Directory UMM :Journals:Journal_of_mathematics:VMJ:

‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
ˆî«ì{ᥭâï¡àì, 2002, ’®¬ 4, ‚ë¯ã᪠3

“„Š 338.24.01

Ž„ŽŽ„“Š’Ž‚€Ÿ „ˆ€Œˆ—…‘Š€Ÿ ŒŽ„…‹œ ‡€Œ…™…ˆŸ
Žˆ‡‚Ž„‘’‚…›• ŒŽ™Ž‘’…‰
‹. €. ¥ª« àï­, ‘. ‚. ®à¨á®¢ 
Ž¯¨á ­  ®¤­®¯à®¤ãªâ®¢ ï ¤¨­ ¬¨ç¥áª ï ¬®¤¥«ì íª®­®¬¨ª¨, ¯®§¢®«ïîé ï ¨áá«¥¤®¢ âì å à ªâ¥à
®¯â¨¬ «ì­ëå áப®¢ ä㭪樮­¨à®¢ ­¨ï ¯à®¨§¢®¤á⢥­­ëå ¬®é­®á⥩.

‘ä®à¬ã«¨à®¢ ­ ¯à¨­-

樯 ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨, ­  ®á­®¢ ­¨¨ ª®â®à®£® ¢ë¡¨à ¥âáï ®¯â¨¬ «ì­ ï ¯®«¨â¨ª 
¢ë¢®¤  áâ àëå ¨ ¢¢®¤  ­®¢ëå, ¡®«¥¥ ᮢ¥à襭­ëå ä®­¤®¢. Ž¯¨á ­ë ¢á¥ ¢®§¬®¦­ë¥ ¢ à¨ ­âë
à §¢¨â¨ï á¨á⥬ë.

‚¢¥¤¥­¨¥

à®¨§¢®¤á⢥­­ ï áâàãªâãà  «î¡®© íª®­®¬¨ç¥áª®© á¨áâ¥¬ë ¯®¤¢¥à¦¥­  ¨§¬¥­¥­¨ï¬ ¢® ¢à¥¬¥­¨, ®¡ãá«®¢«¥­­ë¬ à §«¨ç­ë¬¨ ¯à¨ç¨­ ¬¨, ¢ ⮬ ç¨á«¥ à §¢¨â¨¥¬
­ ãç­®-â¥å­¨ç¥áª®£® ¯à®£à¥áá  ¨ à®á⮬ ä®­¤®¢®®à㦥­­®áâ¨. ‚ á¢ï§¨ á í⨬ ¢áâ îâ á«®¦­ë¥ § ¤ ç¨ ¬®¤¥«¨à®¢ ­¨ï ¨­â¥­á¨¢­®£® à §¢¨â¨ï íª®­®¬¨ç¥áª¨å ¨ ­ ãç­®â¥å­¨ç¥áª¨å á¨á⥬ ¨ ã¯à ¢«¥­¨ï í⨬ à §¢¨â¨¥¬ [1{4]. ®«ì讥 §­ ç¥­¨¥ ¢ í⮬ á«ãç ¥ ¯à¨®¡à¥â ¥â ã«ãç襭¨¥ ¨á¯®«ì§®¢ ­¨ï ®á­®¢­ëå ä®­¤®¢ ¨ ¯à®¨§¢®¤á⢥­­ëå ¬®é­®á⥩. ‚¥«¨ç¨­  ¯à®¨§¢®¤á⢥­­®© ¬®é­®á⨠ä®à¬¨àã¥âáï ¯®¤ ¢«¨ï­¨¥¬ ¬­®¦¥á⢠

¯à®¨§¢®¤á⢥­­®-â¥å­¨ç¥áª¨å ¨ ®à£ ­¨§ æ¨®­­®-íª®­®¬¨ç¥áª¨å ä ªâ®à®¢.  ¨¡®«ì襥 ¦¥ ¢«¨ï­¨¥ ­  ¢¥«¨ç¨­ã ¯à®¨§¢®¤á⢥­­®© ¬®é­®á⨠®ª §ë¢ ¥â ¨á¯®«ì§ã¥¬®¥ ¢
¯à®¨§¢®¤á⢥ ®¡®à㤮¢ ­¨¥, ¥£® ª®«¨ç¥á⢥­­ë© ¨ ª ç¥á⢥­­ë© á®áâ ¢. ‡ ¬¥­  ãáâ à¥¢è¥£® ®¡®à㤮¢ ­¨ï ­  ­®¢®¥, ¡®«¥¥ ᮢ¥à襭­®¥ (®¡« ¤ î饥 ¡®«ì襩 ¯à®¨§¢®¤¨â¥«ì­®áâìî ¨/¨«¨ âà㤮¥¬ª®áâìî) ¯à¨¢®¤¨â ª ­ à é¨¢ ­¨î ¯à®¨§¢®¤á⢥­­®© ¬®é­®áâ¨. ‚ í⮬ á«ãç ¥ ®¤­®© ¨§ ¢ ¦­¥©è¨å ¯à¨ª« ¤­ëå § ¤ ç ï¥âáï ¬®¤¥«¨à®¢ ­¨¥
¨ ®¯â¨¬¨§ æ¨ï áப®¢ ä㭪樮­¨à®¢ ­¨ï ¯à®¨§¢®¤á⢥­­ëå ¬®é­®á⥩ íª®­®¬¨ç¥áª¨å á¨á⥬.
’ ª¨¥ ¬®¤¥«¨ ¨¬¥î⠯ਫ®¦¥­¨ï ­  à §«¨ç­ëå ã஢­ïå ã¯à ¢«¥­¨ï íª®­®¬¨ª®©
(íª®­®¬¨ª  áâà ­ë, ॣ¨®­ , ®âà á«¨, ®â¤¥«ì­®£® ¯à¥¤¯à¨ïâ¨ï ¨ â. ¯.).   ¢ë᮪¨å
ã஢­ïå ã¯à ¢«¥­¨ï â ª¨¥ ¬®¤¥«¨ ®¯â¨¬¨§¨àãîâ ª ª à á¯à¥¤¥«¥­¨¥ à¥áãàᮢ ¬¥¦¤ã
®âà á«ï¬¨, â ª ¨ â¥å­¨ç¥áªãî ¯®«¨â¨ªã.   ¡®«¥¥ ­¨§ª¨å ã஢­ïå (®âà á«ì, ¯à¥¤¯à¨ï⨥) ®­¨ ®¯¨á뢠îâ ¯à®æ¥ááë § ¬¥­ë, ®¡­®¢«¥­¨ï ¨ à á¯à¥¤¥«¥­¨ï ¬ è¨­ ¨ ®¡®à㤮¢ ­¨ï ¢ íª®­®¬¨ç¥áª®© á¨á⥬¥ ¨ ¬®£ãâ ¡ëâì ¨á¯®«ì§®¢ ­ë ¤«ï â ª¨å ª®­ªà¥â­ëå
§ ¤ ç ª ª à áç¥â (¨ ®¯â¨¬¨§ æ¨ï) ¬ â¥à¨ «ì­®-â¥å­¨ç¥áª®£® á­ ¡¦¥­¨ï, ®¯à¥¤¥«¥­¨ï
¯®âॡ­®á⥩ ¢ ¬ è¨­ å ¨ ®¡®à㤮¢ ­¨¨ ¨ ¤à.  ¨¡®«¥¥ ¯¥àᯥªâ¨¢­ë¬ ï¥âáï ¯à¨¬¥­¥­¨¥ ¤ ­­®£® ª« áá  ¬®¤¥«¥© ¤«ï § ¤ ç ¯à®£­®§¨à®¢ ­¨ï ¨ ¤®«£®áà®ç­®£® ¯« ­¨à®¢ ­¨ï ¢ ¬ áèâ ¡ å ­ à®¤­®£® 宧ï©á⢠ ¨«¨ ¢ ä®­¤®¥¬ª¨å ®âà á«ïå ¨ ¯à®¨§¢®¤á⢠å
á ¨­â¥­á¨¢­ë¬ â¥å­¨ç¥áª¨¬ ¯à®£à¥áᮬ.
 ¨¡®«¥¥ ¨§¢¥áâ­ë¬ ¯à¨¬¥à®¬ ¬ ªà®íª®­®¬¨ç¥áª®© ¬®¤¥«¨ á ®¢¥é¥á⢫¥­­ë¬
â¥å­¨ç¥áª¨¬ ¯à®£à¥áᮬ ï¥âáï ¬®¤¥«ì ‘®«®ã [6]. ‘®£« á­® ¬®¤¥«¨ ‘®«®ã â¥å­¨ç¥áª¨© ¯à®£à¥áá ¢®¯«®é¥­ ¢ ®á­®¢­ëå ¯à®¨§¢®¤á⢥­­ëå ä®­¤ å: ä®­¤ë, ᮧ¤ ­­ë¥



c 2002 ¥ª« àï­ ‹. €., ®à¨á®¢  ‘. ‚.

Ž¤­®¯à®¤ãªâ®¢ ï ¤¨­ ¬¨ç¥áª ï ¬®¤¥«ì § ¬¥é¥­¨ï

3{23

­¥¤ ¢­®, ¡®«¥¥ íä䥪⨢­ë ¯® áà ¢­¥­¨î á ¢ë¯ã饭­ë¬¨ ¢ ¡®«¥¥ à ­­¨¥ ¬®¬¥­âë
¢à¥¬¥­¨,   ä®­¤ë, ᮧ¤ ­­ë¥ ¢ ®¤­®¬ £®¤ã, ¨¬¥îâ ®¤¨­ ª®¢ãî íä䥪⨢­®áâì.
‚ ¬®¤¥«¨ ‘®«®ã ®¯¨á뢠¥âáï ⮫쪮 á«ãç © à ¢­®¬¥à­ëå ⥬¯®¢ ¨§­®á  ¨ ®­  ­¥
ãç¨â뢠¥â ¤¥©á⢨⥫쭮 ¤¨­ ¬¨ç¥áª®¥, ­¥à ¢­®¬¥à­®¥ à §¢¨â¨¥ íª®­®¬¨ª¨.
®¢ë¬ íâ ¯®¬ ¢ à §¢¨â¨¨ â ª¨å ¬®¤¥«¥© áâ «¨ ¤¨­ ¬¨ç¥áª¨¥ ¬®¤¥«¨, ¯®§¢®«ïî騥 ã¯à ¢«ïâì ¤¨­ ¬¨ª®© ᢮à ç¨¢ ­¨ï ãáâ à¥¢è¨å ®á­®¢­ëå ä®­¤®¢ [7, 8]. …᫨
à ­ìè¥ ¢à¥¬ï á«ã¦¡ë ä®­¤®¢ ¢ ®á­®¢­®¬ ®¯à¥¤¥«ï«®áì ⮫쪮 «¨èì á â®çª¨ §à¥­¨ï ¨å
䨧¨ç¥áª®£® ¨§­®á , ¨ íâ® ¢à¥¬ï §­ ç¨â¥«ì­® ¯à¥¢ëè «® áப á«ã¦¡ë «î¡®© ¥¤¨­¨æë âà㤮¢ëå à¥áãàᮢ, ⮠⥯¥àì ¢ á¢ï§¨ á à §¢¨â¨¥¬ ­ ãç­®-â¥å­¨ç¥áª®£® ¯à®£à¥áá 
áப á«ã¦¡ë ä®­¤®¢ ¢ ¡®«ì襩 á⥯¥­¨ ®¯à¥¤¥«ï¥âáï ¨å ¬®à «ì­ë¬ ¨§­®á®¬.

®-

¥­¨¥ ­®¢ëå, ¡®«¥¥ ᮢ¥à襭­ëå â¥å­®«®£¨© ¬­®£®ªà â­® ᮪à é ¥â áப á«ã¦¡ë
ä®­¤®¢,   ¢ ®â¤¥«ì­ëå ®âà á«ïå íâ® ¢à¥¬ï á®áâ ¢«ï¥â «¨èì ­¥áª®«ìª® «¥â (­ ¯à¨¬¥à, ¢ ®¡« áâ¨ í«¥ªâà®­­®© ¯à®¬ëè«¥­­®áâ¨, ª®¬¯ìîâ¥à­ëå â¥å­®«®£¨©). ®í⮬ã,
¥á«¨ ¢ ¯¥à¢®¬ á«ãç ¥ ¯à®¨á室¨â ¢á¥£® «¨èì § ¬¥­  áâ à®£® ®¡®à㤮¢ ­¨ï ­  ­®¢®¥, â®
¢® ¢â®à®¬ | ¯®«­ ï ४®­áâàãªæ¨ï ä®­¤®¢, â. ¥. ­ ¨¬¥­¥¥ íä䥪⨢­ë¥ ¢ë¢®¤ïâáï
¨§ ¯à®¨§¢®¤á⢠ (¨ ¢ ¤ «ì­¥©è¥¬ ­¥ ¨á¯®«ì§ãîâáï),   ¢ë᢮¡®¦¤ î騥áï âà㤮¢ë¥
à¥áãàáë ­ ¯à ¢«ïîâáï ­  ¢­®¢ì ᮧ¤ ¢ ¥¬ë¥ ä®­¤ë.
’ ª¨¥ ¬®¤¥«¨ ¢¯¥à¢ë¥ ¡ë«¨ ¢¢¥¤¥­ë ‹. ‚. Š ­â®à®¢¨ç¥¬ ¢ 1959 £. Ž¤­ ª® ¨å
¤ «ì­¥©è¥¥ ¨áá«¥¤®¢ ­¨¥ ­¥ ¯à®¢®¤¨«®áì, ¨ ⮫쪮 ¢ 1973 £.

¯®ï¢«ï¥âáï à ¡®â , ¢

ª®â®à®© ¯à¨¢¥¤¥­  ®¤­®¯à®¤ãªâ®¢ ï ¬ ªà®íª®­®¬¨ç¥áª ï ¤¨­ ¬¨ç¥áª ï ¬®¤¥«ì.
à¨­æ¨¯¨ «ì­® ­®¢ë¬ ¢ ¬®¤¥«¨ Š ­â®à®¢¨ç  ï¥âáï ¢¢¥¤¥­¨¥ ­®¢®© ä㭪樨 |

m(t)

| ¢à¥¬¥­­®© £à ­¨æë ¨á¯®«ì§®¢ ­¨ï ä®­¤®¢:

à ­­¥¥ ¬®¬¥­â 

¢á¥ ä®­¤ë, ᮧ¤ ­­ë¥

m(t), ¢ ¬®¬¥­â t ®ª §ë¢ îâáï ¢ë¢¥¤¥­­ë¬¨ ¨§ ¯à®¨§¢®¤á⢠.

‹. ‚. Š ­â®à®¢¨ç¥¬ ¡ë« ¯à¥¤«®¦¥­ ¯à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨.
‘®£« á­® í⮬㠯ਭ樯ã ᯮᮡ ¢ë¢®¤  ãáâ à¥¢è¨å ¨ áâàãªâãà  ¢­®¢ì ᮧ¤ ­­ëå ®á­®¢­ëå ä®­¤®¢ ïîâáï ®¯â¨¬ «ì­ë¬¨, ¥á«¨ ®­¨ ®¡¥á¯¥ç¨¢ îâ ¬ ªá¨¬ «ì­ë© ⥬¯
à®áâ  ­ æ¨®­ «ì­®£® ¤®å®¤  ¢ ⥪ã騩 ¬®¬¥­â ¢à¥¬¥­¨.
¥§ ¢¨á¨¬® ®â à ¡®âë [9] ‚. Œ. ƒ«ã誮¢ ¯à¥¤«®¦¨« ¢ 1977 £.

¤¢ãå¯à®¤ãªâ®-

¢ãî ¬ ªà®íª®­®¬¨ç¥áªãî ¬®¤¥«ì [8], ¤®¯®«­ïîéãî ¬®¤¥«ì Š ­â®à®¢¨ç . â  ¬®¤¥«ì
®¯¨á뢠¥â ¤¢¥ £àã¯¯ë ¯à®¨§¢®¤á⢠: € | ¯à®¨§¢®¤á⢮ á।á⢠¯à®¨§¢®¤á⢠ ¨  |
¯à®¨§¢®¤á⢮ ¯à¥¤¬¥â®¢ ¯®âॡ«¥­¨ï.
‚ ¬®¤¥«ïå Š ­â®à®¢¨ç  ¨ ƒ«ã誮¢ , ¢ ®â«¨ç¨¥ ®â ¬®¤¥«¨ ‘®«®ã, ­ àï¤ã á ®¢¥é¥á⢫¥­­ë¬ ã஢­¥¬ ®¯¨á ­¨ï ­ ãç­®-â¥å­¨ç¥áª®£® ¯à®£à¥áá , ᮤ¥à¦¨âáï ¢®§¬®¦­®áâì ã¯à ¢«¥­¨ï ¨ ®¯â¨¬¨§ æ¨¨ à §¢¨â¨ï íª®­®¬¨ç¥áª¨å á¨á⥬ §  áç¥â ¨§¬¥­¥­¨ï
¯®«¨â¨ª¨ ®¡­®¢«¥­¨ï ®á­®¢­ëå ä®­¤®¢.
 ãç­®-â¥å­¨ç¥áª¨© ¯à®£à¥áá ¢¥¤¥â ª ¢ë᮪¨¬ ⥬¯ ¬ ®¡­®¢«¥­¨ï â¥å­®«®£¨©
¨ ᮯãâáâ¢ãîé¨å ¨¬ à áè¨à¥­¨î, ४®­áâàãªæ¨¨ ¨ â¥å­¨ç¥áª®¬ã ¯¥à¥¢®®à㦥­¨î
¤¥©áâ¢ãîé¨å ¯à¥¤¯à¨ï⨩. ‚ á¢ï§¨ á í⨬ ¢á⠥⠢®¯à®á ­¥ ⮫쪮 ® ¯®¨áª¥ ®¯â¨¬ «ì­ëå áப®¢ á«ã¦¡ë ä®­¤®¢, ­® ¨ ०¨¬®¢ ¢¢®¤  ­®¢ëå ä®­¤®¢. ¥è¥­¨¥ ¤ ­­®© ¯à®¡«¥¬ë ¯®§¢®«ï¥â ®¡¥á¯¥ç¨âì ­ ãç­® ®¡®á­®¢ ­­ë© ¢ë¡®à ¯®«¨â¨ª¨ à §¢¨â¨ï
¯à®¨§¢®¤á⢠ ¢ ãá«®¢¨ïå ­ ãç­®-â¥å­¨ç¥áª®£® ¯à®£à¥áá  ¨ à®áâ  ä®­¤®¢®®à㦥­­®áâ¨
¯à®¨§¢®¤á⢠.
„ ­­ ï à ¡®â  ¯®á¢ï饭  ¯à®¡«¥¬¥ ¯®¨áª  ®¯â¨¬ «ì­®© ¯®«¨â¨ª¨ ¢ë¢®¤  ¨§ ¯à®¨§¢®¤á⢠ ãáâ à¥¢è¨å ä®­¤®¢ ¨ ¢¢®¤  ­®¢ëå (¡®«¥¥ ᮢ¥à襭­ëå) ¢ à ¬ª å ¯®áâ஥­­®© ®¤­®¯à®¤ãªâ®¢®© ¬®¤¥«¨. „«ï ¤®á⨦¥­¨ï í⮩ 楫¨ ¢ ª ç¥á⢥ ªà¨â¥à¨ï ®¯â¨¬ «ì­®á⨠¨á¯®«ì§ã¥âáï ý¯à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨þ, ¯à¥¤«®¦¥­­ë©
‹. ‚. Š ­â®à®¢¨ç¥¬ [10].

3{24

‹. €. ¥ª« àï­, ‘. ‚. ®à¨á®¢ 

1. Ž¯¨á ­¨¥ å à ªâ¥à¨á⨪ ¬®¤¥«¨

ã¤¥¬ à áᬠâਢ âì á¨á⥬ã, ¯à®¨§¢®¤ïéãî ®¤¨­ ¯à®¤ãªâ ¨ ®¡« ¤ îéãî ¢ë᮪¨¬¨ ⥬¯ ¬¨ ®¡­®¢«¥­¨ï â¥å­®«®£¨©, á ¤¢ã¬ï £« ¢­ë¬¨ ¯à®¨§¢®¤á⢥­­ë¬¨ ä ªâ®à ¬¨ | ¯à®¨§¢®¤á⢥­­ë¥ ä®­¤ë (®¢¥é¥á⢫¥­­ë© ª ¯¨â «), ¤¨ää¥à¥­æ¨à®¢ ­­ë¥
¯® ¬®¬¥­â ¬ ¨å ᮧ¤ ­¨ï ¨ ¨§¬¥àï¥¬ë¥ ¢ ¥¤¨­¨æ å ¯à®¤ãªâ ,   â ª¦¥ âà㤮¢ë¥ à¥áãàáë, ¨§¬¥àï¥¬ë¥ ª®«¨ç¥á⢮¬ âà㤮¢ëå ¥¤¨­¨æ.

t ®¯à¥¤¥«ï¥âáï ¯à®U (x; y;  ), ¢ëà ¦ î饩 ª®«¨ç¥á⢮ ç¨á⮣® ¯à®¤ãªâ , ᮧ¤ -

ãáâì íä䥪⨢­®áâì ¯à®¨§¢®¤á⢠ ¢ «î¡®© ¬®¬¥­â ¢à¥¬¥­¨
¨§¢®¤á⢥­­®© ä㭪樥©

y (¢ ¥¤¨­¨æ㠢६¥­¨) ¯à¨ ¨á¯®«ì§®¢ ­¨¨ ¯à®¨§¢®¤á⢥­­ëå ä®­¤®¢
®¡ê¥¬  x (ᮧ¤ ­­ëå ¢ ¬®¬¥­â  ( < t)). à¥¤¯®« £ ¥âáï, çâ® äã­ªæ¨ï U ¬®­®â®­­®
¢®§à á⠥⠯®  , çâ® ®âà ¦ ¥â 㢥«¨ç¥­¨¥ íä䥪⨢­®á⨠¡®«¥¥ ­®¢ëå ä®­¤®¢ ¯®¤
¢ ¥¬®£® âà㤮¬

¤¥©á⢨¥¬ â¥å­¨ç¥áª®£® ¯à®£à¥áá . Šà®¬¥ ⮣®, ¯à¥¤¯®« £ ¥âáï, çâ® ¯® ¯¥à¢ë¬ ¤¢ã¬
 à£ã¬¥­â ¬ äã­ªæ¨ï

U

ï¥âáï ®¤­®à®¤­®© (íâ® ®âà ¦ ¥â ®âáãâá⢨¥ íää¥ªâ  ¬ á-

èâ ¡  ¯à®¨§¢®¤á⢠) ¨ ¢®£­ã⮩ (â. ¥. äã­ªæ¨ï
¯à®¨§¢®¤á⢠).
—¥à¥§

U ¡ §¨àã¥âáï ­  ®¯â¨¬ «ì­ëå ᯮᮡ å

(t) ®¡®§­ ç¨¬ ¨­â¥­á¨¢­®áâì ¢¢®¤  ª ¯¨â «®¢«®¦¥­¨©, ¨¤ãé¨å ­  㢥«¨-

祭¨¥ ä®­¤®¢ ¨ § ¬¥­ã ¢ë¡ë¢ îé¨å ¨§ ¯à®¨§¢®¤á⢠ ä®­¤®¢, â. ¥.

®¡ê¥¬ ä®­¤®¢,

t; t + dt], à ¢¥­ (t)dt.

¢¢¥¤¥­­ëå ¢ ¯à®¨§¢®¤á⢮ ¢® ¢à¥¬¥­­®¬ ¨­â¥à¢ «¥ [

(t) ¯à¥¤¯®« £ ¥âáï «¨¡® íª§®£¥­­®© ¯¥à¥¬¥­­®© ¬®¤¥«¨, â. ¥.

”ã­ªæ¨ï

§ ¤ ­­®© ¨§­ ç «ì­®,

«¨¡® ¯à¥¤¯®« £ ¥âáï, çâ® ®­  á®áâ ¢«ï¥â ¯®áâ®ï­­ãî ç áâì ¯à®¨§¢®¤¨¬®£® ¢ ¬®¬¥­â
¢à¥¬¥­¨
ä㭪樨

t ç¨á⮣® ¯à®¤ãªâ  (í­¤®£¥­­ ï ¯¥à¥¬¥­­ ï). ‚ á«ãç ¥ íª§®£¥­­® § ¤ ­­®©
(t) ¡ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® ®­  ï¥âáï  ¡á®«îâ­® ­¥¯à¥à뢭®©, á ¯à®¨§-

¢®¤­®© ­¥¯à¥à뢭®© á¯à ¢ .

Š®«¨ç¥á⢮ âà㤮¢ëå à¥áãàᮢ, § ­ïâëå ­  ᮧ¤ ¢ ¥¬ëå ä®­¤ å, ®¯à¥¤¥«ï¥âáï
¨­â¥­á¨¢­®áâìî ¨å ¢¢®¤ 

'(t),

â. ¥. ¯à¥¤¯®« £ ¥âáï, çâ® âà㤮¢ë¥ à¥áãàáë, ¢¢¥¤¥­-

t; t + dt],

­ë¥ ¢® ¢à¥¬¥­­®¬ ¨­â¥à¢ «¥ [

­¥®¡å®¤¨¬® ­ ©â¨. à¥¤¯®« £ ¥âáï, çâ®
­¥¯à¥à뢭 ï á¯à ¢ .

à ¢­ïîâáï

'(t)dt

¥¤¨­¨æ ¬.

”ã­ªæ¨î

'(t)

'(t) | ­¥®âà¨æ â¥«ì­ ï äã­ªæ¨ï ªãá®ç­®-

‚ ¬®¤¥«¨ ¯à¥¤¯®« £ ¥âáï, çâ® â¥å­¨ç¥áª¨© ¯à®£à¥áá ¨ à®áâ ä®­¤®¢®®à㦥­­®áâ¨

¯à¨¢®¤ïâ ª ã᪮७¨î ⥬¯®¢ ᬥ­ï¥¬®á⨠â¥å­®«®£¨© ¯à®¨§¢®¤á⢠, ¨ ª ª á«¥¤á⢨¥
ª ­¥®¡å®¤¨¬®á⨠§ ¬¥­ë ãáâ à¥¢è¨å ä®­¤®¢.

‚ १ã«ìâ â¥ ­ ¨¬¥­¥¥ íä䥪⨢­ë¥

ä®­¤ë ­¥¯à¥à뢭® ¢ë¢®¤ïâáï ¨§ ¯à®¨§¢®¤á⢠ (¨ ¢ ¤ «ì­¥©è¥¬ ­¥ ¨á¯®«ì§ãîâáï),
  ¢ë᢮¡®¦¤ î騥áï âà㤮¢ë¥ à¥áãàáë ­ ¯à ¢«ïîâáï ­  ¢­®¢ì ᮧ¤ ¢ ¥¬ë¥ ä®­¤ë.
 ¨¡®«¥¥ áâ àë¥ ä®­¤ë (¨¬¥î騥 á ¬ë© à ­­¨© ¯¥à¨®¤ ᮧ¤ ­¨ï á।¨ ä®­¤®¢, § -

t

­ïâëå ¢ ¯à®æ¥áᥠ¯à®¨§¢®¤á⢠ ­  ¬®¬¥­â ) ¡ã¤ã⠢뢮¤¨âìáï ¢ ¯¥à¢ãî ®ç¥à¥¤ì. ®«¨â¨ª  ¢ë¢®¤  ¨§ ¯à®¨§¢®¤á⢠ ãáâ à¥¢è¨å ä®­¤®¢ å à ªâ¥à¨§ã¥âáï ä㭪樥©

m(t),

®¯à¥¤¥«ïî饩 ¬®¬¥­â ᮧ¤ ­¨ï ä®­¤®¢, ¢ë¢®¤¨¬ëå ¨§ ¯à®¨§¢®¤á⢠ ¢ ¬®¬¥­â ¢à¥¬¥­¨

t (m(t) < t).

”ã­ªæ¨î

m(t)

­¥®¡å®¤¨¬® ­ ©â¨.

à¥¤¯®« £ ¥âáï, çâ®

m(t)

|

 ¡á®«îâ­® ­¥¯à¥à뢭 ï äã­ªæ¨ï, á ¯à®¨§¢®¤­®© ­¥¯à¥à뢭®© á¯à ¢ . Šà®¬¥ ⮣®,
¯à¨ ¢ë¢®¤¥ áâ àëå ä®­¤®¢ áãé¥áâ¢ãîâ ®£à ­¨ç¥­¨ï, ª®â®àë¥ ¢ëà ¦ îâáï ¢ ᪮à®áâ¨

M (M  1)
t ä®­¤ë ­¥ ¬®£ãâ ¡ëâì
¨§ á¬ëá«  m(t) á«¥¤ã¥â, çâ® m (t) > 0 ¨

§ ¬¥é¥­¨ï ãáâ à¥¢è¨å ä®­¤®¢ ¢® ¢à¥¬¥­¨, â. ¥. áãé¥áâ¢ã¥â ª®­áâ ­â 
â ª ï, çâ®

m (t ) 6 M
0

(â. ¥. ãáâ à¥¢è¨¥ ª ¬®¬¥­â㠢६¥­¨

¢ë¢¥¤¥­ë ¢á¥ ®¤­®¢à¥¬¥­­®). Šà®¬¥ ⮣®,

m(t) < t.

0

ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® ¢® ¢à¥¬¥­¨ áãé¥áâ¢ã¥â ­¥ª¨© àï¤ à §«¨ç­ëå â¥å­®«®£¨©. ‚ ª ¦¤ë© ¬®¬¥­â ¢à¥¬¥­¨

t ¢á⠥⠢®¯à®á ® ¢ë¡®à¥ ¨ ¢­¥¤à¥­¨¨ ¢ ¯à®¨§¢®¤á⢮

Ž¤­®¯à®¤ãªâ®¢ ï ¤¨­ ¬¨ç¥áª ï ¬®¤¥«ì § ¬¥é¥­¨ï

3{25

⮩ ¨«¨ ¨­®© â¥å­®«®£¨¨. ®«¨â¨ª  ¢¢®¤  ¢ ¯à®¨§¢®¤á⢮ ­®¢ëå ¬®é­®á⥩ å à ªâ¥à¨§ã¥âáï ä㭪樥©

(t), ®¯à¥¤¥«ïî饩 ¬®¬¥­â § ¯ã᪠ ­®¢®© â¥å­®«®£¨¨, ¯®«­®áâìî
t
(t) ï¥âáï ­¥¨§¢¥áâ­®©. à¥¤¯®« £ -

®á¢ ¨¢ ¥¬®© ª ¬®¬¥­â㠢६¥­¨ . ”ã­ªæ¨ï

(t) |  ¡á®«îâ­® ­¥¯à¥à뢭 ï äã­ªæ¨ï, á ¯à®¨§¢®¤­®© ­¥¯à¥à뢭®© á¯à ¢ . ˆ§ á¬ëá«  ä㭪権 m(t) ¨ (t) á«¥¤ã¥â, ®­¨ ¤®«¦­ë 㤮¢«¥â¢®àïâì ãá«®¢¨ï¬
m(t) 6 (t) 6 t,  (t) > 0.
—¥à¥§ Ta (t) ¨ Tp (t) ¡ã¤¥¬ ®¡®§­ ç âì, ᮮ⢥âá⢥­­®,  ªâ¨¢­ë¥ ¨ ¯ áᨢ­ë¥ âà㤮¢ë¥ à¥áãàáë ¢ ¬®¬¥­â ¢à¥¬¥­¨ t, â. ¥. Ta (t) | íâ® âà㤮¢ë¥ à¥áãàáë, ãç áâ¢ãî騥
¢ ¯à®¨§¢®¤á⢥ ­  ¤¥©áâ¢ãîé¨å ä®­¤ å ¢ ¬®¬¥­â ¢à¥¬¥­¨ t, a Tp (t) | âà㤮¢ë¥ à¥áãàáë, ª®â®àë¥ § ¤¥©á⢮¢ ­ë ¢ ¬®¬¥­â ¢à¥¬¥­¨ t ­  ¢¢®¤¨¬ëå ä®­¤ å (®á¢ ¨¢ ¥¬ëå

¥âáï, çâ®

0

ä®­¤ å). ã¤¥¬ ¯à¥¤¯®« £ âì, çâ® âà㤮¢ë¥ à¥áãàáë ª ª  ªâ¨¢­ë¥, â ª ¨ ¯ áᨢ­ë¥

­¥ ã¡ë¢ îâ, â. ¥.

Ta (t) > 0, Tp (t) > 0, ¨ § ¤ îâáï  ¡á®«îâ­® ­¥¯à¥à뢭묨 äã­ªæ¨ï0

0

¬¨, á ¯à®¨§¢®¤­®© ªãá®ç­®-­¥¯à¥à뢭®© á¯à ¢ .
Ž¡®§­ ç¨¬ ç¥à¥§

P (t )

ª®«¨ç¥á⢮ ç¨á⮣® ¯à®¤ãªâ , ¯à®¨§¢®¤¨¬®£® ­  ä®­¤ å,

t

ãç áâ¢ãîé¨å ¢ ¯à®¨§¢®¤á⢥ ¢ ¬®¬¥­â ¢à¥¬¥­¨ .
2. ®áâ ­®¢ª  § ¤ ç¨.
à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨

¥à¥©¤¥¬ ⥯¥àì ª ¯®áâ ­®¢ª¥ § ¤ ç¨ ¨ ä®à¬ã«¨à®¢ª¥ ¯à¨­æ¨¯  ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨.

t ¢ëç¨á«ï¥âáï ¯® á«¥¤ãî饩 ä®à-

 æ¨®­ «ì­ë© ¤®å®¤ ¢ ª ¦¤ë© ¬®¬¥­â ¢à¥¬¥­¨
‹Ǵ

P (t) =

Z(t)
m(t)

U (( ); '( );  )d:

:

(2 1)

“à ¢­¥­¨¥ ¡ « ­á   ªâ¨¢­ëå âà㤮¢ëå à¥áãàᮢ:

Z(t)

m(t)

'( )d = Ta (t):

â® ãà ¢­¥­¨¥ ¬®¦¥â ¡ëâì § ¯¨á ­® ¢ ¤¨ää¥à¥­æ¨ «ì­®© ä®à¬¥:

'((t)) (t) , '(m(t))m (t) = Ta (t):
0

0

0

:

(2 2)

“à ¢­¥­¨¥ ¡ « ­á  ¯ áᨢ­ëå âà㤮¢ëå à¥áãàᮢ:

Zt

(t)

'( )d = Tp (t):

‚ ¤¨ää¥à¥­æ¨ «ì­®© ä®à¬¥ íâ® ãà ¢­¥­¨¥ ¡ã¤¥â ¨¬¥âì ¢¨¤:

'(t) = '((t)) (t) + Tp (t):
0

0

:

(2 3)

3{26

‹. €. ¥ª« àï­, ‘. ‚. ®à¨á®¢ 

‘«¥¤ã¥â ®â¬¥â¨âì, çâ® ¢ëà ¦¥­¨ï (2.2), (2.3) ¢®®¡é¥ £®¢®àï, ­¥ íª¢¨¢ «¥­â­ë ãà ¢­¥­¨ï¬ ¡ « ­á   ªâ¨¢­ëå ¨ ¯ áᨢ­ëå âà㤮¢ëå à¥áãàᮢ, § ¯¨á ­­ë¬ ¢ ¨­â¥£à «ì­®©
ä®à¬¥ (á â®ç­®áâìî ¤® ­ ç «ì­ëå §­ ç¥­¨©).
“à ¢­¥­¨¥ ¡ « ­á  ¯® ä®­¤ ¬ ¢ á«ãç ¥, ª®£¤ 
樮­ «ì­®£® ¤®å®¤ , â. ¥.

(t) à ¢­  ¯®áâ®ï­­®© ç á⨠­ -

(t) = P (t) (0 <  < 1 | ¯®áâ®ï­­ ï ­®à¬  ­ ª®¯«¥­¨ï).

ˆá¯®«ì§ãï ä®à¬ã«ã (2.1), ãà ¢­¥­¨¥ ¡ « ­á  ¯® ä®­¤ ¬ ¡ã¤¥â ¨¬¥âì ¢¨¤

(t) = 

Z(t)

m(t)

U (( ); '( );  )d;

  ¢ ¤¨ää¥à¥­æ¨ «ì­®© ä®à¬¥:

 ,




,






 (t) =  U ((t)); '((t)); (t)  (t) , U (m(t)); '(m(t)); m(t) m (t) :
0

0

0

:

(2 4)

Ž£à ­¨ç¥­¨ï ­  ᪮à®áâì ¯à®æ¥¤ãàë § ¬¥é¥­¨ï ãáâ à¥¢è¨å ä®­¤®¢ ¨ ¢¢®¤  ­®¢ëå:
0

6 m (t) 6 M;  (t) > 0:
0

:

0

(2 5)

‚ ¬®¤¥«¨ ‹. ‚. Š ­â®à®¢¨ç  [10] ¨­¥à樮­­®áâì ¯à¨ ¢¢®¤¥ ­®¢ëå ä®­¤®¢ ­¥ ãç¨â뢠¥âáï, áç¨â ¥âáï, çâ® ¢¢®¤¨¬ë¥ ä®­¤ë ¨¬¥îâ ¬£­®¢¥­­ãî ®â¤ çã, â. ¥.

(t)  t.

Šà®¬¥ ⮣®, ¢ ¬®¤¥«¨ Š ­â®à®¢¨ç  ­¥â ®£à ­¨ç¥­¨ï ­  ᪮à®áâì ¯à®æ¥¤ãàë ¢ë¢®¤  ãáâ à¥¢è¨å ä®­¤®¢, â. ¥. ä®­¤ë ¬®£ãâ ¡ëâì ¢ë¢¥¤¥­ë ¨§ ¯à®¨§¢®¤á⢠ ᪮«ì
㣮¤­® ¡ëáâà®.
ãáâì ^ | ­ ç «ì­ë© ¬®¬¥­â ¢à¥¬¥­¨.

t

fm(
); (
); '(
)g,

m(t), (t)

‚¢¥¤¥¬ ¢¥ªâ®à-ä㭪樨

t; 1

(
)

=

|  ¡á®«îâ­® ­¥¯à¥àë¢­ë¥ ­  [^ + [ ä㭪樨 á
­¥¯à¥à뢭®© á¯à ¢  ¯à®¨§¢®¤­®©;
( ),
[^ + [ | äã­ªæ¨ï ®£à ­¨ç¥­­®© ¢ £¤¥

' t t 2 t; 1
(t)

ਠ樨, ­¥¯à¥à뢭 ï á¯à ¢  (¢ á«ãç ¥, ¥á«¨

ï¥âáï íª§®£¥­­®© ¯¥à¥¬¥­-

(t) § ¤ ¥âáï í­¤®£¥­­®, â® ¡ã¤¥¬ à áᬠâਢ âì ¢¥ªâ®à-äã­ªæ¨î
(
) =
fm(
); (
); (
); '(
)g, £¤¥ (t), t 2 [t;^ +1[ |  ¡á®«îâ­® ­¥¯à¥à뢭 ï äã­ªæ¨ï á ¯à®­®©). …᫨

¨§¢®¤­®© ­¥¯à¥à뢭®© á¯à ¢ .
‚¥ªâ®à-ä㭪樨

(
) ¡ã¤¥¬ ­ §ë¢ âì âà ¥ªâ®à¨ï¬¨.

®«¨â¨ª  ¢ë¡®à  ®¯â¨¬ «ì­®© âà ¥ªâ®à¨¨ ®á­®¢ë¢ ¥âáï ­  á«¥¤ãî饬 ¯à¨­æ¨¯¥.
à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨: áâà â¥£¨ï ¢ë¡®à  ¯¥à¥¬¥­­ëå
¬®¤¥«¨ ¤®«¦­  ¡ëâì â ª®¢ , çâ®¡ë ¢ ª ¦¤ë© ¬®¬¥­â ¢à¥¬¥­¨ ®¡¥á¯¥ç¨âì ¬ ªá¨¬ «ì­ë© ⥬¯ à®áâ  ­ æ¨®­ «ì­®£® ¤®å®¤ .

¨¦¥ ¯à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨ ¡ã¤¥â ä®à¬ «¨§®¢ ­ ¤«ï à áᬠâਢ ¥¬®© § ¤ ç¨.

à¨ í⮬ ¯à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨ ¬®¦¥â

¡ëâì áä®à¬ã«¨à®¢ ­ ¢ ¢¨¤¥ ãà ¢­¥­¨ï ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨, ª®â®à®¬ã
¤®«¦­ë 㤮¢«¥â¢®àïâì ¯¥à¥¬¥­­ë¥ ¬®¤¥«¨; ä®à¬ «ì­®, ¢ ª ¦¤ë© ¬®¬¥­â ¢à¥¬¥^ (£¤¥
| ­ ç «ì­ë© ¬®¬¥­â ¢à¥¬¥­¨) ¢¤®«ì âà ¥ªâ®à¨¨, 㤮¢«¥â¢®àïî饩

­¨

t>t

t

¯à¨­æ¨¯ã ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨, ¯à ¢ ï ¯à®¨§¢®¤­ ï ­ æ¨®­ «ì­®£® ¤®-

P
(t) ¤®«¦­  ¯à¨­¨¬ âì ¬ ªá¨¬ «ì­® ¢®§¬®¦­®¥ §­ ç¥­¨¥,   ¨¬¥­­® ¢ ª ¦¤®©
t > t^ ä㭪樮­ « P
(t) ®â­®á¨â¥«ì­®  (t); m (t); '(t) ¤®«¦¥­ ¯à¨­¨¬ âì ¬ ªá¨^ +1[, 㤮¢«¥â¬ «ì­®¥ §­ ç¥­¨¥ ¢¤®«ì âà ¥ªâ®à¨¨
(t) = fm(t); (t); '(t)g ¯à¨ t 2 [t;

室 

â®çª¥

0

0

0

¢®àïî饩 ãá«®¢¨ï¬ (2.2), (2.3), (2.5) (¢ á«ãç ¥ íª§®£¥­­® § ¤ ­­®© ä㭪樨 ª ¯¨â «ì-

­ëå ¢«®¦¥­¨© (t); ¤«ï í­¤®£¥­­® ®¯à¥¤¥«ï¥¬®© ä㭪樨 (t) | ¢¤®«ì âà ¥ªâ®à¨¨
(t) = fm(t); (t); (t); '(t)g ¯à¨ t 2 [t;^ 1[, 㤮¢«¥â¢®àïî饩 ãá«®¢¨ï¬ (2.2){(2.5)).

Ž¤­®¯à®¤ãªâ®¢ ï ¤¨­ ¬¨ç¥áª ï ¬®¤¥«ì § ¬¥é¥­¨ï

3{27

3. ‘«ãç © íª§®£¥­­® § ¤ ­­®© ä㭪樨 ª ¯¨â «®¢«®¦¥­¨©

‚ á«ãç ¥, ª®£¤  ¨­â¥­á¨¢­®áâì ¢¢®¤  ª ¯¨â «®¢«®¦¥­¨© (t) § ¤ ¥âáï  ¯à¨®à¨,
¬®¤¥«ì ®¯¨á뢠¥âáï á«¥¤ãî騬¨ ãà ¢­¥­¨ï¬¨:
­ æ¨®­ «ì­ë© ¤®å®¤ ¢ ¬®¬¥­â ¢à¥¬¥­¨ t:

P (t) =

Z(t)
m(t)

U (( ); '( );  )d ;

(3:1)

ãà ¢­¥­¨ï ¡ « ­á   ªâ¨¢­ëå ¨ ¯ áᨢ­ëå âà㤮¢ëå à¥áãàᮢ ᮮ⢥âá⢥­­®, § ¯¨á ­­ë¥ ¢ ¤¨ää¥à¥­æ¨ «ì­®© ä®à¬¥:

'((t)) (t) , '(m(t))m (t) = Ta (t);
0

0

(3:2)

0

'(t) = '((t)) (t) + Tp (t);

(3:3)
®£à ­¨ç¥­¨ï ­  ᪮à®áâì ¯à®æ¥¤ãàë § ¬¥é¥­¨ï ãáâ à¥¢è¨å ä®­¤®¢ ¨ ¢¢®¤  ­®¢ëå:
0 6 m (t) 6 M;  (t) > 0;
(3:4)
¨­ä®à¬ æ¨ï ® ­ ç «ì­®¬ á®áâ®ï­¨¨ á¨áâ¥¬ë § ¤ ­  ¢ ¢¨¤¥ á«¥¤ãîé¨å ¤ ­­ëå:
1) ­ ç «ì­ë¥ ¤ ­­ë¥ t^, m^ = m(t^), ^ = (t^), ¯à¨ç¥¬ m^ 6 ^ 6 t^, m^ < t^;
2) ­  ¨­â¥à¢ «¥ [m;
^ t^] § ¤ ­  ¯®«®¦¨â¥«ì­ ï ­¥¯à¥à뢭 ï äã­ªæ¨ï
(t) â ª ï,
çâ® '(t) ᮢ¯ ¤ ¥â á
(t) ­  ¯®«ã¨­â¥à¢ «¥ [m;
^ t^];
3) ­  ¯®«ã¯àאַ© [m;
^ +1[ § ¤ ­  äã­ªæ¨ï ª ¯¨â «®¢«®¦¥­¨© (t) | ¯®«®¦¨â¥«ì­ ï, ­¥¯à¥à뢭 ï, ¬®­®â®­­® ¢®§à áâ îé ï.
’¥¯¥àì ¤«ï § ¤ ç¨ (3.1){(3.4) ¤ ¤¨¬ â®ç­®¥ ®¯à¥¤¥«¥­¨¥ ¯à¨­æ¨¯  ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨.
à ¢ ï ¯à®¨§¢®¤­ ï P
(t) ¤«ï âà ¥ªâ®à¨¨
(t) = fm(t); (t); '(t)g, ¢ ¯à®¨§¢®«ì­®©
â®çª¥ t 2 [t^; +1[ ¨¬¥¥â ¢¨¤
0

0

0

0

0

,




,




P
(t) = U ((t)); '((t)); (t)  (t) , U (m(t)); '(m(t)); m(t) m (t):
0

0

0

(3:5)

à¨ ª ¦¤®¬ 䨪á¨à®¢ ­­®¬ [t^; +1[ à áᬮâਬ íªáâ६ «ì­ãî § ¤ çã:

‡ ¤ ç  (€).

,




,




U ((t)); '((t)); (t) u , U (m(t)); '(m(t)); m(t) v ! u;v;w
max ;
'((t))u , '(m(t))v = Ta (t);
w = '((t))u + Tp (t);
0 6 v 6 M; u > 0; w > 0:
(…᫨ ¢ (3.6){(3.8) (t) = t ¨«¨ m(t) = t, â® §­ ç¥­¨¥ '(t) § ¬¥­ï¥âáï ­  w.)
0

0

(3:6)
(3:7)
(3:8)
(3:9)

’¥¯¥àì ¬ë ¬®¦¥¬ ¤ âì ä®à¬ «ì­®¥ ®¯à¥¤¥«¥­¨¥ ¯à¨­æ¨¯  ¤¨ää¥à¥­æ¨ «ì­®©
®¯â¨¬¨§ æ¨¨.

3{28

‹. €. ¥ª« àï­, ‘. ‚. ®à¨á®¢ 

(
) = fm(
); (
)

à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨. ’à ¥ªâ®à¨ï

'(
)g
¥á«¨

¢ â®çª¥
¢

â®çª¥

v ; u ; w  )
w = '(t)
( 

t 2 [t;^ +1[
t 2 [t;^ +1[

,

㤮¢«¥â¢®àï¥â ¯à¨­æ¨¯ã ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨,
®­ 

㤮¢«¥â¢®àï¥â

íªáâ६ «ì­®© § ¤ ç¨

:

:

(3 6){(3 9)

:
:
u = 0 (t) v = m0 (t)

(3 2){(3 4),

®£à ­¨ç¥­¨ï¬

áãé¥áâ¢ã¥â ¨ à ¢­®

 

à¥è¥­¨¥

,

,

.

®«¥¥ ⮣®, ¥á«¨ áâ àë¥ ¯à®¨§¢®¤á⢥­­ë¥ ä®­¤ë ®ª §ë¢ îâáï áà ¢­¨¬ë¬¨ á ¢¢®¤¨¬ë¬¨

( 

¨¬¥­­®, ¨¬¥îâ ®¤¨­ ª®¢ãî ¯à®¨§¢®¤¨â¥«ì­®áâì ), â® ¯à¥¤¯®ç⥭¨¥ ®â¤ ¥â-

áï ­®¢ë¬ ä®­¤ ¬, ¯à¨ í⮬ ãáâ à¥¢è¨¥ ä®­¤ë ¢ë¢®¤ïâáï ¬ ªá¨¬ «ì­® ¡ëáâà®, â. ¥.

v = Œ

.

ã¤¥¬ £®¢®à¨âì, çâ® âà ¥ªâ®à¨ï

(
)

㤮¢«¥â¢®àï¥â ¯à¨­æ¨¯ã ¤¨ää¥à¥­æ¨ «ì­®©

t 2 [t;^ +1[

®¯â¨¬¨§ æ¨¨, ¥á«¨ ®­  㤮¢«¥â¢®àï¥â ¥¬ã ¢ «î¡®© â®çª¥

.

‚ ª ç¥á⢥ ¯à®¨§¢®¤á⢥­­®© ä㭪樨 ¡ã¤¥¬ à áᬠâਢ âì å®à®è® ¨§¢¥áâ­ãî

f (t) (t)' (t),  +  = 1, ¤«ï ª®â®à®© ­¥¯à¥à뢭®-¤¨ää¥à¥­æ¨à㥬 ï äã­ªæ¨ï f (t) 㤮¢«¥â¢®àï¥â ãá«®¢¨ï¬ f (t) > 0,
f 0 (t) > 0 ¨ ®âà ¦ ¥â ã஢¥­ì á®áâ®ï­¨ï ­ ãç­®-â¥å­¨ç¥áª®£® ¯à®£à¥áá  ¢ ¬®¬¥­â t.

äã­ªæ¨î Š®¡¡  | „㣫 á 

U (; '; t)

=

”ã­ªæ¨ï Š®¡¡  | „㣫 á  ï¥âáï ®¤­®à®¤­®© ¨ ¢®£­ã⮩.

à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨ ¯à¨¢®¤¨â ª á«®¦­®© á¨á⥬¥ ä㭪樮­ «ì­®-¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨© (á¨á⥬¥ ãà ¢­¥­¨© ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨). ‚뤥«ïîâáï ¤¢  ®á­®¢­ëå ०¨¬  ¯®¢¥¤¥­¨ï á¨á⥬ë:

(t) = t, â. ¥.
t ä®­¤ë ­ ç¨­ îâ ¤ ¢ âì ®â¤ çã ¬£­®¢¥­­® (¯à¨ í⮬

1. ‘«ãç © ®âáãâáâ¢¨ï ¨­¥à樮­­®á⨠¯à¨ ¢¢®¤¥ ­®¢ëå ä®­¤®¢:
¢¢®¤¨¬ë¥ ¢ ¬®¬¥­â ¢à¥¬¥­¨

०¨¬¥ ®¡ê¥¬ ¯ áᨢ­ëå âà㤮¢ëå à¥áãàᮢ ­¥ ¬¥­ï¥âáï; ¥á«¨ ¦¥ ¯à¨ í⮬ ०¨¬¥
¨§¬¥­ï¥âáï ®¡ê¥¬ ¯ áᨢ­ëå âà㤮¢ëå à¥áãàᮢ
०¨¬ 2);

Tp0 (t) 6= 0, â® ¯à®¨á室¨â ¯¥à¥å®¤ ­ 

2. “ç¥â ¨­¥à樮­­ëå ᢮©á⢠¯à¨ ¢¢®¤¥ ­®¢ëå ä®­¤®¢:

(t) < t, â. ¥.

¢¢®¤¨¬ë¥

t ä®­¤ë ­ ç¨­ îâ ¤ ¢ âì ®â¤ çã ­¥ áà §ã,   âॡãî⠢६¥­¨ ­ 

¢ ¬®¬¥­â ¢à¥¬¥­¨
¢¢®¤ ¨ ®á¢®¥­¨¥.

¥à¥¬¥­­ë¥ á¨á⥬ë ãà ¢­¥­¨© ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨ ¢ ª ¦¤ë© ¬®[^ + [ ¤®«¦­ë 㤮¢«¥â¢®àïâì á«¥¤ãî饩 á¨á⥬¥ ãà ¢­¥­¨©

¬¥­â ¢à¥¬¥­¨

t 2 t; 1

8 0;
>>
>>
>>
Ta t Tp t
t
>> ''max
mt , 'mt ;
>>
>>
>> M;
<
m0 (t) = >
>> 0;
>> 0;
>>
>>
>> M;
>>
>:
(

;

0

( )
( ))

0

( )+
(

0

( )

( ))

¥á«¨

¥á«¨

(t) = t; '(m(t)) 6= 0;

, Ta't m Ttp t 6 0;
(t) = t; '(m(t)) =
6 0;
Ta t Tp t
'max t
< M;
0< 'm t , 'm t
(t) = t; '(m(t)) =
6 0;
T
t
T
'max t
a
p t
' m t , ' m t > M;
'max(t)
'(m(t))

(

¥á«¨

(

¥á«¨
¥á«¨

¥á«¨

¥á«¨

( )
( ))

0

( )+
(

0

( )

( ))

0

( )
( ))

( )+
(

0

( )+
(

0

0

( )

( ))

( )

( ))

(t) = t; '(m(t)) = 0;
(t) < t; '(m(t)) 6= 0; '((t)) 6= 0;
F ((t)) < F (m(t)) ;
'((t)) '(m(t))
(t) < t; '(m(t)) 6= 0; '((t)) 6= 0;
F ((t))
'((t))

> F' mm tt
(
(

( ))
;
( ))

(t) < t; '(m(t)) = 0:

(I)

Ž¤­®¯à®¤ãªâ®¢ ï ¤¨­ ¬¨ç¥áª ï ¬®¤¥«ì § ¬¥é¥­¨ï
8
>> 1 , Ta (tT)+p(tT)p (t) ;
¥á«¨ (t) = t; '(m(t)) 6= 0;
>>
'max (t) Ta (t)+Tp (t)
>>
'(m(t)) , '(m(t)) 6 0;
>> 1 , Tp (t) ;
¥á«¨ (t) = t; '(m(t)) =
6 0;
>> 'max(t)
' (t) , Ta (t)+Tp (t) < M ;
>>
0 < '(max
m(t))
'(m(t))
>>
(
t
)
T
>> 1 , '(m(t))M +pTa (t)+Tp (t) ; ¥á«¨ (t) = t; '(m(t)) =6 0;
>>
'max (t) Ta (t)+Tp (t)
>> 8
'(m(t)) , '(m(t)) > M ;

3{29

0

0

0

0

0

0

0

0

0

0

0

>> > Ta (t)
>> >< Ta(t)+Tp (t) ;
><
 (t) = >
>
>> : 1;
>>
>> 'T(a ((tt))) ;
>>
>>
>> '(m(t)) M + Ta(t) ;
>> '((t)) '((t))
>> 8
>> >> Ta (t)
>> < '((t)) ;
>> >
>> >: 1;
:
8 Ta (t) + Tp (t);
>
>
>
>
>
>
'max (t);
>
>
>
>
>
>
'(m(t))M + Ta (t) + Tp (t);
>
>
<
 (t) = >
Ta (t) + Tp (t);
>
>
>
> Ta (t) + Tp (t);
>
>
>
>
'(m(t))M + Ta (t) + Tp (t);
>
>
>
>
:
0

0

0

0

0

0

0

0

0

0

0

0

¥á«¨

¥á«¨

¥á«¨

F ((t))
'((t))

¥á«¨

¥á«¨

¥á«¨

¥á«¨

¥á«¨

0

¥á«¨

0

0

¥á«¨

¥á«¨

(II)

0

> F'((mm((tt)))) ;

(t) < t; '(m(t)) = 0;
'((t)) 6= 0;
(t) < t; '(m(t)) = 0;
'((t)) = 0;
(t) = t; '(m(t)) 6= 0;
'max (t) Ta (t)+Tp (t)
'(m(t)) , '(m(t))
0

0

0

0

0

0

0

0

(t) = t; '(m(t)) = 0;
Ta (t) + Tp (t) 6= 0;
(t) = t; '(m(t)) = 0;
Ta (t) + Tp (t) = 0;
(t) < t; '(m(t)) 6= 0; '((t)) 6= 0;
F ((t)) < F (m(t)) ;
'((t)) '(m(t))
(t) < t; '(m(t)) 6= 0; '((t)) 6= 0;
0

¥á«¨

0

6 0;

(t) = t; '(m(t)) 6= 0;
' (t) , Ta (t)+Tp (t) < M ;
0 < '(max
m(t))
'(m(t))
(t) = t; '(m(t)) 6= 0;
' (t) , Ta (t)+Tp (t) > M ;
0 < '(max
m(t))
'(m(t))
(t) = t; '(m(t)) = 0;
(t) < t; '(m(t)) 6= 0; '((t)) 6= 0;
F ((t)) < F (m(t)) ;
'((t)) '(m(t))
(t) < t; '(m(t)) 6= 0; '((t)) 6= 0;
F ((t))
'((t))

0

0

0

0

> F'((mm((tt)))) ;

(III)

Ta (t) + Tp (t);
¥á«¨ (t) < t; '(m(t)) = 0;
£¤¥ F (t) = f 1= (t)(t) | ¬®­®â®­­® ¢®§à áâ îé ï äã­ªæ¨ï, 'max (t) | à¥è¥­¨¥ ãà ¢0

0

3{30
­¥­¨ï:

‹. €. ¥ª« àï­, ‘. ‚. ®à¨á®¢ 



0
 (t)  +  Tp (t)
F (t)',max




,
(3:10)
'max(t) , F (m(t))' (m(t)) = 0:
“à ¢­¥­¨¥ (3.10) ¨¬¥¥â ¥¤¨­á⢥­­®¥ ¯®«®¦¨â¥«ì­®¥ à¥è¥­¨¥ [11] 'max(t), ¨¬¥î饥


¯à¥¤áâ ¢«¥­¨¥

'max (t) = y(t; m(t)):

‘ä®à¬ã«¨à㥬 ⥮६ã íª¢¨¢ «¥­â­®áâ¨.
(⥮६  íª¢¨¢ «¥­â­®áâ¨)
m;
^ ^; t^
m^ < ^ 6 t^

(t) > 0 t 2 [m;
^ t^]
(
) =
fm(
); (
); '(
)g
m(t) (t) t 2 [t^; +1[
m(t) 6 (t) 6 t
m(t) < t
m(t^) = t^ (t^) = ^ '(t)
'(t) 
(t) t 2 [m;
^ t^]
^
t 2 [t; +1[
(t)
t
(I){(III)
(⥮६  áãé¥á⢮¢ ­¨ï ¨ ¥¤¨­á⢥­­®áâ¨)
m^ ^ t^
m^ < ^ 6 t^

(t) > 0
t 2 [m;
^ t^]
(I){(III) m(t)
(t) t 2 [t^; +1[
'(t) t 2 [m;
^ +1[
'(t) =
(t) t 2
[m;
^ t^[
m(t^) = m^ (t^) = ^
m(t) (t)
m(t) 6 (t) 6 t m(t) < t
’¥®à¥¬  1

. ãáâì § ¤ ­ë ­ ç «ì­ë¥ ¤ ­­ë¥

, ªà ¥¢ ï ­¥¯à¥à뢭 ï äã­ªæ¨ï
,

£¤¥

,

,

,

¨ âà ¥ªâ®à¨ï

|  ¡á®«îâ­® ­¥¯à¥àë¢­ë¥ ä㭪樨 á

¯à®¨§¢®¤­ë¬¨ ­¥¯à¥à뢭묨 á¯à ¢ , 㤮¢«¥â¢®àïî騥 ãá«®¢¨ï¬
¨ ­ ç «ì­ë¬ §­ ç¥­¨ï¬

,

;

á¯à ¢  äã­ªæ¨ï, 㤮¢«¥â¢®àïîé ï ­ ç «ì­®¬ã ãá«®¢¨î

¤«ï «î¡®£®

âà ¥ªâ®à¨ï

,

| ªãá®ç­®-­¥¯à¥à뢭 ï
,

. ’®£¤ 

㤮¢«¥â¢®àï¥â ¯à¨­æ¨¯ã ¤¨ää¥à¥­æ¨ «ì­®©

®¯â¨¬¨§ æ¨¨ ⮣¤  ¨ ⮫쪮 ⮣¤ , ª®£¤  ¢ â®çª¥

á¯à ¢¥¤«¨¢  á¨á⥬  ãà ¢­¥­¨©

.

’¥®à¥¬  2

¤ ­­ë¥

,

,

. ãáâì § ¤ ­ë ­ ç «ì­ë¥

â ª¨¥, çâ®

¨ ªà ¥¢ ï ­¥¯à¥à뢭 ï äã­ªæ¨ï

. ’®£¤  áãé¥áâ¢ã¥â, ¯à¨ç¥¬ ¥¤¨­á⢥­­®¥, à¥è¥­¨¥ á¨á⥬ë

,

|  ¡á®«îâ­® ­¥¯à¥àë¢­ë¥ ä㭪樨,

çâ®

,

,

. ®«¥¥ ⮣®, ä㭪樨

.

I
m’ (t)=0
α(t)=t
Tp’ (t)=0

Tp’ (t)≠0
II
0