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


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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
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èãðûâñòàíäàðòíîéèãðåñáëèæåíèÿóêëîíåíèÿäëÿâñÿêîéíà÷àëüíîéïîçèöèè
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ñòðàòåãèÿïåðâîãîèãðîêàñîþçíèêà,êîòîðàÿîáåñïå÷èâàåòâñòðå÷óäâèæåíèÿ
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áîñóùåñòâóåòïîçèöèîííàÿñòðàòåãèÿâòîðîãîèãðîêàñîþçíèêà,êîòîðàÿîáåñ-
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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
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íà÷àëüíûõïîçèöèéèïàðàìåòðîâèãðûáûëèïîëó÷åíûäîñòàòî÷íûåóñëîâèÿ
óêëîíåíèÿõîòÿáûîäíîãîóáåãàþùåãîîòãðóïïûïðåñëåäîâàòåëåéèççàäàí-
íûõíà÷àëüíûõïîçèöèéèèçëþáûõíà÷àëüíûõïîçèöèé(âïîñëåäíåìñëó÷àå
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n;m
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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
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1
)
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èäîñòàòî÷íûåóñëîâèÿïîèìêè.
Âðàáîòå[106]Á.Ê.Õàéäàðîâðàññìîòðåëçàäà÷óïîçèöèîííîé
l
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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
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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

]

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
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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
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:
Äîêàæåì,÷òî

+
(
h
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0
;

(
h
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0
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Ïðåäïîëîæèì,÷òî

+
(
h
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:
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
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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
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;
max
s
(
p
s
;v
)
g
=0
:
Ïîýòîìóèçëåììû
2
:
10
ñëåäóåò,÷òî
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2
Intco
f
h
j
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2

0
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r
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2
:
7
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+
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h
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(
h
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0
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h
2
D:
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V
ñòðîãîâûïóêëûéêîìïàêòñãëàäêîéãðàíèöåé,òî
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(
v;

1
;h
)
íåïðåðûâíûïî
(
v;h
)
:
ÎñòàëîñüïðèìåíèòüòåîðåìóÂåéåðøòðàññà.Ëåììàäîêàçàíà.
Ëåììà2.12
Ïóñòüâûïîëíåíûïðåäïîëîæåíèÿ
2
:
6
,
2
:
7
,
r
=1
:
Òîãäàñóùå-
ñòâóåòìîìåíò
T�t
0
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v
(

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óáåãàþùåãî
E;
ëþáîãîíàáîðà
h
2
D
íàéäåòñÿìíîæåñòâî

2
(
m
)
;
÷òî
min
j
2

G
(
T;v
(

)
;h
j
)

1
:
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h
2
D:
Òîãäà,âñèëóëåììû
2
:
11
,
�
0
:
Òàê
êàêóïðàâëåíèå
v
(
t
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E
äîïóñòèìî,òîäëÿâñåõ
t

t
0
(
p
1
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(
t
))
6

(
t
)=

1

(
p
1
;
(
t
))
:
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T
+
(
t
)=
f

:

2
[
t
0
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]
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l

1
(
t;
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0
g
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(
t
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2
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t
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l

1
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t
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(
p
1
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g
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2
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t
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(
p
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g
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t
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p
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;
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2
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g
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t
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l

1
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t;s
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l

1
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2
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1
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t
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:
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max

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(
t;v
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m
n
h
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t
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t
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äëÿâñåõ
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Ëåììà2.13
Ïóñòü
b
1
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n
2
R
k
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k;
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(2.20)
è
b
1
;:::;b
k
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p
2
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k
;
2
R
1
;
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1
=
f
z
j
(
p;z
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g
è
0
2
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f
b
1
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n
;p
g
:
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i

0
;
j

0
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0=

1
b
1
+

+

n
b
n
+

1
p
1
+

+

r
p
r
:
76
Ïóñòü
x
2
R
k
:
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1
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k
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=

1
x
1
+

+

k
x
k
:
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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
)
:
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p
=

1
p
1
+

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r
p
r
:
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d�
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i
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ïîëíÿëèñüíåðàâåíñòâà

i
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i

0
:
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1
b
1
+

+

0
n
b
n
+
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ïðè÷åì

0
i

0
:
Ñëåäîâàòåëüíî[57],
0
2
Intco
f
b
1
;:::;b
n
;p
g
:
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æåñòâî
B
1
=
f
z
j
(
p;z
)
6

g
;
ãäå

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1

1
+
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r
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B

B
1
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Ëåììà
äîêàçàíà.
Ëåììà2.14
Ïóñòü
V
=
D
1
(0)
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1
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n
2
R
k
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m
+
k

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:
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2
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b
j
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2

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1
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r
g
äëÿëþáîãî

2
(
n

m
+1);
2
:
min
v
2
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1
max
s
(
p
s
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0
;
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=
f
v
2
V
j
max
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2
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m
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2
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(
v;
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j
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g
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p
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n

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1
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v
2
co
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1
r
P
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s
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p
s
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p
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1
p
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p
s
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1
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+

s

s
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Ñ÷èòàåì,÷òî
p
6
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1
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z
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(
p;z
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p;v
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v
2
co
V
1
:
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æäåíèåëåììû.Ïðåäïîëîæèì,÷òîñóùåñòâóåòíîìåð

0
2
(
n

m
+1)
äëÿ
êîòîðîãî
0
=
2
Intco
f
b
j
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2

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g
:
Òîãäà,âñèëóëåììû
2
:
10
min
v
2
V
max
f
max
j
2

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(
v;
1
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j
)
;
(
p;v
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g
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77
Ñëåäîâàòåëüíî,ñóùåñòâóåò
v
0
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V
äëÿêîòîðîãî
max
j
2

0

(
v
0
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1
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j
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p;v
0
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v
0
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j
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j
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v
0
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1
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m
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j

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v
0
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p;v
0
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0
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f
t

t
0
j
min
v
(

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min
h
2
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max

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(
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min
j
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t;v
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j
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g
:
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T
0

+
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Ñóùåñòâóþòìîìåíòû

i

T
0
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1
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i
(

i
)
2
D
"
(

i
(

0
i
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;
2
:
inf
v
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max

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m
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min
j
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j
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(

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(
a
)
ñóùåñòâîâàíèå

i
âïóíêòå1ïðåäïîëîæåíèÿ
2
:
8
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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
).
Ïîëó÷åííûåðåçóëüòàòûìîãóòáûòüèñïîëüçîâàíûâäàëüíåéøèõèññëåäîâà-
íèÿõ,íàïðèìåð,äëÿïîëó÷åíèÿäîñòàòî÷íûõóñëîâèé¾ìÿãêîé¿ïîèìêèãðóïïîé
ïðåñëåäîâàòåëåéîäíîãîèëèíåñêîëüêèõóáåãàþùèõâïðèìåðåÀ.Ñ.Ïîíòðÿãè-
íà.
83
Ñïèñîêîáîçíà÷åíèé
Âðàáîòåèñïîëüçóþòñÿñëåäóþùèåîáîçíà÷åíèÿ:
R
k
åâêëèäîâîïðîñòðàíñòâîðàçìåðîì
k
,
jj
x
jj
íîðìàâåêòîðà
x
2
R
k
,
(
x;y
)
ñêàëÿðíîåïðîèçâåäåíèåâåêòîðîâ
x;y
2
R
k
,
D
r
(
a
)=
f
z
:
k
z

a
k
6
r
g
çàìêíóòûéøàððàäèóñà
r
ñöåíòðîìâòî÷êå
a
,
Int
X
âíóòðåííîñòüìíîæåñòâà
X
,
ri
X
îòíîñèòåëüíàÿâíóòðåííîñòüìíîæåñòâà
X
,
co
X
âûïóêëàÿîáîëî÷êàìíîæåñòâà
X
,
@X
ãðàíèöàìíîæåñòâà
X
,

(
X
)
ìåðàËåáåãàìíîæåñòâà
X
.
84
Ëèòåðàòóðà
1.Àçàìîâ,À.Îñóùåñòâîâàíèèñòðàòåãèèñêóñî÷íîïîñòîÿííûìèðåàëèçà-
öèÿìè/À.Àçàìîâ//Ìàòåìàòè÷åñêèåçàìåòêè.1987.Ò.41,5.
C.718723.
2.Àçàìîâ,À.Îáàëüòåðíàòèâåäëÿèãðïðåñëåäîâàíèÿíàáåñêîíå÷íîìèíòåð-
âàëåâðåìåíè/À.Àçàìîâ//Ïðèêëàäíàÿìàòåìàòèêàèìåõàíèêà.1986.
Ò.50,âûï.4.C.561570.
3.Àéçåêñ,Ð.Äèôôåðåíöèàëüíûåèãðû/Ð.Àéçåêñ.Ì.:Ìèð,1967.480ñ.
4.Áàííèêîâ,À.Ñ.Íåñòàöèîíàðíàÿçàäà÷àãðóïïîâîãîïðåñëåäîâà-
íèÿ/À.Ñ.Áàííèêîâ//Èçâåñòèÿâóçîâ.Ìàòåìàòèêà.2009.5.
Ñ.312.
5.Áàííèêîâ,À.Ñ.Îáîäíîéçàäà÷åïðîñòîãîïðåñëåäîâàíèÿ/À.Ñ.Áàííè-
êîâ//ÂåñòíèêÓäìóðòñêîãîóíèâåðñèòåòà.Ìàòåìàòèêà.Ìåõàíèêà.Êîì-
ïüþòåðíûåíàóêè.2009.Âûï.3.Ñ.311.
6.Áàííèêîâ,À.Ñ.Êíåñòàöèîíàðíîéçàäà÷åãðóïïîâîãîïðåñëåäîâà-
íèÿ/À.Ñ.Áàííèêîâ,Í.Í.Ïåòðîâ//ÒðóäûÈíñòèòóòàìàòåìàòèêèèìå-
õàíèêèÓðÎÐÀÍ.2010.Ò.16,1.Ñ.4051.
7.Áåðæ,Ê.Îáùàÿòåîðèÿèãðíåñêîëüêèõëèö/Ê.Áåðæ.Ì.:ÈÔÌË,
1961.126ñ.
8.Áëàãîäàòñêèõ,À.È.Ãðóïïîâîåïðåñëåäîâàíèåâíåñòàöèîíàðíîìïðèìå-
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