1. Wstęp

1.
Wstęp
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JR NLHUXQNX EDGDĔ 3URJQR]\ VWDQyZ SURFHVyZ RSLVXMąF\FK EDGDQH RELHNW\
SURJQRVW\F]QHZ\]QDF]DVLĊZGUXJLP]Z\PLHQLRQ\FKREV]DUyZEDGDĔ]D
SRPRFąPRGHOLPDWHPDW\F]Q\FKW\FKRELHNWyZPRGHOL]QDQ\FKOXEGRSLHUR
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GHMĞFLDPRGHORZHJRLSURJQR]RZDQLDG]LDáDMąF\FKZUyĪQ\FKREV]DUDFKJR
VSRGDUNL6WRVRZDQLHLORĞFLRZ\FKPHWRGZVSRPDJDMąF\FKDQDOL]ĊLUHDOL]DFMĊ
SURFHVXHIHNW\ZQHJRIXQNFMRQRZDQLDWDNLFKV\VWHPyZPDLVWRWQH]QDF]HQLH
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ZVSRVyEFLąJá\QS]DSRPRFąWDĞPRFLąJyZJD]RFLąJyZOLQLLSU]HV\áRZ\FK
VWUXPLHĔSURGXNFML y t UHDOL]RZDQ\SU]H]SRGV\VWHPSURGXNFML P /RVRZH
]PLDQ\ SURFHVX y t RUD] QLHSODQRZDQH SU]HUZ\ Z SUDF\ DZDULH SRGV\V
WHPXWUDQVSRUWRZHJR T VąF]\QQLNDPLZSá\ZDMąF\PLQD]PQLHMV]HQLHHIHN
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7
Wstęp
RGELRUF\EPDJD]\QX]ELRUQLNDMRRNUHĞORQHMREMĊWRĞFLV6WUXPLHĔSURGXN
WX y t MHVWJURPDG]RQ\ZSRGV\VWHPLHMJG\SR]LRPZ\SHáQLHQLDPDJD]\
QXQLHSU]HNUDF]DVRUD]JG\ y t ! a -HĪHOLFKZLORZD]DZDUWRĞüHOHPHQWX
MMHVWUyZQDVRUD] y t ! a WRZLHONRĞüVWUXPLHQLD y t RJUDQLF]DVLĊGR
SR]LRPXa*G\HOHPHQWMMHVWSXVW\L y t a SRZVWDMHV\WXDFMDQLHNRU]\VW
QDGODRGELRUF\ERNUHĞOHQLHSUDZGRSRGRELHĔVWZDWHJR]GDU]HQLDPDZDĪQH
]QDF]HQLHSUDNW\F]QH
:VSyáSUDFĊSRGV\VWHPyZ P T MRUD]ESU]HGVWDZLRQRQDU\V
5\V2JyOQ\VFKHPDWIXQNFMRQRZDQLDUR]ZDĪDQHJRV\VWHPXJRVSRGDUNL]DSDVDPL
3U]\NáDGHPSRGV\VWHPX P PRĪHE\ü]HVSyáNRSDUHN:HMĞFLH wt SRG
V\VWHPXMRSLVXMąFHSURFHVGRVWDZ\SURGXNWXGRMPRĪQDEDGDüZGZyFK
ZDULDQWDFK
‡ ZDULDQW ]DJUHJRZDQ\ wt y t SURFHV wt UHSUH]HQWXMH áąF]QLH
SRGV\VWHPSURGXNFML P RUD]SRGV\VWHPWUDQVSRUWRZ\ T ‡ ZDULDQWVWUXNWXUDOQ\ wt y t vt SURFHV wt XZ]JOĊGQLDH[SOLFLte]DUyZQRSRGV\VWHP VWUXPLHĔSURGXNWX y t MDNLSRGV\VWHP
T SURFHV vt RSLVXMHIXQNFMRQRZDQLHSRGV\VWHPXWUDQVSRUWRZHJR
:FHOX]DSURSRQRZDQLDQDU]ĊG]LZVSRPDJDMąF\FKDQDOL]ĊLSURFHVHIHN
W\ZQHJRIXQNFMRQRZDQLDEDGDQHJRV\VWHPXU\VSU]\MPLHP\ĪH y t RUD]
vt VąSURFHVDPL0DUNRZDRVNRĔF]RQHMOLF]ELHVWDQyZ
6\VWHPSU]HGVWDZLRQ\QDU\VEDGDQ\EĊG]LHZSLHUZ]XZ]JOĊGQLHQLHP
ZDULDQWXSLHUZV]HJR:GDOV]HMNROHMQRĞFLEĊG]LHUR]SDWU\ZDQ\UyZQLHĪZD
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QHMREMĊWRĞFLVJG]LHȦMHVWSDUDPHWUHPORVRZ\FK=DXZDĪP\ĪHSURFHVVWR
FKDVW\F]Q\ z t ZSU]HG]LDOH >D 1 D VWDáRĞFLUHDOL]DFMLSURFHVX wt y t VSHáQLDZDUXQHN
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8
Wstęp
)XQNFMDh]GH¿QLRZDQDMHVWZ]RUHP
h z ­ gdy z d °
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DOL]DFML SURFHVX y t VSHáQLD ZDUXQHN Z\QLND ] ¿]\F]QHM QDWXU\ SUR
FHVX z t -HĪHOLZFKZLOL t1 Z\SHáQLHQLH]ELRUQLNDZ\QRVL z t1 WRLVWRWQLH
ZFKZLOLtJG\ D 1 d t1 t D EĊG]LHZ\QRVLü z t z t1 y t1 a t t1 JG\ z t1 y t1 a t t1 V : SU]\SDGNX z t V i y t t a D ZLĊF JG\ PDJD]\Q MHVW MXĪ SHáQ\ L SRGV\VWHP WUDQVSRUWRZ\ PRĪH GRVWDU
F]Dü ZLĊFHM UH]XOWDWyZ SURGXNFML SRGV\VWHPX P QLĪ SRWU]HEXMH RGELRUFD E
XWU]\PXMHP\ZWHG\]GROQRĞüWUDQVSRUWRZąV\VWHPX]PQLHMV]DMąFMHG\QLHVX
PDU\F]QąZ\GDMQRĞüSRGV\VWHPX P GRSR]LRPXa6WDQ z t Z\VWĊSXMH
W\ONRZWHG\JG\SR]LRPGRVWDZ\ y t MHVWQLHZLĊNV]\QLĪ]DSRWU]HERZDQLH
y t d a 7DNLHZ\UyĪQLDQLHVWDQyZ z t V z t ZVND]XMHQDSRWU]HEĊ
ZSURZDG]HQLDIXQNFMLhRNUHĞORQHMZ]RUHP
9