Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls
We investigate the problem of the existence of a discontinuous feedback that guarantees the stabilization of a nonlinear control system with respect to a part of variables. A solution of the system is defined in the Filippov sense. We establish a necessary condition for stabilization with respect to...
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| Дата: | 2006 |
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| Автори: | , , , , , |
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| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509653788524544 |
|---|---|
| author | Kovalev, A. M. Kravchenko, N. V. Nespirnyi, V. N. Ковалев, А. М. Кравченко, Н. В. Неспирный, В. Н. Ковалев, А. М. Кравченко, Н. В. Неспирный, В. Н. |
| author_facet | Kovalev, A. M. Kravchenko, N. V. Nespirnyi, V. N. Ковалев, А. М. Кравченко, Н. В. Неспирный, В. Н. Ковалев, А. М. Кравченко, Н. В. Неспирный, В. Н. |
| author_sort | Kovalev, A. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:57:10Z |
| description | We investigate the problem of the existence of a discontinuous feedback that guarantees the stabilization of a nonlinear control system with respect to a part of variables. A solution of the system is defined in the Filippov sense. We establish a necessary condition for stabilization with respect to a part of variables in the class of discontinuous controls, which generalizes the Ryan condition to the case of stabilization with respect to a part of variables. An example of a mechanical system that cannot be stabilized with respect to a part of variables is considered. |
| first_indexed | 2026-03-24T02:44:32Z |
| format | Article |
| fulltext |
K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDK 517.93
A. M. Kovalev, N. V. Kravçenko, V. N. Nespyrn¥j
(Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
NEOBXODYMOE USLOVYE STABYLYZYRUEMOSTY
PO ÇASTY PEREMENNÁX NELYNEJNÁX SYSTEM
V KLASSE RAZRÁVNÁX UPRAVLENYJ
We study the existence of a discontinuous feedback, which provides the stabilization of a nonlinear
control system with respect to a part of variables. We determine a solution of the system in the Filippov
sense. We obtain a necessary condition for the stabilizability with respect to a part of variables in the
class of discontinuous controls which generalizes the Ryan condition to the case of stabilizability with
respect to a part of variables. We consider an example of a mechanical system that is not stabilizable
with respect to a part of variables.
DoslidΩeno pytannq isnuvannq rozryvnoho zvorotnoho zv’qzku, qkyj zabezpeçu[ stabilizacig ne-
linijno] systemy keruvannq za çastynog zminnyx. Pry c\omu rozv’qzok systemy vyznaçeno za Fi-
lippovym. OderΩano neobxidnu umovu stabilizaci] za çastynog zminnyx u klasi rozryvnyx keru-
van\, qka uzahal\ng[ umovu Raj[na na vypadok stabilizaci] za çastynog zminnyx. Rozhlqnuto
pryklad mexaniçno] systemy, wo ne moΩe buty stabilizovana za çastynog zminnyx.
1. Vvedenye. Rassmotrym systemu upravlenyq, opys¥vaemug ob¥knovenn¥my
dyfferencyal\n¥my uravnenyqmy
ẋ = f ( x, u ) , (1)
hde x D Rn∈ ⊆ — fazov¥j vektor, u U Rm∈ ⊂ — vektor upravlenyq, mnoΩest-
vo U predpolahaetsq ohranyçenn¥m, soderΩawym toçku nul\ v kaçestve vnut-
rennej. Krome toho, predpolahaetsq f ( , )0 0 0= , çto obespeçyvaet suwestvo-
vanye nulevoho reßenyq system¥ (1). Funkcyq f ( x, u ) qvlqetsq neprer¥vnoj
po sovokupnosty peremenn¥x.
Klassyçeskaq zadaça stabylyzacyy system¥ (1) sostoyt v postroenyy obrat-
noj svqzy u ( x ) ( u ( 0 ) = 0 ) , pry kotoroj reßenye x ≡ 0 avtonomnoj system¥
ẋ = f ( x, u ( x )) (2)
qvlqetsq asymptotyçesky ustojçyv¥m po Lqpunovu.
Yzvestno, çto v sluçae, esly f ( x, u ) — lynejnaq po oboym arhumentam funk-
cyq, systema stabylyzyruema tohda y tol\ko tohda, kohda systema hlobal\no
asymptotyçesky nul\-upravlqema. Analohyçnoj πkvyvalentnosty meΩdu svoj-
stvom asymptotyçeskoj nul\-upravlqemosty y stabylyzyruemost\g dlq nely-
nejn¥x system net. Ymeet mesto sledugwee neobxodymoe uslovye dlq hladkoj
stabylyzacyy [1].
Uslovye Broketta. Pust\ f C∈ 1. Esly systema (1) qvlqetsq C1
-stabyly-
zyruemoj (suwestvuet neprer¥vno dyfferencyruemaq obratnaq svqz\ u ( x ) , ko-
© A. M. KOVALEV, N. V. KRAVÇENKO, V. N. NESPYRNÁJ, 2006
1434 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
NEOBXODYMOE USLOVYE STABYLYZYRUEMOSTY … 1435
toraq delaet nulevoe reßenye asymptotyçesky ustojçyv¥m), to obraz f soder-
Ωyt otkr¥tug okrestnost\ nulq.
Naprymer, systema
ẋ u1 1= ,
ẋ u2 2= ,
ẋ u x u x3 2 1 1 2= −
qvlqetsq upravlqemoj, no ne moΩet b¥t\ hladko stabylyzyrovana, tak kak ona
ne udovletvorqet uslovyg Broketta. (Dlq lgboho ε > 0 toçka ( 0, 0, ε ) ne
prynadleΩyt obrazu f ( x, u ) dlq lgb¥x x R∈ 3
y u R∈ 2 . )
Faktyçesky uslovye Broketta sleduet yz toho, çto topolohyçeskyj yndeks
asymptotyçesky ustojçyvoho reßenyq raven ( )−1 n , hde n — razmernost\ pro-
stranstva [2] (teoremaI52.1).
Rajen [3] pokazal, çto neobxodymoe uslovye Broketta spravedlyvo y dlq
razr¥vn¥x upravlenyj pry uslovyy, çto reßenye opredelqetsq po Fylyppovu
[4]. V nekotor¥x prykladn¥x zadaçax teoryy upravlenyq estestvenn¥m obrazom
voznykaet neobxodymost\ rassmatryvat\ stabylyzacyg po çasty peremenn¥x.
Tohda ymeet mesto vopros o soxranenyy uslovyj Broketta dlq stabylyzacyy po
çasty peremenn¥x. Dlq stabylyzacyy po çasty peremenn¥x hladkoj obratnoj
svqz\g dann¥j vopros reßaetsq poloΩytel\no [5]. Cel\g dannoj stat\y qvlq-
etsq ustanovlenye neobxodym¥x uslovyj stabylyzacyy po çasty peremenn¥x ne-
lynejn¥x system dlq razr¥vn¥x upravlenyj.
2. Opredelenyq y vspomohatel\n¥e utverΩdenyq. Zapyßem fazov¥j
vektor system¥ v vyde x = ( , , , , , )y y z zn n1 11 2
… … = ( y, z ) , n n1 2+ = n. Tohda
systema (1) prymet vyd [6]
˙ ( , , )y Y y z u= ,
(3)
˙ ( , , )z Z y z u= ,
hde Y R R R Rn n m n: 1 2 1× × → , Z R R R Rn n m n: 1 2 2× × → — neprer¥vn¥e funk-
cyy. Predpolahaem, çto Y z( , , )0 0 0= dlq vsex z Rn∈ 2 , funkcyy Y, Z nepre-
r¥vn¥ na mnoΩestve D U× , hde D x y H= ≤{ }: ( )H = >const 0 . Krome
toho, potrebuem, çtob¥ reßenyq system¥ (3) b¥ly z-prodolΩym¥, t. e. dlq lg-
boj ohranyçennoj yzmerymoj funkcyy u t U( ) : [ , )0 +∞ → lgboe reßenye x ( t )
system¥ (3) opredeleno pry vsex t ≥ 0, dlq kotor¥x y t H( ) ≤ .
Budem yspol\zovat\ standartn¥e oboznaçenyq
y = yi
k
n
2
1
1 2
1
=
∑
/
, z = zi
k
n
2
1
1 2
1
=
∑
/
, x = y z2 2 1 2
+( ) /
.
Dlq kratkosty oboznaçym X Rn:= . Íar radyusa r > 0 s centrom v toçke cI∈
∈ X oboznaçym çerez B cr( ), a esly c = 0 — çerez Br . Blyzost\ dvux nepust¥x
zamknut¥x mnoΩestv A y B v metryçeskom prostranstve moΩno oxarakteryzo-
vat\ çyslamy ρ( , )a b = a b− , d a B( , ) = inf ( , )
b B
a b
∈
ρ , dlq lgboho a A∈
β( , )A B = sup ( , )
a A
d a B
∈
.
V kaçestve dopustym¥x upravlenyj rassmatryvaem klass K poluneprer¥v-
n¥x sverxu otobraΩenyj x k x Rm→ ⊂( ) s nepust¥my, v¥pukl¥my y kompakt-
n¥my znaçenyqmy, 0 0∈k z( , ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1436 A. M. KOVALEV, N. V. KRAVÇENKO, V. N. NESPYRNÁJ
Pry takoj dopustymoj obratnoj svqzy k y z( , ) reßenye system¥ (2) oprede-
lqetsq kak reßenye dyfferencyal\noho vklgçenyq [4]
˙ ( , , )y Y y z u∈ ,
(4)
˙ ( , , )z Z y z u∈
s naçal\n¥m uslovyem y y( )0 0= , z z( )0 0= . Reßenyem qvlqetsq absolgtno
neprer¥vnaq funkcyq x t( ) = ( ( ), ( ))y t z t , x( )0 = ( ),y z0 0 , udovletvorqgwaq
dyfferencyal\nomu vklgçenyg (4) poçty vsgdu.
Dlq lgboho k K∈ otobraΩenye x f x k x� ( , ( )) takΩe poluneprer¥vno
sverxu s nepust¥my, v¥pukl¥my y kompaktn¥my znaçenyqmy. Tohda dlq lgboho
x Rn
0 ∈ suwestvuet kak mymymum odno reßenye system¥ (4) [4] (hl.I2, §7, teore-
maI1). Odnako svojstvo edynstvennosty reßenyq ne vsehda v¥polnqetsq dlq
dyfferencyal\n¥x vklgçenyj. V svqzy s πtym voznykaet neobxodymost\ opre-
delenyq ustojçyvosty y prytqΩenyq dlq reßenyq system¥ (4).
Opredelenye��1. Obratnaq svqz\ k y z K( , )∈ naz¥vaetsq πkvyasymptoty-
çesky stabylyzyrugwej dlq system¥ (4) po peremenn¥m y, esly systema (4)
πkvyasymptotyçesky y-ustojçyva, t.(e.
1) ∀ >ε 0 ∃δ ε( ) : y( )0 ≤ δ ⇒ y t( ) < ε ∀ ≥t 0 dlq lgboho maksymal\-
noho reßenyq ( ( ), ( ))y t z t system¥I(4);
2) ∃ >∆ 0 ∀ >ε 0 ∃ >T( )ε 0 : y( )0 < ∆ ⇒ y t( ) < ε ∀ ≥t T( )ε dlq lg-
boho maksymal\noho reßenyq ( ( ), ( ))y t z t system¥ (4).
Opredelenye��2. Obratnaq svqz\ k y z( , ) naz¥vaetsq πkvysΩymagwej dlq
system¥ (4) po peremenn¥m y, esly suwestvugt ρ δ τ> > > 0 y T > 0
takye, çto y y t0 < ⇒ <δ ρ( ) ∀ ≥t 0, y t( ) < τ , t T T∈[ , ]2 , dlq lgboho
maksymal\noho reßenyq ( ( ), ( ))y t z t system¥I(4).
Opredelenye��3. Reßenye zadaçy (4) naz¥vaetsq ravnomerno z-ohranyçen-
n¥m, esly dlq lgboho kompakta K Rn⊂ 2
suwestvuet kompakt K Rn
1
2⊂
takoj, çto esly z K( )0 ∈ y y H( )0 ≤ , to z t K( )∈ 1 ∀ ≥t 0 .
Pod zadaçej stabylyzacyy system¥ (1) po peremenn¥m y v klasse razr¥vn¥x
upravlenyj budem ponymat\ zadaçu naxoΩdenyq obratnoj svqzy k y z K( , )∈ , ko-
toraq obespeçyvaet πkvyasymptotyçeskug ustojçyvost\ reßenyq y = 0 syste-
m¥ (4) po otnoßenyg k peremenn¥m y.
3. Nekotor¥e svedenyq o stepeny otobraΩenyj. Pust\ X Rn:= , Ω ⊂ X
— ohranyçennoe otkr¥toe mnoΩestvo s zam¥kanyem Ω y hranycej ∂Ω ,
M = ( , , ) : : — , , ( )\f p f X X p X fΩ Ω Ω Ω� neprer¥vnoe otobraΩenye ⊂ ∈ ∂{ }.
Tohda stepen\ Brauπra (deg )B qvlqetsq odnoznaçn¥m otobraΩenyem M Z�
so sledugwymy svojstvamy:
1) deg ( , , )B I pΩ = 1 ∀ ∈p Ω ;
2) esly deg ( , , )B I pΩ ≠ 0, to suwestvuet x ∈Ω takoe, çto p f x= ( ) ;
3) esly h X: [ , ]0 1 × Ω � y q X: [ , ]0 1 � neprer¥vn¥, q t h t( ) ( , )( )∉ ⋅ ∂Ω
∀ ∈t [ , ]0 1 , to deg ( ( , ), , ( ))B h t q t⋅ Ω ne zavysyt ot t ∈[ , ]0 1 (ynvaryantnost\ otno-
sytel\no homotopyj);
4) esly Ω — svqznoe y symmetryçnoe otnosytel\no 0 mnoΩestvo v X y
f x f x( ) ( )− = − ∀ ∈∂x Ω, to deg ( , , )B f Ω 0 qvlqetsq neçetnoj y, sootvetstven-
no, ne ravna nulg.
Teorema��1 ([3], teoremaI2). Pust\ F D X: � — poluneprer¥vnoe sverxu
otobraΩenye s nepust¥my, v¥pukl¥my y kompaktn¥my znaçenyqmy v X . Dlq
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
NEOBXODYMOE USLOVYE STABYLYZYRUEMOSTY … 1437
lgboho ε > 0 suwestvuet lokal\no lypßyceva odnoznaçnaq funkcyq fε :
D F D� co( ( )) takaq, çto
β ε( ( ), ( ))graph graphf F < ε .
Opredelym stepen\ mnohoznaçn¥x otobraΩenyj sohlasno Rajenu [3]. Pust\
p X f∈ ∂\ ( )Ω . Oboznaçym F x p F xp( )( ) { : ( )}= − ∈v v . Dlq lgboho ε > 0 so-
hlasno teoremeI1 moΩno v¥brat\ fε dlq F p( ) , pryçem fε ne ymeet nulej na
∂Ω . V [3] pokazano, çto pry dostatoçno malom ε , esly fε y gε — lokal\no
lypßycev¥ pryblyΩenyq, deg ( , , ) deg ( , , )B Bf gε εΩ Ω0 0= . Stepen\ dlq mnoho-
znaçn¥x otobraΩenyj, udovletvorqgwyx teoremeI1, opredelqetsq sledugwym
obrazom:
deg ( , , ) lim deg ( , , )F p fBΩ Ω=
→ε
ε
0
0 .
Teorema��2 ([3] , teoremaI3). Pust\ x F x X� ( ) ⊂ — poluneprer¥vnoe
sverxu otobraΩenye na kompakte Ω ⊂ X s nepust¥my, v¥pukl¥my y kom-
paktn¥my znaçenyqmy.
1. Esly q X F: [ , ] ( )\0 1 � ∂Ω neprer¥vna, to deg( , , ( ))F q tΩ ne zavysyt
ot t ∈[ , ]0 1 .
2. Esly p X F∈ ∂\ ( )Ω y takoe, çto deg( , , )F pΩ ≠ 0, to dlq nekotoroho
x ∈ Ω p ∈ F ( x ) .
4. Osnovnoj rezul\tat.
Teorema��3. Pust\ Y ( y, z, u ) y Z ( y, z, u ) neprer¥vn¥ y udovletvorqgt
uslovyqm:
1) 0 0 0∈Y z( , , ) ;
2) esly K Rm⊂ — v¥pukloe mnoΩestvo, to Y y z K Rn( , , ) ⊂ 1
y
Z y z K Rn( , , ) ⊂ 2
— v¥pukl¥e mnoΩestva.
Esly dlq system¥ (4) suwestvuet πkvysΩymagwee po peremenn¥m y up-
ravlenye s obratnoj svqz\g k y z K( , )∈ y reßenye system¥ (4) ravnomerno z -
ohranyçeno, to obraz Y y z u( , , ) soderΩyt okrestnost\ nulq.
Dokazatel\stvo. PredpoloΩym, çto k y z K( , )∈ qvlqetsq πkvysΩymag-
wej obratnoj svqz\g po peremenn¥m y dlq system¥ (4). Tohda suwestvugt
ρ δ τ> > > 0 y T > 0 takye, çto y y t0 < ⇒ <δ ρ( ) ∀ ≥t 0, y t( ) < τ ,
t T T∈[ , ]2 , dlq lgboho maksymal\noho reßenyq ( ( ), ( ))y t z t zadaçy (4).
Opredelym mnohoznaçnoe otobraΩenye
′Y y z: ( , ) �
Y y z k y z y
Y y
y
z k y
y
z y
( ), , ( , ) , ,
, , , , .
≤
>
ρ
ρ ρ ρ
Predpolahaem, çto z Hz( )0 ≤ . Poskol\ku reßenye ravnomerno z -ohranyçeno,
to suwestvuet kompakt K Rn
1
2⊂ takoj, çto z t K( )∈ 1 ∀ >t 0. Oboznaçym
D B Kρ ρ= × 1, D B Kδ δ= × 1.
Rassmotrym otobraΩenye F y z Y Z: ( , ) ( , )� ′ . Dlq dannoho otobraΩenyq po-
luçym zadaçu
˙ ( )x F x∈ , x x( )0 0= , (5)
hde F qvlqetsq poluneprer¥vn¥m sverxu otobraΩenyem s nepust¥my, v¥puk-
l¥my y kompaktn¥my znaçenyqmy, sledovatel\no, F D F X( ) ( )ρ ≡ y qvlqetsq
kompaktom. Yz kompaktnosty F X( ) moΩno zaklgçyt\, çto dlq lgboho x X0 ∈
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1438 A. M. KOVALEV, N. V. KRAVÇENKO, V. N. NESPYRNÁJ
lgboe reßenye (5) ymeet maksymal\n¥j ynterval suwestvovanyq R+ = ∞[ , )0 .
Dlq x0
takoho, çto y0 ≤ δ , z Hz
0 ≤ , mnoΩestvo maksymal\n¥x reßenyj
(5) qvlqetsq mnoΩestvom maksymal\n¥x reßenyj (4). Oboznaçym Ω0 =
= { }:y y ≤ δ .
Poskol\ku k ( y, z ) — πkvysΩymagwaq obratnaq svqz\ po otnoßenyg k pere-
menn¥m y, dlq lgboho z K∈ 1 0∉ ′Y y z( , ) pry y B∈Ω0 \ τ. Pry kaΩdom fyksy-
rovannom z v¥polnqetsq sootnoßenye 0 0∉ ′ ∂Y z( ),Ω , poπtomu moΩno v¥çys-
lyt\ ′Y y z( , ) kak stepen\ otobraΩenyq, zavysqweho tol\ko ot y, na mnoΩestve
Ω0
v toçke nul\. Tohda deg , ,( )′Y Ω0 0 budet funkcyej ot z.
Tak kak F udovletvorqet uslovyqm teorem¥I1, moΩno v¥brat\ posledova-
tel\nost\ ( )fn n N∈ lokal\no lypßycev¥x pryblyΩenyj dlq F takyx, çto
β( ( ), ( ))graph graphf Fn → 0, n → ∞ ,
y udovletvorqgwyx uslovyg deg , , deg , ,( ) ( )′ ≡Y fn
yΩ Ω0 00 0 ∀n, hde fn
y
— y-
sostavlqgwaq funkcyy fn ( ( ) ),f f fn n
y
n
z T= .
Dlq prostot¥ oboznaçym I T= [ , ]0 2 y M C I X= ( ; ). Na Dδ opredelym
otobraΩenye
F : x0 → x M x t F x t x x∈ ∈ ={ }: ˙( ) ( ( )), ( )0 0 .
Zdes\ F Y Z= ( ), T , hde Y ∈Rn1 , Z ∈Rn2
— sootvetstvenno y- y z-sostavlq-
gwye mnoΩestva reßenyj (5).
Dlq kaΩdoho n opredelym otobraΩenye θ δn D M: � ; θn x( )0
— edynst-
venn¥j πlement x yz M takoj, çto
˙( ) ( ( ))x t f x tn= ∀ ∈t I ,
(6)
x x( )0 0= .
Yz klassyçeskoj teoryy ob¥knovenn¥x dyfferencyal\n¥x uravnenyj yzvest-
no, çto otobraΩenye ( , ) ( ) ( )( )t x x tn
0 0� θ qvlqetsq neprer¥vn¥m. Oboznaçym
çerez ϕn y ψn y - y z -sostavlqgwye edynstvennoho reßenyq θn x( )0 =
= ( )( ), ( )ϕ ψn n
Tx x0 0 .
V [3] pokazano, çto dlq lgboho ε > 0 suwestvuet n takoe, çto
β θ ε( ( ), ( ))graph graphn F < .
Yz πtoho sleduet, çto dlq lgboho ε > 0 suwestvuet n takoe, çto
β ϕ ε( ( ), ( ))graph graphn Y < .
Pust\ 0 < < −ε δ τ y m takovo, çto
β ϕ ε( ( ), ( ))graph graphm Y < .
Tohda dlq lgboho y0 0∈Ω suwestvuet z y0 0= ξ( ) takoe, çto
ϕ ξm y y t( ), ( ) ( )0 0 0∈Ω ∀ ∈t T T[ , ]2 . (7)
Opredelym funkcyg h Y: [ , ]0 1 0× ′Ω � sledugwym obrazom:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
NEOBXODYMOE USLOVYE STABYLYZYRUEMOSTY … 1439
h s y( , )0 =
T f y y s
s
y y sT y s
m
y
m
( )
( ( ))
, ( ) , ,
, ( ) ( ) , .
0 0
0 0 0
0
1 0 1
ξ
ϕ ξ
=
−[ ] < ≤
Dannaq funkcyq neprer¥vna. PokaΩem, çto h s y( , )0 0≠ dlq lgb¥x
( , ) [ , ]s y0 00 1∈ × ∂Ω . PredpoloΩym, çto h y( , )0 00 = dlq nekotoroho y0 0∈∂Ω .
Tohda f y ym
y( ), ( )0 0 0ξ = y, sledovatel\no, ϕ ξm y y t y( ), ( ) ( )0 0 0= ∀ ∈t I ,
y0 0∈∂Ω , çto protyvoreçyt (7). PredpoloΩym, çto h s y( , )0 0= dlq nekoto-
r¥x ( , ) ( , ]s y0 00 1∈ × ∂Ω . Tohda ϕ ξm y y sT y( ), ( ) ( )0 0 0= , otkuda ϕ ξm y y( ), ( )0 0
I×
× ( )nsT y= 0
∀ ∈n N , ns ≤ 2, y0 0∈∂Ω . Ytak, suwestvuet n ∈ N takoe, çto
1 ≤ ns ≤ 2 y ϕ ξm y y nsT y( ), ( ) ( )0 0 0 0= ∈∂Ω , çto protyvoreçyt (7). Takym ob-
razom, ustanovleno, çto h s y( , )0
qvlqetsq homotopyej dlq otobraΩenyj
f y ym
y( ), ( )0 0ξ y g y y y T ym m( ) ( ), ( ) ( )0 0 0 0= −ϕ ξ .
Rassmotrym funkcyg h s y s g y sym1
0 0 01( , ) ( ) ( )= − − . Dannaq funkcyq qvlq-
etsq homotopyej dlq otobraΩenyj g ym( )0
y l y ym( )0 0= − . Yz svojstv stepeny
otobraΩenyj poluçaem
deg , , deg , , deg , , deg , ,( ) ( ) ( ) ( )′ = = = ≠Y f g lB m
y
B m B mΩ Ω Ω Ω0 0 0 00 0 0 0 0.
Takym obrazom, 0 0∉ ′ ∂Y ( )Ω y d Y y y( ( )), , ( )0 00 0′ >ξ dlq lgboho y∈∂Ω0 .
Dalee, pokaΩem, çto otobraΩenye y d Y y y� ( ( )), , ( )0 ′ ξ qvlqetsq polunepre-
r¥vn¥m snyzu na ∂Ω0 . Pust\ y∈∂Ω0
budet proyzvol\n¥m y ( )yn ⊂ ∂Ω0
— po-
sledovatel\nost\, sxodqwaqsq k y. V¥berem podposledovatel\nost\ ( )ynk
ta-
kug, çto
lim , , ( )( ( ))
k
n nd Y y y
k k→∞
′0 ξ = lim inf , , ( )( ( ))
n
n nd Y y y
→∞
′0 ξ ,
y ( )qk takye, çto q d Y y yk n nk k
= ′( ( )), , ( )0 ξ . Yz poluneprer¥vnosty sverxu
′Y y y( ), ( )ξ dlq vsex ε > 0 ymeem q Y y yk n nk k
∈ ′( ), ( )ξ ⊂ ′ +Y y y B( ), ( )ξ ε dlq
lgboho dostatoçno bol\ßoho k . Yz kompaktnosty ′Y y y( ), ( )ξ sleduet, çto
( )qk ymeet sxodqwugsq podposledovatel\nost\ s predelom q Y y y∈ ′( ), ( )ξ .
Otsgda
d Y y y Y y y( ( )), , ( ) min ( , ( ))0 ′ = ∈ ′ξ ξv v ≤
≤ q q d Y y y
k
k
n
n n= = ′
→∞ →∞
lim lim inf , , ( )( ( ))0 ξ .
Takym obrazom, y d Y y y� ( ( )), , ( )0 ′ ξ — poloΩytel\no opredelennoe y po-
luneprer¥vnoe snyzu otobraΩenye na kompakte ∂Ω0 . Poπtomu suwestvuet
skalqr µ > 0 takoj, çto p Y∉ ′ ∂( )Ω0
∀ ∈p Bµ . Yz teorem¥I2 sleduet, çto
deg , ,( )′Y pΩ0 = deg , ,( )′Y Ω0 0 ≠ 0 ∀ ∈p Bµ . Ytak, dlq lgboho p B∈ µ suwest-
vuet y∈Ω0
takoe, çto y Y y y∈ ′( ), ( )ξ . Sledovatel\no, p B∈ µ qvlqetsq ob-
razom nekotoroj toçky ( , )y u B Rm∈ ×δ .
Teorema dokazana.
Sledstvye. Esly obratnaq svqz\ k ( y, z ) πkvystabylyzyrugwaq po otno-
ßenyg k peremenn¥m y, to ona takΩe πkvysΩymagwaq po otnoßenyg k pere-
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1440 A. M. KOVALEV, N. V. KRAVÇENKO, V. N. NESPYRNÁJ
menn¥m y. Takym obrazom, teoremaI3 qvlqetsq neobxodym¥m uslovyem dlq
stabylyzacyy po çasty peremenn¥x nelynejn¥x system v klasse razr¥vn¥x up-
ravlenyj.
Zameçanye�1. Po sravnenyg s rabotoj [5] v πtoj teoreme ne trebuetsq dyf-
ferencyruemosty prav¥x çastej system¥ (2), dostatoçno lyß\ yx neprer¥vno-
sty. PredpoloΩenye o z-prodolΩymosty reßenyj soxranqetsq, a trebovanye
y-stabylyzyruemosty oslablqetsq do y-sΩymaemosty. Odnako poqvlqetsq us-
lovyeI2 soxranenyq v¥puklosty po upravlenyg, analohyçnoe odnomu yz uslovyj
teorem¥ Rajena [3]. ∏to uslovye zavedomo v¥polnqetsq dlq affynn¥x system
s razr¥vn¥m upravlenyem, kohda pravaq çast\ doopredelqetsq po Fylyppovu
[4]. Takym obrazom, teoremaI3 obobwaet rezul\tat¥ rabot¥ [5] na sluçaj raz-
r¥vn¥x upravlenyj.
5. Prymer. Po ßeroxovatoj ploskoj poverxnosty dvyΩetsq trexkolesnaq
teleΩka. Ee poloΩenye opredelqetsq tremq obobwenn¥my koordynatamy: x, y
— dekartov¥ koordynat¥ toçky pereseçenyq osy symmetryy teleΩky s os\g, na
kotorug nasaΩen¥ kolesa, θ — uhol meΩdu os\g symmetryy teleΩky y os\g
Ox . TeleΩka upravlqetsq s pomow\g peremenn¥x u1, u2 (uhlov¥e skorosty
vrawenyq sootvetstvenno pravoho y levoho kolesa). Predpolahaem, çto kuzov
teleΩky ne ysp¥t¥vaet vertykal\n¥x peremewenyj. V toçke ( x, y ) na ne-
kotoroj v¥sote s pomow\g cylyndryçeskoho ßarnyra, os\ kotoroho paral-
lel\na osy symmetryy teleΩky, podvesym nevesom¥j y nerastqΩym¥j ster-
Ωen\, a k eho koncu prykrepym massyvn¥j ßaryk. Oboznaçym uhol meΩdu ver-
tykal\noj os\g y sterΩnem çerez α. Poluçennaq model\ qvlqetsq nekotor¥m
usloΩnenyem mexanyçeskoj modely teleΩky, rassmotrennoj v rabotax [7, 8].
Pry opredelennom sootnoßenyy meΩdu radyusom koles y razmeramy te-
leΩky ee dvyΩenye budet podçynqt\sq uravnenyqm
˙ ( ) cosx u u= +1 2 θ,
˙ ( ) siny u u= +1 2 θ , (8)
θ̇ = −u u1 2 ,
kotor¥e sledugt yz neholonomn¥x svqzej dlq koles teleΩky (uslovyq kaçenyq
bez proskal\z¥vanyq y otsutstvye bokovoho proskal\z¥vanyq; perednee koleso
moΩet skol\zyt\ po poverxnosty bez trenyq).
Uravnenye LahranΩa 2-ho roda dlq peremennoj α (pry sootvetstvugwym
obrazom v¥brannoj dlyne sterΩnq) ymeet vyd
˙̇ ( ˙ cos ˙ sin ˙ sin ) ˙ cosα θ θ θ α θ α− + +x y = – sin α . (9)
Oboznaçym uhlovug skorost\ vrawenyq sterΩnq çerez ω α= ˙ . V¥polnym v
systeme (8), (9) zamenu peremenn¥x
z1 := θ,
z x y2 : cos sin= +θ θ ,
z x y3 : sin cos= −θ θ,
z4 := α ,
z5 := ω ,
v1 1 2:= −u u ,
v2 1 2 1 2 3: ( ) ( )= + − −u u u u z .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
Tohda dynamyka system¥ budet opys¥vat\sq uravnenyqmy
̇z1 1= v ,
̇z2 2= v ,
̇z z3 1 2= v , (10)
ż z4 5= ,
̇ ( sin ) cos sinz z z z z5 2 1 3 1 4 1 4 4= + + −v v v v .
Uslovye Broketta po peremenn¥m z z z1 2 3, , dlq system¥ (10) ne v¥polnqet-
sq. Sledovatel\no, sohlasno teoremeI3 systema ne moΩet b¥t\ stabylyzyrovana
po πtym peremenn¥m daΩe s pomow\g razr¥vnoho upravlenyq.
Zameçanye�2. Dlq ysxodnoj system¥ (8), (9) dlq peremenn¥x x, y, θ uslo-
vye Broketta ne narußeno, no yz πtoho nel\zq sdelat\ v¥vod o stabylyzyruemo-
sty, poskol\ku ono qvlqetsq lyß\ neobxodym¥m. Lyß\ rassmotrenye system¥
(10) pozvolqet utverΩdat\, çto ysxodnaq systema ne ymeet svojstva stabylyzy-
ruemosty po çasty peremenn¥x.
1. Brockett R. W. Asymptotic stability and feedback stabilization // Diffferential Geometric Control
Theory / R. W. Brockett, R. S. Millman, H. J. Sussman. – Boston: Birkhauser, 1983. – P. 181 – 191.
2. Krasnosel\skyj M. A., Zabrejko P. P. Heometryçeskye metod¥ nelynejnoho analyza. – M.:
Nauka, 1975. – 511 s.
3. Ryan E. P. On Brockett’s condition for smooth stabilizability and its necessity in a context of
nonsmooth feedback // SIAM J. Control and Optim. – 1994. – 32, # 6. – P. 1597 – 1604.
4. Fylyppov A. F. Dyfferencyal\n¥e uravnenyq s razr¥vnoj pravoj çast\g. – M.: Nauka,
1985. – 224 s.
5. Zuiev A. L. On Brockett’s condition for smooth stabilization with respect to a part of the variables
// Proc. ECC’99. – Karlsruhe, 1999.
6. Rumqncev V. V., Ozyraner A. S. Ustojçyvost\ y stabylyzacyq dvyΩenyq po otnoßenyg k
çasty peremenn¥x. – M.: Nauka, 1987. – 253 s.
7. Nejmark G. Y., Fufaev N. A. Dynamyka neholonomn¥x system. – M.: Nauka, 1967. – 519 s.
8. Sontag E. D. Stability and stabilization: discontinuities and the effect of disturbances // Nonlinear
Analysis, Differential Equations and Control / F. H. Clarke, R. Stern. – Dordrecht: Kluwer, 1998. –
P. 551 – 598.
Poluçeno 19.08.2005
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| id | umjimathkievua-article-3544 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:44:32Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/57/5ecfee82511297d270a0c7813cd71557.pdf |
| spelling | umjimathkievua-article-35442020-03-18T19:57:10Z Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls Необходимое условие стабилизируемости по части переменных нелинейных систем в классе разрывных управлений Kovalev, A. M. Kravchenko, N. V. Nespirnyi, V. N. Ковалев, А. М. Кравченко, Н. В. Неспирный, В. Н. Ковалев, А. М. Кравченко, Н. В. Неспирный, В. Н. We investigate the problem of the existence of a discontinuous feedback that guarantees the stabilization of a nonlinear control system with respect to a part of variables. A solution of the system is defined in the Filippov sense. We establish a necessary condition for stabilization with respect to a part of variables in the class of discontinuous controls, which generalizes the Ryan condition to the case of stabilization with respect to a part of variables. An example of a mechanical system that cannot be stabilized with respect to a part of variables is considered. Досліджено питання існування розривного зворотного зв'язку, який забезпечує стабілізацію нелінійної системи керування за частиною змінних. При цьому розв'язок системи визначено за Філіпповим. Одержано необхідну умову стабілізації за частиною змінних у класі розривних керувань, яка узагальнює умову Райєна на випадок стабілізації за частиною змінних. Розглянуто приклад механічної системи, що не може бути стабілізована за частиною змінних. Institute of Mathematics, NAS of Ukraine 2006-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3544 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 10 (2006); 1434–1440 Український математичний журнал; Том 58 № 10 (2006); 1434–1440 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3544/3824 https://umj.imath.kiev.ua/index.php/umj/article/view/3544/3825 Copyright (c) 2006 Kovalev A. M.; Kravchenko N. V.; Nespirnyi V. N. |
| spellingShingle | Kovalev, A. M. Kravchenko, N. V. Nespirnyi, V. N. Ковалев, А. М. Кравченко, Н. В. Неспирный, В. Н. Ковалев, А. М. Кравченко, Н. В. Неспирный, В. Н. Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls |
| title | Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls |
| title_alt | Необходимое условие стабилизируемости по части переменных нелинейных систем в классе разрывных управлений |
| title_full | Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls |
| title_fullStr | Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls |
| title_full_unstemmed | Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls |
| title_short | Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls |
| title_sort | necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3544 |
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