Segundo Metodo de Liapunov
August 28, 2022 | Author: Anonymous | Category: N/A
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PXH HV HL AHTNDN DH LE@]XINW
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THNZHA@V
PXH HV HL AHTNDN DH LE@]XINW
T`jl` dh Fnithiedn E
PXH HV HL AHTNDN DH LE@]XINW
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THNZHA@V
PXH HV HL AHTNDN DH LE@]XINW
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AHTNDN DH LE@]XINW Hs ui` khihr`lez`fe´ni ni dh dns preifepens g´ g´İsefns dh lns sestha`s fnishrv`tnrens3 • Xi` pnsefe´ nni i dh rhpnsn hs hst`jlh se l` hihrk´ hihrk´İ` pnthife`l hs ui a´İiean lnf`l, hi f`sn fnitr`ren, hs eihst`jlh • L` hihrk´ hihrk´İ` tnt`l hs ui` fnist`ith dur`ith fu`lquehr anveaehitn
]`r` vhr ahcnr hstns fnifhptns, sh fnisedhr` ui p´hiduln ]`r` hiduln in `anrtek `anrteku`dn u`dn hs dhfer ui sestha` ahf´ `iefn `iefn fnishrv`tevn, quh sh rekh pnr l` hfu`fe´nni3 i3 d : β
k
dt : + l sei β 1 =
(0)
Hl sestha` fnrrhspnidehith dh hfu`fenihs dh preahr nrdhi hs3 dx dt 1 y , y ,
dy
k
dt 1 √ l sei x
(:)
PXH HV HL AHTNDN DH LE@]XINW
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´ AHTNDN DH LE@]XINW d β dnidh x 1 β y y 1 dt ..
Hi Ve sh naeth ui` fnist`ith `rjetr` `rjetr`re`,l` re`,l` hihrk hihrk´ ´İ` pnthife`l X hs hl tr`j`cn hghftu`dn h ghftu`dn `l hhlhv`r lhv`r hl p´hhiduln iduln p pnr nr `rrej` dh su pnsefe´ni ni a`s j`c`, ` s`jhr3 X (x , y ) 1 akl (0 √ fns x )
(6)
PXH HV HL AHTNDN DH LE@]XINW
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´ AHTNDN DH LE@]XINW
Lns puitns fr´ fr´İtefns dh l` hfu`fe´ hfu`fe ni ´ni : sni x 1 °iύ y y 1 1 = fni i 1 =, 0, :, 6, .. .... .. β fnrrhspnidehiths ` β 1 ύ , d dt 1 =. G´İsef`ahith, hs dh hsphr`r quh lns puitns x 1 =, y 1 = x 1 °:ύ, y 1 = fnrrhspnidehiths ` β 1 = ° :ύ, .. .... .. p`r` lns quh l` lhithc` dhl p´ hiduln hiduln hst` vhrtef`l fni hl phsn b`fe` `j`cn, sh`i hst`jlhs y quh lns puitns x 1 ύ , y 1 =? x 1 °6ύ , y 1 =,...., .... ..,, p`r` lns quh l` lhithc` dhl p´ fnrrhspnidehiths ` β 1 °ύ, °6ύ, .. hiduln hiduln hst` vhrtef`l fni hl phsn b`fe` `rrej`, sh`i eihst`jlhs.Hstn fnifuhrd` fni l` fni l` prnpnsefe´ni ni 0 pnrquh hi lns preahrns puitns X hs a´İiean eku`l ` fhrn y hi lns shkuidns X hs ui a´``xean xean eku eku`l `l ` :akl .
PXH HV HL AHTNDN DH LE@]XINW
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´ AHTNDN DH LE@]XINW Fnisedh r`ans Fnisedhr`ans hihrk´ ´İ` tnt`l W , quh hs l` sua` dh l` hihrk´ hihrk´İ` pnthife`l X y l` hihr hi hrk k´İ` l`hihrk 0 : d β : fei´htef` : a dt hi t´hhraei raeins ns dh x y y W (x , y ) 1 akl ((0 0 √ fns x ) +
0
al : y :
(4)
: Vnjrh ui` tr`yhftnre` fnrrhspnidehith ` ui` snlufe´ni ni x 1 χ (t ) , y 1 ς (t ) dh l` hfu`fe´ni ni : , W sh puhdh fnisedhr`rsh fnan ui` guife´nni i dh t . L` dhrev`d` dh W _χ(t ), ς (t )Q )Q sh ll`a` r`z´nni i dh f`ajen dh W `l shkuer l` tr`yhftnre`, b`fehidn rhkl` dh l` f`dhi` sh tehih3 dW _χ(t ), ς (t )Q d χ d χ(t ) d d ς )Q ς(t ) 1 W xx _χ(t ), ς(t )Q )Q + W y y _χ(t ), ς (t ))QQ dt dt dt dx : dy 1 (akl sei sei x ) dt + al dt
(8) (7)
PXH HV HL AHTNDN DH LE@]XINW
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´ AHTNDN DH LE@]XINW Fnisedhr`ans hl sestha` `ut´nninan3 inan3 dx dy 1 G (x , y ) 1 K (x , y ) dt dt
(;)
sh supnih quh hl puitn x 1 =, y 1 1 = hs ui puitn fretefn `seitntef`ahith hst`jlh, hitnifhs hxesth `lk´ u ui i dnaeien D quh fnitehih ` (=,=) t`l quh tnd` tr`yhftnre` quh sh eiefeh hi D dhjh thihr `l nrekhi fu`idn t ↚ ∙. Vh supnih quh hxesth ui` guife´ ni ni dh hihrk´ hihrk´İ` W t`l quh W (x , y ) ≨ = p`r` (x,y) hi D, fni W 1 = snln hi hl nrekhi.Fnan f`d` tr`yhftnre` hi D tehidh `l nrekhi fu`idn t ↚ ∙, se sh sekuh fu`lquehr tr`yhftnre` p`rteful`r, W dhfrhfh ` fhrn fu`idn t tehih `l eiffietn. Hl tepn dh rhsult`dn quh sh dhsh` dhanstr`r hs hi hshife` hl eivhrsn , se snjrh tnd` tr`yhftnre` W, dhfrhfh ` fhrn fu`idn t frhfh, hitnifhs l`s tr`yhftnre`s dhjhi thihr `l nrekhi fu`idn t ↚ ∙ y dh dnidh hl nrekhi hs `seitntef`ahith hst`jlh.
PXH HV HL AHTNDN DH LE@]XINW
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´ AHTNDN DH LE@]XINW Vh` W dhffied` snjrh ui `lk´u ui i dnaeien D quh fnitehih `l nrekhi. Hitnifhs sh defh3 • W hs dhffied` pnsetev` snjrh D se W (=, =) 1 = y W (x , y ) 9 = p`r` p `r` tndns tn dns lns dha´`s `s
puitns hi D. • W hs dhffied` ihk`tev` snjrh D se W (=, =) 1 = y W (x , y ) < = p`r` p `r` tndns tn dns lns dha´``ss puitns hi D. • W hs shae-dhffied` pnsetev` snjrh D se W (=, =) 1 = y W (x , y ) ≨ = p`r` tndns lns
dha´`s `s pui puitns tns hi D. • W hs shae-dhffied` ihk`tev` snjrh D se W (=, =) 1 = y W (x , y ) ≤ = p`r` tndns lns dha´`s `s pui puitns tns hi D.
PXH HV HL AHTNDN DH LE@]XINW
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HCHA]LN Vh` l` guife´ nni i
W (x , y ) 1 sei(x : + y : )
Hs dhffied` pnsetev` snjrh x : + y : < ύ: y` quh W (=, =) 1 = y W (x , y ) 9 = p`r` = <
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