =1 : Ñ÷èòàåì,÷òî  ( v ( t ) ; ( t ) w 0 s )=0 äëÿâñåõ s è t 2 [ t 1 ;T 0 ] : 39 Èçñèñòåìû( 1 : 8 )èìååì z i 1 ( t )=( t ) z 0 i 1 h i ( t ) ;i 2 I 0 ; z q + � 1 ; 1 ( t )+  ( t ) c 1 =( t )( z q + � 1 ; 1 + c 1 ) h q + � 1 ( t ) ; 2 J 0 ; 6 =1 ; ãäå h s ( t )=1 � t Z t 0  ( v (  ) ; (  ) w 0 s ) ds: Èç( 1 : 11 )ñëåäóåò,÷òîñóùåñòâóåòíîìåð r òàêîé,÷òî h r ( T 0 )=0 : Åñëè r 2 I 0 ; òîâ èãðå � 2 ïðîèñõîäèòïîèìêà.Åñëè h q + r � 1 ( T 0 )=0 ; òî z q + r � 1 ; 1 ( T 0 )= �  ( T 0 ) c r 1 : Ïóñòü T 0 òàêîé,÷òî t 0 + T 0 �T 0 : Òàêêàê ( t ) ÿâëÿåòñÿðåêóððåíòíîé ôóíêöèåé,òîïî "; âûáðàííîìóðàíåå,ñóùåñòâóåò÷èñëî T ( " ) òàêîå,÷òîíà ïðîìåæóòêå [ T 0 ;T 0 + T ( " )) íàéäåòñÿ÷èñëî  ( t 0 ) ; äëÿêîòîðîãîñïðàâåäëèâî íåðàâåíñòâî k ( t 0 +  ( t 0 )) � ( t 0 ) k " M ; ãäå M =max fk z 0 ij k ; k z 0 q + s � 1 ; 1 k ;s 2 J 0 ; k c 1 k ; 2 J 0 ; 6 =1 g : Çàäàåìóïðàâëåíèÿïðåñëåäîâàòåëåéíàîòðåçêå [ T 0 ;T 1 ] ; ãäå T 1 = t 0 +  ( t 0 ) ; ïîëàãàÿ u i ( t )= v ( t ) äëÿâñåõ i èâñåõ t 2 [ T 0 ;T 1 ] : Òîãäàáóäåìèìåòü z i 1 ( T 1 )=( T 1 ) z 0 i 1 h i ( T 0 ) ;i 2 I 0 ; (1.12) z q + r � 1 ; 1 ( T 1 )= �  ( T 1 ) c r 1 : Ïîêàæåì,÷òî rico f x i ( T 1 ) ;i 2 I 0 g\ rico f y j ( T 1 ) ;j 2 J 0 g6 = ; (1.13) 40 Èç( 1 : 12 )èìååì ( T 1 ) z 0 i 1 = z i 1 ( T 1 ) h i ( T 1 ) : Êðîìåòîãî,äëÿâñåõ 2 J 0 ; 6 =1 ñïðàâåä- ëèâîðàâåíñòâî z i ( T 1 )= z i 1 ( T 1 )+( T 1 ) c 1 = z i 1 ( T 1 )+( T 1 )( z 0 i � z 0 i 1 ) : Ïîýòîìóäëÿâñåõ ; 6 =1 ( T 1 ) z 0 i = z i ( T 1 ) � z i 1 ( T 1 )+( T 1 ) z 0 i 1 = z i ( T 1 )+ z i 1 ( T 1 )  1 � h i ( T 1 ) h i ( T 1 )  : Òàêêàê ( T 1 ) z 0 i 2 D 2 " ( z 0 i ) äëÿâñåõ i; ; òîñèñòåìà f ( T 1 ) z 0 ij ;i 2 I 0 ;j 2 J 0 g îáðàçóåòïîëîæèòåëüíûéáàçèñ R k : Ñëåäî- âàòåëüíî,ïîëîæèòåëüíûéáàçèñîáðàçóåòñèñòåìà f z ij ( T 1 ) ;i 2 I 0 ;j 2 J 0 g : Îòñþäàïîëó÷àåì( 1 : 13 ).Òàêêàê ( T 1 ) c r 1 2 D 2 " ( c r 1 ) ; òîèñïîëüçóÿëåììó9 ðàáîòû[19],ïîëó÷àåì rico f x i ( T 1 ) ;i 2 I 0 ;x q + r � 1 ( T 1 ) g\ rico f y j ( T 1 ) ;j 6 =1 ;j 2 J 0 g6 = ; : Ñ÷èòàåì,÷òî r =2 : Äàëååïîëàãàåì I 0 = f 1 ; 2 ;:::;q +1 g ;J 0 = f 2 ;:::;l g : Äëÿïîëó÷åííûõìíîæåñòâ I 0 ;J 0 ñïðàâåäëèâîóñëîâèå( 1 : 9 )ïðèýòîì÷èñëîóáå- ãàþùèõ,ó÷àñòâóþùèõâäàííîìóñëîâèè,óìåíüøèëîñüíà1.Ïðèíèìàÿìîìåíò T 1 çàíà÷àëüíûé,áóäåìïîâòîðÿòüíàøèðàññóæäåíèÿäîòåõïîð,ïîêà÷èñëî óáåãàþùèõíåñòàíåòðàâíûì1.Ïîëó÷èì,÷òî rico f x i (  ) ;i 2 I 0 g\ rico f y j (  ) ;j 2 J 0 g6 = ; ; âíåêîòîðûéìîìåíò � 0 ; ïðè÷åì j I 0 j = k +1 ; j J 0 j =1 : Òåïåðüïîèìêàñëåäóåò èçòåîðåìû 1 : 1 .Òåîðåìàäîêàçàíà. Ñëåäñòâèå1.4 ([19]) Ïóñòü A ( t )  0 äëÿâñåõ t � t 0 , Intco f x 0 1 ;:::;x 0 n g\ co f y 0 1 ;:::;y 0 m g6 = ; : 41 Òîãäàâèãðå � 2 ïðîèñõîäèòïîèìêà. Ïðèìåð1.2 Ðèñóíêè,ïðèâåäåííûåíèæåèëëþñòðèðóþòòåîðåìó 1 : 2 . v P 1 v P 2 v P 3 v E 3 v E 1 v E 2 Ðèñ.1.Ïîèìêàãðóïïûñêîîðäèíèðîâàííûõóáåãàþùèõ (íà÷àëüíûåïîçèöèè). v P 1 v P 2 v P 3 v E 3 v E 1 v E 2 Ðèñ.2.Ïîèìêàãðóïïûñêîîðäèíèðîâàííûõóáåãàþùèõ(øàã1). 42 v P 1 v P 2 v P 3 v E 3 v E 1 v E 2 Ðèñ.3.Ïîèìêàãðóïïûñêîîðäèíèðîâàííûõóáåãàþùèõ(øàã2). Ïðèìåð1.3 Ïóñòü k =2 ;n =4 ;m =2 ;t 0 =0 ;A ( t )  0 ;V = D 1 (0) . x 0 1 = 0 @ 3 2 1 A ;x 0 2 = 0 @ 2 � 2 1 A ;x 0 3 = 0 @ 4 � 1 1 A ;y 0 1 = 0 @ 0 0 1 A ;y 0 2 = 0 @ 5 0 1 A : 6 - v X 0 1 v X 0 3 v X 0 2 v Y 0 1 v Y 0 2 Ðèñ.4.Ïîèìêàäâóõæåñòêîñîåäèíåííûõóáåãàþùèõ. 43 ( t )= E ðåêóððåíòíàÿôóíêöèÿè Intco f x 0 1 ;x 0 2 ;x 0 3 g\ co f y 0 1 ;y 0 2 g6 = ; ,òî åñòüâûïîëíåíûâñåóñëîâèÿòåîðåìû 1 : 2 .Âàæíîîòìåòèòü,÷òîíåâûïîëíåíû óñëîâèÿòåîðåìûÑàòèìîâàÌàìàòîâà[19].Îäíàêîïîòåîðåìå 1 : 2 âèãðå � 2 ïðîèñõîäèòïîèìêà. Ïðèìåð1.4 Ïóñòü k =2 ;t 0 =0 ; ìàòðèöà A ( t ) èìååòâèä A ( t )= 8 � � � � � � � � � � � � � � � � : 0 B @ 00 cos t 0 1 C A ;t 2 [0;4  ) 0 B @ sin t 0 cos t sin t 1 C A ;t � 4  Òîãäàôóíäàìåíòàëüíàÿìàòðèöà ( t ) èìååòâèä ( t )= 8 � � � � � � � � � � � � � � � � : 0 B @ 10 sin t 1 1 C A ;t 2 [0;4  ) 0 B @ e 1 � cos t 0 sin t  e 1 � cos t e 1 � cos t 1 C A ;t � 4  Ìàòðèöà ( t ) ÿâëÿåòñÿðåêóððåíòíîé. Óòâåðæäåíèå1.2 Ïóñòü Intco f x 0 1 ;:::;x 0 n g\ co f y 0 1 ;:::;y 0 m g6 = ; : Òîãäàâèãðå � 2 ïðîèñõîäèòïîèìêà. 44 1.3.Ïîèìêàçàäàííîãî÷èñëàóáåãàþùèõ Âïðîñòðàíñòâå R k ( k � 2) ðàññìàòðèâàåòñÿäèôôåðåíöèàëüíàÿèãðà n + m ëèö: n ïðåñëåäîâàòåëåé P 1 ;:::;P n è m óáåãàþùèõ E 1 ;:::E m . Çàêîíäâèæåíèÿêàæäîãîèçïðåñëåäîâàòåëåé P i èìååòâèä _ x i = A ( t ) x i + u i ;u i 2 V: (1.14) Çàêîíäâèæåíèÿêàæäîãîèçóáåãàþùèõ E j èìååòâèä _ y j = A ( t ) y j + v j ;v j 2 V: (1.15) Çäåñüèäàëåå x i ;y j ;u i ;v j 2 R k ;i 2 I = f 1 ;:::;n g ;j 2 J = f 1 ;:::;m g ;A ( t )  íåïðåðûâíàÿíà [ t 0 ; 1 ) êâàäðàòíàÿìàòðèöàïîðÿäêà k , V ñòðîãîâûïóêëûé êîìïàêò R k ñãëàäêîéãðàíèöåé.Ïðè t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x i ( t 0 )= x 0 i ;y j ( t 0 )= y 0 j ; (1.16) ïðè÷åì x 0 i 6 = y 0 j äëÿâñåõ i;j: Öåëüãðóïïûïðåñëåäîâàòåëåéïîéìàòüíåìåíåå÷åì q (1 6 q 6 m ) óáå- ãàþùèõ,ïðèóñëîâèè,÷òîñíà÷àëàóáåãàþùèåâûáèðàþòñâîèóïðàâëåíèÿ,à çàòåìïðåñëåäîâàòåëè,çíàÿèíôîðìàöèþîâûáîðåóáåãàþùèõ,âûáèðàþòñâîè óïðàâëåíèÿ,ïðè÷åìêàæäûéïðåñëåäîâàòåëüìîæåòïîéìàòüíåáîëååîäíîãî óáåãàþùåãî. Ñ÷èòàåì,÷òî n � q . Âìåñòîñèñòåì (1 : 14) , (1 : 15) , (1 : 16) ðàññìîòðèìñèñòåìóñíà÷àëüíûìèóñëî- âèÿìè _ z ij = A ( t ) z ij + u i � v;u i ;v j 2 V;z ij ( t 0 )= z 0 ij = x 0 i � y 0 j : (1.17) Îòìåòèì,÷òî z 0 ij 6 =0 . Îáîçíà÷èìäàííóþèãðó÷åðåç � 3 . 45 Îïðåäåëåíèå1.8 Âèãðå � 3 ïðîèñõîäèòïîèìêàíåìåíåå q óáåãàþùèõ,åñëè ñóùåñòâóåòìîìåíò T 0 = T ( z 0 ) òàêîé,÷òîäëÿëþáîãîëþáîéñîâîêóïíîñòè äîïóñòèìûõóïðàâëåíèé v j ( t ) óáåãàþùèõ E j , t 2 [ t 0 ;T 0 ] íàéäóòñÿäîïóñòèìûå óïðàâëåíèÿïðåñëåäîâàòåëåé P 1 ;:::;P n u i ( t )= u i ( t;z 0 ij ;v j ( s ) ;s 2 [ t 0 ;T 0 ]) ; òàêèå,÷òîñóùåñòâóþòìíîæåñòâà N  I;M  J; j N j = j M j = q èäëÿ êàæäîãîíîìåðà 2 M íàéäóòñÿíîìåð = ( ) 2 N èìîìåíò  2 [ t 0 ;T 0 ] , äëÿêîòîðûõâûïîëíåíî z (  )=0 èïðèýòîì ( 1 ) 6 = ( 2 ) äëÿëþáûõ 1 6 = 2 . ×åðåç ( t ) îáîçíà÷èìôóíäàìåíòàëüíóþìàòðèöóñèñòåìû _ ! = A ( t ) !; ãäå ( t 0 )  E , E åäèíè÷íàÿìàòðèöà. Òåîðåìà1.3 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ìàòðèöà ( t ) ðåêóððåíòíàíà [ t 0 ; 1 ) ,àååïåðâàÿïðîèçâîäíàÿðàâíî- ìåðíîîãðàíè÷åíàíà [ t 0 ; 1 ) ; 2 : Äëÿêàæäîãî s 2f 0 ; 1 ;:::;q � 1 g âåðíîñëåäóþùåå:äëÿëþáîãîìíîæå- ñòâà N  I , j N j = n � s íàéäåòñÿòàêîåìíîæåñòâî M  J; j M j = q � s , ÷òîäëÿâñåõ 2 M 0 2 Intco f z 0 ; 2 N g : Òîãäàâèãðå � 3 ïðîèñõîäèòïîèìêàíåìåíåå q óáåãàþùèõ. Äîêàçàòåëüñòâî.Äîêàæåì,÷òîëþáûå n � s ïðåñëåäîâàòåëåéëîâÿòíå ìåíåå q � s óáåãàþùèõ,ãäå s 2f 0 ; 1 ;:::;q � 1 g .Ïðè s =0 ïîëó÷èìóòâåðæäåíèå òåîðåìû. 46 Ïóñòü s = q � 1 è N  I , j N j = n � s .Âñèëóóñëîâèÿ2ïî N ñóùåñòâóåò 2 J òàêîé,÷òî 0 2 Intco f z 0 ; 2 N g .Èçòåîðåìû1.1ñëåäóåò,÷òîïðåñëåäîâàòåëè P ; 2 N ëîâÿòóáåãàþùåãî E . Ïðåäïîëîæèì,÷òîóòâåðæäåíèåäîêàçàíîäëÿâñåõ s � s 0 +1 . Äîêàæåìóòâåðæäåíèåïðè s = s 0 .Ïóñòü N  I , j N j = n � s 0 .Òîãäàñó- ùåñòâóåò M  J , j M j = q � s 0 òàêîå,÷òî 0 2 Intco f z 0 ; 2 N g äëÿëþáîãî 2 M . Äëÿâñåõ 2 M îïðåäåëèììíîæåñòâà J = f 2 N : ïðåñëåäîâàòåëü P ëîâèòóáåãàþùåãî E g : Áåçîãðàíè÷åíèÿîáùíîñòèáóäåìñ÷èòàòü,÷òî M = f 1 ; 2 ;:::;q � s 0 g è j J 1 j 6 j J 2 j 6 ::: 6 j J q � s 0 j : Âñèëóòåîðåìû1.1 J 6 = ; äëÿâñåõ 2 M .Âîçìîæíûäâàñëó÷àÿ. 1. j n 1 S =1 J j � n 1 äëÿëþáîãî n 1 =1 ; 2 ;:::;q � s 0 : ÒîãäàïîòåîðåìåÕîëëà äëÿìíîæåñòâ J ñóùåñòâóåòñèñòåìàðàçëè÷íûõïðåäñòàâèòåëåé,òîåñòüñóùå- ñòâóþòïîïàðíîðàçëè÷íûå 2 N; 2 M òàêèå,÷òî 2 J .Òàêèìîáðàçîì, äîêàçàíî,÷òîïðåñëåäîâàòåëü P ëîâèòóáåãàþùåãî E ; 2 M èóòâåðæäåíèå âýòîìñëó÷àåñïðàâåäëèâî. 2.Ñóùåñòâóåò n 1 2f 1 ; 2 ;:::;q � k 0 g ,ïðèêîòîðîì j n 1 S =1 J j n 1 .Ïóñòü n 1 íàèìåíüøååèçíàòóðàëüíûõ÷èñåë,óäîâëåòâîðÿþùèõäàííîìóñâîéñòâó. Îòìåòèì,÷òî n 1 � 1 è j n 2 [ =1 J j � n 2 äëÿâñåõ n 2 2f 1 ; 2 ;:::;n 1 � 1 g : 47 Ïðè n 2 = n 1 � 1 èìååìñèñòåìóíåðàâåíñòâ j n 1 [ =1 J j n 1 ; j n 1 � 1 [ =1 J j � n 1 � 1 ; âñèëóêîòîðîé,ïîëó÷àåì j n 1 S =1 J j = n 1 � 1 .Ðàññìîòðèììíîæåñòâî N 1 = N n n 1 [ =1 J Ìíîæåñòâî N 1 íåïóñòî,òàêêàê j N j = n � s 0 ; j n 1 [ =1 J j = n 1 � 1 ;n 1 2f 1 ; 2 ;:::;q � s 0 g ;n � q: Ïîïðåäïîëîæåíèþäëÿ÷èñëà s = s 0 + n 1 � 1 ñóùåñòâóåòìíîæåñòâî M 1  J , j M 1 j = q � ( s 0 + n 1 � 1) òàêîå,÷òîïðåñëåäîâàòåëè P ; 2 N 1 ëîâÿòóáåãàþùèõ E ; 2 M 1 ,ïðè÷åì f 1 ; 2 ;:::;n 1 � 1 g\ M 1 = ; ,èáîâïðîòèâíîìñëó÷àåñóùå- ñòâîâàëáûíîìåð 2 N 1 òàêîé,÷òîïðåñëåäîâàòåëü P ëîâèòóáåãàþùåãî E , ãäå 2f 1 ; 2 ;:::;n 1 � 1 g ,÷òîïðîòèâîðå÷èòïîñòðîåíèþìíîæåñòâà N 1 . j n 2 S =1 J j � n 2 äëÿâñåõ n 2 2f 1 ; 2 ;:::;n 1 � 1 g ,ïðèìåíÿÿòåîðåìóÕîëëà, ïîëó÷èì,÷òîäëÿ J ñóùåñòâóåòñèñòåìàðàçëè÷íûõïðåäñòàâèòåëåé,òîåñòü ñóùåñòâóþòïîïàðíîðàçëè÷íûå 2 J ,ãäå =1 ; 2 ;:::;n 1 � 1 .Çíà÷èòïðåñëå- äîâàòåëè P ,ãäå 2 n 1 � 1 S =1 J ëîâÿòíåìåíåå n 1 � 1 óáåãàþùèõ.Òàêèìîáðàçîì, âñåïðåñëåäîâàòåëèëîâÿòíåìåíåå ( q � ( s 0 + n 1 � 1))+( n 1 � 1)= q � s 0 óáåãàþùèõ. Îãðàíè÷åííîñòüâðåìåíèïðåñëåäîâàíèÿñëåäóåòíåïîñðåäñòâåííîèçòåîðå- ìû1.1.Òåîðåìàäîêàçàíà. 48 Ïðèìåð1.5 Ïóñòü k =2 ;n =4 ;m = q =2 ;t 0 =0 ; ìàòðèöà A ( t ) èìååò âèä A ( t )= 0 @ cos t sin t; � cos t 2cos t � cos 3 t; � cos t sin t 1 A ; y 0 1 = 0 @ 1 0 1 A ;y 0 2 = 0 @ � 1 0 1 A ;x 0 1 = 0 @ 2 2 1 A ; x 0 2 = 0 @ 2 � 2 1 A ;x 0 3 = 0 @ � 2 � 2 1 A ;x 0 4 = 0 @ � 2 2 1 A : Òîãäàôóíäàìåíòàëüíàÿìàòðèöà ( t ) èìååòâèä ( t )= 0 @ 1 � sin t sin t cos 2 t 1 A Ìàòðèöà ( t ) ÿâëÿåòñÿðåêóððåíòíîéèâûïîëíåíîóñëîâèå2òåîðåìû1.3.Òîãäà âèãðå � 3 ïðîèñõîäèòïîèìêàíåìåíåå q óáåãàþùèõ. 6 - v X 0 1 v X 0 4 v X 0 3 v X 0 2 v Y 0 1 v Y 0 2 Ðèñ.5.Ïîèìêàäâóõóáåãàþùèõ. Óòâåðæäåíèå1.3 ([63]) Ïóñòü A ( t )  0 èâûïîëíåíîóñëîâèå 2 òåîðåìû 1.3 . Òîãäàâèãðåïðîèñõîäèòïîèìêàíåìåíåå q óáåãàþùèõ. 49 Ãëàâà2.ÏðèìåðË.Ñ.Ïîíòðÿãèíàñîìíîãèìè ó÷àñòíèêàìè 2.1.ÏîèìêàîäíîãîóáåãàþùåãîâðåêóððåíòíîìïðèìåðåË.Ñ.Ïîíò- ðÿãèíà Âïðîñòðàíñòâå R k ( k � 2) ðàññìàòðèâàåòñÿäèôôåðåíöèàëüíàÿèãðà n +1 ëèö: n ïðåñëåäîâàòåëåé P 1 ;:::;P n èóáåãàþùåãî E . Çàêîíäâèæåíèÿêàæäîãîèçïðåñëåäîâàòåëåé P i èìååòâèä x ( l ) i + a 1 ( t ) x ( l � 1) i +  + a l ( t ) x i = u i ;u i 2 V: (2.1) Çàêîíäâèæåíèÿóáåãàþùåãî E èìååòâèä y ( l ) + a 1 ( t ) y ( l � 1) +  + a l ( t ) y = v;v 2 V: (2.2) Çäåñüèäàëåå x i ;y;u i ;v 2 R k ;a 1 ( t ) ;:::;a l ( t ) íåïðåðûâíûåíà [ t 0 ; 1 ) ôóíê- öèè, V ñòðîãîâûïóêëûéêîìïàêòâ R k ñãëàäêîéãðàíèöåé. Ïðè t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x ( q ) i ( t 0 )= x q i ;y ( q ) ( t 0 )= y q ; ïðè÷åì x 0 i 6 = y 0 äëÿâñåõ i: (2.3) Çäåñüèäàëåå i 2 I = f 1 ; 2 ;:::;n g ;q =0 ; 1 ;:::;l � 1 : Îáîçíà÷èìäàííóþèãðó÷åðåç � 4 . Âìåñòîñèñòåì(2.1)(2.3)ðàññìîòðèìñèñòåìó z ( l ) i + a 1 ( t ) z ( l � 1) i +  + a l ( t ) z i = u i � v;u i ;v 2 V: (2.4) z ( q ) i ( t 0 )= z q i = x q i � y q ;q =0 ; 1 ;:::;l � 1 : (2.5) Íàçîâåìïðåäûñòîðèåéóïðàâëåíèÿ v ( t ) óáåãàþùåãî E âìîìåíòâðåìåíè 50 t;t 2 [ t 0 ; 1 ) ìíîæåñòâî v t (  )= f v ( s ) ;s 2 [ t 0 ;t ] ;v � èçìåðèìàÿôóíêöèÿ. g Îïðåäåëåíèå2.1 Áóäåìãîâîðèòü,÷òîçàäàíàêâàçèñòðàòåãèÿ U i ïðåñëå- äîâàòåëÿ P i ; åñëèîïðåäåëåíîîòîáðàæåíèå U i ( t;z 0 ;v t (  )) ; ñòàâÿùååâñîîò- âåòñòâèåíà÷àëüíîìóñîñòîÿíèþ z 0 =( z 0 1 ;:::;z 0 n ) ,ìîìåíòó t èïðîèç- âîëüíîéïðåäûñòîðèèóïðàâëåíèÿ v t (  ) óáåãàþùåãî E èçìåðèìóþôóíêöèþ u i ( t )= U i ( t;z 0 ;v t (  )) ñîçíà÷åíèÿìèâ V: Îáîçíà÷èìäàííóþèãðó÷åðåç � 4 . Îïðåäåëåíèå2.2 Âèãðå � 4 ïðîèñõîäèòïîèìêà,åñëèñóùåñòâóåòìîìåíò T 0 = T ( z 0 ) èêâàçèñòðàòåãèè U 1 ( t;z 0 ;v t (  )) ;:::; U n ( t;z 0 ;v t (  )) ïðåñëåäîâàòå- ëåé P 1 ;:::;P n òàêèå,÷òîäëÿëþáîéèçìåðèìîéôóíêöèè v (  ) , v ( t ) 2 V , t 2 [ t 0 ;T ( z 0 )] íàéäóòñÿíîìåð 2f 1 ;:::;n g èìîìåíò  2 [ t 0 ;T ( z 0 )] òàêèå, ÷òî z (  )=0 : Îáîçíà÷èì÷åðåç ' q ( t;s ) ;q =0 ;:::;l � 1 ; ( t � s � t 0 ) ðåøåíèÿóðàâíåíèÿ ! ( l ) + a 1 ( t ) ! ( l � 1) +  + a l ( t ) ! =0 ñíà÷àëüíûìèóñëîâèÿìè ! ( j ) ( s )=0 ;j =0 ;:::;q � 1 ;q +1 :::;l � 1 ;! ( q ) ( s )=1 : Ïóñòüäàëåå  i ( t )= ' 0 ( t;t 0 ) z 0 i + ' 1 ( t;t 0 ) z 1 i + ::: + ' l � 1 ( t;t 0 ) z l � 1 i : Îáîçíà÷èì H i = f  i ( t ) ;t 2 [ t 0 ; 1 ) g . 51 Ëåììà2.1 Ïóñòüäëÿâñåõ i 2 I = f 1 ; 2 ;:::;n g ñóùåñòâóþò h 0 i 2 H i , h 0 i 6 =0 òàêèå,÷òî 0 2 Intco f h 0 i g èôóíêöèè  i ( t ) ÿâëÿþòñÿðåêóððåíòíûìè.Òîãäà ñóùåñòâóþò "� 0 è T ( " ) � 0 òàêèå,÷òîñïðàâåäëèâûñëåäóþùèåóòâåðæäå- íèÿ: 1 : 0 = 2 D " ( h 0 i ) èäëÿëþáûõ h i 2 D " ( h 0 i ) âûïîëíåíî 0 2 Intco f h i g ,ãäå D " ( a )= f z : k z � a k 6 " g ; 2 : Äëÿêàæäîãî t � t 0 íàéäóòñÿòàêèåìîìåíòû  i 2 [ t;t + T ( " )] ,÷òî jj  i (  i ) � h 0 i k ": Äîêàçàòåëüñòâî.Ìíîæåñòâî co f h 0 i g ÿâëÿåòñÿâûïóêëûììíîãîãðàí- íèêîìñâåðøèíàìèâòî÷êàõ h 0 j ;j 2 K  I .Èçóñëîâèÿëåììûñëåäóåò, ÷òî 0 2 Intco f h 0 j g .Ìíîæåñòâî Intco f h 0 j g îòêðûòî.Ñëåäîâàòåëüíî,ñóùåñòâó- åò "� 0 òàêîå,÷òîäëÿëþáûõ h j 2 D " ( h 0 j ) ñïðàâåäëèâî 0 2 Intco f h j g .Òàêêàê Intco f h j g Intco f h i g ,òîïîëó÷àåìóòâåðæäåíèå1ëåììû. Òàêêàê  i ( t ) ÿâëÿþòñÿðåêóððåíòíûìè,òîäëÿëþáîãî "� 0 ñóùå- ñòâóåò T ( " ) � 0 òàêîå,÷òîäëÿêàæäîãî t � t 0 íàéäóòñÿòàêèåìîìåíòû  i 2 [ t;t + T ( " )] ,÷òî jj  i (  i ) � h 0 i jj ": Ëåììàäîêàçàíà. Âäàëüíåéøåìñ÷èòàåì,÷òî "� 0 è T âûáðàíîâñîîòâåòñòâèèñóñëîâèÿìè ëåììû2.1. Îïðåäåëèìôóíêöèè:  ( t;s )= 8 : 1 ; åñëè ' l � 1 ( t;s ) � 0 � 1 ; èíà÷å,  ( v;;h i )=sup f  :  � 0 ;v � h i 2 V g ; G ( t;h i )= Z t t 0 j ' l � 1 ( t;s ) j  ( v ( s ) ; ( t;s ) ;h i ) ds: 52 Ïîëàãàåìäàëåå h =( h 1 ;h 2 ;:::;h n ) ;D = D " ( h 0 1 )  D " ( h 0 2 )  D " ( h 0 n ) : Ëåììà2.2 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ôóíêöèè  i ( t ) ðåêóððåíòíûíà [ t 0 ; 1 ) ; 2 : lim t !1 R t t 0 j ' l � 1 ( t;s ) j ds =+ 1 ; 3 : Äëÿâñåõ i 2 I = f 1 ; 2 ;:::;n g ñóùåñòâóþò h 0 i 2 H i ;h 0 i 6 =0 òàêèå,÷òî 0 2 Intco f h 0 i g . Òîãäàñóùåñòâóåòìîìåíò T 1 òàêîé,÷òîäëÿëþáîãîäîïóñòèìîãîóïðàâ- ëåíèÿ v ( t ) èäëÿëþáîãî h 2 D ñóùåñòâóåò 2 I òàêîå,÷òî G ( T 1 ;h ) � 1 Äîêàçàòåëüñòâî.Èçóñëîâèéëåììûñëåäóåò,÷òîäëÿëþáîãî h 2 D ñïðàâåäëèâîíåðàâåíñòâî   1 ( h )=min v 2 V max i 2 I  ( v;  1 ;h i ) � 0 : Ïîëåììå1.3.13[110]ôóíêöèÿ  íåïðåðûâíàíàêàæäîìèçìíîæåñòâ V f 1 g D " ( h 0 i ) ,ïîýòîìó lim h  ! h   1 ( h  )=lim h  ! h min v 2 V max i 2 I  ( v;  1 ;h  i )=min v 2 V max i 2 I  ( v;  1 ;h i )=   1 ( h ) ; ñëåäîâàòåëüíî,èôóíêöèè   1 ÿâëÿþòñÿíåïðåðûâíûìèíà D .Ó÷èòûâàÿ,÷òî ìíîæåñòâî D êîìïàêò,ïîëó÷èì  =min h 2 D min  2f� 1 ; 1 g min v 2 V max i 2 I  ( v;;h i )=min h 2 D f  +1 ( h ) ; � 1 ( h ) g � 0 : Äàëåå max i 2 I G ( t;h i )=max i 2 I Z t t 0 j ' l � 1 ( t;s ) j  ( v ( s ) ; ( t;s ) ;h i ) ds � 53 � 1 n Z t t 0 j ' l � 1 ( t;s ) j X i 2 I  ( v ( s ) ; ( t;s ) ;h i ) ds �  n Z t t 0 j ' l � 1 ( t;s ) j ds: Òàêèìîáðàçîì,äëÿìîìåíòà T 1 ,îïðåäåëÿåìîãîèçóñëîâèÿ  n R T 1 t 0 j ' l � 1 ( T 1 ;s ) j ds � 1 èíåêîòîðîãî 2 I âûïîëíåíîíåðàâåíñòâî G ( T 1 ;h ) � 1 .Ëåììàäîêàçàíà. Ïóñòü T ( z 0 )=min f t � 0:inf v (  ) min h 2 D max i 2 I G ( t;h i ) � 1 g : Âñèëóëåììû2.2âûïîëíåíîíåðàâåíñòâî T ( z 0 ) 1 : Òåîðåìà2.1 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ôóíêöèè  i ( t ) ðåêóððåíòíûíà [ t 0 ; 1 ) ; 2 : Ñóùåñòâóþò h 0 i 2 H i ;h 0 i 6 =0 òàêèå,÷òî 0 2 Intco f h 0 1 ;h 0 2 ;:::;h 0 n g ; 3 : Ñóùåñòâóþòìîìåíòû  i � T ( z 0 ) òàêèå,÷òî ( a )  i (  i ) 2 D " ( h 0 i ) ; ( b )inf v (  ) max i G (  i ; i (  i )) � 1 . Òîãäàâèãðå � 4 ïðîèñõîäèòïîèìêà. Äîêàçàòåëüñòâî.ÏîôîðìóëåÊîøèðåøåíèåçàäà÷è(2.4)(2.5)ïðè ëþáûõäîïóñòèìûõóïðàâëåíèÿõèìååòâèä z i ( t )=  i ( t )+ Z t t 0 ' l � 1 ( t;s )( u i ( s ) � v ( s )) ds: Ïóñòü  i ìîìåíòû,óäîâëåòâîðÿþùèåóñëîâèþòåîðåìû, v ( s ) ;s 2 [ t 0 ;T 0 ]  ïðîèçâîëüíîåäîïóñòèìîåóïðàâëåíèåóáåãàþùåãî E ,ãäå T 0 =max i  i . Ðàññìîòðèìôóíêöèþ f ( t )=1 � max i 2 I Z t t 0 j ' l � 1 (  i ;s ) j  ( v ( s ) ; (  i ;s ) ; i (  i )) ds: Îáîçíà÷èì÷åðåç t 1 � t 0 íàèìåíüøèéêîðåíüäàííîéôóíêöèè.Îòìåòèì,÷òî ìîìåíò t 1 ñóùåñòâóåò,âñèëóóñëîâèÿ3òåîðåìû,è t 1 6  i ïîêðàéíåéìåðåäëÿ 54 îäíîãî i . Êðîìåòîãî,ñóùåñòâóåòíîìåð l 2 I òàêîé,÷òî 1 � Z t 1 t 0 j ' l � 1 (  l ;s ) j  ( v ( s ) ; (  l ;s ) ; l (  l )) ds =0 : Çàäàåìóïðàâëåíèåïðåñëåäîâàòåëåé P i ñëåäóþùèìîáðàçîì: u i ( t )= v ( t ) �  ( v ( t ) ; (  i ;t ) ; i (  i ))  (  i ;t )  i (  i ) äëÿâñåõ t 2 [ t 0 ;t 1 ] : ãäåñ÷èòàåì,÷òî  ( v ( t ) ; (  i ;t ) ; i (  i ))=0 ïðè t 2 [ t 1 ;T 0 ] : Òîãäà,ñó÷åòîìôîðìóëûÊîøè,èìååì z i (  i )=  i (  i )  1 � Z t 1 t 0 j ' l � 1 (  i ;s ) j  ( v ( s ) ; (  i ;s ) ; i (  i )) ds  : Âñèëóîïðåäåëåíèÿ t 1 ,äëÿíîìåðà l 2 I âûðàæåíèåâñêîáêàõîáðàùàåòñÿâ íîëü,ïîýòîìó z l (  l )=0 : Òåîðåìàäîêàçàíà. Ñëåäñòâèå2.1 Ïóñòüâûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : Ôóíêöèè  i ( t ) ðåêóððåíòíûíà [ t 0 ; 1 ) ; 2 : 0 2 Intco f z 0 1 ;:::;z 0 n g : Òîãäàâèãðå � 4 ïðîèñõîäèòïîèìêà. Ñïðàâåäëèâîñòüäàííîãîóòâåðæäåíèÿñëåäóåòèçëåììû2.2èòåîðåìû2.1. 2.2.Ãðóïïîâîåïðåñëåäîâàíèåñôàçîâûìèîãðàíè÷åíèÿìèâðåêóð- ðåíòíîìïðèìåðåË.Ñ.Ïîíòðÿãèíà Âïðîñòðàíñòâå R k ( k � 2) ðàññìàòðèâàåòñÿäèôôåðåíöèàëüíàÿèãðà n +1 ëèö: n ïðåñëåäîâàòåëåé P 1 ;P 2 ;:::;P n èóáåãàþùèé E: Äâèæåíèåêàæäîãîïðå- ñëåäîâàòåëÿ P i îïèñûâàåòñÿóðàâíåíèåì x ( l ) i + a 1 ( t ) x ( l � 1) i + a 2 ( t ) x ( l � 2) i + ::: + a l ( t ) x i = u i ;u i 2 V; (2.6) 55 çàêîíäâèæåíèÿóáåãàþùåãî E èìååòâèä y ( l ) + a 1 ( t ) y ( l � 1) + a 2 y ( l � 2) + ::: + a l ( t ) y = v;v 2 V; (2.7) ãäå x i ;y j ;u i ;v j 2 R k ; ôóíêöèè a 1 ( t ) ;a 2 ( t ) ;:::;a l ( t ) íåïðåðûâíûíàïðîìåæóòêå [ t 0 ; 1 ) ;V ñòðîãîâûïóêëûéêîìïàêòâ R k ñãëàäêîéãðàíèöåé. Âìîìåíò t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x ( q ) i ( t 0 )= x q i ;y ( q ) ( t 0 )= y q ; ïðè÷åì x 0 i � y 0 = 2 M i äëÿâñåõ i; (2.8) ãäå M i çàäàííûåâûïóêëûåêîìïàêòû.Çäåñüèäàëåå i 2 I = f 1 ; 2 ;:::;n g ; q =0 ; 1 ;:::;l � 1 : Äîïîëíèòåëüíîïðåäïîëàãàåòñÿ,÷òîóáåãàþùèéíåïîêèäàåòïðåäåëûâû- ïóêëîãîìíîæåñòâà B = f y : y 2 R k ; ( p c ;y ) 6  c ;c =1 ; 2 ;:::;r g ; ñíåïóñòîéâíóòðåííîñòüþ,ãäå ( a;b ) ñêàëÿðíîåïðîèçâåäåíèåâåêòîðîâ a è b; p 1 ;:::;p r åäèíè÷íûåâåêòîðû R k ; 1 ;:::; r âåùåñòâåííûå÷èñëà. Îáîçíà÷èìäàííóþèãðó÷åðåç �( n;B ) . Âìåñòî(2.6)(2.8)ðàññìîòðèìóðàâíåíèå z ( l ) i + a 1 ( t ) z ( l � 1) i + a 2 ( t ) z ( l � 2) i + ::: + a l ( t ) z i = u i � v; (2.9) ñíà÷àëüíûìèóñëîâèÿìè z ( q ) i ( t 0 )= z q i = x q i � y q : (2.10) ×åðåç ' q ( t;s )( t � s � t 0 ) îáîçíà÷èìðåøåíèåóðàâíåíèÿ ! ( l ) + a 1 ( t ) ! ( l � 1) + a 2 ( t ) ! ( l � 2) + ::: + a l ( t ) ! =0 ; 56 ñíà÷àëüíûìèóñëîâèÿìè ! ( s )=0 ;:::;! ( q � 1) ( s )=0 ;! ( q ) ( s )=1 ;! ( q +1) ( s )=0 ;:::;! ( l � 1) ( s )=0 : Ïóñòüäàëåå  i ( t )= ' 0 ( t;t 0 ) z 0 i + ' 1 ( t;t 0 ) z 1 i + ::: + ' l � 1 ( t;t 0 ) z l � 1 i ;  ( t )= ' 0 ( t;t 0 ) y 0 + ' 1 ( t;t 0 ) y 1 + ::: + ' l � 1 ( t;t 0 ) y l � 1 : Ñ÷èòàåì,÷òî  i ( t ) = 2 M i äëÿâñåõ i;t � t 0 ; èáîåñëè  i (  ) 2 M i ïðèíåêîòîðûõ i;; òîïðåñëåäîâàòåëü P i ëîâèòóáåãàþùåãî E; ïîëàãàÿ u i ( t )= v ( t ) : Íàçîâåìïðåäûñòîðèåéóïðàâëåíèÿ v ( t ) óáåãàþùåãî E âìîìåíòâðåìåíè t;t 2 [ t 0 ; 1 ) ìíîæåñòâî v t (  )= f v ( s ) ;s 2 [ t 0 ;t ] ;v � èçìåðèìàÿôóíêöèÿ. g Îïðåäåëåíèå2.3 Áóäåìãîâîðèòü,÷òîçàäàíàêâàçèñòðàòåãèÿ U i ïðåñëå- äîâàòåëÿ P i ; åñëèîïðåäåëåíîîòîáðàæåíèå U i ( t;z 0 ;v t (  )) ; ñòàâÿùååâñîîò- âåòñòâèåíà÷àëüíîìóñîñòîÿíèþ z 0 =( z 0 1 ;:::;z 0 n ) ,ìîìåíòó t èïðîèç- âîëüíîéïðåäûñòîðèèóïðàâëåíèÿ v t (  ) óáåãàþùåãî E èçìåðèìóþôóíêöèþ u i ( t )= U i ( t;z 0 ;v t (  )) ñîçíà÷åíèÿìèâ V: Îïðåäåëåíèå2.4 Âèãðå �( n;B ) ïðîèñõîäèòïîèìêà,åñëèñóùåñòâóåòìî- ìåíò T ( z 0 ) ; êâàçèñòðàòåãèè U 1 ( t;z 0 ;v t (  )) ;:::;U n ( t;z 0 ;v t (  )) ïðåñëåäîâàòåëåé P 1 ;:::;P n òàêèå,÷òîäëÿëþáîéèçìåðèìîéôóíêöèè v (  ) , v ( t ) 2 V , y ( t ) 2 B , t 2 [ t 0 ;T ( z 0 )] ñóùåñòâóþòìîìåíò  2 [ t 0 ;T ( z 0 )] èíîìåð 2 I; ÷òî z (  ) 2 M : 57 Ïóñòü  ( t;s )= 8 � � : 1 ; åñëè ' l � 1 ( t;s ) � 0 ; � 1 ; åñëè ' l � 1 ( t;s ) 0 ( t 0 6 s 6 t ) ;  ( v;;b i )=sup f  j�  ( b i � M i ) \ ( V � v ) 6 = ;g ; G ( t;v (  ) ;b i )= t Z t 0 j ' l � 1 ( t;s ) j  ( v ( s ) ; ( t;s ) ;b i )) ds; F ( t )= t Z t 0 j ' l � 1 ( t;s ) j ds: Ïðåäïîëîæåíèå2.1 1 : Ôóíêöèè  i ( t ) ÿâëÿþòñÿðåêóððåíòíûìèíà [ t 0 ; 1 ); 2 : Ôóíêöèÿ  ( t ) îãðàíè÷åíàíà [ t 0 ; 1 ); 3 : lim t !1 F ( t )= 1 ; Ïðåäïîëîæåíèå2.2 Ñóùåñòâóþòìîìåíòû  0 i � t 0 ; ïîëîæèòåëüíûå÷èñëà "; òàêèå,÷òî 1 : Äëÿâñåõ i èäëÿâñåõ h i 2 D " (  i (  0 i )) âûïîëíåíî h i = 2 M i ; 2 : Äëÿâñåõ h i 2 D " (  i (  0 i )) ñïðàâåäëèâûíåðàâåíñòâà min v max  max i  ( v; +1 ;h i ) ; max j ( p j ;v ) � ; min v max  max i  ( v; � 1 ;h i ) ; max j ( � p j ;v ) � : Îáîçíà÷èì: h =( h 1 ;h 2 ;:::;h n ) ;D = D " (  1 (  0 1 ))  D " (  2 (  0 2 ))  :::  D " (  n (  0 n )) : Ëåììà2.3 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 1 , 2 : 2 , r =1 : Òîãäàñóùå- ñòâóåòìîìåíò T � t 0 òàêîé,÷òîäëÿëþáîãîäîïóñòèìîãîóïðàâëåíèÿ v (  ) óáåãàþùåãî E; ëþáîãî h 2 D ñóùåñòâóåòíîìåð m 2 I äëÿêîòîðîãî G ( T;v (  ) ;h m ) � 1 : Äîêàçàòåëüñòâî.Òàêêàêóïðàâëåíèå v ( t ) óáåãàþùåãî E äîïóñòèìî,òî 58 äëÿâñåõ t � t 0 ( p 1 ;y ( t )) 6  ( t )=  1 � ( p 1 ; ( t )) : Îïðåäåëèììíîæåñòâà T + ( t )= f  :  2 [ t 0 ;t ] ;' l � 1 ( t; ) � 0 g ;T � ( t )= f  :  2 [ t 0 ;t ] ;' l � 1 ( t; ) 0 g ; T + 1 ( t )= f  :  2 T + ( t ) ; ( p 1 ;v (  )) �  g ; T + 2 ( t )= f  :  2 T + ( t ) ; ( p 1 ;v (  ))  g ; T � 1 ( t )= f  :  2 T � ( t ) ; ( � p 1 ;v (  )) �  g ; T � 2 ( t )= f  :  2 T � ( t ) ; ( � p 1 ;v (  ))  g : Òîãäà t Z t 0 ' l � 1 ( t;s )( p 1 ;v ( s )) ds = Z T + ( t ) ' l � 1 ( t;s )( p 1 ;v ( s )) ds + + Z T � ( t ) ( � ' l � 1 ( t;s ))( � p 1 ;v ( s )) ds = Z T + 1 ( t ) ' l � 1 ( t;s )( p 1 ;v ( s )) ds + + Z T + 2 ( t ) ' l � 1 ( t;s )( p 1 ;v ( s )) ds + Z T � 1 ( t ) ( � ' l � 1 ( t;s ))( � p 1 ;v ( s )) ds + + Z T � 2 ( t ) ( � ' l � 1 ( t;s ))( � p 1 ;v ( s )) ds �  Z T + 1 ( t ) ' l � 1 ( t;s ) ds � Z T + 2 ( t ) ' l � 1 ( t;s ) ds + +  Z T � 1 ( t ) ( � ' l � 1 ( t;s )) ds � Z T � 2 ( t ) ( � ' l � 1 ( t;s )) ds = =  Z T + 1 ( t ) [ T � 1 ( t ) j ' l � 1 ( t;s ) j ds � Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds: 59 Ïîëó÷àåì  Z T + 1 ( t ) [ T � 1 ( t ) j ' l � 1 ( t;s ) j ds � Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds 6  ( t ) ; Z T + 1 ( t ) [ T � 1 ( t ) j ' l � 1 ( t;s ) j ds + Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds = F ( t ) : Èçïîñëåäíèõäâóõñîîòíîøåíèéñëåäóåò,÷òî Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds � F ( t ) �  ( t ) 1+  : Äàëååèìååì max i 2 I G ( t;v (  ) ;h i )=max i 2 I t Z t 0 j ' l � 1 ( t;s ) j  ( v ( s ) ; ( t;s ) ;h i ) ds � � 1 n t Z t 0 j ' l � 1 ( t;s ) j X i 2 I  ( v ( s ) ; ( t;s ) ;h i ) ds � �  n Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds �  n  F ( t ) �  ( t ) 1+   : Òàêêàê F ( t ) !1 ïðè t !1 ; à  ( t ) îãðàíè÷åíà,òîïîëó÷àåìòðåáóåìîå óòâåðæäåíèå.Ëåììàäîêàçàíà. Îïðåäåëèì÷èñëî T 0 : T 0 =min f t � t 0 :inf v (  ) min h 2 D max i 2 I G ( t;v (  ) ;h i ) � 1 g : (2.11) Ïðåäïîëîæåíèå2.3 Ñóùåñòâóþòìîìåíòû  i � T 0 òàêèå,÷òî 1 : i (  i ) 2 D " (  i (  0 i )) äëÿâñåõ i ; 2 : inf v (  ) max i G (  i ;v (  ) ; i (  i )) � 1 : 60 Çàìå÷àíèå2.1 ( a ) ñóùåñòâîâàíèå  i âïóíêòå1)ïðåäïîëîæåíèÿ 2 : 3 ãàðàí- òèðîâàííîïðåäïîëîæåíèåìîðåêóððåíòíîñòèôóíêöèé  i ( t ); ( b ) åñëèâïðåäïîëîæåíèè 2 : 3 âñå  i = ; òîïóíêò2)äàííîãîïðåäïîëîæå- íèÿâûïîëíåíàâòîìàòè÷åñêèâñèëóëåììû 2 : 3 : Òåîðåìà2.2 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 1 , 2 : 2 , 2 : 3 , r =1 : Òîãäàâ èãðå �( n;B ) ïðîèñõîäèòïîèìêà. Äîêàçàòåëüñòâî.ÏîôîðìóëåÊîøèäëÿâñåõ t � t 0 ðåøåíèåçàäà÷è (2.9)(2.10)ïðèëþáûõäîïóñòèìûõóïðàâëåíèÿõèìååòâèä z i ( t )=  i ( t )+ t Z t 0 ' l � 1 ( t;s )( u i ( s ) � v ( s )) ds: Ïóñòü  i ìîìåíòûâðåìåíè,óäîâëåòâîðÿþùèåïðåäïîëîæåíèþ 2 : 3 , v ( s ) ; s 2 [ t 0 ;T 1 ] ïðîèçâîëüíîåäîïóñòèìîåóïðàâëåíèåóáåãàþùåãî E; ãäå T 1 =max i  i : Ðàññìîòðèìôóíêöèþ H ( t )=1 � max i t Z t 0 j ' l � 1 (  i ;s ) j  ( v ( s ) ; (  i ;s ) ; i (  i )) ds: Îáîçíà÷èì÷åðåç  0 � t 0 ïåðâûéêîðåíüäàííîéôóíêöèè.Îòìåòèì,÷òî ìîìåíò  0 ñóùåñòâóåòâñèëóïðåäïîëîæåíèÿ 2 : 2 ,ïðè÷åì  0 6  i õîòÿáûäëÿ îäíîãî i: Êðîìåòîãî,ñóùåñòâóåòíîìåð m òàêîé,÷òî 1 �  0 Z t 0 j ' l � 1 (  m ;s ) j  ( v ( s ) ; (  m ;s ) ; m (  m )) ds =0 : (2.12) Äëÿ j 6 = m òàêæåîáîçíà÷èì÷åðåç t j ìîìåíòûâðåìåíèäëÿêîòîðûõâûïîë- íåíîóñëîâèå (2 : 12) ; åñëèòàêèåìîìåíòûñóùåñòâóþò.ÂñèëóëåììûÔèëèïïî- âà[105]äëÿêàæäîãî i ñóùåñòâóþòèçìåðèìûåôóíêöèè m i ( s ) ;u i ( s ) ;s 2 [ t 0 ;T 1 ] 61 ÿâëÿþùèåñÿïðèêàæäîìôèêñèðîâàííîì s ðåøåíèåìóðàâíåíèÿ  ( v ( s ) ; (  j ;s ) ; i (  j ))(  i (  i ) � m i )= u i � v ( s ) : Çàäàäèìóïðàâëåíèåïðåñëåäîâàòåëåé P i ; ïîëàãàÿ u i ( t )= v ( t ) �  ( v ( t ) ; (  i ;t ) ; i (  i ))(  i (  i ) � m i ( t )) ;t 2 [ t 0 ; min f t i ;T 1 g ] ; u i ( t )= v ( t ) ;t 2 (min f t i ;T 1 g ;T 1 ] : Òîãäà z i (  i )=  i (  i )+  i Z t 0 ' (  i ;s )( u i ( s ) � v ( s )) ds = =  i (  i ) �  i Z t 0 j ' l � 1 (  i ;s ) j  ( v ( s ) ; (  i ;s ) ; i (  i ))(  i (  i ) � m i ( s )) ds = =  i (  i )(1 � t i Z t 0 j ' l � 1 (  i ;s ) j  ( v ( s ) ; (  i ;s ) ; i (  i )) ds )+ + t i Z t 0 j ' l � 1 (  i ;s ) j  ( v ( s ) ; (  i ;s ) ; i (  i )) m i ( s ) ds: Èç (2 : 12) ñëåäóåò,÷òî z m (  m )= t m Z t 0 j ' l � 1 (  m ;s ) j  ( v ( s ) ; (  m ;s ) ; m (  m )) m m ( s ) ds 2 M m Òåîðåìàäîêàçàíà. Îáîçíà÷èì  ( M i ;v )=sup   � 0 ;M i \ ( V � v ) 6 = ; : 62 Ëåììà2.4 Ïóñòü V = D 1 (0) ;Q i ;i 2 I âûïóêëûåêîìïàêòû R k ; 0 = 2 Q i äëÿâñåõ i: Òîãäà  + =min v max  max i ; ( Q i ;v ) ; max j ( p j ;v ) � 0 òîãäàèòîëüêîòîãäà,êîãäà 0 2 Intco f Q 1 ;:::;Q n ;p 1 ;:::;p r g : Äîêàçàòåëüñòâî.Îòìåòèì,÷òî([110,c.46])  ( Q i ;v )=max q i 2 Q i ( q i ;v )+ p ( q i ;v ) 2 + k q i k 2 (1 �k v k 2 ) k q i k 2 : Ïðåäïîëîæèì,÷òî  + =0 : Òîãäàñóùåñòâóåò v 0 ; k v 0 k =1 òàêîé,÷òî  ( Q i ;v 0 )=0 äëÿâñåõ i; ( p j ;v 0 ) 6 0 äëÿâñåõ j: Ñëåäîâàòåëüíî,ñïðàâåäëèâûñëåäóþùèåíåðàâåíñòâà ( q i ;v 0 ) 6 0 äëÿâñåõ i;q i 2 Q i ; ( p j ;v 0 ) 6 0 äëÿâñåõ j: (2.13) Ïîýòîìó0è co f Q 1 ;:::;Q n ;p 1 ;:::;p r g îòäåëèìû.Çíà÷èò 0 = 2 Intco f Q 1 ;:::;Q n ;p 1 ;:::;p r g : Ïðåäïîëîæèìòåïåðü,÷òî 0 = 2 Intco f Q 1 ;:::;Q n ;p 1 ;:::;p r g : Òîãäà0è co f Q 1 ;:::;Q n ;p 1 ;:::;p r g îòäåëèìû.Ñëåäîâàòåëüíî,ñóùåñòâóåò v 0 ; k v 0 k =1 òàêîé,÷òîñïðàâåäëèâûíåðàâåíñòâà (2 : 13) : Çíà÷èò  ( Q i ;v 0 )=0 äëÿâñåõ i: Ïîýòîìó  + =0 : Ëåììàäîêàçàíà. 63 Ñëåäñòâèå2.2 Ïóñòü V = D 1 (0) ;Q i ;i 2 I âûïóêëûåêîìïàêòû R k ; 0 = 2 Q i äëÿâñåõ i: Òîãäà  � =min v max  max i ; ( � Q i ;v ) ; max j ( � p j ;v ) � 0 òîãäàèòîëüêîòîãäà,êîãäà 0 2 Intco f Q 1 ;:::;Q n ;p 1 ;:::;p r g : Ïðåäïîëîæåíèå2.4 Ñóùåñòâóþò  0 i � t 0 òàêèå,÷òî 0 2 Intco   i (  0 i ) � M i ;i 2 I;p 1 ;:::;p r Ëåììà2.5 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 1 , 2 : 4 .Òîãäàñóùåñòâóþòïî- ëîæèòåëüíûå÷èñëà ";T ( " ) äëÿêîòîðûõñïðàâåäëèâûñëåäóþùèåóòâåðæäå- íèÿ: 1 : Äëÿâñåõ h i 2 D " (  i (  0 i )) èìåþòìåñòîñëåäóþùèåâêëþ÷åíèÿ h i = 2 M i ; 0 2 Intco  h i � M i ;i 2 I;p 1 ;:::;p r ; 2 : Äëÿêàæäîãî t � t 0 íàéäóòñÿìîìåíòû  i 2 [ t;t + T ( " )] òàêèå,÷òî  i (  i ) 2 D " (  i (  0 i )) : Ñïðàâåäëèâîñòüïåðâîãîóòâåðæäåíèÿñëåäóåòèçñâîéñòâàñòðîãîéîòäåëè- ìîñòèâûïóêëûõìíîæåñòâ,àñïðàâåäëèâîñòüâòîðîãîóòâåðæäåíèÿñëåäóåòèç ñâîéñòâàðåêóððåíòíûõôóíêöèé. Âûáåðåìèçàôèêñèðóåì "� 0 è T ( " ) � 0 òàê,÷òîáûèìåëèìåñòîóòâåð- æäåíèÿëåììû 2 : 5 . 64 Ëåììà2.6 Ïóñòü V = D 1 (0) èâûïîëíåíîïðåäïîëîæåíèå 2 : 4 .Òîãäàäëÿëþ- áîãî h 2 D ñïðàâåäëèâûíåðàâåíñòâà  + ( h )=min v max f max i  ( h i ; 1 ;v ) ; max j ( p j ;v ) g � 0 ;  � ( h )=min v max f max i  ( h i ; � 1 ;v ) ; max j ( � p j ;v ) g � 0 ;  =min h 2 S min f  + ( h ) ; � ( h ) g � 0 : Äîêàçàòåëüñòâî.Ïóñòü h 2 D: Òîãäàäëÿìíîæåñòâ Q i = h i � M i âû- ïîëíåíûóñëîâèÿëåìì 2 : 4 è 2 : 5 .Ïîýòîìó  + ( h ) � 0 : Âñèëóëåììû1.3.13 [110]ôóíêöèÿ  + ( h ) íåïðåðûâíàíà D: ÏîýòîìóïîòåîðåìåÂåéåðøòðàññà min h 2 S  + ( h ) � 0 : Àíàëîãè÷íî min h 2 S  � ( h ) � 0 : Ñëåäîâàòåëüíî, � 0 : Ëåììàäî- êàçàíà. Òåîðåìà2.3 ([57]) Âåêòîðû a 1 ;:::;a s îáðàçóþòïîëîæèòåëüíûéáàçèñ R k òîãäàèòîëüêîòîãäà,êîãäà 0 2 Intco f a 1 ;:::;a s g : Ëåììà2.7 Ïóñòü Q i ;i 2 I âûïóêëûåêîìïàêòû R k , 0 = 2 Q i äëÿâñåõ i è âûïîëíåíûñëåäóþùèåóñëîâèÿ: 1 : 0 2 Intco f Q 1 ;:::;Q n ;p 1 ;:::p r g ; 2 : Êîëè÷åñòâîýëåìåíòîâìíîæåñòâà n S i =1 Q i íåìåíåå k ; 3 : Âìíîæåñòâå n S i =1 Q i ñóùåñòâóåò k ëèíåéíîíåçàâèñèìûõâåêòîðîâ. Òîãäàñóùåñòâóþò p 2 R k ; 2 R 1 òàêèå,÷òî 1 :B  B 1 = f z z 2 R k ; ( p;z ) 6  g ; 2 : 0 2 Intco f Q 1 ;:::;Q n ;p g : Äîêàçàòåëüñòâî.Èçóñëîâèÿëåììûñëåäóåò,÷òîñóùåñòâóþò q 1 ;:::;q s 2 n S i =1 Q i òàêèå,÷òî 0 2 Intco f q 1 ;:::;q s ;p 1 ;:::;p r g : 65 Ìîæíîñ÷èòàòü,÷òî s � k èâåêòîðû q 1 ;:::;q k ëèíåéíîíåçàâèñèìû.Âñèëóòåî- ðåìû 2 : 3 âåêòîðû q 1 ;:::;q s ;p 1 ;:::;p r îáðàçóþòïîëîæèòåëüíûéáàçèñ.Ïîýòîìó ñóùåñòâóþòïîëîæèòåëüíûå÷èñëà 1 ;:::; s ; 1 ;:::; r òàêèå,÷òî 0= 1 q 1 +  + s q s + 1 p 1 +  + r p r : (2.14) Ðàññìîòðèìâåêòîð p = 1 p 1 +  + r p r : Ïîêàæåì,÷òîíàáîð q 1 ;:::;q s ;p îáðà- çóåòïîëîæèòåëüíûéáàçèñ. Ïóñòü x 2 R k : Òàêêàê q 1 ;:::q k îáðàçóþòáàçèñïðîñòðàíñòâà R k ; òîñóùå- ñòâóþò÷èñëà 1 ;:::; k òàêèå,÷òî x = 1 q 1 +  + k q k : Âñèëóñîîòíîøåíèÿ (2 : 14) ïîëó÷àåì,÷òîäëÿëþáîãî d 2 R 1 ñïðàâåäëèâîðà- âåíñòâî x = 1 q 1 +  + k q k + d � 1 q 1 +  + s q s + 1 p 1 +  + r p r  : Âçÿâ d� 0 òàê,÷òîáûâûïîëíÿëèñüíåðàâåíñòâà c + d c � 0 äëÿâñåõ c =1 ;:::;k ,ïîëó÷èì,÷òî x = 0 1 q 1 + 0 s q s + dp èïðèýòîìâñåêîýôôèöèåíòûïîëîæèòåëüíû.Ñëåäîâàòåëüíî, q 1 ;:::;q s ;p îáðàçóþòïîëîæèòåëüíûéáàçèñ.Çíà÷èò 0 2 Intco f q 1 ;:::;q s ;p g èïîýòîìó 0 2 Intco f Q 1 ;:::;Q s ;p g : Ðàññìîòðèììíîæåñòâî B 1 = f z z 2 R k ; ( p;z ) 6  g ; 66 ãäå  = 1  1 +  + r  r : Òîãäà B  B 1 : Îòìåòèì,÷òîåñëè p =0 ; òî B 1 = R k : Ëåììàäîêàçàíà. Ïðåäïîëîæåíèå2.5 Äëÿëþáîãî h 2 D âìíîæåñòâå n S i =1 ( h i � M i ) ñóùåñòâó- åò k ëèíåéíîíåçàâèñèìûõâåêòîðîâ. Ñëåäñòâèå2.3 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 1 , 2 : 4 , 2 : 5 .Òîãäàäëÿëþ- áîãî h 2 D ñóùåñòâóþòâåêòîð p ( h ) 2 R k è÷èñëî  ( h ) 2 R 1 òàêèå,÷òî 1 : 0 2 Intco f h i � M i ;i 2 I;p ( h ) g ; 2 :B  B 1 = f z z 2 R k ; ( p ( h ) ;z ) 6  ( h ) g : Ëåììà2.8 Ïóñòü V = D 1 (0) èâûïîëíåíûïðåäïîëîæåíèÿ 2 : 1 , 2 : 4 , 2 : 5 .Òîãäà ñóùåñòâóåòìîìåíò T � t 0 òàêîé,÷òîäëÿëþáîãîäîïóñòèìîãîóïðàâëåíèÿ v (  ) óáåãàþùåãî E âèãðå �( n;B 1 ) ; ëþáîãî h 2 D ñóùåñòâóåòíîìåð m 2 I äëÿ êîòîðîãî G ( T;v (  ) ;h m ) � 1 ; ãäå B 1 îïðåäåëåíîâñëåäñòâèè 2 : 3 . Äîêàçàòåëüñòâî.Ïóñòü h 2 D: Âñèëóëåììû 2 : 6 èìååì � 0 : Ïîýòîìó âûïîëíåíûóñëîâèÿïðåäïîëîæåíèÿ 2 : 2 .Ñëåäîâàòåëüíî,ïðèìåíèìàëåììà 2 : 3 , îòêóäàèñëåäóåòòðåáóåìîåóòâåðæäåíèå.Ëåììàäîêàçàíà. Òåîðåìà2.4 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 1 , 2 : 4 , 2 : 5 èñóùåñòâóþò  i � T 0 òàêèå,÷òî 1 : i (  i ) 2 D " (  i (  0 i )); 2 : inf v (  ) max i G (  i ;v (  ) ; i (  i )) � 1 âèãðå �( n;B 1 ) : Òîãäàâèãðå �( n;B 1 ) ïðîèñõîäèòïîèìêà. Ñïðàâåäëèâîñòüäàííîãîóòâåðæäåíèÿñëåäóåòèçòåîðåìû 2 : 2 . Ñëåäñòâèå2.4 Ïóñòüâûïîëíåíûâñåóñëîâèÿòåîðåìû 2 : 4 .Òîãäàâèãðå �( n;B ) ïðîèñõîäèòïîèìêà. 67 Ïðèìåð2.1 Ïóñòüñèñòåìà (2.9) , (2.10) èìååòâèä _ z i = u i � v;z i (0)= z 0 i : Äëÿäàííîãîïðèìåðàâûïîëíåíîïðåäïîëîæåíèå 2 : 1 . Óòâåðæäåíèå2.1 Ïóñòü V = D 1 (0) ; âìíîæåñòâå n S i =1 ( z 0 i � M i ) ñóùåñòâóåò k ëèíåéíîíåçàâèñèìûõâåêòîðîâè 0 2 Intco f z 0 i � M i ;i 2 I;p 1 ;:::;p r g : Òîãäàâèãðå �( n;B ) ïðîèñõîäèòïîèìêà. Óòâåðæäåíèå2.2 ([39]) Ïóñòü V = D 1 (0) ;B ìíîãîãðàííèê, M i = f 0 g äëÿ âñåõ i;n � k: Òîãäàâèãðå �( n;B ) ïðîèñõîäèòïîèìêà. Ïðèìåð2.2 Ïóñòü V = D 1 (0) ; ñèñòåìà (2.9) , (2.10) èìååòâèä  z i + 2 3 t _ z i + 1 9 t 4 = 3 z i = u i � v; ïðè÷åì t 0 =8  2 : Òîãäà ' 0 ( t;s )=cos( 3 p t � 3 p s ) ;' 1 ( t;s )=3 s 2 = 3 sin( 3 p t � 3 p s ) ;  i ( t )= z 0 i cos( 3 p t )+12  2 z 0 i sin( 3 p t ) : Ðåêóððåíòíîñòüôóíêöèè  i ( t ) ñëåäóåòèçðåçóëüòàòîâðàáîòû[33]. Óòâåðæäåíèå2.3 Ïóñòüñóùåñòâóåòìîìåíò  2 [ t 0 ; 1 ) òàêîé,÷òîâ ìíîæåñòâå n S i =1 (  i (  ) � M i ) ñóùåñòâóåò k ëèíåéíîíåçàâèñèìûõâåêòîðîâè 0 2 Intco f  i (  ) � M i ;i 2 I;p 1 ;:::;p r g : Òîãäàâèãðå �( n;B ) ïðîèñõîäèòïîèìêà. 68 Âçÿââêà÷åñòâå  0 i = t 0 =8  2 ; ñïðàâåäëèâî Óòâåðæäåíèå2.4 Ïóñòüíà÷àëüíûåïîçèöèè z 0 i òàêîâû,÷òîâìíîæåñòâå n S i =1 ( z 0 i � M i ) ñóùåñòâóåò k ëèíåéíîíåçàâèñèìûõâåêòîðîâè 0 2 Intco f z 0 i � M i ;i 2 I;p 1 ;:::;p r g : Òîãäàâèãðå �( n;B ) ïðîèñõîäèòïîèìêà. Îòìåòèì,÷òîäëÿäàííîãîïðèìåðàíåâûïîëíåíûóñëîâèÿðàáîòû[6]. 2.3.ÌíîãîêðàòíàÿïîèìêàâðåêóððåíòíîìïðèìåðåË.Ñ.Ïîíòðÿãè- íàñôàçîâûìèîãðàíè÷åíèÿìè Âïðîñòðàíñòâå R k ( k � 2) ðàññìàòðèâàåòñÿäèôôåðåíöèàëüíàÿèãðà n +1 ëèö: n ïðåñëåäîâàòåëåé P 1 ;P 2 ;:::;P n èóáåãàþùèé E: Äâèæåíèåêàæäîãîïðåñëåäîâàòåëÿ P i îïèñûâàåòñÿóðàâíåíèåì x ( l ) i + a 1 ( t ) x ( l � 1) i + a 2 ( t ) x ( l � 2) i + ::: + a l ( t ) x i = u i ;u i 2 V; (2.15) çàêîíäâèæåíèÿóáåãàþùåãî E èìååòâèä y ( l ) + a 1 ( t ) y ( l � 1) + a 2 y ( l � 2) + ::: + a l ( t ) y = v;v 2 V; (2.16) ãäå x i ;y j ;u i ;v j 2 R k ; ôóíêöèè a 1 ( t ) ;a 2 ( t ) ;:::;a l ( t ) íåïðåðûâíûíàïðîìåæóòêå [ t 0 ; 1 ) ;V ñòðîãîâûïóêëûéêîìïàêòâ R k ñãëàäêîéãðàíèöåé. Âìîìåíò t = t 0 çàäàíûíà÷àëüíûåóñëîâèÿ x ( q ) i ( t 0 )= x q i ;y ( q ) ( t 0 )= y q ; ïðè÷åì x 0 i � y 0 6 =0 äëÿâñåõ i: (2.17) Çäåñüèäàëåå i 2 I = f 1 ; 2 ;:::;n g ;q =0 ; 1 ;:::;l � 1 : 69 Äîïîëíèòåëüíîïðåäïîëàãàåòñÿ,÷òîóáåãàþùèéíåïîêèäàåòïðåäåëûâû- ïóêëîãîìíîæåñòâà B = f y : y 2 R k ; ( p c ;y ) 6  c ;c =1 ; 2 ;:::;r g ; ñíåïóñòîéâíóòðåííîñòüþ,ãäå ( a;b ) ñêàëÿðíîåïðîèçâåäåíèåâåêòîðîâ a è b; p 1 ;:::;p r åäèíè÷íûåâåêòîðû R k ; 1 ;:::; r âåùåñòâåííûå÷èñëà. Âìåñòî(2.15)(2.17)ðàññìîòðèìóðàâíåíèå z ( l ) i + a 1 ( t ) z ( l � 1) i + a 2 ( t ) z ( l � 2) i + ::: + a l ( t ) z i = u i � v; (2.18) ñíà÷àëüíûìèóñëîâèÿìè z ( q ) i ( t 0 )= z q i = x q i � y q : (2.19) ×åðåç ' q ( t;s )( t � s � t 0 ) îáîçíà÷èìðåøåíèåóðàâíåíèÿ ! ( l ) + a 1 ( t ) ! ( l � 1) + a 2 ( t ) ! ( l � 2) + ::: + a l ( t ) ! =0 ; ñíà÷àëüíûìèóñëîâèÿìè ! ( s )=0 ;:::;! ( q � 1) ( s )=0 ;! ( q ) ( s )=1 ;! ( q +1) ( s )=0 ;:::;! ( l � 1) ( s )=0 : Ïóñòüäàëåå  i ( t )= ' 0 ( t;t 0 ) z 0 i + ' 1 ( t;t 0 ) z 1 i + ::: + ' l � 1 ( t;t 0 ) z l � 1 i ;  ( t )= ' 0 ( t;t 0 ) y 0 + ' 1 ( t;t 0 ) y 1 + ::: + ' l � 1 ( t;t 0 ) y l � 1 : Ñ÷èòàåì,÷òî  i ( t ) 6 =0 äëÿâñåõ i;t � t 0 ; èáîåñëè  i (  )=0 ïðèíåêîòîðûõ i;; òîïðåñëåäîâàòåëü P i ëîâèòóáåãàþùåãî E; ïîëàãàÿ u i ( t )= v ( t ) : 70 Îïðåäåëåíèå2.5 Áóäåìãîâîðèòü,÷òîçàäàíàêâàçèñòðàòåãèÿ U i ïðåñëå- äîâàòåëÿ P i ; åñëèîïðåäåëåíîîòîáðàæåíèå U i ( t;z 0 ;v t (  )) ; ñòàâÿùååâñîîò- âåòñòâèåíà÷àëüíîìóñîñòîÿíèþ z 0 =( z 0 1 ;:::;z 0 n ) ,ìîìåíòó t èïðîèç- âîëüíîéïðåäûñòîðèèóïðàâëåíèÿ v t (  ) óáåãàþùåãî E èçìåðèìóþôóíêöèþ u i ( t )= U i ( t;z 0 ;v t (  )) ñîçíà÷åíèÿìèâ V: Îïðåäåëåíèå2.6 Âèãðå �( n;B ) ïðîèñõîäèò m -êðàòíàÿïîèìêà(ïðè m =1 ïîèìêà),åñëèñóùåñòâóþòìîìåíò T ( z 0 ) ; êâàçèñòðàòåãèè U 1 ( t;z 0 ;v t (  )) ;:::;U n ( t;z 0 ;v t (  )) ïðåñëåäîâàòåëåé P 1 ;:::;P n òàêèå,÷òîäëÿëþ- áîéèçìåðèìîéôóíêöèè v (  ) ;v ( t ) 2 V;y ( t ) 2 B;t 2 [ t 0 ;T ( z 0 )] ñóùåñòâóþòìî- ìåíòû  1 ;:::; m 2 [ t 0 ;T ( z 0 )] ; ïîïàðíîðàçëè÷íûåèíäåêñû i 1 ;:::;i m 2 I; ÷òî z i s (  s )=0 ;s =1 ;:::;m: ( p )= f ( i 1 ;:::;i p ) j i 1 ;:::;i p 2 I èïîïàðíîðàçëè÷íû g ;  ( t;s )= 8 � � : 1 ; åñëè ' l � 1 ( t;s ) � 0 ; � 1 ; åñëè ' l � 1 ( t;s ) 0 ( t 0 6 s 6 t ) ;  ( v;;b i )=sup f  � 0 j� b i \ ( V � v ) 6 = ;g ; G ( t;v (  ) ;b i )= t Z t 0 j ' l � 1 ( t;s ) j  ( v ( s ) ; ( t;s ) ;b i )) ds; F ( t )= t Z t 0 j ' l � 1 ( t;s ) j ds: Ïðåäïîëîæåíèå2.6 1 :n � m + k � 1; 2 : Ôóíêöèè  i ( t ) ÿâëÿþòñÿðåêóððåíòíûìèíà [ t 0 ; 1 ); 3 : Ôóíêöèÿ  ( t ) îãðàíè÷åíàíà [ t 0 ; 1 ); 4 : lim t !1 F ( t )= 1 ; 5 :V = D 1 (0) ; ãäå D r ( a )= f z 2 R k jk z � a k 6 r g : 71 Ïðåäïîëîæåíèå2.7 Ñóùåñòâóþòìîìåíòû  0 i 2 [ t 0 ; 1 ) òàêèå,÷òîäëÿ âñåõ  2 ( n � m +1) âûïîëíåíîâêëþ÷åíèå 0 2 Intco f  j (  0 j ) ;j 2  ;p 1 ;:::;p r g : Îòìåòèì,÷òîïðåäïîëîæåíèå 2 : 6 áóäåò,â÷àñòíîñòè,âûïîëíåíîåñëèôóíê- öèè a i ( t ) ÿâëÿþòñÿïîñòîÿííûìè,àêîðíèõàðàêòåðèñòè÷åñêîãîóðàâíåíèÿ(2.18) ÿâëÿþòñÿïðîñòûìèè÷èñòîìíèìûìè. Ëåììà2.9 Ïóñòüâûïîëíåíîïðåäïîëîæåíèå 2 : 7 .Òîãäàñóùåñòâóþò "� 0 ; T ( " ) � 0 äëÿêîòîðûõñïðàâåäëèâûñëåäóþùèåóòâåðæäåíèÿ: 1 : 0 = 2 D " (  i (  0 i )) èêàæäûéíàáîð h =( h 1 ;:::;h n ) ;h i 2 D " (  i (  0 i )) îáëàäàåò ñâîéñòâîì 0 2 Intco f h j ;j 2  ;p 1 ;:::;p r g äëÿâñåõ  2 ( n � m +1); 2 : Äëÿêàæäîãî t � t 0 íàéäåòñÿìîìåíò  i ( t ) 2 [ t;t + T ( " )] òàêîé,÷òî  i (  i ( t )) 2 D " (  i (  0 i )) : Ñïðàâåäëèâîñòüïåðâîãîóòâåðæäåíèÿñëåäóåòèçñâîéñòâàîòêðûòûõìíî- æåñòâ,àñïðàâåäëèâîñòüâòîðîãîóòâåðæäåíèÿ-èçñâîéñòâàðåêóððåíòíûõ ôóíêöèé. Âûáåðåìèçàôèêñèðóåì "� 0 ;T ( " ) � 0 òàê,÷òîáûèìåëèìåñòîóòâåðæäå- íèÿëåììû 2 : 9 . Îáîçíà÷èì÷åðåç D = D " (  1 (  0 1 ))  D " (  2 (  0 2 ))  D " (  n (  0 n )) ;  =min h 2 D min r 2f� 1 ; 1 g min v 2 V max f max  2 ( m ) min j 2   ( v;r;h j ) ; max s ( p s ;v ) g : 72 Ëåììà2.10 Ïóñòü b 1 ;:::;b n 2 R k ;b j 6 =0 ;V = D 1 (0) : Òîãäà 0 2 Intco f b 1 ;:::;b n ;p 1 ;:::;p r g (2.20) òîãäàèòîëüêîòîãäàêîãäà  0 =min v 2 V max f max j  ( v; 1 ;b j ) ; max s ( p s ;v ) g � 0 : Äîêàçàòåëüñòâî.Ïóñòüâûïîëíåíîóñëîâèå(2.20) : Ïðåäïîëîæèì,÷òî  0 =0 : Òîãäàñóùåñòâóåò v 0 2 V äëÿêîòîðîãî  ( v 0 ; 1 ;b j )=0 ; ( p s ;v 0 ) 6 0 äëÿ âñåõ j;s: Òàêêàê[110,c.56]  ( v; 1 ;b j )= ( b j ;v )+ p ( b j ;v ) 2 + k b j k 2 (1 �k v k 2 ) k b j k 2 ; òîïîëó÷àåì,÷òî k v 0 k =1 è ( b j ;v 0 ) 6 0 äëÿâñåõ j: Ñëåäîâàòåëüíî,íàáîð co f b 1 ;:::;b n ;p 1 ;:::;p r g îòäåëèìîòíóëÿ.Ïîëó÷èëèïðîòèâîðå÷èå. Ïóñòü  0 � 0 : Äîêàæåì(2.20) : Ïðåäïîëîæèì,÷òîóñëîâèå(2.20)íåâûïîë- íÿåòñÿ.Òîãäàìíîæåñòâî co f b 1 ;:::;b n ;p 1 ;:::;p r g îòäåëèìîîòíóëÿ.Ïîýòîìó ñóùåñòâóåò v 0 2 V; k v 0 k =1 èòàêîé,÷òî ( b j ;v 0 ) 6 0 ; ( p s ;v 0 ) 6 0 äëÿâñåõ j;s: Îòñþäà  0 =0 : Ïîëó÷èëèïðîòèâîðå÷èå.Ëåììàäîêàçàíà. Ëåììà2.11 Ïóñòü V = D 1 (0) èâûïîëíåíîïðåäïîëîæåíèå 2 : 7 .Òîãäà � 0 : Äîêàçàòåëüñòâî.Âîçüìåì h 2 D: Ïóñòü  + ( h )=min v 2 V max  2 ( m ) min j 2   ( v; +1 ;h j ) ;  � ( h )=min v 2 V max  2 ( m ) min j 2   ( v; � 1 ;h j ) : Äîêàæåì,÷òî  + ( h ) � 0 ; � ( h ) � 0 : Ïðåäïîëîæèì,÷òî  + ( h )=0 : Òîãäàñóùå- ñòâóåò v 2 V; ÷òîäëÿêàæäîãî  2 ( m ) íàéäåòñÿíîìåð p 2  äëÿêîòîðîãî  ( v; +1 ;h p )=0 è,êðîìåòîãî, max s ( p s ;v ) 6 0 : 73 Ïîñòðîèììíîæåñòâî  0 2 ( n � m +1) ïîñëåäóþùåìóïðàâèëó.Âûáåðåì p 1 2 L 1 = f 1 ; 2 ;:::;m g2 ( m ) è h p 1 èçóñëîâèÿ  ( v; 1 ;h p 1 )=0 : Äàëååâûáåðåì p 2 2 L 2 = � L 1 [f m +1 g  nf p 1 g è h p 2 òàêèå,÷òî  ( v; 1 ;h p 2 )=0 èòàêäàëåå.Íà ïîñëåäíåìøàãåïîñòðîèììíîæåñòâî L n � m +1 = � L n � m [f n g  nf p n � m g èâûáåðåì p n � m +1 2 L n � m +1 ;h p n � m +1 ,äëÿêîòîðûõ  ( v; 1 ;h n � m +1 )=0 : Ïîïîñòðîåíèþìíî- æåñòâà  0 èìååì max f max j 2  0  ( v; 1 ;h j ) ; max s ( p s ;v ) g =0 : Ïîýòîìóèçëåììû 2 : 10 ñëåäóåò,÷òî 0 = 2 Intco f h j ;j 2  0 ;p 1 ;:::p r g ; ÷òîïðîòèâîðå÷èòïðåäïîëîæåíèþ 2 : 7 .Çíà÷èò  + ( h ) � 0 äëÿâñåõ h 2 D: Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî  � ( h ) � 0 äëÿâñåõ h 2 D: Òàêêàê V ñòðîãîâûïóêëûéêîìïàêòñãëàäêîéãðàíèöåé,òî âñèëóëåììû1.3.13([110,c.30])ôóíêöèè  ( v;  1 ;h ) íåïðåðûâíûïî ( v;h ) : ÎñòàëîñüïðèìåíèòüòåîðåìóÂåéåðøòðàññà.Ëåììàäîêàçàíà. Ëåììà2.12 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 6 , 2 : 7 , r =1 : Òîãäàñóùå- ñòâóåòìîìåíò T�t 0 òàêîé,÷òîäëÿëþáîãîäîïóñòèìîãîóïðàâëåíèÿ v (  ) óáåãàþùåãî E; ëþáîãîíàáîðà h 2 D íàéäåòñÿìíîæåñòâî  2 ( m ) ; ÷òî min j 2  G ( T;v (  ) ;h j ) � 1 : Äîêàçàòåëüñòâî.Ïóñòü h 2 D: Òîãäà,âñèëóëåììû 2 : 11 , � 0 : Òàê êàêóïðàâëåíèå v ( t ) óáåãàþùåãî E äîïóñòèìî,òîäëÿâñåõ t � t 0 ( p 1 ;y ( t )) 6  ( t )=  1 � ( p 1 ; ( t )) : Îïðåäåëèììíîæåñòâà T + ( t )= f  :  2 [ t 0 ;t ] ;' l � 1 ( t; ) � 0 g ;T � ( t )= f  :  2 [ t 0 ;t ] ;' l � 1 ( t; ) 0 g ; T + 1 ( t )= f  :  2 T + ( t ) ; ( p 1 ;v (  )) �  g ; T + 2 ( t )= f  :  2 T + ( t ) ; ( p 1 ;v (  ))  g ; T � 1 ( t )= f  :  2 T � ( t ) ; ( � p 1 ;v (  )) �  g ; 74 T � 2 ( t )= f  :  2 T � ( t ) ; ( � p 1 ;v (  ))  g : Òîãäà t Z t 0 ' l � 1 ( t;s )( p 1 ;v ( s )) ds = Z T + ( t ) ' l � 1 ( t;s )( p 1 ;v ( s )) ds + + Z T � ( t ) ( � ' l � 1 ( t;s ))( � p 1 ;v ( s )) ds = Z T + 1 ( t ) ' l � 1 ( t;s )( p 1 ;v ( s )) ds + + Z T + 2 ( t ) ' l � 1 ( t;s )( p 1 ;v ( s )) ds + Z T � 1 ( t ) ( � ' l � 1 ( t;s ))( � p 1 ;v ( s )) ds + + Z T � 2 ( t ) ( � ' l � 1 ( t;s ))( � p 1 ;v ( s )) ds �  Z T + 1 ( t ) ' l � 1 ( t;s ) ds � Z T + 2 ( t ) ' l � 1 ( t;s ) ds + +  Z T � 1 ( t ) ( � ' l � 1 ( t;s )) ds � Z T � 2 ( t ) ( � ' l � 1 ( t;s )) ds = =  Z T + 1 ( t ) [ T � 1 ( t ) j ' l � 1 ( t;s ) j ds � Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds: Ïîëó÷àåì  Z T + 1 ( t ) [ T � 1 ( t ) j ' l � 1 ( t;s ) j ds � Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds 6  ( t ) ; Z T + 1 ( t ) [ T � 1 ( t ) j ' l � 1 ( t;s ) j ds + Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds = F ( t ) : Èçïîñëåäíèõäâóõñîîòíîøåíèéñëåäóåò,÷òî Z T + 2 ( t ) [ T � 2 ( t ) j ' l � 1 ( t;s ) j ds � F ( t ) �  ( t ) 1+  : 75 Äàëååèìååì max  2 ( m ) min j 2  G ( t;v (  ) ;h j )=max  2 ( m ) min j 2  t Z t 0 j ' l � 1 ( t;s ) j  ( v ( s ) ; ( t;s ) ;h j ) ds � � max  2 ( m ) t Z t 0 j ' l � 1 ( t;s ) j min j 2   ( v ( s ) ; ( t;s ) ;h j ) ds � � 1 C m n t Z t 0 j ' l � 1 ( t;s ) j X  2 ( m )  min j 2   ( v ( s ) ; ( t;s ) ;h j )  ds � � 1 C m n t Z t 0 j ' l � 1 ( t;s ) j max  2 ( m ) min j 2   ( v ( s ) ; ( t;s ) ;h j ) ds � � 1 C m n Z T + 1 ( t ) [ T � 1 ( t ) j ' l � 1 ( t;s ) j max  2 ( m ) min j 2   ( v ( s ) ; ( t;s ) ;h j ) ds � �  C m n Z T + 1 ( t ) [ T � 1 ( t ) j ' l � 1 ( t;s ) j ds �  C m n h F ( t ) �  ( t ) 1+  i : Òàêêàê F ( t ) !1 ; ( t ) îãðàíè÷åíà,òîñóùåñòâóåò T òàêîé,÷òî  C m n h F ( t ) �  ( t ) 1+  i � 1 äëÿâñåõ t � T; îòêóäàïîëó÷àåìòðåáóåìîåóòâåðæäåíèå.Ëåììàäîêàçàíà. Ëåììà2.13 Ïóñòü b 1 ;:::;b n 2 R k ;n � k; âûïîëíåíîâêëþ÷åíèå (2.20) è b 1 ;:::;b k ëèíåéíîíåçàâèñèìû.Òîãäàñóùåñòâóþò p 2 R k ; 2 R 1 ; ÷òî B  B 1 = f z j ( p;z ) 6  g è 0 2 Intco f b 1 ;:::;b n ;p g : Äîêàçàòåëüñòâî.Èçñîîòíîøåíèÿ(2.20)[57]ñëåäóåò,÷òîñóùåñòâóþò i � 0 ; j � 0 òàêèå,÷òî 0= 1 b 1 +  + n b n + 1 p 1 +  + r p r : 76 Ïóñòü x 2 R k : Òîãäàñóùåñòâóþò 1 ;:::; k òàêèå,÷òî x = 1 x 1 +  + k x k : Ïîýòîìóïðèëþáîì d 2 R 1 ñïðàâåäëèâîðàâåíñòâî x = 1 x 1 +  + k x k + d ( 1 b 1 +  + n b n + 1 p 1 +  + r p r ) : Ïîëàãàåì p = 1 p 1 +  + r p r : Âîçüìåì d� 0 òàêèì,÷òîáûäëÿâñåõ i âû- ïîëíÿëèñüíåðàâåíñòâà i + d i � 0 : Ïîëó÷àåì x = 0 1 b 1 +  + 0 n b n + dp; ïðè÷åì 0 i � 0 : Ñëåäîâàòåëüíî[57], 0 2 Intco f b 1 ;:::;b n ;p g : Ðàññìîòðèììíî- æåñòâî B 1 = f z j ( p;z ) 6  g ; ãäå  = 1  1 + ::: r  r : Òîãäà B  B 1 : Ëåììà äîêàçàíà. Ëåììà2.14 Ïóñòü V = D 1 (0) ;b 1 ;:::;b n 2 R k ;n � m + k � 1 èâûïîëíåíû ñëåäóþùèåóñëîâèÿ 1 : 0 2 Intco f b j ;j 2  ;p 1 ;:::;p r g äëÿëþáîãî  2 ( n � m +1); 2 : min v 2 coV 1 max s ( p s ;v ) � 0 ; ãäå V 1 = f v 2 V j max J 2 ( m ) min j 2 J  ( v; 1 ;b j )=0 g : Òîãäàñóùåñòâóþò p 2 R k ; 2 R 1 òàêèå,÷òî 1 :B  B 1 = f z 2 R k j ( p;z ) 6  g ; 2 : 0 2 Intco f b j ;j 2  ;p g äëÿâñåõ  2 ( n � m +1) : Äîêàçàòåëüñòâî.ÏîòåîðåìåÁîííåáëàñòà,Êàðëèíà,Øåïëè[51]ñóùå- ñòâóþò s � 0 ; 1 +  + s =1 òàêèå,÷òî inf v 2 co V 1 r P s =1 s ( p s ;v ) � 0 : Ïîëàãàåì p = 1 p 1 +  + s p s ; = 1  1 +  + s  s : Ñ÷èòàåì,÷òî p 6 =0 : Ïîëó÷àåì B  B 1 = f z j ( p;z ) 6  g ; ( p;v ) � 0 äëÿâñåõ v 2 co V 1 : Äîêàæåìâòîðîåóòâåð- æäåíèåëåììû.Ïðåäïîëîæèì,÷òîñóùåñòâóåòíîìåð  0 2 ( n � m +1) äëÿ êîòîðîãî 0 = 2 Intco f b j ;j 2  0 ;p g : Òîãäà,âñèëóëåììû 2 : 10 min v 2 V max f max j 2  0  ( v; 1 ;b j ) ; ( p;v ) g =0 : 77 Ñëåäîâàòåëüíî,ñóùåñòâóåò v 0 2 V äëÿêîòîðîãî max j 2  0  ( v 0 ; 1 ;b j )=0 ; ( p;v 0 ) 6 0 : Ïîýòîìó  ( v 0 ; 1 ;b j )=0 äëÿâñåõ j 2  0 : Îòñþäàäëÿ ëþáîãî J 2 ( m ) áóäåòâûïîëíÿòüñÿ min j 2 J  ( v 0 ; 1 ;b j )=0 : Ñëåäîâàòåëüíî, max J 2 ( m ) min j  ( v 0 ; 1 ;b j )=0 : Ïîëó÷èëè,÷òî v 0 2 V 1 èïîýòîìó ( p;v 0 ) � 0 : Ïî- ëó÷èëèïðîòèâîðå÷èå.Ëåììàäîêàçàíà. Îïðåäåëèì÷èñëî T 0 =min f t � t 0 j min v (  ) min h 2 D max  2 ( m ) min j 2  G ( t;v (  ) ;h j ) � 1 g : Âñèëóëåììû 2 : 12 T 0 + 1 : Ïðåäïîëîæåíèå2.8 Ñóùåñòâóþòìîìåíòû  i � T 0 òàêèå,÷òî 1 : i (  i ) 2 D " (  i (  0 i )) äëÿâñåõ i ; 2 : inf v (  ) max  2 ( m ) min j 2  G (  j ;v (  ) ; j (  j )) � 1 : Çàìå÷àíèå2.2 ( a ) ñóùåñòâîâàíèå  i âïóíêòå1ïðåäïîëîæåíèÿ 2 : 8 ãàðàí- òèðîâàíîïðåäïîëîæåíèåìîðåêóððåíòíîñòèôóíêöèé  i ( t ); ( b ) åñëèâïðåäïîëîæåíèè 2 : 8 âñå  i = ; òîïóíêò2äàííîãîïðåäïîëîæå- íèÿâûïîëíåíàâòîìàòè÷åñêèâñèëóëåììû 2 : 9 . Òåîðåìà2.5 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 6 , 2 : 7 , 2 : 8 , r =1 : Òîãäàâ èãðå �( n;B ) ïðîèñõîäèò m -êðàòíàÿïîèìêà. Äîêàçàòåëüñòâî.Ïóñòü  j ìîìåíòû,óäîâëåòâîðÿþùèåïðåäïîëîæå- íèþ 2 : 8 , v ( s ) ;s 2 [ t 0 ;T 1 ] ïðîèçâîëüíîåäîïóñòèìîåóïðàâëåíèåóáåãàþùåãî E; ãäå T 1 =max i  i : Ðàññìîòðèìôóíêöèþ H ( t )=1 � max  2 ( m ) min j 2  t Z t 0 j ' l � 1 (  j ;s ) j  ( v ( s ) ;r (  j ;s ) ; j (  j )) ds: Îáîçíà÷èì÷åðåç  0 � t 0 ïåðâûéêîðåíüäàííîéôóíêöèè.Îòìåòèì,÷òî 78 ìîìåíò  0 ñóùåñòâóåòâñèëóïðåäïîëîæåíèÿ 2 : 8 .Êðîìåòîãî,ñóùåñòâóåòìíî- æåñòâî  0 2 ( m ) òàêîå,÷òî  0 6  j äëÿâñåõ j 2  0 è 1 � min j 2   0 Z t 0 j ' l � 1 (  j ;s ) j  ( v ( s ) ;r (  j ;s ) ; j (  j )) ds 6 0 : äëÿâñåõ j 2  0 : Ïîýòîìóñóùåñòâóþòìîìåíòû t j 6  0 ;j 2  0 äëÿêîòîðûõ 1 � t j Z t 0 j ' l � 1 (  j ;s ) j  ( v ( s ) ;r (  j ;s ) ; j (  j )) ds =0 : (2.21) Äëÿ j= 2  0 òàêæåîáîçíà÷èì÷åðåç t j ìîìåíòûâðåìåíèäëÿêîòîðûõâûïîë- íåíîóñëîâèå (2 : 21) ; åñëèòàêèåìîìåíòûñóùåñòâóþò.ÂñèëóëåììûÔèëèï- ïîâà[105]äëÿêàæäîãî i ñóùåñòâóþòèçìåðèìûåôóíêöèè u i ( s ) ;s 2 [ t 0 ;T 1 ] , ÿâëÿþùèåñÿïðèêàæäîìôèêñèðîâàííîì s ðåøåíèåìóðàâíåíèÿ  ( v ( s ) ;r (  i ;s ) ; i (  i ))  i (  i )= u i � v ( s ) : Çàäàåìóïðàâëåíèÿïðåñëåäîâàòåëåé P i ; ïîëàãàÿ u i ( t )= v ( t ) �  ( v ( t ) ;r (  i ;t ) ; i (  i ))   i (  i ) ;t 2 [ t 0 ; min f t i ;T 1 g ] ; u i ( t )= v ( t ) ;t 2 (min f t i ;T 1 g ;T 1 ] : Òîãäà z i (  i )=  i (  i )+  i Z t 0 ' (  i ;s )( u i ( s ) � v ( s )) ds = =  i (  i ) �  i Z t 0 j ' l � 1 (  i ;s ) j  ( v ( s ) ;r (  i ;s ) ; i (  i ))   i (  i ) ds = 79 =  i (  i )(1 � t i Z t 0 j ' l � 1 (  i ;s ) j  ( v ( s ) ;r (  i ;s ) ; i (  i )) ds: Èç (2 : 21) ñëåäóåò,÷òî z j (  j )=0 äëÿâñåõ j 2  0 : Òåîðåìàäîêàçàíà. Òåîðåìà2.6 Ïóñòü m =1 èâûïîëíåíûïðåäïîëîæåíèÿ 2 : 6 , 2 : 7 , 2 : 8 .Òîãäàâ èãðå �( n;B ) ïðîèñõîäèòïîèìêà. Äîêàçàòåëüñòâî.Èçïðåäïîëîæåíèÿ 2 : 7 èëåììû 2 : 13 ñëåäóåò,÷òîñó- ùåñòâóþò p 2 R k ; 2 R 1 òàêèå,÷òî 0 2 Intco f  i (  0 i ) ;i 2 I;p g è B  B 1 = f z j ( p;z ) 6  g : Èçòåîðåìû 2 : 5 ñëåäóåò,÷òîâèãðå �( n;B 1 ) ïðîèñõîäèòïîèìêà.Ïîýòîìóïîèìêà ïðîèñõîäèòèâèãðå �( n;B ) : Òåîðåìàäîêàçàíà. Ïðåäïîëîæåíèå2.9 Ñóùåñòâóþò p 2 R k ; 2 R 1 ; ìîìåíòû  0 i 2 [ t 0 ; 1 ) òàêèå,÷òî 1 :B  B 1 = f z j ( p;z ) 6  g ; 2 : Äëÿâñåõ  2 ( n � m +1) âûïîëíåíîâêëþ÷åíèå 0 2 Intco f  j (  0 j ) ;j 2  ;p g : Çàìå÷àíèå2.3 Åñëèâûïîëíåíûïðåäïîëîæåíèå 2 : 7 èóñëîâèÿëåììû 2 : 14 , òîïðåäïîëîæåíèå 2 : 9 âûïîëíåíî. Òåîðåìà2.7 Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ 2 : 6 , 2 : 8 , 2 : 9 .Òîãäàâèãðå �( n;B ) ïðîèñõîäèò m -êðàòíàÿïîèìêà. Äîêàçàòåëüñòâî.Èçóñëîâèÿòåîðåìûèòåîðåìû 2 : 5 ñëåäóåò,÷òî m - êðàòíàÿïîèìêàïðîèñõîäèòâèãðå �( n;B 1 ) : Ñëåäîâàòåëüíî,ïîèìêàïðîèçîéäåò èâèãðå �( n;B ) : 80 Ïðèìåð2.3 Ïóñòü r =1 ; ñèñòåìà (2.18) ; (2.19) èìååòâèä _ z i = u i � v;z i (0)= z 0 i : Óòâåðæäåíèå2.5 Ïóñòü V = D 1 (0) ; è 0 2 Intco f z 0 j ;j 2  ;p 1 g äëÿâñåõ  2 ( n � m +1) : Òîãäàâèãðå �( n;B ) ïðîèñõîäèò m -êðàòíàÿïîèìêà. Ïðèìåð2.4 Ïóñòüâñèñòåìå (2.18) ; (2.19) l =1 ;t 0 =0 ; ôóíêöèÿ a 1 ( t ) èìååò âèä a 1 ( t )= 8 � � : 0 ; åñëè t 2 [0 ; 2  ] ; sin t; åñëè t� 2  Òîãäàôóíêöèÿ ' 0 ( t ) èìååòâèä ' 0 ( t )= 8 � � : 1 ; åñëè t 2 [0 ; 2  ] ; e 1 � cos t ; åñëè t� 2 : Ôóíêöèÿ ' 0 ( t ) ÿâëÿåòñÿðåêóððåíòíîé,íîíåÿâëÿåòñÿïî÷òè-ïåðèîäè÷åñêîé ([33]).Ïðåäïîëîæåíèå 2 : 6 âûïîëíåíî. Óòâåðæäåíèå2.6 Ïóñòü V = D 1 (0) ;m =1 ;n � k; 0 2 Intco f z 0 i ;i 2 I;p 1 ;:::;p r g : Òîãäàâèãðå �( n;B ) ïðîèñõîäèòïîèìêà. Ïðèìåð2.5 Ïóñòüñèñòåìà (2.18) ; (2.19) èìååòâèä  z i + 2 3 t _ z i + 1 9 t 4 = 3 z i = u i � v; 81 ïðè÷åì t 0 =8  2 : Òîãäà ' 0 ( t;s )=cos( 3 p t � 3 p s ) ;' 1 ( t;s )=3 s 2 = 3 sin( 3 p t � 3 p s ) ;  i ( t )= z 0 i cos( 3 p t )+12  2 z 1 i sin( 3 p t ) : Ðåêóððåíòíîñòüôóíêöèé  i ( t ) ñëåäóåòèçðåçóëüòàòîâðàáîòû([33]). Óòâåðæäåíèå2.7 Ïóñòü V = D 1 (0) ;n � m + k � 1 èâûïîëíåíûïðåäïîëî- æåíèÿ 2 : 8 ; 2 : 9 .Òîãäàâèãðå �( n;B ) ïðîèñõîäèò m -êðàòíàÿïîèìêà. Âçÿââêà÷åñòâå  0 i = t 0 =8  2 ; ïîëó÷àåì,÷òîñïðàâåäëèâî Óòâåðæäåíèå2.8 Ïóñòü V = D 1 (0) ;m =1 ;n � k; è 0 2 Intco f z 0 i ;i 2 I;p 1 ;:::;p r g : Òîãäàâèãðå �( n;B ) ïðîèñõîäèòïîèìêà. 82 Çàêëþ÷åíèå Îñíîâíûåðåçóëüòàòû,ïîëó÷åííûåâäèññåðòàöèè: 1 : Äîñòàòî÷íûåóñëîâèÿïîèìêèãðóïïîéïðåñëåäîâàòåëåéîäíîãîóáåãàþ- ùåãîâëèíåéíûõíåñòàöèîíàðíûõäèôôåðåíöèàëüíûõèãðàõâïðåäïîëîæåíèè, ÷òîôóíäàìåíòàëüíàÿìàòðèöàîäíîðîäíîéñèñòåìûÿâëÿåòñÿðåêóððåíòíîéïî Çóáîâó(òåîðåìà 1 : 1 ); 2 : Äîñòàòî÷íûåóñëîâèÿïîèìêèõîòÿáûîäíîãîóáåãàþùåãîäëÿëèíåéíîé íåñòàöèîíàðíîéçàäà÷èïðåñëåäîâàíèÿãðóïïîéïðåñëåäîâàòåëåéãðóïïûóáåãà- þùèõ,ïðèóñëîâèè,÷òîôóíäàìåíòàëüíàÿìàòðèöàîäíîðîäíîéñèñòåìûÿâëÿ- åòñÿðåêóððåíòíîéïîÇóáîâóôóíêöèåé,àâñåóáåãàþùèåèñïîëüçóþòîäíîèòî æåóïðàâëåíèå(òåîðåìà 1 : 2 ); 3 : Äîñòàòî÷íûåóñëîâèÿïîèìêèçàäàííîãî÷èñëàóáåãàþùèõäëÿëèíåéíîé íåñòàöèîíàðíîéçàäà÷èïðåñëåäîâàíèÿãðóïïîéïðåñëåäîâàòåëåéãðóïïûóáåãà- þùèõ,ïðèóñëîâèè,÷òîôóíäàìåíòàëüíàÿìàòðèöàîäíîðîäíîéñèñòåìûÿâëÿåò- ñÿðåêóððåíòíîéïîÇóáîâóôóíêöèåéèêàæäûéïðåñëåäîâàòåëüìîæåòïîéìàòü íåáîëååîäíîãîóáåãàþùåãî(òåîðåìà 1 : 3 ); 4 : Äîñòàòî÷íûåóñëîâèÿðàçðåøèìîñòèçàäà÷èïðåñëåäîâàíèÿâîáîáùåííîì íåñòàöèîíàðíîìïðèìåðåÀ.Ñ.Ïîíòðÿãèíàñîìíîãèìèó÷àñòíèêàìèâïðåäïî- ëîæåíèèðåêóððåíòíîñòèïîÇóáîâóíåêîòîðûõôóíêöèéâòåðìèíàõíà÷àëüíûõ ïîçèöèéèïàðàìåòðîâèãðû(òåîðåìû 2 : 1 ; 2 : 2 ; 2 : 4 ). 5 : Äîñòàòî÷íûåóñëîâèÿìíîãîêðàòíîéïîèìêèâïðèìåðåÀ.Ñ.Ïîíòðÿãèíà âïðåäïîëîæåíèèðåêóððåíòíîñòèïîÇóáîâóíåêîòîðûõôóíêöèéâòåðìèíàõ íà÷àëüíûõïîçèöèéèïàðàìåòðîâèãðû(òåîðåìû 2 : 5 ; 2 : 6 ; 2 : 7 ). 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