Моделирование сложных информационных систем: Методические указания по изучению спецкурса

FBGBKL?JKL...

Author: Сирота А.А.

59 downloads 169 Views 514KB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

FBGBKL?JKL

F?LH>BQ?KDB?MD:A:GBY

ihbamq_gbxki_pdmjkZ«Fh^_ebjh\Zgb_keh`guobgnhjfZpbhg guokbkl_f»^eyklm^_glh\dmjkZ^g_\gh]hhl^_e_gbyihki_p jZ^bhnbabdZbwe_dljhgbdZ

KhklZ\bl_evijhn::KbjhlZ

<?GB?

2 Fh^_ebjh\Zgb_y\ey_lkyh^gbfbafhsguokh\j_f_gguobgkljmf_g lh\bkke_^h\Zgbykbkl_fbbo\aZbfh^_ckl\byf_`^mkh[hc.Zgguc dmjk y\ey_lky _kl_kl\_gguf ijh^he`_gb_f dmjkZ ©Fh^_eb jh\Zgb_ jZ^bhl_ogbq_kdbo dZgZeh\ i_j_^Zqb bgnhjfZpbbª qblZ_fh]h klm^_glZf dmjkZ ^g_\gh]h hl^_e_gby \ k_f_klj_ \ dhlhjhf hkgh\gh_ \gbfZgb_m^_ey_lkyjZkkfhlj_gbx\hijhkh\fh^_ebjh\Zgby\aZbfh^_ckl \by\bgnhjfZpbhgguokbkl_fZoBK b]eZ\gufh[jZahf\jZ^bhl_ogb q_kdbo kbkl_fZo G_h[oh^bfhklv ki_pbZevgh]h jZkkfhlj_gby \hijhkh\ fh^_ebjh\Zgby dZgZeh\ i_j_^Zqb bgnhjfZpbb DIB hij_^_ey_lkyg_h[ oh^bfhklvx Z[kljZ]bjh\Zlvky hl jy^Z lhgdbo wnn_dlh\ ijb \hkijhba\_ ^_gbb [hevrh]h dhebq_kl\Z \hagbdZxsbo \aZbfh^_ckl\bc k h[t_dlZfb \g_rg_ckj_^uqlhoZjZdl_jgh^eymkeh\bcnmgdpbhgbjh\Zgbykeh`guo b[hevrbokbkl_fDqbkemlZdbojZg__jZkkfhlj_gguo\hijhkh\hlghkyl ky\hijhkuZgZebaZij_h[jZah\Zgbckb]gZeh\bbgnhjfZpbhgguoijhp_k kh\ \ ebg_cguo b g_ebg_cguo a\_gvyo jZ^bhljZdlh\ gZ ljZkk_ jZkijh kljZg_gby jZ^bh\heg DIB b l^ < j_amevlZl_ kha^ZgZhkgh\Ziha\heyx sZy \ khhl\_lkl\bb k hkgh\guf ijbgpbihf nmgdpbhgZevgh]h fh^_ebjh \Zgby \ jZfdZo ^Zggh]h dmjkZ i_j_clb d jZkkfhlj_gbx ijbgpbih\ ih kljh_gbyfZl_fZlbq_kdbobpbnjh\uofh^_e_cBK\\h^ymijhs_ggu_^_ l_jfbgbjh\Zggu_ beb \_jhylghklgu_ nmgdpbhgZevgu_ wd\b\Ze_glu \aZbfhk\ya_ckbkl_fuk\g_rg_ckj_^hcZlZd`_f_`^mwe_f_glZfb\gml jb kZfhc kbkl_fu < ^Zgguo f_lh^bq_kdbo mdZaZgbyo jZkkfZljb\Zxlky jZaghh[jZagu_f_lh^ubih^oh^ukbkl_fgh]hfh^_ebjh\Zgbyh[t_^bg_g gu_ \ jZfdZo i_j\hc - h[s_c qZklb qblZ_fh]h dmjkZ Ih l_dklm ^Zxlky ijbf_ju b dhgljhevgu_ aZ^Zgby kihkh[kl\mxsb_ m]em[e_gghfm bamq_ gbxij_^f_lZ

3 Bkoh^gu_iheh`_gbyf_lh^heh]bbkbkl_fgh]hfh^_ebjh\Zgby Hkgh\gu_ihgylbyB_jZjobq_kdb_hibkZgbykeh`guokbkl_f Kbkl_fZ - p_ehklgh_ hj]Zgbqgh _^bgh_ h[jZah\Zgb_ khklhys__ ba fgh`_kl\Zwe_f_glh\,gZoh^ysboky\hlghr_gbyobebk\yayo^jm] k^jm ]hf P_e_\h_ gZagZq_gb_ kbkl_fu - `_eZ_fuc j_amevlZl dhlhjuc ^he`_g [ulv ^hklb]gml\ijhp_kk_nmgdpbhgbjh\Zgbykbkl_fu KljmdlmjZ - hibkZgb_ kh\hdmighklb mklhcqb\uo k\ya_c f_`^m we_ f_glZfbkbkl_fu ey hibkZgbb j_Zevghc kbkl_fufh]mlbkihevah\Zlvky\k_lbiub_jZjobb Ijb jZkkfhlj_gbb \hijhkh\ kbgl_aZ b ZgZebaZ keh`guo BK h[uqgh ijh\h^blky bo ^_dhfihabpby gZ kbkl_fu gbar_]h mjh\gy dhlhju_gZau \Zxlih^kbkl_fZfbbdhlhju_\k\hxhq_j_^vh[eZ^ZxlnmgdpbhgZevghc p_ehklghklvx \uihegyy hij_^_e_ggmx hi_jZpbx h[_ki_qb\Zxsmx j_Z ebaZpbx h[s_c p_e_\hc nmgdpbb kbkl_fu >Zevg_crZy ^_dhfihabpby ih^kbkl_fijb\h^bldh[jZah\Zgbxb_jZjobb\dhlhjhch[uqgh\u^_ey xl q_luj_ mjh\gy: mjh\_gv kZfhc kbkl_fu; mjh\_gv ih^kbkl_f; mjh\_gv l_ogbq_kdbo mkljhckl\ beb nmgdpbhgZevguo a\_gv_\; mjh\_gv ko_fguo we_f_glh\ y\eyxsboky \ ^Zgghf kemqZ_ ij_^_evgufb _^bgbpZfb b_jZj obq_kdh]hhibkZgby
4 e_gb_ kbkl_f ih^kbkl_f b l^ gZ qZklb gh b bkihevah\Zgb_ jZaebqguo Zki_dlh\hibkZgbykbkl_fuihgbfZgby__dZdh[t_dlZfZl_jbZevgh]hfb jZGZkZfhf\_jog_fmjh\g_qZklhjZkkfZljb\Z_lkyijhp_kknmgdpbhgb jh\Zgby kbkl_fu dZd km[t_dlZ wdhghfbdb b li Ke_^mxsbc mjh\_gv ih a\hey_l hibku\Zlv ijhp_kk ij_h[jZah\Zgby bgnhjfZpbb ijb nmgdpbhgb jh\Zgbb ih^kbkl_f Mjh\_gv a\_gv_\ h^gh\j_f_ggh ij_^iheZ]Z_l hibkZ gb_ij_h[jZah\Zgbckb]gZeh\-ghkbl_e_cbgnhjfZpbbGbarbcmjh\_gvmjh\_gv hibkZgby nbabq_kdbo ijhp_kkh\ ijhl_dZxsbo \ we_f_glZo kbk l_fu FZl_fZlbq_kdb_hij_^_e_gbykbkl_fbfh^_e_ckbkl_f BKdZdex[Zykeh`gZykbkl_fZhij_^_ey_lkylj_fyhkgh\gufbdZl_ ]hjbyfb: we_f_glu hlghr_gby k\hckl\Z H^ghagZqgh_ b ihegh_ aZ^Zgb_ wlbodZl_]hjbcnZdlbq_kdbhij_^_ey_lkbkl_fm:__kljmdlmjmp_e_\h_gZ agZq_gb_ dZq_kl\Z b l^ JZkkfhlj_gb_ \hafh`guo kihkh[h\ nhjfZebah \Zggh]h hibkZgby kbkl_fu h^ghagZqgh k\yaZgh k ihgylb_f fh^_eb Fh ^_ev eZl ©f_jZª wlh h[t_dl - aZf_gbl_ev h[t_dlZ hjb]bgZeZ h[_ki_qb \Zxsbc bamq_gb_ gZb[he__ kms_kl\_gguo \ bgl_j_kmxs_f Zki_dl_ k\hckl\ ihke_^g_]h b gZh[hjhl iha\heyxsbc Z[kljZ]bjh\Zlvky hl _]h g_kms_kl\_gguo oZjZdl_jbklbd IjbfZl_fZlbq_kdhcnhjfZebaZpbbhib kZgby kbkl_f jZaebqZxl ^\Z ih^oh^Z: kljmdlmjguc \ jZfdZo dhlhjh]h hkh[h_ \gbfZgb_ m^_ey_lky ZgZebam \gmlj_ggbo khklhygbc kbkl_fu b nmgdpbhgZevguc\jZfdZodhlhjh]hkhkj_^hlZqb\ZxlkygZbkke_^h\Zgbb \g_rgboijhy\e_gbckbkl_fudZdij_h[jZah\Zl_ey©\oh^-\uoh^ªZkZfZ kbkl_fZjZkkfZljb\Z_lkydZd©q_jgucysbdª ;bgZjgufhlghr_gb_f RgZfgh`_kl\Zoijhba\hevghcijbjh^uX,Y gZau\Z_lky ih^fgh`_kl\h ^_dZjlh\Z ijhba\_^_gby wlbo fgh`_kl\ 5 ⊂ ; × < ?keb iZjZ we_f_glh\ [ ∈ ;\ ∈ < lZdh\Z qlh ([ \) ∈ 5 lh wlhl nZdl h[hagZqZxl lZd`_ ke_^mxsbf h[jZahf: xRy. Imklv ; = {; L ∈, } k_f_ckl\h fgh`_kl\ ]^_ In fgh`_kl\h p_ehqbke_gguo bg^_dkh\ Lh]^Z \ khhl\_lkl\bbkl_hj_lbdh-fgh`_kl\_ggufih^oh^hfh[sZykbkl_fZS,aZ ^ZggZy gZ wlhf k_f_ckl\_ hij_^_ey_lky dZd n-Zjgh_ hlghr_gb_ beb ih^ fgh`_kl\h^_dZjlh\Zijhba\_^_gby ; L ∈, , 6 ⊂ ; × ; ×× ; . >Zggh_h[s__hij_^_e_gb_\aZ\bkbfhklbhllh]hdZdbfkh^_j`Zgb _fgZ^_eyxlkyfgh`_kl\Z ; L ∈ , iha\hey_lihemqblvhij_^_e_gb_kbk l_fu \ khhl\_lkl\bb k l_f beb bguf ih^oh^hf <\_^_f ihgylb_ mgb\_j kmfZ ih^ dhlhjuf ihgbfZ_lky [Zah\h_ fgh`_kl\h kh^_j`Zs__ \k_ we_ f_gludZdhc-eb[hij_^f_lghch[eZklbImklvM -mgb\_jkmfwe_f_glh\; T - mgb\_jkmfhlghr_gbcf_`^mgbfb; V - mgb\_jkmfk\hckl\j_Zebam_fuo gZwlbohlghr_gbyolh]^Zkbkl_fZS 6 ⊂ 0 × 7 × 9
L

L

Q

Q

L

L

L

Q

Q

5 ]^_Ys - mgb\_jkmfkbkl_f; W = M×T - mgb\_jkmfkljmdlmjguooZjZdl_jb klbdZV - mgb\_jkmfiZjZf_ljh\kbkl_fh[_ki_qb\Zxsbog_h[oh^bfuc mjh\_gv^_lZebaZpbbhibkZgbywe_f_glh\bk\ya_cf_`^mgbfb Hij_^_ebf l_i_jv X dZd fgh`_kl\h \oh^guo h[t_dlh\ Z Y - \uoh^ guo h[t_dlh\. Lh]^Z ijb j_ZebaZpbb nmgdpbhgZevgh]h ih^oh^Z kbkl_fZ hij_^_ey_lkydZd[bgZjgh_hlghr_gb_ 6 ⊂ ; × < QZklh\f_klh[bgZjgh]h hlghr_gbyijbwlhfbkihevam_lkyhlh[jZ`_gb_bebnmgdpby 6 ; → < I; → < ( [ \) ∈ 6 → \ I[ , lh_klvh^ghagZqgh_ij_h[jZah\Zgb_\oh^-\uoh^^eydhlhjh]hijbex[uo x1, x2, x1Sy1 Λ x1Sy2 → y1=y2. >ey hij_^_e_gby fh^_e_c kbkl_f bkihevamxlky ihgylby fhjnbafh\ Fhjnbaf hagZqZ_l khhl\_lkl\b_ lh _klv ijZ\beh ih dhlhjhfm dZ`^hfm we_f_glm ijhkljZgkl\Z K' klZ\blky \ khhl\_lkl\bb we_f_gl ijhkljZgkl\Z K"ey h[s_]h hij_^_e_gby kbkl_f \\h^blky ke_^mxs__ ihgylb_ fh^_eb Imklv 6 ⊂ ; × < b 6 ⊂ ; × < b hx:X→X hy: Y→Y hlh[jZ`_gby Lh]^Z hlh[jZ `_gb_ K ([ \) (K ([) K (\)) gZau\Z_lky]hfhjnbafhfhlghr_gbcba S \

[\

[

\

S ZS fh^_evxkbkl_fu_keb ∀([ \) ([ \) ∈6 → K ([) K (\) ∈ 6 .

(

[

\

)

Imklv kbkl_fZ S⊂ W×V _klv ih^fgh`_kl\h ^_dZjlh\Z ijhba\_^_gby fgh`_kl\ kljmdlmjguo oZjZdl_jbklbd b iZjZf_ljh\ kbkl_fu DZ`^uc we_f_gl S [m^_fgZau\Zlv\ZjbZglhfkbkl_fuhij_^_ey_fuciZjhc (w,v) bh[hagZqZlv 6 6(Z Y) ∈ 6 AZ^Zqbkbgl_aZbZgZebaZkeh`guokbkl_f Kh\hdmighklvaZ^Zq\hagbdZxsbo\k\yabkbkke_^h\Zgb_fkeh`guo kbkl_fjZa[b\Z_lkygZ^\ZdeZkkZ:ZgZebabkbgl_a AZ^ZqZZgZebaZkhklhbl\bamq_gbbk\hckl\kbkl_fu_kebaZ^Zgu: oZ jZdl_jbklbdb\g_rg_ckj_^uZlZd`_ex[uokljmdlmjguobiZjZf_ljbq_ kdbo g_hij_^_e_gghkl_c \ \b^_ mihjy^hq_ggh]h gZ[hjZ ^Zgguo ω ∈Ω ; kljmdlmjZbiZjZf_lju Z ∈ : Y ∈ 9 \ZjbZglZkbkl_fu 6 ∈6 ;hp_ghqgh_ hlh[jZ`_gb_\b^Z .6 × Ω → 8 ,]^_ 8 -fgh`_kl\hagZq_gbchp_ghq ghc nmgdpbb hibku\Zxsbo gZ[hj ba n ihdZaZl_e_c wnn_dlb\ghklb kbk l_fu ZY

ZY

Q

Q

6 < oh^_ j_r_gby aZ^Zqb kbgl_aZ lj_[m_lky gZclb \ZjbZgl 6 ∈ 6 , m^h\e_l\hjyxsbc j_kmjkguf 4 6 = 4 Z Y ≤ 4 b djbl_jbZevguf

( )

(

(

)

YZ

. 6 ω ≥ . (ω ) h]jZgbq_gbyflZdhcqlh^ey

YZ

∀ω ∈Ω ∀6

ZY

(

)

ZY

)

(

)

. 6 ω 5 i . 6 ω ,

∈ 6 :

YZ

YZ

(1)

]^_ 5 i ijZ\beh ij_^ihql_gby \ \b^_ [bgZjgh]h hlghr_gby ihjy^dZ ©[ulv emqr_ª gZ 8 Fgh`_kl\h 6 ^ ⊆ 6 6 ^ {6 4(6 ) ≤ 4 } gZau\Z _lky fgh`_kl\hf ^himklbfuo kbkl_f Fgh`_kl\h 6 k^ ⊆ 6 ^ ⊂ 6 6 k^ {6 .(6 ω ) ≥ . (ω ) ω ∈ Ω 6 ∈6 ^ } gZau\Z_lkyfgh`_kl\hfkljh ]h^himklbfuokbkl_f >ey ih^Z\eyxs_]h [hevrbgkl\Z keh`guo kbkl_f j_r_gb_ aZ^Zqb kbgl_aZ \ ihklZgh\d_ g_ fh`_l [ulv ihemq_gh iml_f h^ghc nhjfZevgh hij_^_e_gghc hi_jZpbb hilbfbaZpbb Ihwlhfm \ oh^_ j_Zevgh]h kbgl_aZhkms_kl\ey_lkykbkl_fgucih^oh^]eZ\gZyb^_ydhlhjh]hkhklhbl \ khq_lZgbb nhjfZevguo b g_nhjfZevguo w\jbklbq_kdbo ijb_fh\ gZ ijZ\e_gguo gZ jZkqe_g_gb_ ^_dhfihabpbx ijhp_kkZ j_r_gby keh`ghc ijh[e_fugZhl^_evgu_khklZ\eyxsb_ghijbwlhflZdbfh[jZahfqlh[u gZklhevdhgZkdhevdhwlh\hafh`ghmq_klvk\yabf_`^mgbfbbkhojZgblv p_ehklgh_ij_^klZ\e_gb_hkbkl_f_Ijbwlhf\\h^blkyke_^mxsZyb_jZj obykeh_\ijbgylbyj_r_gbc\jZfdZodZ`^h]hbadhlhjuonbdkbjmxlky iZjZf_ljuj_r_gbyh[s_caZ^ZqbihqZklyf: 1) kehc\u[hjZfgh`_kl\Zg_hij_^_e_gghkl_cbZevl_jgZlb\guokljZ l_]bc gZ dhlhjhf aZ^Z_lky ^bkdj_lgh_ fgh`_kl\h \ZjbZglh\ kbkl_fu b bkoh^guo^Zgguo $ 6 = {6 Y ∈ 9 Z ∈ : ω ∈ Ω } ; Q

ZY

ZY

ZY

VZ

ZY

D

ZY

D

D

D

2) kehc h[mq_gby beb Z^ZilZpbb aZ^Zq_c dhlhjh]h y\ey_lky km`_gb_ fgh`_kl\Zg_hij_^_e_gghkl_caZkq_lbamq_gby\g_rg_ckj_^ubgZdhlh jhfnZdlbq_kdbijh\h^blkyZgZebadZ`^h]hbaZevl_jgZlb\guo\ZjbZglh\ kbkl_fu 2 . 6 ω 4 6 6 ∈ 6 ω ∈Ω ;

{(

D

D

ZY

) ( ) D

D

ZY

ZY

D

D

D

}

3) kehc kZfhhj]ZgbaZpbb gZ dhlhjhf hdhgqZl_evgh \u[bjZxlky kljmdlmjubkljZl_]bbbkke_^h\Zggu_gZgb`_klhysbomjh\gyolZdqlh [uih\hafh`ghklbijb[ebablvkyd]eh[Zevghcp_eb % . 6 ω → opt, 6 ∈ 6 k^ ω ∈ Ω .

(

D

ZY

D

)

D

5

Q

ZY

D

D

B_jZjobx ih^ijhp_kkh\ ijbgylby j_r_gbc ij_^klZ\ey_l ko_fZ jbk NZdlbq_kdb hgZ j_Zebam_l j_r_gb_ aZ^Zqb kbgl_aZ ©q_j_a ZgZebaª K bk

7 ihevah\Zgb_fw\jbklbdbijb]_g_jZpbbfgh`_kl\ZZevl_jgZlb\guo\Zjb Zglh\^Zevg_crZy^_lZebaZpbyijhp_kkZijbgylbyj_r_gbcihkbgl_am keh`ghc kbkl_fu hkms_kl\ey_lkygZhk gh\_ kljZlbnbdZpbb kbkl_fu lh _klv < \u^_e_gbyjZaebqguomjh\g_cZ[kljZ]b jh\Zgbyijb__hibkZgbbZlZd`_wlZih\ kha^Zgbykbkl_fu\h\j_f_gbIjbwlhf H nZdlbq_kdb hkms_kl\ey_lky ijh_dpby fgh]hkehcgh]h ijhp_kkZ ijbgylby j_ r_gbc gZ ihke_^h\Zl_evgh ^_lZebabjm_ : fh_ b mlhqgy_fh_ ij_^klZ\e_gb_ kbkl_ Jbk fu ]^_ i_j_oh^ gZ dZ`^uc wlZi khijh \h`^Z_lkyi_j_oh^hfgZgh\mxkljZlmbf_xsmx[he__gbadbcmjh\_gv Z[kljZdpbbBeexkljbjm_lwlhlijhp_kkko_fZjbk

Jbk
8 kh\hdmighklb_^bgbpwe_f_glghc[ZauK >ey H>K aZ^Zxlky ijh kljZgkl\Z \oh^guo ; = ; ×× ; \ha^_ckl\bc \ha^_ckl\bc \g_rg_c kj_^u Ω = Ω ××Ω Ω \gmlj_ggbo iZjZf_ljh\ kbkl_fu + = + ×× + ,

1;

1

1+

y\eyxsboky g_aZ\bkbfufb wdah]_ggufb i_j_f_ggufb bf_xsbfb dZd ^_l_jfbgbjh\Zggu_lZdbklhoZklbq_kdb_khklZ\eyxsb_bijhkljZgkl\h \uoh^guobaZ\bkbfuowg^h]_gguo i_j_f_gguo < = < ×× <

[ W = [ W [

1;

ω W = ω W ω

W ∈ ; 7

W ∈ Ω 7

1

Ω

K W = K W K

1+

\ W = \ W \

W

1<

1<

7

W

7

∈ + ∈ <

Lh]^Zijhp_kknmgdpbhgbjh\ZgbyH>Khibku\Z_lky\h\j_f_gb W ∈ 7 hi_ jZlhjhf y(t)= Fs(xt,ωt,ht,t), ]^_ xt,ωt,ht j_ZebaZpbb ijhp_kkh\ x(t), ω(t), h(t) gZ \k_f ij_^r_kl\mxs_f bgl_j\Ze_ \j_f_gb QZklh \\h^yl lZd`_ ijh kljZgkl\h\gmlj_ggbokhklhygbckbkl_fu = = = ×× = ,bf_y\\b^mqlh

1=

\_dlhj\gmlj_ggbokhklhygbc ] W = ] W ] W ∈ = \ex[hcfhf_glt 7

1=

9 : t0≤t iheghklvxhij_^_ey_lkyagZq_gbyfb ] = ] W ]

W ZlZd 7

1=

`_j_ZebaZpbyfbx(t),ω(t),h(t)gZbgl_j\Ze_[t0,t]. Km^h[ gh hibku\Zlv ^\mfy hi_jZlhjZfb khklZ\eyxsbfb p_ihqdm ©\oh^khklhygb_-\uoh^ª ] W = N ] [ W K W ω W W \W W W W

N ] W W .

LZdbf h[jZahf H>K dZd fZl_fZlbq_kdZy fh^_ev we_f_glZ keh`ghc kbk l_fu \ h[s_f kemqZ_ hij_^_ey_lky \hkvf_jdhc fgh`_kl\ b hlh[jZ`_gbc H>K .Fh^_evH>Ky\ey_lky^hklZlhqghZ[kljZdl ghcIhwlhfm^eyihkljh_gbyfZl_fZlbq_kdbofh^_e_cwe_f_glh\bkihev amxl [he__ ijhklu_ fZl_fZlbq_kdb_ ko_fu hlh[jZ`Zxsb_ jZaebqgu_ ih^oh^udmijhs_gbxhibkZgby 1. G_ij_ju\gh - ^_l_jfbgbjh\Zgguc ih^oh^ D - ko_fu < jZfdZo ^Zggh]h ih^oh^Z \ha^_ckl\by klhoZklbq_kdh]h oZjZdl_jZ g_ mqblu\Zxlky b \uoh^gu_ \ha^_ckl\by h^ghagZqgh hij_^_eyxlky \oh^gufb \ha^_ckl \byfbaZ^Zggufb\g_ij_ju\ghf\j_f_gb>eyhibkZgby\gmlj_ggbokh klhygbc \ D -ko_f_lbibqghbkihevah\Zgb_h[udgh\_gguo^bnn_j_gpb ZevguomjZ\g_gbc G] = JW ]W [W ]W GW

]

+

W ∫ J τ ]τ [τ Gτ , y(t)=f(z(t)). W

LZdbfh[jZahfD -ko_fZhij_^_ey_lkyr_kl_jdhc.FZl_fZ lbq_kdZyfh^_evD -ko_fu\hkgh\ghfhij_^_ey_lbfh^_ebjmxsbcZe]h jblf__BF. 2. >bkdj_lgh-^_l_jfbgbjh\Zggucih^oh^F -ko_fu Ijbj_ZebaZ pbb ih^oh^Z oZjZdl_jgh hlkmlkl\b_ \hafms_gbc klhoZklbq_kdh]h oZjZd l_jZbaZ^Zgb_nmgdpbc\oh^Zkhklhygbyb\uoh^Z\^bkdj_lghf\j_f_gb Lbih\hcfZl_fZlbq_kdhcko_fhc\^ZgghfkemqZ_y\ey_lkyko_fZdhg_qgh ]h Z\lhfZlZ m dhlhjh]h fgh`_kl\Z X,Y,Z bf_xldhg_qgh_qbkehwe_f_g lh\ >ey aZ^Zgby Z[kljZdlgh]h dhg_qgh]h Z\lhfZlZ lj_[m_lky hij_^_eblv hlh[jZ`_gbyi_j_oh^h\b\uoh^h\ ψ : X×Z → Y. ϕ : X×Z → Z; GZijbf_j^eyF -Z\lhfZlZi_j\h]hjh^Zijhp_kknmgdpbhgbjh\Zgbyhi j_^_ey_lky\ujZ`_gb_f z(t+1)=ϕ[z(t),x(t)], y(t)=ψ[z(t),x(t)], [ W ∈; [ [ . ]W ∈ ] ] \W ∈ < \ \ , t=0,1,2.

{

,

}

{

-

}

{

.

}

< h[s_f kemqZ_ F - ko_fZ hij_^_ey_lky iylvx khklZ\eyxsbfb <X,Z,Y,ϕ,ψ>. AZ^Zgb_hlh[jZ`_gbc^eyF -ko_fhkms_kl\eyxl\\b^_lZ[ ebpkhhl\_lkl\byi_j_oh^h\b\uoh^h\beb]jZnh\i_j_oh^h\FZl_fZlb

10 q_kdZyfh^_ev^eyF - ko_funZdlbq_kdb^Z_lbihegh_hibkZgb_fh^_eb jmxs_]hZe]hjblfZ 3. >bkdj_lgh-klhoZklbq_kdbc ih^oh^ P-ko_fu >Zgguc ih^oh^ oZ jZdl_jbam_lky klhoZklbq_kdbf oZjZdl_jhf ij_h[jZah\Zgby bgnhjfZpbb npb aZ^Zgbbkbkl_fu\^bkdj_lghf\j_f_gbLbih\hcko_fhca^_kvy\ey _lkyko_fZ\_jhylghklgh]hZ\lhfZlZmdhlhjh]hfgh`_kl\ZX, Y, Z, dZdb\ ij_^u^ms_fkemqZ_y\eyxlkydhg_qgufb, ]^_Bk_f_ckl\hK×I fZljbpjZaf_jgh klbJ×J % = % S .U , , ijbq_fwe_f_glZfbfZljbpuBp,r y\ey

{

}

S U

xlky\_jhylghklbi_j_oh^h\ E =P(yp, zj/zi,xr), L M = - . NZdlbq_kdb ^Zggh_ hibkZgb_ wd\b\Ze_glgh aZ^Zgbx kemqZcguo aψ a , hij_^_e_gguogZl_o`_fgh`_kl\Zoqlhbnmgdpbbϕ, ψ\ nmgdpbc ϕ F-ko_f_ghk\\_^_gb_fkhhl\_lkl\mxsbo\_jhylghklguof_jAZ^Zgb_Pko_fZgZeh]bqghfh`_lhkms_kl\eylvky\\b^_lZ[ebpbeb]jZnh\^hihe g_gguo agZq_gbyfb \_jhylghkl_c i_j_oh^h\ A^_kv dZd b \ ij_^u^msbo kemqZyo fZl_fZlbq_kdZy fh^_ev ^Z_l ihegh_ hibkZgb_ fh^_ebjmxs_]h Ze]hjblfZBFdhlhjuc^he`_g[ulvlhevdh^hiheg_g]_g_jZlhjZfbklh oZklbq_kdhcdhfihg_glui_j_oh^h\ G_ij_ju\gh-klhoZklbq_kdbcih^oh^Q-ko_fu >ey^Zggh]hdeZkkZ fh^_e_cbf__lf_klhklhoZklbq_kdbcoZjZdl_jkf_gukhklhygbcijbhib kZgbb nmgdpbc ″\oh^-khklhygb_-\uoh^″ \ g_ij_ju\ghf \j_f_gb < dZq_ kl\_ klZg^Zjlgh]h h[t_dlZ \ ^Zgghf kemqZ_ jZkkfZljb\Z_lky we_f_glZj gucijb[hjh[kem`b\ZgbyWIH ihlhdZaZy\hdbebQ-ko_fZLZdhcijb [hjkfjbk khklhblbagZdhibl_eyaZy\hdH\dhlhjhffh`_l gZoh^blvky O = / aZy\hd b U dZgZeZh[kem`b\ZgbyDGZdZ`^uc K Y WIH ihklmiZ_l ihlhd aZy\hd W H Z = {(W I ) Q = } ∈ : b ihlhd S U LM

+

{(W

Q

Q

}

Jbk h[kem`b\Z gby. AZy\db h[kem`_ggu_ dZgZehf K, b aZy\db ihdbgm\rb_ ijb[hj ih jZaguf ijbqbgZf gZijbf_j ba-aZ i_j_iheg_gby gZdhibl_ey h[jZamxl \uoh^ghcihlhd \ = W T M = ∈ < A^_kvt∗ - fhf_glu\j_f_gbkh X=

N

J ) N = ∈ 8 N

{(

M

M

)

}

[ulbc Z f∗, g∗, q∗ - ijbagZdb kh[ulbc <_dlhj \gmlj_ggbo khklhygbc ^ey

(

WIHbf__l\b^ ] = ] g ] d

)

L

∈ = ]^_ ]g ∈ { / g } -khklhygb_gZdhibl_ey

ih qbkem ojZgysboky aZy\hd; zd - khklhygb_ dZgZeZ h[kem`b\Zgby zd=1 dZgZek\h[h^_g; zd=0 - dZgZeaZgyl >eyWIHlZd`_aZ^Z_lkykh\hdmighklv \gmlj_ggbo iZjZf_ljh\ C b Ze]hjblf nmgdpbhgbjh\Zgby : hij_^_eyx

11 sbcgZ[hjijZ\beihdhlhjufaZy\dbihdb^ZxlGbDLZdbfh[jZahfQ ko_fZhij_^_ey_lkygZ[hjhf^Zgguo<W,U,Y,Z,C,A>. Kbkl_fuwe_f_glu dhlhjuohibku\ZxlkyQ -ko_fZfbbdhlhju_gZau\Zxlkykbkl_fZfbfZk kh\h]hh[kem`b\Zgbymqblu\Zybo\Z`gh_agZq_gb_[m^ml^Ze__jZkkfhl j_guih^jh[gh H[h[s_ggucih^oh^:-ko_fu >Zggucih^oh^\[bjZ_l\k_[y\k_ jZkkfhlj_ggu_\ur_by\ey_lkyboh[h[s_gb_fIjbwlhf\\h^blkyih gylb_ Z]j_]ZlZ beb :-ko_fZ ij_^klZ\eyxsZy kh[hc fZl_fZlbq_kdmx fh ^_ev we_f_glZ keh`ghc kbkl_fu b \\_^_ggZy GI ;mke_gdh ki_pbZevgh ^eykha^Zgbymgb\_jkZevguoZ\lhfZlbabjh\ZgguobfblZpbhgguofh^_e_c keh`guokbkl_fHkh[_gghklvx:-ko_fuy\ey_lkylhqlhbag__dZdqZ klgu_ kemqZb fh]ml [ulv ihemq_gu D, F, P, Q - ko_fu FZl_fZlbq_kdZy fh^_evZ]j_]ZlZ[m^_lih^jh[ghjZkkfhlj_gZ\jZa^_e_ihk\ys_gghf\h ijhkZffh^_ebjh\ZgbyZ]j_]Zlb\guokbkl_f 2.2 FZl_fZlbq_kdb_fh^_eb\aZbfh^_ckl\by\keh`ghckbkl_f_ IjbnhjfZebaZpbb\aZbfh^_ckl\bywe_f_glh\keh`ghckbkl_fuijb ^_j`b\Zxlky ij_^iheh`_gby kh]eZkgh dhlhjhfm hgh ^hklZlhqgh ihegh hibku\Z_lky \ jZfdZo f_oZgbafZ h[f_gZ © kb]gZeZfbª Kb]gZeu i_j_^Z xlkyihdZgZeZfk\yabijheh`_gguff_`^mwe_f_glZfbGZqZehf^Zggh ]hdZgZeZy\ey_lky\uoh^\uoh^ghcihexk we_f_glZ\u^Zxs_]hkb]gZe Zdhgphf-\oh^\oh^ghcihexk we_f_glZijbgbfZxs_]hkb]gZeDZgZe i_j_^Zxsbckb]gZeuf]gh\_gghb[_abkdZ`_gbcgZau\Z_lkyb^_Zevguf :^_d\ZlghnhjfZebah\ZggZykeh`gZykbkl_fZbf__llhevdhb^_Zevgu_dZ gZeu k\yab Nbabq_kdb_ dZgZeu \ghkysb_ bkdZ`_gby \ i_j_^Z\Z_fu_ kb]gZeu jZkkfZljb\Zxlky dZd kZfhklhyl_evgu_wd\b\Ze_glgu_we_f_glu kbkl_fu Ijb lZdhf ih^oh^_ hkgh\guf \ hibkZgbb f_oZgbafZ h[f_gZ kb]gZeZfb klZgh\blky hij_^_e_gb_ ko_fu khijy`_gby we_f_glh\ ih \oh ^Zfb\uoh^Zf Imklvkeh`gZykbkl_fZSkh^_j`blwe_f_gluC1,…,CN.>eyex[h]hba we_f_glh\ Cj \oh^ghckb]gZe [ W = [ [ ij_^klZ\ey_lkh\hdmi ghklv we_f_glZjguo \oh^guo kb]gZeh\ ih^dexq_gguo dh \oh^guf dhg lZdlZf ; L P Z\uoh^ghc \ W = \ \ -kh\hdmighklvwe_ f_glZjguo \uoh^guo kb]gZeh\ \uoh^ysbo ba \uoh^guo dhglZdlh\ < O U
M

M

M

PM

M

L

M

7

M

M

UM

7

M

O

M

C0 kh^_j`Zs_]h \oh^gu_ dhglZdlu ; L P b \uoh^gu_ dhglZdlu < O U LZdbfh[jZahfdZ`^ucwe_f_glkeh`ghckbkl_fuCjoZjZd L

O

l_jbam_lky fgh`_kl\hf \oh^guo dhglZdlh\ ; dhglZdlh\ <

M

{ }

= <

U

M

O

O

M

M

=

M

{ }

= ;

P

M

L

L

M

=

b \uoh^guo

1 Hkgh\ghc ijbgpbi ihkljh_gby ko_fu

12 khijy`_gby we_f_glh\ hij_^_ey_lky ke_^mxsbf h[jZahf: dh \oh^ghfm dhglZdlm ex[h]h we_f_glZ ih^dexqZ_lky g_ [he__ h^gh]h we_f_glZjgh]h dZgZeZ k\yab; d \uoh^ghfm dhglZdlm ex[h]h we_f_glZ fh`_l [ulv ih^ dexq_ghex[h_dhg_qgh_qbkehwe_f_glZjguodZgZeh\Lh]^Zfh`gh\\_ klbh^ghagZqgh_hlh[jZ`_gb_ 1

5 ; → < ;

1

; <

<

M

M

M

M

.

Kh\hdmighklv <X,Y,R> gZau\Z_lky ko_fhc khijy`_gby we_f_glh\ \ kbkl_f_S.Hi_jZlhjkhijy`_gby RaZ^Z_lky\\b^_lZ[ebpu\dhlhjhcgZ i_j_k_q_gbbkljhdkghf_jZfbwe_f_glh\j bklhe[ph\kghf_jZfbdhglZd lh\ijZkiheZ]Z_lkyiZjZqbk_ek,l), mdZau\Zxsboghf_jwe_f_glZbghf_j dhglZdlZ < kdhlhjufkh_^bg_g ; . N

M

O

{ }

Imklv ; = ;

M L

L

{ }

⊂ ; b < = <

N

O

⊂ < g_dhlhju_ ih^fgh`_kl\Z

ijbq_f < = 5 ; ; ∈ ; lh _klv < = 5 ; Ijb wlhf hq_\b^gh qlh khhl\_lkl\b_ f_`^m < b ; g_ y\ey_lky \aZbfgh h^ghagZqguf JZkkfhljbfl_i_jvhi_jZlhjRj gZfgh`_kl\_X(j). Lh]^Z\kbemhkgh\gh]h ijbgpbiZ khijy`_gby ^ey ex[h]h 5 M 1 kms_kl\m_l h^ghagZqguc N

O

M

M

L

L

N

M

O

L

M

−

hi_jZlhj 5 khihklZ\eyxsbc dZ`^hfm dhglZdlm ba fgh`_kl\Z

M

< = 5 ; khhl\_lkl\mxsbcdhglZdl ;

M

M

< <

<

∧

;

∧

∧

;

<

;

;

C1

;

<

<

<

;

;

S1

<

;

∧

L

= 5 − <

M

N

O

<

;

N

O

< ∧

M

C3

∈< .

;

<

;

<

<

;

S2

C2

;

C4 <

;

∧

<

;

<

Jbk Imklv l_i_jv keh`gZy kbkl_fZ jZkqe_g_gZ gZ g_dhlhjh_ fgh`_kl\h ih^kbkl_f 6µ µ 0 (kf jbk DZ`^Zy ba ih^kbkl_f ^he`gZ bf_lv dhglZdlu ; µ b < µ oZjZdl_jbamxsb_ nbdlb\guc we_f_gl &µ ij_^ klZ\eyxsbc^eyg__\g_rgxxkj_^mZlZd`_\uoh^gu_dhglZdlu ; µ b L

O

L

13 µ ^eyk\yabk^jm]bfbih^kbkl_fZfbLZdih^kbkl_fZS1 bf__ldhglZd < lu < − ; oZjZdl_jbamxsb_ bklhqgbdb b ihlj_[bl_eb __ \g_rg_c kj_^u Z lZd`_ dhglZdlu ; − < − ^ey k\yab k ^jm]bfb ih^kbkl_fZfb Khhl\_lkl\mxsb_ dhglZdlu ; µ b < µ ijb i=l, Z lZd`_ ; µ b < µ ijb i=lh[t_^bgyxlky\nbdlb\gu_^\hcgu_dhglZdlugZ]jZgbpZoih^kbkl_f <\_^_f fgh`_kl\Z \oh^guo b \uoh^guo dhglZdlh\ \k_o we_f_glh\ & ∈ 6µ kh_^bg_gguokh\k_fbwe_f_glZfb & ∉6 µ

O

L

L

O

M

O

N

{ }

;µ = ;

]^_ ;

M N

M L

M

µ

[

= 5− <

M

  = ; ;  , <µ = < ∈µ ∉µ  M N

{ }

M

&

6

M

&

]

5 ; <

N

M

6

N

= < 5 ;

MN

M

N

  = < <  , ∈µ ∉µ  M

µ

O

&

M N

6

M

&

N

6

fgh`_kl\Z dhglZdlh\ kh

_^bgyxsbo ih \oh^Zf b \uoh^Zf we_f_glu & b & DZ`^hfm dhglZdlm N

M

<

M

O

∈ <µ fh`ghihklZ\blv\khhl\_lkl\b_dhglZdlu ; µ = 4 µ < , < µ = 4 µ < ;

<

M

L

;

M

L

O

O

]^_hi_jZlhju 4µ b 4µ aZ^ZxlkylZ[ebpZfbkhhl\_lkl\bcbbf_xlh^gh <

;

agZqgu_hi_jZlhju 4 µ − b 4 µ − LhqghlZd`_^eydZ`^hckh\hdmighklb <

{; } ⊂ ; M L

µ

;

bf_xs_ch[sbc <

N

O

;

µ

O

(

= 3µ ; ;

M L

) <

µ

O

( ) \\h^ylkyhi_jZlhju ( ; ) {;( )5( ; ) = < } .

=5;

= 3µ

<

M L

M

M L

M

L

N

L

O

Hi_jZlhju 3µ b 3µ g_bf_xlh^ghagZqguoh[jZlguohi_jZlhjh\gh^ey ;

<

µ fh`gh\u[jZlvijhba\hevghh^bgba\hafh`guo ; gZijb ^Zggh]h ; µ . f_jkfbgbfZevgufi bh[hagZqblv_]hkbf\hehf 3µ − ; M

O

;

(

O

)

L

GZhkgh\_\\_^_gguohi_jZlhjh\fh`ghbah^ghmjh\g_\hcko_fukh ijy`_gby<X, Y, R>ihkljhblv^\momjh\g_\mxko_fmkhklhysmxbako_fu < 5 bko_fkhijy`_gbywe_f_glh\\ih^kbk khijy`_gbyih^kbkl_f ; ,,

l_fZo 6µ : ; µ <µ 5 µ µ = 0 ]^_ 5 µ hij_^_ey_lky\\b^_

( )

<

N

O

(

= 5µ ;

M L

)

5 ; ; ∈ ;  ∈µ ∈µ  = < µ = 3µ ; ; ∈ ; ∈µK ∉µ  4 − ; µ ; = ; µ ∈ ; µ  µ µ → < = < µ hij_^_ey_lkykhhlghr_gb_f ;

,,

M L

M N

&

<

O

M

M

L

L

0

{ }

µ

L

6

M

M

6

M

L

L

L

0

µ=

{

O

}

&

N

6

M N

&

;

Hi_jZlhj 5 ;

M

L

N

6

{

L

}

14 <

ν

O

]^_ <

N

O

[

= 5 3µ

;

−

;

L

= 5 ; ,,

µ

]

4 < N ≠ & ∈6 ν =  ν < = < N <

L

µ

N

O

O

N

O

Lhqgh lZd `_ fh`_l [ulv j_r_gZ b h[jZlgZy aZ

^ZqZ gZoh`^_gby hi_jZlhjZ R ijb aZ^Zgghc ^\momjh\g_\hc ko_f_ khijy `_gby:gZeh]bqgh_ihkljh_gb_ko_fkhijy`_gbydZdfh^_e_c\aZbfhk\y a_cwe_f_glh\keh`ghckbkl_fufh`_lijh\h^blvky^eyex[h]hdhebq_kl \Zmjh\g_cb_jZjobq_kdh]hhibkZgbykeh`ghckbkl_fu Dhgljhevgu_\hijhkubaZ^Zgby 1. Hij_^_eblv kljmdlmjm nhjfZebah\Zggh]h lZ[ebqgh]h hibkZgby hi_jZ lhjZi_j_oh^Zkhklhygbcdhg_qgh]hZ\lhfZlZ 2. Hij_^_eblvkljmdlmjmnhjfZebah\Zggh]hhibkZgbyhi_jZlhjZi_j_oh^Z khklhygbcdhg_qgh]hZ\lhfZlZ\\b^_]jZnZ 3.
Jbk \a]ey^Zfb gZ kljmdlmjm kbkl_fu < h[h[s_gghc kljmdlmj_ KFH jbk \k_]^Zfh`gh\u^_eblvljbdhfihg_glu:bklhqgbdbnhjfbjmxsb_ihlhd aZy\hd; gZdhibl_eb hij_^_eyxsb_ \hafh`ghklb kha^Zgby hq_j_^_c aZy \hdgZh[kem`b\Zgb_;dZgZeuh[kem`b\Zgbyhij_^_eyxsb_\uoh^ghcih

15 lhd h[kem`_gguo aZy\hd beb aZy\hd ihdb^Zxsbo kbkl_fm g_h[kem`_g gufbDhgdj_lbaZpbyh[h[s_gghcko_fuKFHfh`_l[ulvijh\_^_gZim l_fdeZkkbnbdZpbb Ij_`^_\k_]h\k_KFH^_eylgZ^\Z[hevrbodeZkkZ:jZahfdgmlu_\ dhlhjuo \oh^ghc ihlhd g_ aZ\bkbl hl ijhp_kkZ h[kem`b\Zgby b aZfdgm lu_\dhlhjuokms_kl\m_lh[jZlgZyk\yavb\oh^ghcihlhdaZy\hdaZ\bkbl hlijhp_kkZh[kem`b\ZgbyIhfbfhwlh]h:ihgZebqbx\hafh`ghklbh`b ^ZgbyaZy\hd\hq_j_^b\u^_eyxlKFH[_ah`b^ZgbyKFHkhldZaZfbZ lZd`_kf_rZggh]hlbiZ;ihqbkemiZjZee_evgh\dexq_gguowe_f_glZjguo ijb[hjh\ h[kem`b\Zgby \u^_eyxl h^ghdZgZevgu_ b fgh]hdZgZevgu_ KFH;ihqbkemwlZih\nZa h[kem`b\Zgbyijbihke_^h\Zl_evghf\dex q_gbb ijb[hjh\ h[kem`b\Zgby hij_^_eyxl h^ghnZagu_ b fgh]hnZagu_ KFHbebk_lbfZkkh\h]hh[kem`b\Zgby Ex[Zy KFH ij_^gZagZq_gZ ^ey h[kem`b\Zgby \oh^gh]h ihlhdZ k aZ ^Zgghc wnn_dlb\ghklvx < dZq_kl\_ hkgh\guo ihdZaZl_e_c wnn_dlb\gh klb ^ey KFH k hldZaZfb bkihevamxl Z[khexlgmx ijhimkdgmx kihkh[ ghklvQ (kj_^g__qbkehh[kem`_gguoaZy\hdaZ\j_fyL bhlghkbl_evgmx ijhimkdgmx kihkh[ghklv beb \_jhylghklv h[kem`b\Zgby T = 4 λ ]^_ λ kj_^g__qbkehihklmib\rboaZy\hdaZ\j_fyL>eyKFHkh`b^Zgb_f^h ihegbl_evgh \ dZq_kl\_ ihdZaZl_e_c bkihevamxl \j_f_ggu_ oZjZdl_jb klbdbij_[u\ZgbyaZy\hd\hq_j_^bb\kbkl_f_\p_ehf 3.2. Hkh[_gghklbihkljh_gbybfblZpbhgguofh^_e_cKFH =_g_jZpbyihlhdh\kh[ulbc Kmsghklv f_lh^Z bfblZpbhggh]h fh^_ebjh\Zgby ijbf_gbl_evgh d KFHkhklhbl: 1)\ihkljh_gbbZe]hjblfh\nhjfbjh\Zgbyj_ZebaZpbcih lhdh\kh[ulbcaZy\hd lh_klvZe]hjblfh\©bklhqgbdh\ª; \ihkljh_gbb Ze]hjblfh\ ijhp_kkZ h[kem`b\Zgby hl^_evgufb we_f_glZfb dZgZeZfb gZdhibl_eyfb b KFH \ p_ehf; fgh]hdjZlghf \hkijhba\_^_gbb ijh p_kkZ h[kem`b\Zgby j_ZebaZpbc \oh^guo ihlhdh\ ijb nbdkbjh\Zgguo \gmlj_ggboiZjZf_ljZoKFHkp_evxklZlbklbq_kdhchp_gdbihdZaZl_e_c wnn_dlb\ghklb < jZfdZo bfblZpbhgghc fh^_eb KFH ijbgylh \u^_eylv ljb lbiZ we_f_glh\ fh^_eb: Zdlb\gu_ iZkkb\gu_ b Zdlb\gh-iZkkb\gu_ Baf_g_gb_ khklhygbc Zdlb\guo we_f_glh\ h[mkeh\e_gh lhevdh bo \gml j_ggbfb k\hckl\Zfb lh]^Z dZd iZkkb\gu_ we_f_glu baf_gyxl k\hb kh klhygby lhevdh ih^ \ha^_ckl\b_f \g_rgbo nZdlhjh\ D Zdlb\ghiZkkb\guf we_f_glZf hlghkylky lZdb_ dhlhju_ \ h^ghf ba \hafh`guo khklhygbcy\eyxlkyZdlb\gufbZ\^jm]boiZkkb\gufb Ih\_^_gb_kbkl_fijbBF\hkijhba\h^blkygZhij_^_e_gghfhlj_ad_ \j_f_gb[0,T]
16 ^mxsb_ hkgh\gu_ kihkh[u aZ^Zgby fh^_evgh]h \j_f_gb: kihkh[ © ∆W », ijbdhlhjhf\fh^_ebjmxs_fZe]hjblf_ijbjZs_gb_\j_f_gbhkms_kl\ eyxlknbdkbjh\Zggufbgl_j\Zehf ∆W ; kihkh[© ∆] ªijbdhlhjhfijbjZ s_gb_ fh^_evgh]h \j_f_gb hkms_kl\eyxl lhevdh \ fhf_gl gZklmie_gby hq_j_^gh]h kh[ulby k\yaZggh]h k baf_g_gb_f khklhygby ex[h]h ba we_ f_glh\kbkl_fub__\g_rg_ckj_^u;kihkh[© ∆W × ∆] ªj_Zebamxsbcdhf [bgZpbxwlbokihkh[h\\jZaebqguodhfihg_glZoBF g_aZ\bkbfu gZau\Z_lky ihlhdhf k h]jZgb q_gguf ihke_^_ckl\b_f Ihlhdhf [_a ihke_^_ckl\by [he__ kbevgh_ mkeh\b_ gZau\Z_lky ihlhd \ dhlhjhf \_jhylghklv 3 W τ gZklmie_gby k kh[ulbcgZbgl_j\Ze_ [W W + τ] g_aZ\bkblhlihy\e_gbykh[ulbcgZ^jm]bo g_i_j_k_dZxsboky k ^Zgguf bgl_j\ZeZo Ihlhd kh[ulbc gZau\Z_lky hj ^bgZjguf _keb \_jhylghklv ihy\e_gby [he__ h^gh]h kh[ulby \ fZehc hdj_klghklbfhf_glZtij_g_[j_`bfhfZeZihkjZ\g_gbxk\_jhylghklvx \uiZ^_gbyh^gh]hkh[ulby 3 (W ∆W ) >> 3 > W ∆W = Θ ∆W WlhhagZqZ_lqlh kh[ulby \ hkgh\ghf ijhbkoh^yl ih hl^_evghklb [_a h[jZah\Zgby k]mkl dh\ Ihlhd kh[ulbc gZau\Z_lkyklZpbhgZjguf\h\j_f_gb_keb\_jhyl ghklvihy\e_gbylh]hbebbgh]hqbkeZkh[ulbcgZnbdkbjh\Zgghfbgl_j \Ze_g_aZ\bkblhl\j_f_gb 3 (W τ) = 3 (τ) N = >ey hj^bgZjguo ihlhdh\ \\h^blky ihgylb_ bgl_gkb\ghklb ihlhdZ λ(W ) = OLP 3 (W ∆W ) ∆W . >ey klZpbhgZjguo ihlhdh\ λ W = λ = FRQVW Z _keb Q

Q

Q

Q

Q

Q

L

Q

L

L

N

N

N

∆→ W

N

wlh _s_ b ihlhd k h]jZgbq_gguf ihke_^_ckl\b_f lh ^ey fgh]hf_jghc iehlghklbbgl_j\Zeh\f_`^mkh[ulbyfb\uihegy_lky I (τ τ ) = I (τ )×× I (τ ) , Q

I (τ



) = λ − 

 τ τ I G ; ( ) ∫ 

τ

Q

Q

I (τ) = = I (τ) = I (τ) ;

Q

Q

∞

λ = ∫ τI (τ )Gτ .

17 H[sbcih^oh^dfh^_ebjh\Zgbxhj^bgZjguoklZpbhgZjguoihlhdh\ k h]jZgbq_gguf ihke_^_ckl\b_f ihlhdb IZevfZ aZdexqZ_lky \ ke_^mx s_f KqblZ_lky ba\_klguf aZdhg jZkij_^_e_gby bgl_j\ZeZ f_`^m kh[u lbyfb I τ Lh]^ZkgZqZeZfh^_ebjm_lkyfhf_glgZklmie_gbyi_j\h]hkh [ulby\khhl\_lkl\bbkkhhlghr_gb_f τ =ϕ

W = W +τ ,

−

ϕ (τ

([ ) ,

)=

τ

∫ I (X)GX ,

ZaZl_f\k_oihke_^mxsbo\khhl\_lkl\bbkkhhlghr_gb_f W =W L

L

+ τ τ = ϕ

−

L

τ

−

L

([ ) ϕ(τ ) = ∫ I (X)GX L = Q , L

L

L

]^_ [ L = Q ihke_^h\Zl_evghklvjZ\ghf_jghjZkij_^_e_gguokemqZcguo qbk_egZbgl_j\Ze_[0,1]. Ijbf_j Ihlhd ImZkkhgZ beb ijhkl_crbc hj^bgZjguc [_a ih ke_^_ckl\byklZpbhgZjguc ihlhdk I (τ) = λH − λτ Lh]^Z L

 τ  I (τ) = λ  − ∫ λH − λ GX = λH − λτ = I (τ ) ,   X

lh _klv \ ^Zgghf kemqZ_ i_j\uc bgl_j\Ze jZkij_^_e_g lZd `_ dZd b \k_ hklZevgu_Hlkx^Z^eyihke_^h\Zl_evghklbfhf_glh\kh[ulbc: + τ τ = − OQ [ L = Q . λ

τ

ϕ (τ) = ϕ(τ ) = ∫ I (X)GX = − H − λτ = [ , W = W

L

L

−

L

L

L

Ijbf_jIhlhdWjeZg]ZIhlhdWjeZg]Zm-]hihjy^dZh[jZam_lky ba ijhkl_cr_]h \ j_amevlZl_ _]h ©ijhj_`b\Zgbyª dh]^Z \u[jZku\Zxlky \k_lhqdbaZbkdexq_gb_fm-hc>eylZdh]hihlhdZ

Imklvm=2, lh]^Z

(λP) I (τ ) = (P − )

P

τH − Pλτ =

∗ λ∗  − λ∗τ + λ∗ τ H − λ τ I (τ ) =  H 

( )

λ∗ λ∗ τ

P

−

H − λ τ λ =

(P − )

 ϕ (τ 

λ∗ . P

 λ∗ τ  − λ∗τ H ) = −  +  

b^eygZoh`^_gby τ lj_[m_lkyj_rZlvmjZ\g_gb_

H−

]

(] + ) = [ ] = λ∗ τ

.

>eyi>1 bgl_j\Zef_`^mkh[ulbyfb\ihlhd_nhjfbjm_lkykmq_lhflh]h qlh τ = τ + τ _klvkmffZ^\moh^bgZdh\hjZkij_^_e_gguoihihdZaZl_ev ghfmaZdhgm\_ebqbgkiZjZf_ljhf λ∗ dhlhjucihkmlbhij_^_ey_lbgl_g kb\ghklvbkoh^gh]h©ijhj_`b\Z_fh]hªihlhdZImZkkhgZLh]^Z L

L

L

W =W L

L

−

+ τ τ = − L

L

OQ [ [ L = Q . λ∗

L

L

18 GZoh`^_gb_Ze]hjblfZ^eyfh^_ebjh\Zgbyfhf_glZi_j\h]hkh[ulby ihlhdZ m-]h ihjy^dZ k\yaZgh k j_r_gb_f _s_ [he__ keh`guo ljZgkp_g ^_glguo mjZ\g_gbc Ihwlhfm _keb j_ZebaZpby \h \j_f_gb ^hklZlhqgh ^ebggZylhfh`ghihevah\Zlvkyke_^mxsbfmgb\_jkZevgufZe]hjblfhf: W =W L

L

−

+ τ τ = − L

L

P OQ ∏ [ L = Q . λ = P

M L

M

Ijbgpbiu fh^_ebjh\Zgby g_klZpbhgZjguo ihlhdh\ fh]ml [ulv ijh beexkljbjh\ZgugZke_^mxs_fijbf_j_ Ijbf_j G_klZpbhgZjguc ihlhd ImZkkhgZ bgl_gkb\ghklv dhlh jh]h λ = λ W aZ\bkbl hl \j_f_gb Z iehlghklv jZkij_^_e_gby bgl_j\ZeZ f_`^mkh[ulbyfbijhbkoh^ysbfb\fhf_glu W ∗ b W ∗ + W bf__l\b^ I (W ∗ W ) = Λ (W ∗ W )H

−Λ

W

∗+

Λ(W W ) = ∫ λ(X)GX , W

W

W

W

∗

]^_ Λ (W W ) = λ(W ∗ + W ) Hlkx^Z^eyfhf_glZihy\e_gbyi_j\h]hkh[ulby

τ

W = τ ∫ I (X)GX = Λ( τ

τ

) = ∫ λ(X)GX = − OQ [ I (X) = Λ ( X)H − Λ X .

>Zevg_crZyf_lh^bdZgZoh`^_gbyfhf_glh\kh[ulbcZgZeh]bqgZ W =W L

L

−

+ τ Λ(W L

L

−

τ ) =

W L

−

L

W

L

+τ

∫

L

λ(X)GX = − OQ [ L = Q .

−

>ey

fh^_ebjh\Zgby g_hj^bgZjguo ihlhdh\ \ Q-ko_fZo OLP 3 W ∆W ≠ djhf_aZ^ZgbyaZdhgh\jZkij_^_e_gbyfhf_glh\ihy\e_ ∆→ W

P

gby W L = Q g_h[oh^bfhhij_^_eylvjZkij_^_e_gb_qbkeZkh[ulbc\jZk kfZljb\Z_fuc fhf_gl \j_f_gb :gZeh]bqgh ^ey g_h^ghjh^guo ihlhdh\ lj_[m_lkyaZ^Zgb_aZdhgZjZkij_^_e_gbyijbagZdh\kh[ulbc I L = Q Ihkljh_gb_fh^_ebjmxs_]hZe]hjblfZKFH\p_ehf Djhf_ aZ^Zgby ihlhdh\ kh[ulbc ijb nhjfZebaZpbb ijhp_kkZ nmgd pbhgbjh\Zgby kbkl_fu k ihfhsvx Q-ko_f b ihkljh_gbb __ fh^_ebjmx s_]hZe]hjblfZg_h[oh^bfhaZ^ZlvkljmdlmjmlZdhckbkl_fuHgZij_^\Z jbl_evgh hij_^_ey_lky kh[kl\_ggufb \gmlj_ggbfb iZjZf_ljZfb KFH: dhebq_kl\hf nZa / n ; dhebq_kl\hf dZgZeh\ h[kem`b\Zgby dZ`^hc nZau L

L

/ D M = / n ;dhebq_kl\hfgZdhibl_e_cdZ`^hcnZau / M = /n ;_fdhklvx M

+M

dZ`^h]h gZdhibl_ey / L = / M = / n Z lZd`_ k\yayfb f_`^m BG b D lh _klv hi_jZlhjhf khijy`_gbyR we_f_glh\kbkl_fuDjhf_lh]haZ ^Zxlky^bkpbiebguh`b^ZgbcaZy\hd\Hb\u[hjZbogZh[kem`b\Zgb_\ D>eyjZajZ[hldbfh^_ebjmxs_]hZe]hjblfZp_e_khh[jZaghbkihevah\Zlv ki_pbZevgu_]jZnbq_kdb_bah[jZ`_gbykfjbk kljmdlmjuKFH +LM

+M

H31

D11

D31

19 H11

H21 µ11

b1 λ1

l11

µ21

l21

D12

µ31

l31 H32

D21

D32

µ12

µ32

l32

Jbk A^_kvh[hagZq_gby λ ∗ hij_^_eyxlbgl_gkb\ghklb\oh^guoihlhdh\Z µ ∗∗ bgl_gkb\ghklb ihlhdh\ h[kem`b\Zgby Rljboh\ufb ebgbyfb ihdZaZgu \hafh`gu_[ehdbjh\db
{

L

L

P

L

P

}

L P

( ) bgl_gkb\ghklvihlhdZhl B .

e_gbyhq_j_^ghcaZy\dbZ λ W L

L

L

P

L

2. We_f_glu lbiZ H, y\eyxsb_ky iZkkb\gufb DZ`^uc i-uc gZdhib l_ev k-hc nZau + fh`_l gZoh^blvky \ l_dms_f khklhygbb ] = {O ∈ O } I_j_oh^ ba h^gh]h khklhygby \ ^jm]h_ ijhbkoh^bl LN

L

L

+N

N

+LN

f]gh\_gghijbihklmie_gbbmoh^_ aZy\dbbkhijh\h`^Z_lkym\_ebq_gb _fmf_gvr_gb_f O gZ_^bgbpmaZbkdexq_gb_f]jZgbqguokhklhygbc 3. We_f_glulbiZKy\eyxsb_kyZdlb\gh-iZkkb\gufbHgbfh]mlgZ oh^blvky \ khklhygbb ©aZgylª ]^_ y\eyxlky Zdlb\gufb b \ khklhygbb ©k\h[h^_gª]^_y\eyxlkyiZkkb\gufbL_dms__khklhygb_i-]hdZgZeZj-hc nZau . hij_^_ey_lky ljhcdhc \_ebqbg ] D = α W µ W ]^_ L N

L M

LM

L M

L M U

L M

LM

U

}

- \j_fy h[kem`b\Zgby hq_j_^ghc aZ y\dbZ µ W -bgl_gkb\ghklvihlhdZh[kem`b\ZgbyI_j_oh^bah^gh]h khklhygby\^jm]h_ijhbkoh^blf]gh\_gghijbihklmie_gbbaZy\dbgZh[ kem`b\Zgb_bebhdhgqZgbbh[kem`b\Zgby Djhf_ lh]h \ BF p_e_khh[jZagh \\_klb ^\Z lbiZ \kihfh]Zl_evguo we_f_glh\: hq_j_^v aZy\hd b hq_j_^v k\h[h^guo dZgZeh\ Hq_j_^v aZy\hd HA gZ\oh^_dZ`^hcnZauh[kem`b\Zgbyy\ey_lkyiZkkb\gufwe_f_glhf, b __ khklhygb_ hij_^_ey_lky \_ebqbghc h[s_c hq_j_^b α

L M

= ©aZgylª ©k\h[h^_gª W

{

L M U

L M

LM

U

 ](ha ) = O (ha ) O (ha ) = ∑ O ( ) O ( ) ∈ O =  /

N

N

N

+N

L

N

L

L

N

(

+LN

  Khklhygby baf_gyxlky f]gh\_ggh 

)

ijbihklmie_gbbgh\hcaZy\db O (ha ) = O (ha ) + bebijbhk\h[h`^_gbbdZgZeZ

(

)

nZau O (ha ) = O (ha ) − N

N

N

N

Hq_j_^v k\h[h^guo dZgZeh\ HD dZ`^hc j-hc nZau

y\ey_lky iZkkb\guf we_f_glhf b oZjZdl_jbam_lky qbkehf b \hafh`gh

20 lbihf k\h[h^guodZgZeh\>eyHD__khklhygb_hij_^_ey_lky\_ebqbghc

D   ](hd) = O (hd) O (hd) = ∑ α ( ) O (hd) ∈ / D  Khklhygb_ baf_gy_lky f]gh\_ggh =   /

M

M

M

L M

M

M

M

L

(

)

ijbihklmie_gbbgh\hcaZy\db O (hd) = O (hd) − bebijbhk\h[h`^_gbbex[h]h

(

)

dZgZeZ O (hd) = O (hd) + . M

M

M

M

LZdbf h[jZahf ijb bfblZpbb ijhp_kkZ nmgdpbhgbjh\Zgby Q-ko_fu gZW
{

{

L

Q

}

}

+

{

{

: D = ](d ) L / D M /n : ha = ](ha ) N L

M

M

N

}

L

+N

+N

{

}

}

/ n : hd = ](hd) M / n M

dhlhju_bkihevamxlkyijbj_ZebaZpbbex[h]hlbiZfh^_ebjmxs_]hZe]h jblfZ H[sZy [ehd-ko_fZ lZdh]h Ze]hjblfZ ijb j_ZebaZpbb f_lh^Z∆zaZ^Z gbyfh^_evgh]h\j_f_gbbf__l\b^ihdZaZggucgZjbk<[ehdZoko_fu: lbiZ O (ha) = O (ha) + aZihfbgZ_lkyfhf_glihklmie_gbyaZy\db\h^bgbagZdh ibl_e_c + b __ ijbhjbl_l \ fZkkb\_ hq_j_^b aZy\hd nZau; lbiZ M

M

LM

O (ha) = O (ha) − \u[bjZ_lky aZy\dZ ba gZdhibl_ey + nbdkbjm_lky \j_fy \u M

M

LM

[hjZ b \j_fyijh\_^_gbyaZy\db\hq_j_^b;lbiZ O (hd) = O (hd) − ijhba\h^blky \u[hj dZgZeZ D ba fZkkb\Z hq_j_^b dZgZeh\ \uqbkey_lky ^ebl_evghklv h[kem`b\Zgby b \j_fy hk\h[h`^_gby dZgZeZ hl ^Zgghc aZy\db; lbiZ O (hd) = O (hd) + ijhba\h^blkyaZkuedZfhf_glZhk\h[h`^_gbydZgZeZKijb_]h ijbhjbl_lZ \ fZkkb\ hq_j_^b k\h[h^guo dZgZeh\ nZau dZgZe klZgh\blky iZkkb\gufbbkdexqZ_lkyba\j_f_gghcihke_^h\Zl_evghklbkh[ulbc Ij_^klZ\e_ggZy ko_fZ ihkljh_gby BF y\ey_lky ^hklZlhqgh h[s_c ohly_kl_kl\_ggh\dZ`^hfdhgdj_lghfkemqZ_hgZfh`_l[ulvmlhqg_gZ GZijbf_j\kemqZ__kebj_Zebam_lkyf_lh^- ∆t aZ^Zgbyfh^_evgh]h\j_ f_gb,\g_c^he`_g[ulv^hihegbl_evgh\\_^_gpbdeihqbkemhlkq_lh\gZ \j_f_gghc hkb n=T/∆t, Z kjZ\g_gb_ \j_f_gguo bgl_j\Zeh\ kh[ulbc ijh \h^blky k lhqghklvx ∆t =eZ\gh_ `_ \ ij_^klZ\e_gghc ko_f_ lh, qlh hgZ beexkljbjm_l \Z`g_crbc ijbgpbi ihkljh_gby BF ijb dhlhjhf hkgh\ gu_ [ehdbfh^_ebbf_xsb_ZgZeh]bqgu_nmgdpbb\uihegyxlky\\b^_ hl^_evguofh^me_cbebih^ijh]jZffklZg^Zjlgh]h\b^Z Dhgljhevgu_\hijhkubaZ^Zgby 1. AZibkZlv Ze]hjblf fh^_ebjh\Zgby ihlhdZ kh[ulbc^hjh`gh]hljZnbdZ kjZ\gh\_jhylgufaZdhghfjZkij_^_e_gbyf(τ) gZbgl_j\Ze_[a,b]. 2. AZibkZlv Ze]hjblf fh^_ebjh\Zgby ihlhdZ kh[ulbc \ dhlhjhf τ=Zξ, ]^_ξ -jZkij_^_e_gZihihdZaZl_evghfmaZdhgm M

LM

M

M

M

21 3. Hij_^_eblvZe]hjblffh^_ebjh\ZgbyKFHbf_xs_ckljmdlmjm\khhl \_lkl\bbkjbk 4. DZdbaf_gblkykljmdlmjZbkoh^guo^Zgguobfh^_ebjmxs_]hZe]hjbl fZ\p_ehf^eyKFHkg_bkijZ\ghklyfbdZgZeh\h[kem`b\Zgby? 5. DZdbaf_gblkykljmdlmjZbkoh^guo^Zgguobfh^_ebjmxs_]hZe]hjbl fZKFHijbgZebqbbg_h^ghjh^gh]hihlhdZaZy\hd? GZqZeh mklZgh\dZgZqmkeh\bc\k_cfh ^_ebgZqZehpbdeZih , = 1 F mklZgh\dZgZqmkeh\bcgZh^gm j_ZebaZpbx E ihbkd

W = PLQPLQ W

L M U

PLQ W L P

i_j_oh^dbfblZpbbkhhl\Zd lb\gh]hwe_f_glZ ihklmie_gb_ aZy\dbhlBi

-

hk\h[h`^_gb_ dZgZeZg_ihkenZau

+

hq_j_^v k\dZg

-

O ha = O ha + O hd = O hd − O

O

O

O

M

O hd

M = O hd +

-

nhjfbjh\Zgb_ gh\hcaZy\db W = W +τ

hq_j_^v aZy\

hk\h[h`^_gb_ dZgZeZihkenZau

+

-

M O M ha = O ha −

hq_j_^v k\dZg

+

M

E

-

M

M

= O hd+ − M

t*>T?

+ F

-

I>N

+

/

/

/

O hd n = O hd n + O ha n = Oha n −

nbdkZpbyiZjZ f_ljh\h[kem `_gghcaZy\db

O ha+ = O ha+ + O hd

/

+

hq_j_^v aZy\

22 h[jZ[hldZj_amevlZlh\BF

dhg_p Jbk

DZdbaf_gblkykljmdlmjZbkoh^guo^Zgguobfh^_ebjmxs_]hZe]hjblfZ ko_fujbkijbj_ZebaZpbbf_lh^Z∆t? Hij_^_eblv bkoh^gu_ hibkZgby ^ey fh^_ebjh\Zgby bgnhjfZpbhggh \uqbkebl_evghck_lbij_^ijbylbydZdKFHklZg^Zjlghckljmdlmju Fh^_ebjh\Zgb_Z]j_]Zlb\guokbkl_f Ihgylb_Z]j_]ZlZb_]hfZl_fZlbq_kdZyfh^_ev GZb[he__ h[sbf ih^oh^hf d nhjfZebaZpbb hibkZgby ijhp_kkh\ nmgdpbhgbjh\Zgbykeh`guokbkl_fy\ey_lkyZ]j_]Zlb\gucih^oh^\jZf dZo dhlhjh]h we_f_glu hibku\Zxlky gZ hkgh\_ : - ko_f beb Z]j_]Zlh\ Ex[hcZ]j_]Zljbk oZjZdl_jbam_lkyfgh`_kl\hfTxfhf_glh\\j_f_gb \oh^guo X mijZ\eyxsbo G b \uoh^guo Y \ha^_ckl\bc fgh`_kl\hf \gmlj_ggbokhklhygbcZZlZd`_kemqZcgufbhi_jZlhjZfbH bF - i_j_ oh^h\ b \uoh^h\ >ey nmgdpbhgbjh\Zgby Z]j_]ZlZ oZjZdl_jgh qlh \oh^ gu_x(t),mijZ\eyxsb_g(t)b\uoh^gu_\ha^_ckl\byihklmiZxl\hij_^_ e_ggu_ fhf_glu \j_f_gb b h[jZamxl iZju kemqZcguo ihke_^h\Zl_evgh kl_c W [ L W J M W \ N KqblZ_lkyqlhaZ

(

[

L

L

)

(

J

M

M

)

(

\

N

N

)

dhg_qgucbgl_j\Ze\j_f_gb\Z]j_]ZlihklmiZ_ldhg_qgh_qbkeh\ha^_ckl \bc Kh\hdmighklb \ha^_ckl\bc gZ ihembgl_j\Ze_ t0,t] gZau\Zxlky khhl \_lkl\_ggh \oh^guf - [ mijZ\eyxsbf - J b \uoh^guf khh[s_gb_f W

W

W

W

\
\h \j_f_gb ihklhyggh Khklhygby g(t) Z]j_]ZlZ \ fhf_glu ihklmie_gby x(t) y(t) \oh^guo b mijZ\eyxsbo \ha^_ckl H,F \bc Z lZd`_ \u^Zqb \uoh^gh]h z(t) \ha^_ckl\by gZau\Zxlky hkh[ufb Ba hkh[uo khklhygbc Z]j_]Zl i_j_ Jbk oh^bl \ gh\u_ kdZqdhf lh _klv aZ [_kdhg_qgh fZeuc bgl_j\Ze \j_f_gb Ihwlhfm ^ey Z]j_]ZlZ ijh\h^blky dhgdj_lbaZpby \b^Z hi_jZlhjZH^eyjZaebqguokblmZpbc\hagbdgh\_gby hkh[uo khklhygbc Imklv W fhf_gl \hagbdgh\_gby hkh[h]h khklhygby z( W ) k\yaZggh]h k gZklmie_gb_f \oh^gh]h \ha^_ckl\by [ ∈X Z J ∈ * ihke_^g__mijZ\eyxs__\ha^_ckl\b_ W ≤ W lh]^Zhi_jZlhjHbf__l\b^ [

[

J

[

V

V

(

23

)

(( )

)

] W + = + ] W [ J W W ≤ W . [

[

[

V

J

[

V

?keb W - fhf_gl ihklmie_gby mijZ\eyxs_]h \ha^_ckl\by b \hagbdgh\_ gbykhhl\_lkl\mxs_]hhkh[h]hkhklhygbylh J

(

)

(( )

)

] W + = + ] W J W J

J

J

?keb W -fhf_gl\u^Zqb\uoh^gh]hkb]gZeZlh \

(

)

(( )

)

] W + = + ] W J W W ≤ W \

\

(

\

GZdhg_p gZ bgl_j\Ze_ \j_f_gb W W

V

J

\

V

] ]^_ g_l hkh[uo khklhygbc ih\_

^_gb_Z]j_]ZlZhibku\Z_lkyhi_jZlhjhf

((

)

)

(

](W ) = + ] W + J W W ≤ W W ∈ W W

V

J V

]

>ey hij_^_e_gby hi_jZlhjZFgZfgh`_kl\_Z\gmlj_ggbokhklhygbc Z]j_]ZlZ \u^_eyxl k_f_ckl\h ih^fgh`_kl\ = = {= } h[eZ^Zxs__ ke_ \

\

^mxsbfb k\hckl\Zfb
( ) \

( ) ∉=

]W

\

(

\

\

\

)

(

\

\

)

\

Ijbwlhfhi_jZlhjF ij_^klZ\ey_lky\\b^_

 ](W ) ∈ =( ) ( ) \(W ) = ) (](W ) J W )) ](W ) W = W ≤ W ) (W ) =   ](W ) ∉ =( ) Jhevhi_jZlhjZF2 a^_kvk\h^blkydnbdkZpbbfhf_glZ W ^hklb`_gby]jZ

V

(

\

)

\

J

\

V

\

\

gbph^gh]hbaih^fgh`_kl\k_f_ckl\Z = ( ) LZdbfh[jZahfih^Z]j_]Zlhf ihgbfZ_lky h[t_dl hij_^_eyxsbcky kh\hdmighklvx fgh`_kl\ b kemqZc guohi_jZlhjh\ : =< 7 ; * < = = ( ) {+ } {) } > Ijbf_jJZkkfhljbfh^ghdZgZevgmxKFHwe_f_glZjgh]hlbiZ hibku\Zxsmxijhp_kkj_r_gbyaZ^ZqgZWebl_evghklv j_r_gby aZ^Zqb hij_^_ey_lky \_ebqbghc ηj=ψ(aj,β) Ih hdhgqZgbb j_r_ gbyaZ^Zqb\uqbkebl_evijbklmiZ_ldj_r_gbxke_^mxs_c
\

[

L

L

24 zr(t), r=2+2k, k= ] W − - hklZ\r__ky \j_fy gZoh`^_gby k - hc aZ^Zqb \ hq_j_^b ^h fhf_glZ ihemq_gby hldZaZ
[ M

hij_^_ey_lky iZjZf_ljhf aZ^Zqb x( W )=aj. Imklv W fhf_gl ihklmie_gby gh\hc aZ^Zqb lh]^Z hi_jZlhj H=H1=H2 mijZ\eyxsbo \ha^_ckl\bc g_l aZ^Z_lkyijbz2( W )>0 khhlghr_gbyfb [

[

M

M

[ M

(

) ( ) ( ) ( ) + ( W + ) = ] ( W ) ] ( W + ) = ] ( W ) ≤ N ≤ ] ( W ) − ] ( W + ) = ϕ([( W ) β) , ( W + ) = [( W ) ] ] W + = ] W ] W + = ] W

]

+

[

N

W

 

[ M

[

M

+

M

+ ] 

[

N

M

[

M

+

M

[

[

M

M

M

+ ] 

[

[

N

( ) = khhlghr_gbyfb\b^Z ] ( W + ) = ψ ([( W ) β) ] ( W

Zijb ] W

[

W

[ M

+

M

[

N

[

M

[

[

M

M

 

M

[ M

[

[

M

M

)

( ) + =

+ =] W

[ M

[ M

A^_kvb^Ze__ijbz2(t)=0 khklhygbyz1+2k(t), z2+2k(t), a k ≥ 1 g_hij_^_e_gu K_f_ckl\hih^fgh`_kl\ Z ( ) hij_^_ey_lkyiZjhc = = {= = } , = = {] W ] W = } ; = = {] W ] + W = N ∈ ] W − }. \

<

<

<

<

<

N

M

Lh]^Z _keb W = W b wlhl fhf_gl khhl\_lkl\m_l hdhgqZgbx j_r_gby hq_ j_^ghcaZ^Zqblhhi_jZlhj + jZkdju\Z_lkylZdbfh[jZahf

(

)

(

)

(

)

( ) −

   ] + W + = ] + + W ] + W + = ] + + W ] W > ;  ≤ N ≤ ] W −  ] W + = ] W = ] W + = ] W = . ] W + = ψ ] W β ] W + = ] W

\

N

(

)

\

\

( )

\

\

N

N

(

)

\

\

( ) ( ) \

N

\

\

(

)

\

[

( ) \

(

} ( )

)

\

[

\

[

?keb`_ W = W fhf_gl\j_f_gbdh]^Zbkl_dZ_l\j_fyh`b^Zgbyh^ghcba aZy\hd\hq_j_^bgZijbf_jl-hclhhi_jZlhj + hij_^_ey_lkylZd \

(

] ( W + ) = ] ( W ) − ( ) ] ( W + ) = ] ( W ) ] ( W + ) = ] ( W ) N < O ( W + ) = ] ( W ) ] ( W + ) = ] ( W ) O ≤ N ≤ ] ( W ) − . ] W

+

]

+

\

N

+

\

N

)

+ = ] W

\

+

N

\

\

\

N

\

+

+

+

\

\

N

\

N

+

N

+

\

N

\

+

\

GZdhg_p \ bgl_j\ZeZo f_`^m fhf_glZfb gZklmie_gby hkh[uo khklhygbc ih\_^_gb_Z]j_]ZlZhibku\Z_lkyhi_jZlhjhf + \b^Z

] (W ) = ] (W + ) − (W − W ) ] (W ) = ] (W + ) ] + (W ) = ] + (W + ) ] + (W + ) − (W − W ) .

N

N

N

25

(

\ = D β W

M

\

)

l

- _klv kh\hdmighklv k\_^_gbc h aZ^Zq_ hij_^_eyxsbo lbi

aZ^Zqb iZjZf_lj kbkl_fu b \j_fy \uoh^Z ba kbkl_fu Khhl\_lkl\_ggh hi_jZlhjF^eyfhf_glh\\j_f_gb W b W hij_^_ey_lkykhhlghr_gbyfb

(

\ = D β W M

\

) ] ( W ) = ;

\

\

(

\ = D β W

\

M

\

) ] ( W ) = . \

+

O

< ^Zgghf ijbf_j_ jhev hi_jZlhjZ ) nZdlbq_kdb k\h^blky d ijh\_jd_ ]jZgbqguo mkeh\bc ^ey \j_f_gb gZoh`^_gby aZ^Zq \ kbkl_f_ \ hq_j_^b bebgZh[kem`b\Zgbb Ko_fZdmkhqgh-ebg_cgh]hZ]j_]ZlZb__ijbf_g_gb_ ey DE: hl^_evgh g_ \u^_eyxl mijZ\eyxsb_ \ha^_ckl\by gh ]eZ\gZy hkh[_gghklvaZdexqZ_lky\hkh[hc kljmdlmj_ fgh`_kl\Z \gmlj_ggbo khklhygbc AZ^Z_lky g_dhlhjh_ kq_lgh_ fgh`_kl\h I hkgh\guo khklhygbc gZijbf_j , = {} ) We_f_gl ν ∈, gZau\Z_lkyhkgh\gufkhklhygb_fdhlhjhfmklZ\blky\khhl\_lkl\b_p_eh_ g_hljbpZl_evgh_ qbkeh ν gZau\Z_fh_ jZg]hf khklhygby DZ`^hfm ν ∈, klZ\blky\khhl\_lkl\b_ih^fgh`_kl\hZν ^hihegbl_evguokhklhygbch[ jZamxs__ \ _\deb^h\hf ijhkljZgkl\_ jZaf_jghklb Q ν = ν \uimdeuc fgh]h]jZggbd Lh]^Z h[s__ fgh`_kl\h khklhygbc ij_^klZ\ey_lky \ \b^_ h[t_^bg_gby kq_lgh]h gZ[hjZ g_i_j_k_dZxsboky ih^fgh`_kl\ b khklhbl ba\k_\hafh`guoiZj (ν ] ν ) .

= = =ν ν∈

(

(

)

(ν) ](W ) = ν ] ( ν ) ∈ = ν ∈ , ] ( ν) = ] ] ( νν)

,

Q

)

L

∈=ν

?keb \ gZqZevguc fhf_gl t0 z(t0)=(ν,z(ν)(0)), z(ν)(0)∈Zν, lh gZ bgl_j\ZeZo \j_f_gbg_kh^_j`Zsbohkh[uokhklhygbcijb W > W lhqdZ ] ν (W ) i_j_ f_sZ_lky\gmljb = ν ^hl_oihjihdZg_^hklb]g_l_]h]jZgbpu\hihjguc fhf_gl \j_f_gb W Lh]^Z ijb W < W ≤ W ih\_^_gb_ DE: hibku\Z_lky mjZ\g_gb_f

ν(W ) = ν = FRQVW ] ν (W ) = ]

]^_ D

ν

ν

() + (W − W )D

ν

D

-ihklhygguc\_dlhjjZaf_jghklb Q ν = ν D

ν

ν

=

(

= D

G] ν , GW

ν

D

ν ν

Q

)

l

>ey

DE: agZq_gb_ hihjgh]h fhf_glZ W hij_^_ey_lky ljZ_dlhjb_c ] ν (W ) b nhjfhcfgh]h]jZggbdZ = ν W = LQI W ] ν () + (W − W )D ν ∉ ] ν W > W .

{

}

26 H[s__ qbkeh ]jZg_c = ν jZ\gh P ν ,ijbq_fdZ`^Zy]jZgv =(ν ) hij_^_ ey_lkymjZ\g_gb_f M

ν ν ν ν =(ν ) ∑ γ ( ) ] ( ) + γ ( ) = M P ν . Q

M

L

ML

L

M

Lh]^Zfhf_gl^hklb`_gby]jZgbpu = ν hij_^_ey_lkymjZ\g_gb_f t1=t0+min τj, ν  ν  ν ν τ = −  γ ( ) + ∑ γ ( ) ] ( )    = Q

M

M

ML

L

Q

ν

∑γ

L

L

=

( ν ) ( ν)

D

ML

M P ν .

L

Ke_^mxs_chkh[_gghklvxDE:y\ey_lkylhqlh\fhf_glt1_]hkhklhygb_ baf_gy_lkykdZqdhh[jZagh\khhl\_lkl\bbkaZdhghfjZkij_^_e_gby P1aZ \bkys_flhevdhhlkhklhygbyz(t1) by\ey_lkyhkh[ufkhklhygb_fZ]j_]ZlZ khijh\h`^Zxsbfky\u^Zq_c\uoh^gh]hkb]gZeZ

(

](W + ) 3 (] ] W ) = 3 ν ](W

)) 3

 ] 

ν

ν ](W ) . 

]^_ ν ∈, gh\h_agZq_gb_hkgh\gh]hkhklhygbyOZjZdl_jbamyijhp_kk\u ^Zqb \uoh^gh]h kb]gZeZ ijb ^hklb`_gbb hkh[h]h khklhygby \ ^Zgghf kemqZ_ ke_^m_l hlf_lblv qlh k_f_ckl\h ih^fgh`_kl\ = \\_^_ggh_ \ h[s_f hij_^_e_gbb Z]j_]ZlZ \ ^Zgghf kemqZ_ kh\iZ^Z_l k ih\_joghklvx

<

\k_o ]jZg_c DE: =

<

∞

P

ν

= ∑ = ν LZdbf h[jZahf ihke_ kdZqdZ ih ^hklb M

ν= = M

`_gbb ]jZgbpu = ν DE:ihiZ^Z_l\khklhygb_ ] =  ν ] ν W +  dhlhjh_ fh`ghkqblZlvgh\ufgZqZevguf>Ze__ijbt>t1 ^\b`_gb_DE:hkms_kl \ey_lky\gmljbh[eZklb = ihZgZeh]bqghfmaZdhgm

ν

(

(

jh]h kq_lgh]h fgh`_kl\Z Z [ µ = [ µ [ µµ

Q

)

)

7

- \_dlhj ijbgbfZxsbc

agZq_gby ba _\deb^h\Z ijhkljZgkl\Z ;µ jZaf_jghklb Q µ = µ
(

\ha^_ckl\byij_^klZ\eyxlkykhhl\_lkl\_ggh\\b^_iZju \ = λ \ λ λ - we_f_gl kq_lgh]h fgh`_kl\Z Z \

λ

(

= \

λ

\

λ λ

Q

)

L

)

,

]^_

-^_ckl\bl_evguc

\_dlhj Q λ = λ HibkZgb_ih\_^_gbyDE:ijbihklmie_gbb\oh^guo\ha ^_ckl\bcgZijbf_j [ ∗µ \fhf_gl W ∗ (hi_jZlhjH1 \h[s_cko_f_Z]j_]ZlZ hij_^_ey_lky aZdhghf jZkij_^_e_gby P2 aZ^Zxsbf kdZqhd ba lhqdb

( ) (

] W ∗ = ν ] [

ν

[

) \lhqdm ]( W

[

∗

)

+ =  ν ] 

ν

 bf_xsbf\b^ 

27

(

)

( ( ) )

] W ∗ + 3 ] ] W ∗ [ ∗ = 3 [

[

(

( ) )

ν ] W ∗ [ ∗ 3  ] 

[

ν

( )

ν ] W ∗ [ ∗  . 

[

Djhf_ lh]h \ fhf_gl ihklmie_gby \oh^gh]h kb]gZeZ b kdZqdZ z(t) DE: fh`_l\u^Z\Zlv\uoh^ghckb]gZekh^_j`Zgb_dhlhjh]hklZlbklbq_kdbaZ \bkbl hl ](W ∗ ) b [ µ W ∗ < h[s_f kemqZ_ \uoh^gu_ khh[s_gby DE:

( ) [

nmgdpbhgZehlijhp_kkZz(t). LZdbf h[jZahf DE: ij_^klZ\ey_l kh[hc dhfihabpbx kbgojhgguo \_jhylghklguo Z\lhfZlh\ ^\b`_gb_ dhlhjuo khklhbl bakdZqdh\ijbih klmie_gbb \oh^guo kb]gZeh\ b kbkl_f hibku\Z_fuo h[udgh\_ggufb ^bnn_j_gpbZevgufb mjZ\g_gbyfb lZf ]^_ kdZqdb hlkmlkl\mxl
D

D

khklhygb_za(t+0)=j\u[bjZ_lkykemqZcghk\_jhylghklvx 3

1

N

LM

≥ ∑ 3 =

N

LM

=

M

^eyex[h]h≤k≤K.Lh_klv^ey\oh^gh]hkhklhygbyxa(t) gZ^haZ^ZlvfZl jbpm 3 ( ) Z^eydhgdj_lgh]hkhklhygbyza(t) kljhdm 3 ( ) 3 ( )
(

D

LM

[

]

D

[

D

]

D

D

1

)

oh^ \u^Z\Z_fuc \ wlhl fhf_gl h^ghagZqgh hij_^_ey_lky hi_jZlhjhf ya(t+0)=Nj)=m.JZkkfZljb\ZyZ\lhfZldZdDE:fh`gh\\_klbZa dZdfgh `_kl\hhkgh\guokhklhygbckqblZyqlh Q ν = ν = ijb ν ∈, <_dlhj^h ihegbl_evguo khklhygbc \ ^Zgghf kemqZ_ g_ hij_^_ey_lky lZd `_ dZd b fgh]h]jZggbd = ν
{

3 3

[

LM

D

[ ∈; D

D

}.

?keb iheh`blv ν = µ = λ = ^ey \k_o ν µ λ lh DE: ij_\jZsZ_lky \ \_jhylghklgucZ\lhfZl^hklZlhqghh[s_]h\b^Z

28 Ijbf_j.2.2.JZkkfhljbfD-ko_fmhibku\Z_fmxmjZ\g_gb_f G] = I (W ]W [W ) GW

ijbbgl_]jbjh\Zgbbj_r_gbyf_lh^hfWce_jZRZ]bgl_]jbjh\Zgbyh\u [bjZ_lkybamkeh\by[ebahklbmjZ\g_gbyb_]hdmkhqgh-ebg_cghcZiijhd kbfZpbb Ijb wlhf djb\Zy z(t) aZf_gy_lky dmkhqgh-ebg_cghc nmgdpb_c a] W a\_gvydhlhjhc\lhqdZo W = W + NK N = V bf_xlm]he α gZdeh gZhij_^_ey_fucdZd WJα = I (W a] W [W ) , Z^bnn_j_gpbZevgh_mjZ\g_gb_aZf_gy_lkymjZ\g_gb_f\dhg_qguojZagh klyo a] W = a] W + KI (W a] W [W ) . + Ij_^klZ\eyy lZdmx ^bgZfbq_kdmx kbkl_fm \ \b^_ DE:, \ dZq_kl\_ kh klhygby \u[_j_f iZjm z(t)=(ν,z(ν)) ]^_ ν - ghf_j bgl_j\ZeZ \j_f_gb \b^Z [t0+νh, t0+(ν+1)h]y\eyxsbckyhkgh\gufkhklhygb_fZ ] ( υ ) = (] W ] W ) N

N

N

N

N

N

N

N

N

N

7

= a] W ]

= W + νK − W = τ ν - \_dlhj ^hihegbl_evguo khklhygbc LZ dbf h[jZahf, ]W = (ν a] W ] W ) IZjZ z1(t),z2(t) baf_gy_lkyebg_cghijb ]

ν

ν

7

nbdkbjh\Zgghf ν gh ijb z2(t)=0 ijhbkoh^bl kdZqhd \ fhf_gl W ν = W + νK ν = V hij_^_ey_fucmjZ\g_gbyfb

ν ′ = ν + ] (W ν + ) = ] W ν ] (W ν + ) = K , ](W ν + ) = (ν + ] W ν K) . Ihke_ kdZqdZ ijb W ν < W ≤ W ν+ ^\b`_gb_kbkl_fuijhbkoh^bl\khhl\_lkl 7

\bbkkhhlghr_gb_f

] W = ] W ν + W − W ν I W ν ] W ν [ W ν ] W = K − W − W ν = W + ν + K − W = W ν+ − W

LZdbfh[jZahffh`ghkqblZlvqlh\^ZgghfkemqZ_^eyDE:

{

, = { V} Q ν = ν = = ν = ]

{

P ν = = ν = ]

ν

]

ν

}

ν

]

ν

}

≥

=

29 ^hc ijhbkoh^bl ih dZgZeZf k\yab q_j_a Z]j_]Zlu gZau\Z_fu_ ihexkZfb :1:2:4 gZ jbk9 >\Z Z]j_]ZlZ Ai b Aj gZau\Zxlky g_ihkj_^kl\_ggh k\y aZggufb _keb f_`^m gbfb kms_kl\m_l ijyfZy i_j_^ZqZ bgnhjfZpbb b k\yaZggufb _keb f_`^m gbfb kms_kl\m_l kh\hdmighklv Z]j_]Zlh\ Ai,Ak,...,An,AjlZdZyqlhdZ`^u_^\Zbakhk_^gboZ]j_]Zlh\g_ihkj_^kl\_g ghk\yaZguIjbwlhfZ]j_]ZlAj g_ihkj_^kl\_gghke_^m_lke_^m_l aZZ] j_]Zlhf Ai Z Ai g_ihkj_^kl\_ggh ij_^r_kl\m_l ij_^r_kl\m_l Aj _keb g_dhlhjZy qZklv \uoh^ghc bgnhjfZpbb Ai y\ey_lky \oh^ghc ^ey Aj Ih ZgZeh]bb ^ey mijZ\eyxsbo \ha^_ckl\bc bkihevamxlky hlghr_gby lbiZ ©Z]j_]ZlAig_ihkj_^kl\_gghmijZ\ey_lmijZ\ey_l ZAjg_ihkj_^kl\_ggh ih^qbg_g ih^qbg_g Z]j_]Zlm Ajª _keb g_dhlhjZy qZklv mijZ\eyxs_c bgnhjfZpbb^eyAj y\ey_lky\uoh^ghc^eyAi.

Y(0)

X(0) A1

A3

A0

A0 A2

A4

Jbk9 Ex[Zyih^kbkl_fZ:-kbkl_fulZd`_y\ey_lky:-kbkl_fhc:-kbkl_fZ gZau\Z_lkydhfie_dkhf_keb__ex[hcZ]j_]Zlk\yaZgohly[ukh^gbfZ] j_]Zlhfwlhc:-kbkl_fu
30 Hgbiha\heyxl\\_klbke_^mxs__hij_^_e_gb_:ex[Zykh\hdmighklvZ]j_ ]Zlh\gZau\Z_lkyZ]j_]Zlb\ghckbkl_fhc_kebi_j_^ZqZbgnhjfZpbbf_` ^m Z]j_]ZlZfb ijhbkoh^bl f]gh\_ggh b [_a bkdZ`_gby FZl_fZlbq_kdh_ hibkZgb_ kljmdlmju :-kbkl_fu ijhba\h^blky iml_f aZ^Zgby hi_jZlhjZ khijy`_gby 5 8 = ; * → < < = 5 8 , O

( ) M

N

V

V

L

kihkh[uihkljh_gbydhlhjh]hjZkkfhlj_gu\Ijbfh^_ebjh\ZgbbZ] j_]Zlb\ghc kbkl_fu kqblZxl ba\_klgufb: hi_jZlhj khijy`_gby bkoh^ gu_^Zggu_g_h[oh^bfu_^eyZ\lhghfgh]hfh^_ebjh\ZgbydZ`^h]hZ]j_ ]ZlZ kbkl_fu \j_f_ggmx ^bZ]jZffm ihklmie_gby \ha^_ckl\bc \g_rg_c kj_^ufh^_evZ]j_]ZlZ:0). H[sbc ih^oh^ d ihkljh_gbx fh^_ebjmxsbo Ze]hjblfh\ :-kbkl_f aZdexqZ_lky\k\_^_gbbaZ^Zqbdihke_^h\Zl_evghfmfh^_ebjh\ZgbxZ]j_ ]Zlh\\oh^ysbo\kbkl_fmIjbwlhfbkihevamxlkyf_lh^udZd ∆] lZdb ∆W ?keb\kbkl_f_hlkmlkl\mxlh[jZlgu_k\yablhj_Zebam_lkyke_^mx sbc ih^oh^ Fh^_ebjh\Zgb_ gZqbgZ_lky k ex[h]h \oh^gh]h ihexkZ b ijh^he`Z_lkyihoh^mi_j_^Zqb\ha^_ckl\bc^hl_oihjihdZg_^hoh^bl ^h\uoh^gh]hihexkZbebZ]j_]ZlZgZdhlhjucihklmiZxl\ha^_ckl\byhl g_kdhevdbo Z]j_]Zlh\ < ihke_^g_f kemqZ_ fh^_ebjm_lky ke_^mxsZy p_ ihqdZZ]j_]Zlh\hl\oh^gh]hihexkZ^hwlh]hZ]j_]ZlZlZd^hl_oihjih dZg_[m^_lihemq_gZihegZy^bZ]jZffZ\oh^guobmijZ\eyxsbo\ha^_c kl\bc ihklmiZxsbo gZ wlhl Z]j_]Zl >ZggZy ^bZ]jZffZ y\ey_lky ijhf_ `mlhqghc^bZ]jZffhc\oh^guo\ha^_ckl\bcihhlghr_gbxdijZ\hcqZk lb:-kbkl_fue_\Zy]jZgbpZdhlhjhckfjbk hij_^_ey_lkyijbi_j_ f_s_gbb e_\hc ]jZgbpu kbkl_fu q_j_a dhlhjmx ijhoh^yl \oh^gu_ \ha ^_ckl\by\g_rg_ckj_^uL_fkZfufkblmZpbyihkms_kl\m\ha\jZsZ_lky d gZqZevghc, b ijhp_kk fh^_ebjh\Zgby ijh^he`Z_lky ZgZeh]bqguf h[jZ ahfFh^_ebjh\Zgb_hdZgqb\Z_lkylh]^Zdh]^Z[m^mljZkkfhlj_gu\k_p_ ihqdb hl \oh^guo ^h \uoh^guo ihexkh\ DZ`^uc Z]j_]Zl a^_kv fh^_eb jm_lky\lhqdZohkh[uokhklhygbc bijbwlhfihke_^h\Zl_evghijhke_`b \Z_lky imlv dZ`^h]h \ha^_ckl\by hl \oh^gh]h ihexkZ ^h \uoh^gh]h H[ sZy kljmdlmjZ fh^_ebjmxs_]h Ze]hjblfZ j_Zebah\Zggh]h f_lh^hf ©∆z» \jZfdZo^Zggh]hih^oh^Zij_^klZ\e_gZgZjbkZ ?keb \ : - kbkl_f_ ijbkmlkl\mxl h[jZlgu_ k\yab lh hq_gv keh`gh hij_^_eblv fhf_glu ihklmie_gby \ha^_ckl\bc ih dZgZem h[jZlghc k\yab [_afh^_ebjh\Zgby\k_oho\Zq_gguo__Z]j_]Zlh\Ihwlhfm\^Zgghfkem qZ_wnn_dlb\gufy\ey_lkyj_ZebaZpbyZe]hjblfZihf_lh^m©∆tªijbq_f dhgdj_lgh_ agZq_gb_ bgl_j\ZeZ f_`^m khk_^gbfb lhqdZfb nbdkZpbb kh klhygbcwe_f_glh\:-kbkl_fuih^[bjZ_lkygZw\jbklbq_kdhchkgh\_GZ

31 jbk[^ZgZh[sZykljmdlmjZfh^_ebjmxs_]hZe]hjblfZhl\_qZxs_]haZ bfblZpbx:-kbkl_fuihf_lh^m∆t. :]j_]Zlb\gucih^oh^kha^Z_lnhjfZevgmxhkgh\m^eyZ\lhfZlbaZpbb fZrbgguo wdki_jbf_glh\ b iha\hey_l ihkljhblv mgb\_jkZevgmx Z\lhfZ lbabjh\Zggmx bfblZpbhggmx fh^_ev M:BF j_Zebamxsmx klZg^Zjl gmxnhjfmhibkZgbykeh`ghckbkl_fu.WlZklZg^ZjlgZynhjfZnZdlbq_ kdbhl\_qZ_ljZkkfhlj_gguf\ur_ihgylbyfZ]j_]ZlZbko_f__khijy`_ gbyM:BFhnhjfey_lky\\b^_iZd_lZijbdeZ^guoijh]jZffhkgh\gZy

⊗

Nhjfbjh\Zgb_ ^bZ]jZff \oh^guo \ha^_ckl\bc ^ey :- kbkl_fu

k=k+1

⊗ GZqZeh pbdeZ ih qbkem rZ]h\ \ ^bkdj_lghf \j_f_gb t1=0+∆t

i=i+1

GZqZeh pbdeZ ijhkfhljZ \oh^guo beb ijhf ihexkh\ gZ khhl\ ]jZgbp_ :-kbkl_fu

Hij_^_e_gb_ kibkdZ \oh^guo b mijZ\eyxsbo \ha^_ckl\bc :-kbkl_fu hlg_k_gguo d fhf_glm tk GZqZeh pbdeZ ih qbkem Z]j_]Zlh\ :-kbkl_fu

j=j+1

i=i+1

Hij_^_e_gb_ kibkdZ \o b mijZ\eyxsbo \ha^_ckl\bc ^ey i-]h Z]j_]ZlZ

Fh^_ebjh\Zgb_ (∆z) j-]h Z]j_]ZlZ \ p_ihqd_ hl i-]h ihexkZ NbdkZpby \uoh^guo \ha ^_ckl\bc j-]h Z]j_]ZlZ

Fh^_ebjh\Zgb_ i-]h Z]j_]ZlZ gZ bgl_j\Ze_

(tk, tk+∆t] j-c Z]j_]Zl y\ey_lky \uo ihexkhf?

Hij_^_e_gb_ \uoh^guo \ha^_ckl\bc i-]h Z]j_]ZlZ hlg_k_gguo d fhf_glm

+

tk+∆t

-

-

j+1-c bf__l g_kdhevdh bklhqgbdh\ gZ \oh^_?

-

+

+

IjbjZs_gb_ fh^_evgh]h \j_f_gb

Ijhkf \k_ \oh^gu_ ihexku?

-

i = Na?

tk=tk+∆t

+

Nhjfbjh\Zgb_ ^bZ]jZffu +

\oh^guo \ha^_ckl\bc ^ey ijhf_`ml ihexkZ b khhl\ ]jZgbpu :-kbkl_fu

32

tk=T?

Nhjfbjh\Zgb_ ^bZ]jZffu \uoh^guo \ha^_ckl\bc

Nhjfbjh\Zgb_ \uoh^guo \ha^_ckl\bc hlg_k_gguo d fhf_glm tk=T

⊗

⊗

JbkZ

Jbk[

qZklv dhlhjuo hl\_qZ_l aZ nhjfbjh\Zgb_ [Zau ^Zgguo h[ h[t_dl_ bkke_ ^h\Zgbybi_j_\h^Z^Zgguo\klZg^Zjlguc\b^ Dhgljhevgu_\hijhkubaZ^Zgby 1. HibkZlvD -ko_fmijbf_jZ\\b^_ DE:ijbbkihevah\Zgbb^jm]bo ba\_klguof_lh^h\qbke_ggh]hbgl_]jbjh\Zgby 2. JZkkfhlj_lv hibkZgb_ D - ko_fu kbkl_fu h[udgh\_gguo ^bnn_j_gpb ZevguomjZ\g_gbc\\b^_DE: 3. HibkZlv kbkl_fm (Q-ko_fm jZkkfhlj_ggmx \ ijbf_j_ , \ \b^_ DE: 4. Hij_^_eblv kljmdlmjm bkoh^guo ^Zgguo ^ey fh^_ebjh\Zgby keh`ghc kbkl_fugZhkgh\_:-ko_f 5. Hij_^_eblvhkgh\gu_fh^mebM:BFgZhkgh\_Z]j_]Zlb\gh]hih^oh^Z KH> ? J @ : GB? <?GB? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Bkoh^gu_iheh`_gbyf_lh^heh]bbkbkl_fgh]hfh^_ebjh\Zgby 1.1. Hkgh\gu_ihgylbyB_jZjobq_kdb_hibkZgbykeh`guokbkl_f 1.2. FZl_fZlbq_kdb_hij_^_e_gbykbkl_fbfh^_e_ckbkl_f 1.3. AZ^Zqbkbgl_aZbZgZebaZkeh`guokbkl_f 2. FZl_fZlbq_kdb_fh^_ebwe_f_glh\kbkl_fbbo\aZbfh^_ckl\by 2.1. Lbih\u_fZl_fZlbq_kdb_ko_fuhibkZgbywe_f_glh\kbkl_fu 2.2. FZl_fZlbq_kdb_fh^_eb\aZbfh^_ckl\by\keh`ghckbkl_f_ 3. Fh^_ebjh\Zgb_kbkl_ffZkkh\h]hh[kem`b\Zgby 3.1. DeZkkbnbdZpbykbkl_ffZkkh\h]hh[kem`b\Zgby 3.2. Hkh[_gghklbihkljh_gbybfblZpbhgguofh^_e_cKFH=_g_ jZpbyihlhdh\kh[ulbc 3.3. Ihkljh_gb_fh^_ebjmxs_]hZe]hjblfZKFH\p_ehf 4. Fh^_ebjh\Zgb_Z]j_]Zlb\guokbkl_f 4.1. Ihgylb_Z]j_]ZlZb_]hfZl_fZlbq_kdZyfh^_ev 4.2. Ko_fZdmkhqgh-ebg_cgh]hZ]j_]ZlZb__ijbf_g_gb_ 4.3. :]j_]Zlb\gucih^oh^ijbfh^_ebjh\Zgbbkeh`guokbkl_f

2 3 3 4 5 8 8 11 14 14 15 18 22 22 25 28

KIBKHDBKIHEVAH<:GGUOBKLHQGBDH< 1. ;mke_gdhGIFh^_ebjh\Zgb_keh`guokbkl_f-F:GZmdZ-k 2. ;mke_gdh <G :\lhfZlbaZpby bfblZpbhggh]h fh^_ebjh\Zgby keh`guo kbkl_f-F:GZmdZ-k

33 3. Kh\_lh\;YYdh\e_\K:Fh^_ebjh\Zgb_kbkl_f-F:

Моделирование сложных информационных систем: Методические указания по изучению спецкурса

Recommend Documents