Controllability problems for the string equation
For the string equation controlled by boundary conditions, we establish necessary and sufficient conditions for 0-and ε-controllability. The controls that solve such problems are found in explicit form. Moreover, using the Markov trigonometric moment problem, we construct bangbang controls that solv...
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| Дата: | 2007 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3358 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509435960492032 |
|---|---|
| author | Fardigola, L. V. Khalina, K. S. Фардигола, Л. В. Халіна, К. С. |
| author_facet | Fardigola, L. V. Khalina, K. S. Фардигола, Л. В. Халіна, К. С. |
| author_sort | Fardigola, L. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:14Z |
| description | For the string equation controlled by boundary conditions, we establish necessary and sufficient conditions for 0-and ε-controllability. The controls that solve such problems are found in explicit form. Moreover, using the Markov trigonometric moment problem, we construct bangbang controls that solve the problem of ε-controllability. |
| first_indexed | 2026-03-24T02:41:04Z |
| format | Article |
| fulltext |
UDK 517.9
L. V. Fardyhola, K. S. Xalina (Fiz.-texn. in-t nyz\kyx temperatur, Xarkiv)
PROBLEMY KEROVANOSTI DLQ RIVNQNNQ STRUNY
Necessary and sufficient conditions of the null-controllability and approximate null-controllability are
obtained for the string equation controlled by boundary conditions. Controls solving these problems are
found explicitly. Moreover, bang-bang controls solving the approximate null-controllability problem are
constructed with the use of the Markov trigonometric moment problem.
Poluçen¥ neobxodym¥e y dostatoçn¥e uslovyq 0- y ε-upravlqemosty dlq uravnenyq strun¥,
upravlqemoho kraev¥my uslovyqmy. Upravlenyq, reßagwye πty zadaçy, najden¥ v qvnom vyde.
Bolee toho, s pomow\g tryhonometryçeskoj problem¥ momentov Markova postroen¥ relejn¥e
upravlenyq, reßagwye zadaçu ε-upravlqemosty.
0. Vstup. Problemy kerovanosti dlq hiperboliçnyx rivnqn\ vyvçagt\sq zaraz
bahat\ma matematykamy (dyv., napryklad, bibliohrafig v [1]).
U cij roboti my doslidΩu[mo krajovu kerovanist\ xvyl\ovoho rivnqnnq na
skinçennomu vidrizku za prostorovog zminnog za dopomohog obmeΩenyx napered
zadanog konstantog keruvan\. Slid vidmityty, wo v bil\ßosti robit, v qkyx
vyvçalosq take rivnqnnq, doslidΩu[t\sq L2
-kerovanist\, abo, qk uzahal\nennq,
Lp
-kerovanist\ ( 2 ≤ p ≤ ∞ ) [2 – 6]. Ale lyße L∞
-keruvannq moΩna praktyçno
realizuvaty. Bil\ß toho, z praktyçnyx mirkuvan\ taki keruvannq magt\ buty
obmeΩeni napered zadanog konstantog (qk v (0.3)). Okrim toho, pry zarodΩenni
teori] keruvannq panuvaly taki pohlqdy, wo lyße keruvannq v formi peremyka-
ça moΩna realizuvaty praktyçno. Poßuk takyx keruvan\ dlq rozv’qzannq prob-
lem kerovanosti [ aktual\nog zadaçeg i zaraz. Same tomu v p. 2 budut\ pobudo-
vani relejni keruvannq, qki rozv’qzugt\ problemu ε-kerovanosti dlq rivnqnnq,
wo rozhlqda[t\sq.
Problemy krajovo] kerovanosti dlq xvyl\ovoho rivnqnnq na pivosi z keruvan-
nqmy, obmeΩenymy napered zadanog stalog, bulo vyvçeno v [9, 10].
Rozhlqnemo xvyl\ove rivnqnnq
∂
∂
= ∂
∂
2
2
2
2
w
t
w
x
, x ∈ ( 0, π ), t ∈ ( 0, T ), (0.1)
kerovane krajovymy umovamy
w ( 0, t ) = u0 ( t ), w ( π, t ) = uπ ( t ), t ∈ ( 0, T ), (0.2)
de T > 0. My vvaΩa[mo, wo keruvannq uj , j = 0, π, zadovol\nqgt\ umovu
uj ∈ B ( 0, T ) = { v ∈ L∞
( 0, T ) | | v ( t ) | ≤ 1 majΩe skriz\ na ( 0, T ) }. (0.3)
Usi funkci], wo rozhlqdagt\sq v rivnqnni (0.1) ta krajovyx umovax (0.2), vyzna-
çeni na skinçennomu vidrizku. Dali skriz\ budemo vvaΩaty, wo taki funkci] vyz-
naçeni na R ta nabuvagt\ znaçennq 0 na dopovnenni v R svo[] oblasti vyzna-
çennq.
Rozhlqnemo funkcional\ni prostory, wo vykorystovugt\sq v roboti. Nexaj
S — prostir Ívarca [7]:
S = { ϕ ∈ C
∞
( R ) : ∀m ∈ N0 ∀l ∈ N0 ∃ Cml > 0 ∀x ∈ R :
| ϕ(
m
)
( x ) ( 1 + | x | 2 )
l
| ≤ Cml },
de N0 ∈ N ∪ 0, ta S ′ — prostir uzahal\nenyx funkcij nad S. Poslidovnist\
{ } =
∞ϕn n 1 ⊂ S nazyva[t\sq zbiΩnog v S, qkwo dlq bud\-qkyx m ∈ N i l ∈ N
© L. V. FARDYHOLA, K. S. XALINA, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7 939
940 L. V. FARDYHOLA, K. S. XALINA
isnu[ Cml take, wo dlq bud\-qkoho n ∈ N x xl
n
mϕ( )( ) ≤ Cml ta x xl
n
mϕ( )( ) ⇒ 0
pry n → ∞ v R. Pid zbiΩnistg v S ′ rozumi[t\sq slabka zbiΩnist\. Nexaj
takoΩ
D ( a, b ) = { ϕ ∈ C
∞
( R ) : supp ϕ ∈ ( a, b ) },
D ′ ( a, b ) — prostir uzahal\nenyx funkcij nad D ( a, b ) ta D ( – ∞, ∞ ) = D ,
D ′ ( – ∞, ∞ ) = D ′. Poslidovnist\ { } =
∞ϕn n 1 ⊂ D ( a, b ) nazyva[t\sq zbiΩnog v
D ( a, b ), qkwo dlq bud\-qkoho n ∈ N supp ϕn ⊂ [ α, β ] ⊂ ( a, b ) ta dlq bud\-
qkoho m ∈ N ϕn
m x( )( ) ⇒ 0 pry n → ∞ v R. Pid zbiΩnistg v D ′ ( a, b ) rozu-
mi[t\sq slabka zbiΩnist\. Poslidovnist\ { } =
∞δn n 1 ⊂ D nazyva[t\sq δ-poslidov-
nistg, qkwo supp δn ⊂ [ 0, 1 / n ] ta δn x dx( )
− ∞
∞
∫ = 1. Nexaj f ∈ D ′ ( a, b ). Budemo
hovoryty, wo f ( a + 0 ) = α ∈ R, qkwo dlq koΩno] δ-poslidovnosti { } =
∞δn n 1
( f ( x ), δn ( x – a ) ) → α pry n → ∞, ta f ( b – 0 ) = β ∈ R, qkwo dlq koΩno] δ-pos-
lidovnosti { } =
∞δn n 1 ( f ( x ), δn ( b – x ) ) → β pry n → ∞.
Poznaçymo çerez Hl
s
nastupni prostory Soboleva [8], (hl. 1):
Hl
s = { ϕ ∈ S ′ : ( 1 + | x | 2 )
l
/
2 ( 1 + | D |
2
)
s
/
2
ϕ ∈ L2
( R ) },
H a bl
s( ), = { ϕ ∈ D ′ ( a, b ) ∩ Hl
s
: ∃ ϕ ( a + 0 ) ∈ R ∃ ϕ ( b – 0 ) ∈ R },
ϕ ϕl
s l s
x D x dx= +( ) +( ) ( )
− ∞
+ ∞
∫ 1 12 2 2 2 2
1 2
/ /
/
,
de D = – i d / dx.
U prostori Hl
s
razom z wojno vvedenog normog budemo rozhlqdaty we j
normu
[] ϕ []l
s = 1 12 2 2 2 2
1 2
+( ) +( ) ( )
− ∞
+ ∞
∫ D x x dx
s l/ /
/
ϕ .
Vidomo (dyv. [8], hl. 1), wo dlq bud\-qkyx s i l isnu[ stala Kl
s > 0 taka, wo
1
Kl
s l
sϕ ≤ [] ϕ []l
s ≤ Kl
s
l
sϕ . (0.4)
Nexaj F : S ′ → S ′ — operator peretvorennq Fur’[. Ma[mo
( F ϕ ) ( σ ) = 1
2π
( )−
− ∞
+ ∞
∫ e x dxi xσ ϕ , ϕ ∈ S,
ta
( F f, ϕ ) = ( f, F
–
1
ϕ ), f ∈ S ′, ϕ ∈ S.
Vidomo, wo F S = S, F S ′ = S ′. Okrim toho (dyv. [8], hl. 1), F Hl
s = Hs
l
ta ϕ l
s =
= [] F ϕ []s
l
, ϕ ∈ Hl
s
. U cij roboti my zavΩdy budemo vvaΩaty l < – 1 / 2, s ≤ 0.
Dali budemo vykorystovuvaty takoΩ prostory
H̃l
s = { ϕ ∈ Hl
s × Hl
s −1
: ϕ — neparna, 2π-periodyçna },
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PROBLEMY KEROVANOSTI DLQ RIVNQNNQ STRUNY 941
˜ , , ,H a b H a b H a bl
s
l
s
l
s( ) = ( ) × ( )−1
z normog ϕ ϕ ϕl
s
l
s
l
s= ( ) + ( )( )−
0
2
1
1 2 1 2/
, ϕ =
ϕ
ϕ
0
1
, ta
Ĥ H Hs
l
s
l
s
l= × −1
z normog [][] ϕ [][]s
l = [] []( ) + [] []( )( )−ϕ ϕ0
2
1 1
2 1 2
s
l
s
l
/
, ϕ =
ϕ
ϕ
0
1
.
U p. 1 oderΩano kryteri] 0- ta ε-kerovanosti systemy (0.1), (0.2) z obmeΩen-
nqmy na keruvannq (0.3). Keruvannq, wo rozv’qzugt\ problemu 0-kerovanosti,
znajdeno v qvnomu vyhlqdi, ale ci keruvannq moΩut\ buty dosyt\ skladnymy
dlq praktyçno] realizaci]. Tomu holovnog metog p. 2 [ pobudova relejnyx ke-
ruvan\, wo rozv’qzugt\ problemu ε-kerovanosti. Cg problemu zvedeno do sys-
temy tryhonometryçnyx problem momentiv Markova. Okrim toho, otrymano
ocinku toçnosti vluçennq dlq pobudovano] systemy relejnyx keruvan\, wo roz-
v’qzugt\ problemu ε-kerovanosti.
1. Problemy kerovanosti. Rozhlqnemo kerovanu systemu (0.1), (0.2) z po-
çatkovymy umovamy
w ( x, 0 ) = w x0
0( ),
x ∈ ( 0, π ), (1.1)
∂ ( )
∂
w x
t
, 0
= w x1
0( ),
de w0 =
w
w
0
0
1
0
∈
˜ ,Hs
0 0[ π].
Nexaj Ω : S ′ → S ′ — operator neparnoho prodovΩennq, tobto ( Ω f ) ( x ) =
= f ( x ) – f ( – x ), f ∈ S ′, supp f ⊂ [ 0, + ∞ ). Rozhlqnemo neparni 2π-periodyçni pro-
dovΩennq (po x ) funkcij, wo zv’qzani kerovanog systemog (0.1), (0.2), (1.1):
W
0 =
T
2
0
π
∈
∑ k
k
wΩ
Z
,
W ( ⋅, t ) =
T
2π
∈
∑
(⋅ )
∂ (⋅ )
∂
k
k
w t
w t
t
Ω
Z
,
, , t ∈ ( 0, T ),
de Th — operator zsuvu: ( Th ϕ ) ( x ) = ϕ ( x + h ), ϕ ∈ S, ta ( Th f, ϕ ) = ( f, T– h ϕ ), f ∈
∈ S ′, ϕ ∈ S.
Oskil\ky dlq bud\-qkoho N ∈ N
( ) ≤ + − π( ) ≤ ( )
∈
∑c x Nk Cl
N l
k
l
N2 2 21
Z
, x ∈ R,
de cl
N = (π ) −N Sl
l2 1, Cl
N = 3 2 2+ (π )N Sl
l , Sl = k l
k
2
1=
∞∑ , to
c g g C gl
N s
Nk
k l
s
l
N s
0 0≤ ≤π
∈
∑T
Z
, g ∈ H Hs s
0 0
1× −
. (1.2)
Okrim toho, Ω f s
0 ≤ 2 0f s , f ∈ Hs
0 , supp f ∈ [ 0, + ∞ ). Tomu W
0 ∈ H̃l
s , W ( ⋅,
t ) ∈ H̃l
s , t ∈ ( 0, T ).
Qk lehko baçyty, zadaçu (0.1), (0.2), (1.1) moΩna zvesty do zadaçi
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
942 L. V. FARDYHOLA, K. S. XALINA
∂
∂
= ∂
∂
W
t
x
W
0 1
0
2 – 2
0
20u t
x kk
( )
′( + π )
∈
∑ δ
Z
+
+ 2
0
2
u t
x kk
π
∈
( )
′( − π + π )
∑ δ
Z
, x ∈ R, t ∈ ( 0, T ), (1.3)
W ( x, 0 ) = W
0
( x ), x ∈ R, (1.4)
de δ — funkciq Diraka.
Nexaj wT =
w
w
T
T
0
1
∈
˜ ,Hs
0 0[ π], WT =
T
2π∈∑ k
T
k
wΩ
Z
. Rozhlqnemo dlq zadaçi
(1.3), (1.4) umovy vluçennq
W ( x, T ) = WT
( x ), x ∈ R. (1.5)
Nexaj T > 0, w0 ∈
˜ ,Hs
0 0[ π]. Poznaçymo çerez R T ( w0
) mnoΩynu staniv wT ∈
∈ ˜ ,Hs
0 0[ π], dlq qkyx isnu[ keruvannq u ∈ B ( 0, T ) take, wo zadaça (1.3) – (1.5) z
Wβ = T
2π∈∑ kk
wΩ β
Z
, β = 0, T, ma[ [dynyj rozv’qzok v H̃l
s
.
Oznaçennq 1.1. Stan w0 ∈
˜ ,Hs
0 0[ π] nazyva[t\sq 0-kerovanym za ças T,
qkwo 0 naleΩyt\ R T ( w0
), t a ε -kerovanym za ças T, qkwo 0 naleΩyt\
zamykanng R T ( w0
) v
˜ ,Hs
0 0[ π].
Zastosuvavßy do (1.3) – (1.5) peretvorennq Fur’[ po x, oderΩymo
d
dt
i e u t e u tk i i
k
v
v=
−
−
π
( ) − ( )π − π
∈
( )∑
0 1
0
2
2
2
0σ
σ σ σ
π
Z
, σ ∈ R, t ∈ ( 0, T ),
(1.6)
v ( σ, 0 ) = v0
( σ ), (1.7)
v ( σ, T ) = vT
( σ ), (1.8)
de v ( ⋅, t ) = F W ( ⋅, t ) ∈ Ĥs
l
, v0 = F W
0 ∈ Ĥs
l
, vT = F W T ∈ Ĥs
l
. OtΩe,
vT
( σ ) = Σ Σ( ) ( ) −
π
( − )
( ) − ( )
π
∈
− π∑ ∫σ σ σ σσ
σ
π
, ,T i e t
u t e u t
dtk i
k
i
T
v0 2
00
2 0
Z
,
de
Σ ( σ, t ) =
cos
sin
sin cos
( ) ( )
− ( ) ( )
σ σ
σ
σ σ σ
t
t
t t
.
Tomu
W T ( x ) = E ( x, T ) *
W x
u x u x
u x u xk
k
0
2
0
0
( ) −
( )( ) − ( )( − π)
( ′ ) ( ) − ( ′ ) ( − π)
π
∈
π
π
∑ ( ) ( )T
Z
Ω Ω
Ω Ω
, (1.9)
de
E ( x, T ) =
1
2
1
π
( )−F Σ σ, T =
= 1
2
1
2
δ δ
δ δ δ δ
( + ) + ( − ) ( + ) − ( − )( )
− ′( + ) + ′( − ) ( + ) + ( − )
x T x T x T x T
x T x T x T x T
sign sign
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PROBLEMY KEROVANOSTI DLQ RIVNQNNQ STRUNY 943
Tut i dali * oznaça[ zhortku po x.
Dlq W
0 ∈ H̃l
s
poznaçymo
RT ( W
0
) =
E ( x, T ) *
W x
u x u x
u x u xk
k
0
2
0
0
( ) −
( )( ) − ( )( − π)
( ′ ) ( ) − ( ′ ) ( − π)
π
∈
π
π
∑ ( ) ( )T
Z
Ω Ω
Ω Ω
| u0 ∈ B ( 0, T ) ∧ uπ ∈ B ( 0, T )
.
Todi oznaçennq 1.1 ekvivalentne nastupnomu oznaçenng.
Oznaçennq 1.2. Stan W
0 ∈ H̃l
s
nazyva[t\sq 0-kerovanym za ças T ,
qkwo 0 naleΩyt\ RT ( W
0
), t a ε -kerovanym za ças T, qkwo 0 naleΩyt\
zamykanng RT ( W
0
) v H̃l
s
.
Oçevydno, spravedlyvymy [ nastupni tverdΩennq.
TverdΩennq 1.1. Stan w0 ∈
˜ ,Hs
0 0[ π] [ 0-kerovanym za ças T todi i ly-
ße todi, koly stan W0 = T
2
0
π∈∑ kk
wΩ
Z
, wo naleΩyt\ H̃l
s
, [ 0 -kerovanym
za ças T.
TverdΩennq 1.2. Qkwo stan w0 ∈
˜ ,Hs
0 0[ π] [ ε-kerovanym za ças T, to
stan W
0 = T
2
0
π∈∑ kk
wΩ
Z
, wo naleΩyt\ H̃l
s
, [ ε-kerovanym za ças T.
Vyznaçymo K ∈ N take, wo π ( K – 1 ) < T ≤ π K, ta poznaçymo
u tk
α( ) = uα ( t ) ( H ( t – π ( k – 1 ) ) – H ( t – π k ) ), k = 1, K , α = 0, π, (1.10)
de H — funkciq Xevisajda: H ( t ) = 1, qkwo t > 0, ta H ( t ) = 0, qkwo t ≤ 0.
OtΩe,
uα ( t ) = u tk
k
K
α( )
=
∑
1
, supp uk
α ⊂ [ π ( k – 1 ), π k ], k = 1, K , α = 0, π.
(1.11)
Z (1.9) oderΩu[mo
W
T
( x ) = E ( x, T ) *
T
2
0
0
1
0π
∈
∑
( )
( )
k
k
w x
w xZ
Ω –
–
Ω
Ξ
(− ) π
+ (− )
+ π
+ (− ) ( − π)
π
+ (− )
− π
+ (− ) (
+ +
π
=
+
π
∑ 1 2
2
1 2
2
1
2
2
1 2
2
1
1
0
1
1
0
1
k k k k k
k
K
k k k k
u
k
x u
k
x
u
k
x u
k
x −− π)
′
=
∑
k
K
1
, (1.12)
de Ξ : S ′ → S ′ — operator parnoho prodovΩennq: ( Ξ f ) ( x ) = f ( x ) + f ( – x ), f ∈ S ′,
supp f ⊂ [ 0, + ∞ ).
Poznaçymo H s
0 [− π π], = { ϕ ∈ Hs
0 : supp ϕ ∈ [ – π, π ] }. Qsno, wo H s
0 [− π π], —
kompaktnyj pidprostir Hs
0 .
Lema 1.1. Nexaj g ∈ H s
0
1− [− π π], [ neparnog. Todi isnu[ g̃ ∈ H s
0 [− π π],
parna i taka, wo ˜ ′g = g.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
944 L. V. FARDYHOLA, K. S. XALINA
Dovedennq. Oskil\ky g ∈ S ′ ta [ finitnog ( supp g ⊂ [ – π, π ] ), to za uza-
hal\nenog teoremog Peli – Vinera [11] (hl. 3) G = F g ∈ S ′ [ rehulqrnym funk-
cionalom, G zrosta[ na R ne ßvydße, niΩ polinom, ta prodovΩu[t\sq do cilo]
funkci] porqdku ≤ 1 ta typu ≤ π. Z neparnosti g vyplyva[ neparnist\ G.
Tomu G ( 0 ) = 0 ta
G s
is
( )
[ cilog funkci[g porqdku ≤ 1 ta typu ≤ π , wo zros-
ta[ na R ne ßvydße, niΩ polinom. OtΩe, za ti[g Ω teoremog g̃ =
=
F − ( )
1 G
i
σ
σ
∈ S ′, supp g ⊂ [ – π, π ] ta ˜ ′g = g. Oçevydno, wo g̃ [ parnog.
Ma[mo G ∈ Hs −1
0
. OtΩe,
G
i
( )σ
σ
s
0
= 1 2
2 1 2
+( ) ( )
− ∞
+ ∞
∫ σ σ
σ
σ
s G
i
d
/
≤ [] G [] −s 1
0 + 2 1 2
1 1
( + )
∈[− ]
′( )s G/
,
sup
σ
σ .
Tomu
G
i
( )σ
σ
∈ Hs
0
, a g̃ ∈ H s
0 [− π π], .
Lemu dovedeno.
Poznaçymo çerez ∂ operator ∂ : H s
0 [− π π], → H s
0
1− [− π π], z oblastg vyzna-
çennq D ( ∂ ) = { ϕ ∈ H s
0 [− π π], : ϕ — parna }. Todi Im ( ∂ ) = { ϕ ∈ H s
0
1− [− π π], : ϕ
— neparna }. Ma[mo ∂ −g s
0
1 =
i g sσF −1
0 ≤ Fg
s
0
= g s
0 , g ∈ D ( ∂ ), otΩe, ∂ —
linijnyj neperervnyj operator. Za lemog 1.1 cej operator ma[ obernenyj. Todi
∂–
1
: H s
0
1− [− π π], → H s
0 [− π π], , D ( ∂–
1
) = Im ( ∂ ), Im ( ∂–
1
) = D ( ∂ ). Za teoremog
Banaxa pro obernenyj operator ∂–
1
[ linijnym ta neperervnym, otΩe, dlq koΩ-
noho s ≤ 0 isnu[ Ms > 0 take, wo
∂ ≤− −1
0 0
1g M g
s
s
s , g ∈ D ( ∂–
1
). (1.13)
Lehko baçyty, wo Ms ≥ 1.
Lema 1.2. Nexaj g ∈
˜ ,Hs
0[− π π]. Todi
E T( ) ≤π
∈
∑x t g C M g
k
k l
s
l s
s, * 2
2
02 2Ω
Z
, t ∈ R, (1.14)
de Cl
2 > 0 — stala z nerivnosti (1.2), a Ms ≥ 1 — stala z nerivnosti (1.13).
Dovedennq. Vraxovugçy (1.2), ma[mo
E T E( ) ≤ ( )π
∈
∑x t g C x t g
k
k l
s
l
s, * , *2
2
0Ω Ω
Z
, t ∈ R. (1.15)
Poznaçagçy G = F Ω g, oderΩu[mo
E( )x t g s, * Ω 0 = [][] Σ Ω( )σ, t gF [][]s
0 ≤
≤
cos
sin
( )
− ( )
( )
σ
σ
σ
t
t
G0
s
0
+
sin
cos
( )
( )
( )
σ
σ
σ
σ
t
t
G1
s
0
≤
≤ 2 0
0 1
0 2
1 1
0 2
1 2
G
G
i
Gs
s
s= ( )
+ ( )
−
σ
σ
/
, t ∈ R.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PROBLEMY KEROVANOSTI DLQ RIVNQNNQ STRUNY 945
Oskil\ky
G
i s
1
0( )σ
σ
= ∂−1
1 0
Ωg
s
, to, vykorystovugçy (1.13), ma[mo
E( ) ≤ + ( ) + ( )( )− − −
x t g g M g gs s
s
s s, *
/
Ω 0 0 0 1 0
1 2
1 0
1 2 1 2
2 2 2 ≤
≤ 2 2 0M gs
s , t ∈ R.
ProdovΩugçy ocinku (1.15), zvidsy oderΩu[mo (1.14).
Lemu dovedeno.
Lema 1.3. Nexaj g ∈ Ĥs
l
. Todi
E( )x t g l
s, * = [][] Σ( )σ, t gF [][]s
l ≤
≤ 4 62t + [][] F g [][]s
l = 4 62t g l
s+ , t ∈ R. (1.16)
Dovedennq. Dlq koΩnoho t ∈ R ma[mo
E( )x t g l
s, * = [][] Σ( )σ, t gF [][]s
l ≤
≤
cos
sin
( )
− ( )
σ
σ
t
t
gF 0
s
l
+
sin
cos
( )
( )
σ
σ
σ
t
t
gF 1
s
l
≤
≤ 2 [] F g0 []s
l +
( )
+ [] []( )
−
sin
/
σ
σ
t
g g
s
l
s
lF F1
2
1 1
2
1 2
.
Vraxovugçy, wo 1 2
2
+( ) ( )σ σ
σ
sin t
≤ 2 12( + )t , oderΩu[mo (1.16).
Lemu dovedeno.
Teorema 1.1. Nexaj T > 0, w0 ∈
˜ ,Hs
0 0[ π], K ∈ N — take çyslo, wo π ( K –
– 1 ) < T ≤ π K, ta isnu[ µ ∈ R take, wo
∂ ( ) ± ( ) +
π
≤−1
1
0
0
0 2Ω Ωw x w x T
K
Kµ majΩe skriz\ na [ – π, π ] , (1.17)
supp{ }( )∂ ( ) + ( )−H x w x w x1
1
0
0
0Ω ⊂ [ 0, T – π ( K – 1 ) ], (1.18)
supp{ }( )∂ ( ) − ( )−H x w x w x1
1
0
0
0Ω ⊂ [ π K – T, π ]. (1.19)
Todi stan w0
[ 0-kerovanym za ças T. Okrim toho, qkwo
w̃ x w x H x T
K
H t H t T1
0 1
1
0( ) = ∂ ( ) ( ) +
π
( ) − ( − )− ( )Ω µ ,
u t
K
w t k w t kk
0
2 1
1
0
0
01
2
2 2+ ( ) = ( − π ) + ( − π )[ ]˜ , k = 0 1 2, /[ ]( − )K ,
u t
K
w k t w k tk
0
2
1
0
0
01
2
2 2( ) = ( π − ) − ( π − )[ ]˜ , k = 1 2, /[ ]K ,
u t
K
w k t w k tk
π
+ ( ) = − ( π + π − ) − ( π + π − )[ ]2 1
1
0
0
01
2
2 2˜ , k = 0 1 2, /[ ]( − )K ,
u t
K
w k t w k tk
π ( ) = − (− π + π + ) + (− π + π + )[ ]2
1
0
0
01
2
2 2˜ , k = 1 2, /[ ]K ,
to keruvannq
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
946 L. V. FARDYHOLA, K. S. XALINA
uα ( t ) = u tk
k
K
α( )
=
∑
1
, α = 0, π,
zadovol\nqgt\ umovy supp uα ⊂ [ 0, T ], α = 0, π, ta rozv’qzugt\ problemu 0-
kerovanosti za ças T.
Dovedennq. Z (1.17) vyplyva[, wo uα ∈ B ( 0, T ), α = 0, π. Za formulog
(1.12) W T = 0, tobto stan W
0
[ ε-kerovanym za ças T (dyv. (1.18), (1.19)). Na
pidstavi tverdΩennq 1.1 robymo vysnovok, wo stan w0
[ takoΩ 0-kerovanym za
ças T.
Teoremu dovedeno.
Teorema 1.2. Nexaj T > 0, w0 ∈
˜ ,Hs
0 0[ π], K ∈ N — take çyslo, wo π ( K –
– 1 ) < T ≤ π K. Todi qkwo stan w0
[ 0-kerovanym za ças T, to umovy (1.17) –
(1.19) vykonano.
Dovedennq. Za tverdΩennqm 1.2 stan W
0 = Ωw
k
0
∈∑ Z
[ ε-kerovanym za
ças T . Tomu dlq koΩnoho n ∈ N isnu[ Wn ∈ RT ( W
0
) take, wo W n
l
s
< 1
n
.
OtΩe, isnugt\ un
α ∈ B ( 0, T ), α = 0, π, taki, wo
W
n = E ( x, T ) * W x
u x u x
u x u xk
k
n n
n n
0
2
0
0
( ) −
( )( ) − ( )( − π)
( ′) ( ) − ( ′) ( − π)
π
∈
π
π
∑ ( ) ( )
T
Z
Ω Ω
Ω Ω
.
Zastosovugçy lemu 1.3 ta formuly (1.10) – (1.13), oderΩu[mo
Ω{ }( ) − ( )w x u xn
0
0
0ˆ → 0 ta { }∂ ( ) − ( )−1
1
0
1Ω Ξw x u xnˆ → 0 pry n → ∞ (1.20)
v H s
0 [− π π], , a otΩe, i v S ′. Tut
ˆ , ,u t u
k
t u
k
tn k n k k n k k
k
K
0
1
0
1
0
1 2
2
1 2
2
1( ) = (− ) π
+ (− )
+ π
+ (− ) ( − π)
+ +
π
=
∑ ,
ˆ , ,u t u
k
t u
k
tn n k k n k k
k
K
1 0
1
0
2
2
1 2
2
1( ) = π
+ (− )
− π
+ (− ) ( − π)
+
π
=
∑ ,
u u H x p H x pn p n
α α
, = ( − π( − )) − ( − π )( )1 , p = 1, K , α = 0, π.
Oskil\ky un
α ∈ B ( 0, T ), α = 0, π, n ∈ N, to
û xj
n( ) ≤ 2K majΩe skriz\ na [ 0, π ], j = 0, 1. (1.21)
Vykorystovugçy te, wo S [ wil\nym u L2
( R ) ta supp û j
n ⊂ [ 0, π ], n ∈ N, j = 0,
1, z (1.20) oderΩu[mo Ωû xn
0 ( ) → Ωw x0
0( ) ta Ξû xn
1 ( ) → ∂ ( )−1
1
0Ωw x pry n → ∞
v ( L2
( – π, π ) ) ′. Za teoremog Rissa Ωw0
0 ∈ L2
( – π, π ), ∂−1
1
0Ωw ∈ L2
( – π, π ).
OtΩe, z (1.20), (1.21) vyplyva[ (1.17) – (1.19).
Teoremu dovedeno.
Takym çynom, teoremy 1.1 ta 1.2 dagt\ nastupnyj kryterij 0- ta ε-kerova-
nosti.
Vysnovok 1.1. Nexaj T > 0, w0 ∈
˜ ,Hs
0 0[ π] t a K ∈ N — take çyslo, wo
π ( K – 1 ) < T ≤ π K. Todi nastupni tverdΩennq [ rivnosyl\nymy:
i) stan w0
[ 0-kerovanym za ças T;
ii) stan w0
[ ε-kerovanym za ças T;
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PROBLEMY KEROVANOSTI DLQ RIVNQNNQ STRUNY 947
iii) vykonano try umovy:
∂ ( ) ± ( ) +
π
≤−1
1
0
0
0 2Ω Ωw x w x T
K
Kµ majΩe skriz\ na [ – π, π ]
dlq deqkoho µ ∈ R,
supp{ }( )∂ ( ) + ( )−H x w x w x1
1
0
0
0Ω ⊂ [ 0, T – π ( K – 1 ) ],
supp{ }( )∂ ( ) − ( )−H x w x w x1
1
0
0
0Ω ⊂ [ π K – T, π ].
Proilgstru[mo cej kryterij prykladamy.
Pryklad 1.1. Nexaj w x0
0( ) = 2 sin ( x ), w x1
0( ) = 2 sin ( 2x ) na [ 0, π ]. Todi
dlq bud\-qkoho s ≤ 0 w0 ∈
˜ ,Hs
0 0[ π]. Ma[mo
˜ sinw x H x w x H x w x dx x H x H x
x
1
0 1
1
0
1
0 22( ) = ( )∂ ( ) = ( ) ( ) = ( ) ( ) − ( − π)−
− ∞
∫ [ ]Ω Ω .
Oskil\ky sup ˜ : ,w x w x x1
0
0
0 0( ) ± ( ) ∈[ π]{ } = 4, to za teoremog 1.1 keruvannq
u0 ( t ) = 1
2
2( + )sin sint t , uπ ( t ) = 1
2
2( )(π − ) + (π − )sin sint t , t ∈ [ 0, 2π ], rozv’qzu-
gt\ problemu 0-kerovanosti systemy (0.1), (0.2), (1.1) za ças T = 2π. Oskil\ky
supp ˜( )( ) + ( )w x w x1
0
0
0 = supp ˜( )( ) − ( )w x w x1
0
0
0 = [ 0, π ], to stan w0
ne [ 0-kerova-
nym ( ε-kerovanym) za ças T < 2π.
Pryklad 1.2. Nexaj
w x0
0( ) = x H x H x( ) − − π
4
+ π − π
− − π
4 4
3
4
H x H x +
+ (π − ) − π
− ( − π)
x H x H x3
4
,
w x1
0( ) = H x H x H x H x( ) − − π
+ − π
− ( − π)2
4
2 3
4
.
Todi dlq bud\-qkoho s ≤ 0 w0 ∈
˜ ,Hs
0 0[ π]. Ma[mo
w̃ x H x w x H x w x dx
x
1
0 1
1
0
1
0( ) = ( )∂ ( ) = ( ) ( )−
− ∞
∫Ω Ω =
= x H x H x x H x H x( ) − − π
+ π −
− π
− − π
4 2 4
3
4
+
+ ( − π) − π
− ( − π)
x H x H x3
4
.
Oskil\ky sup ˜ : ,w x w x x1
0
0
0 0( ) ± ( ) ∈[ π]{ } = π
2
, to za teoremog 1.1 keruvannq
u0 ( t ) = uπ ( t ) = t H t H t t H t H t( ) − − π
+ π −
− π
− − π
4
1
2
3
4 4
3
4
rozv’qzugt\ problemu 0-kerovanosti systemy (0.1), (0.2), (1.1) za ças T = 3
4
π
.
Oskil\ky supp ˜( )( ) + ( )w x w x1
0
0
0 = 0 3
4
, π
, supp ˜( )( ) − ( )w x w x1
0
0
0 = π π
4
, , to
stan w0
ne [ 0-kerovanym ( ε-kerovanym) za ças T < 3
4
π
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
948 L. V. FARDYHOLA, K. S. XALINA
2. Relejni keruvannq ta tryhonometryçna problema momentiv Markova.
Rozv’qzok problemy 0-kerovanosti (keruvannq), znajdenyj u p. 1, moΩe buty do-
syt\ skladnym dlq praktyçnoho vykorystannq. U c\omu punkti my budu[mo re-
lejni keruvannq, wo rozv’qzugt\ problemu ε-kerovanosti (relejnym nazyva[t\-
sq keruvannq u ( t ), t ∈ ( 0, T ), take, wo | u ( t ) | = 1 majΩe skriz\ na ( 0, T ), u ( t ) =
= 0 majΩe skriz\ na R \ ( 0, T ) ta çyslo toçok rozryvu [ skinçennym). Budemo
rozhlqdaty systemu tryhonometryçnyx problem momentiv Markova, qki pobudo-
vani za danymy kerovano] systemy, ta dovedemo, wo deqki ]xni rozv’qzky [ relej-
nymy keruvannqmy, wo rozv’qzugt\ problemu ε-kerovanosti.
Skriz\ u c\omu punkti budemo vvaΩaty, wo T > 0, w0 ∈
˜ ,Hs
0 0[ π], K ∈ N —
take çyslo, wo π ( K – 1 ) < T ≤ π K. Okrim toho, budemo vvaΩaty, wo umovu iii)
vysnovku 1.1 vykonano, ta poznaçymo w̃ x H x w x T
K
H x1
0 1
1
0( ) = ( )∂ ( ) +
π
( )− (Ω µ –
– H x T( − )), W
0 =
T
2
0
π∈∑ kk
wΩ
Z
,
˜ ˜W x w x
kk1
0
2 1
0( ) = ( )π∈∑ T
Z
Ξ .
Vykorystovugçy tverdΩennq 1.1 ta 1.2, my moΩemo zamist\ kerovano] sys-
temy (0.1), (0.2), (1.1) rozhlqdaty kerovanu systemu (1.3) – (1.5), de W ( ⋅, t ) =
= T
2π∈ (⋅ )∑ kk
w tΩ ,
Z
, t ∈ ( 0, T ), W T = T
2π∈∑ k
T
k
wΩ
Z
.
Rozhlqnemo zvuΩennq operatora neparnoho prodovΩennq Ω̃ : H s
0 [− π π], z
oblastg vyznaçennq D( )Ω̃ = H s
0 0[ π], ∩ L2
( 0, π ). Ma[mo Ω̃ f = Ω f, f ∈ D( )Ω̃ .
Todi Im ( )Ω̃ = { ϕ ∈ Hs
0(− π π), ∩ L2
( – π, π ) : ϕ — neparna }. Lehko pobaçyty, wo
Ω̃ vza[mno odnoznaçno vidobraΩu[ D( )Ω̃ na Im ( )Ω̃ ta ( )( )−Ω̃ 1g x = g ( x ) H ( x ),
g ∈ Im ( )Ω̃ . Vidmitymo, wo nemoΩlyvo dlq dvox dovil\nyx funkcij f1 i f2 , wo
naleΩat\ Hs
0 , vyznaçyty f1 f2 (dobutok cyx dvox funkcij), ale v rozhlqduva-
nomu vypadku odna z funkcij [ finitnog ta naleΩyt\ L2
( R ), a inßa [ obmeΩe-
nog, tomu naße oznaçennq Ω̃−1g [ korektnym dlq g ∈ D( )Ω̃ . Takym çynom, Ω̃
[ oborotnym linijnym operatorom, Ω̃−1
: H Hs s
0 0 0[− π π] → [ π], , , D( )−Ω̃ 1 =
= Im ( )Ω̃ , Im ( )−Ω̃ 1 = D( )Ω̃ .
Lema 2.1. Operator Ω̃−1
[ linijnym obmeΩenym, tobto dlq bud\-qkoho
s ≤ ≤ 0 isnu[ Ps > 0 take, wo
Ω̃− ≤1
0 0g P g
s
s
s , g ∈ Im ( )Ω̃ = D( )−Ω̃ 1
. (2.1)
Dovedennq. Zaznaçymo, wo oskil\ky Hs
0 [ banaxovym prostorom [8] (hl. 1),
to H s
0 0[ π], ta H s
0 [− π π], takoΩ [ banaxovymy prostoramy. Dovedemo, wo
D( )−Ω̃ 1
ta Im ˜( )Ω [ kompaktnymy linijnymy pidprostoramy H s
0 0[ π], ta
H s
0 [− π π], vidpovidno.
Nexaj [ a, b ] — sehment [ 0, π ] abo [ – π, π ]. Nexaj takoΩ { } =
∞gn n 0 ⊂
⊂ H a bs
0 [ ], ∩ L2
[ a, b ] ta gn → g pry n → ∞ v H a bs
0 [ ], . PokaΩemo, wo g ∈
∈ L2
[ a, b ]. Ma[mo gn → g pry n → ∞ v S ′. Oskil\ky S [ wil\nym v L2
( R ),
to gn → g pry n → ∞ v ( L2
( R ) ) ′. Tomu za teoremog Rissa g ∈ L2
[ a, b ]. OtΩe,
D( )Ω̃ [ kompaktnym linijnym pidprostorom H s
0 0[ π], . Qkwo dodatkovo prypus-
tyty, wo u vypadku [ a, b ] = [ – π, π ] gn [ neparnog ( n ∈ N ), to g takoΩ [ ne-
parnog. Tomu Im ( )Ω̃ [ kompaktnym linijnym pidprostorom H s
0 0[ π], . Ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PROBLEMY KEROVANOSTI DLQ RIVNQNNQ STRUNY 949
Ω̃g g
s s
0 02≤ , g ∈ D( )Ω̃ . (2.2)
Zastosovugçy teoremu Banaxa pro obernenyj operator, zvidsy oderΩu[mo ocin-
ku (2.1).
Lemu dovedeno.
ZauvaΩennq 2.1. Zaznaçymo, wo z toho, wo supp f ⊂ [ 0, + ∞ ), vzahali kaΩu-
çy, ne vyplyva[, wo supp ( 1 + | D |
2
)
s
f ⊂ [ 0, + ∞ ). Tomu oderΩaty qvnu ocinku
zverxu dlq stalo] Ps v (2.1) dosyt\ skladno. Skorystavßys\ lemog 2.1 ta vys-
novkom 1.1, my moΩemo utoçnyty tverdΩennq 1.2.
TverdΩennq 2.1. Stan w0 ∈
˜ ,Hs
0 0( π) [ ε-kerovanym za ças T todi i ly-
ße todi, koly stan W0 =
T
2
0
π∈∑ kk
wΩ
Z
, wo naleΩyt\ H̃l
s
, [ ε -kerovanym
za ças T.
Poznaçagçy
ˆ ˜w x
K
W x W x H x H x K0 1
0
0
01
2
( ) = ( ) + ( ) ( ) − ( − π )( )( ) ,
ˆ ˜w x
K
W x W x H x H x Kπ( ) = (π − ) − (π − ) ( ) − ( − π )( )( )1
2 1
0
0
0
ta zastosovugçy formulu (1.9), oderΩu[mo
W T ( x ) = E ( x, T ) *
T T2
0
2 1
2 0 0
1
π
=
−
π
∈
∑ ∑
( ) − ( )( )m
m
K
Kk
k d dx
w x u xΩ
/
ˆ
Z
+
+
T
2
1
π + π
∈
π π
( ) − ( )
∑ ( )
Kk
k d dx
w x u xΩ
/
ˆ
Z
. (2.3)
Z formuly (1.2) vyplyva[, wo dlq bud\-qkoho g ∈ Hs
0 , supp g ⊂ [ 0, + ∞ ), ma[mo
T
2
2
0
1 1
π
∈
≤
∑ Kk
k l
s
l
K
s
d dx
g C
d dx
gΩ Ω
/ /
Z
≤
≤
2
22
0
2
C g
C
c
gl
K s l
K
l
K Kk
k l
s
≤ π
∈
∑T
Z
.
Zvidsy na pidstavi lem 1.3, 2.1 ta formul (1.2), (2.3) oderΩu[mo
w
P
c
W K
P C
c c
w x u xT s s
l
T
l
s s l
K
l l
K Kk
k l
s
0 2
2
2
2 0 010≤ ≤ π
( ) − ( )π
∈
∑ ( )T
Z
ˆ +
+
Tπ + π
∈
π π∑ ( )( ) − ( )
Kk
k l
s
w x u x
Z
ˆ . (2.4)
Rozhlqnemo rozvynennq funkcij
Tπ∈∑ ( )( ) − ( )
Kkk
w x u x
Z
ˆα α , α = 0, π, v rq-
dy Fur’[ po e mx K−2 /
. Ma[mo
*
ˆ ˆ /ωα α
m i mx K
K
w x e dx= ( )
π
∫ 2
0
, m ∈ Z, α = 0, π, (2.5)
να α
m i mx K
K
u x e dx= ( )
π
∫ 2
0
/ , m ∈ Z, α = 0, π. (2.6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
950 L. V. FARDYHOLA, K. S. XALINA
Todi
Tπ
∈
−
∈
∑ ∑( )( ) − ( ) =
π
( − )
Kk
k
m m i mx K
m
w x u x
K
e
Z Z
ˆ ˆ /
α α α αω ν1 2
v Hs
0 , α = 0, π. (2.7)
Oskil\ky uα ∈ B ( 0, T ), α = 0, π, a w0
zadovol\nq[ umovu iii) vysnovku 1.1, to
ω̂α
m ≤ 1, να
m ≤ 1, m ∈ Z, α = 0, π. (2.8)
Tomu dlq s < – 1 / 2 otrymu[mo
Tπ
∈
∑ ( )( ) − ( )
Kk
k l
s
w x u x
Z
ˆα α ≤
C
K
m
K
l
K s
m m
mπ
+
−
∈
∑ 1 2 2 2
ω̂ να α
Z
, α = 0, π.
ProdovΩugçy (2.4), zvidsy znaxodymo
w KP
C C
c c
m
K
T s
s
l
K
l
K
l l
K
s
m m
m
0
2
2
2
0 0
2
10 1 2≤
+
−
∈
∑ ω̂ ν
Z
+
+ 1 2 2 2
+
−
π π
∈
∑ m
K
s
m m
m
ω̂ ν
Z
. (2.9)
Takym çynom, dovedeno nastupnu teoremu (dyv. (2.4), (2.7), (2.9)).
Teorema 2.1. Nexaj T > 0, dlq w0 ∈
˜ ,Hs
0 0[ π] vykonano umovu iii) vysnovku
1.1 ta ω̂α
m
vyznaçeno za formulog (2.5). Todi vykonugt\sq tverdΩennq:
i) stan w0
[ 0-kerovanym za ças T todi i lyße todi, koly isnugt\ uα ∈
∈ B ( 0, T ), α = 0, π, taki, wo ω̂α
m = να
m
, m ∈ Z , α = 0, π, de να
m
vyznaça[t\-
sq formulog (2.6);
ii) stan w0
[ ε-kerovanym za ças T todi i lyße todi, koly dlq koΩnoho
ε > 0 isnugt\ uα, ε ∈ B ( 0, T ), α = 0, π, taki, wo
1
2 2 2
+
−
∈
∑ m
K
s
m m
m
ˆ ,ω να α ε
Z
< ε2
, α = 0, π, (2.10)
de να ε,
m
vyznaça[t\sq formulog (2.6) z zaminog uα n a uα , ε
. Okrim toho,
qkwo (2.10) vykonano, to
w
KP C C
c c
T s s l
K
l
K
l l
K0
2
2
20≤ ε .
Zafiksu[mo α = 0, π ta rozhlqnemo problemu poßuku takoho uα ∈ B ( 0, T ),
wo
u x e dxi mx K
T
α( )∫ 2
0
/ = ω̂α
m
, m ∈ Z. (2.11)
Taka problema nazyva[t\sq tryhonometryçnog problemog momentiv Markova
dlq neskinçenno] poslidovnosti { } = − ∞
+ ∞ω̂α
m
m . Za teoremog 2.1 uα ∈ B ( 0, T ), α =
= 0, π, [ rozv’qzkamy problemy 0-kerovanosti za ças T dlq systemy (0.1), (0.2),
(1.1) v tomu i lyße v tomu vypadku, koly uα
, α = 0, π, [ rozv’qzkamy tryhono-
metryçno] problemy momentiv Markova dlq neskinçenno] poslidovnosti
{ } = − ∞
+ ∞ω̂α
m
m , vyznaçeno] formulamy (2.5).
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
PROBLEMY KEROVANOSTI DLQ RIVNQNNQ STRUNY 951
Rozhlqnemo dlq fiksovanoho α = 0, π , N ∈ N problemu poßuku uα ∈ B ( 0,
T ) takoho, wo
u x e dxi mx K
T
α( )∫ 2
0
/ = ω̂α
m
, m = − N N, . (2.12)
Taka problema nazyva[t\sq tryhonometryçnog problemog momentiv Markova
dlq skinçenno] poslidovnosti { } = −ω̂α
m
m N
N
.
Vyqvlq[t\sq, wo rozv’qzky takyx tryhonometryçnyx problem momentiv dlq
riznyx N ∈ N dagt\ nam rozv’qzky problemy ε-kerovanosti za ças T systemy
(0.1), (0.2), (1,1).
Teorema 2.2. Nexaj T > 0, s < – 1 / 2, dlq w0 ∈
˜ ,Hs
0 0[ π] vykonano umovu iii)
vysnovku 1.1 i K ∈ N take, wo π ( K – 1 ) < T ≤ π K. Nexaj takoΩ { } = − ∞
+ ∞ω̂α
m
m
vyznaçeno formulog (2.5). Todi qkwo uN
α ∈ B ( 0, T ), α = 0, π, [ rozv’qzkamy
tryhonometryçnyx problem momentiv Markova (2.12) dlq deqkoho N ∈ N, to
kincevyj stan wT
kerovano] systemy (0.1), (0.2), (1.1) zadovol\nq[ umovu
w
P C C N
c c K s
T s s
s l
K
l
K s
l l
K s0
6 2 1 2
2 1
2
2 1
≤
− −
+ +
−
/
→ 0 pry N → ∞. (2.13)
Dovedennq. Vraxovugçy umovu iii) (vysnovok 1.1), te, wo uN
α ∈ B ( 0, T ), α =
= 0, π, ta (2.5), (2.6), (2.8), (2.9), odrazu oderΩu[mo (2.13).
Teoremu dovedeno.
Poznaçymo
B N ( 0, T ) = { u ∈ B ( 0, T ) | ∃ T* ( 0, T ) ( | u ( t ) | = 1 majΩe skriz\ na ( 0, T* ) )
∧ ( u ( t ) = 0 majΩe skriz\ na ( T*
, T ) )
∧ ( u ( t ) ma[ ne bil\ße niΩ 2N toçok rozryvu na ( 0, T* ) ) }.
Vidomo [12] (hl. 7), wo qkwo skinçenna tryhonometryçna problema momentiv
(2.12) [ rozv’qznog, to vona ma[ rozv’qzok, wo naleΩyt\ B N ( 0, T ). Na pidstavi
teoremy 2.2 robymo vysnovok, wo za umov ci[] teoremy moΩna znajty rozv’qzky
uN
α ∈ B ( 0, T ), α = 0, π , problemy momentiv (2.12), i ci rozv’qzky dagt\ nam re-
lejni keruvannq, wo rozv’qzugt\ problemu ε-kerovanosti, a formula (2.13) vyz-
naça[ ocinku toçnosti vluçennq v 0. OtΩe, my moΩemo sformulgvaty takyj
vysnovok.
Vysnovok 2.1. Nexaj T > 0, s < – 1 / 2, dlq w0 ∈
˜ ,Hs
0 0[ π] vykonano umovu
iii) vysnovku 1.1 i K ∈ N take, wo π ( K – 1 ) < T ≤ π K . Nexaj takoΩ
{ } = − ∞
+ ∞ω̂α
m
m vyznaçeno formulog (2.5). Todi dlq koΩnoho N ∈ N isnugt\
uN
α ∈ B N ( 0, T ), α = 0, π, — rozv’qzky tryhonometryçnyx problem momentiv
Markova (2.12) dlq c\oho N, ta dlq koΩnoho ε > 0 isnu[ N ∈ N take, wo
kincevyj stan w T
kerovano] systemy (0.1), (0.2), (1.1) zadovol\nq[ umovu
wT s
0
≤ ε, pryçomu N vyznaça[t\sq umovog
2
2 1
6 2 1 2
2 1
s
s l
K
l
K s
l l
K s
P C C N
c c K s
+ +
− − −
/
< ε.
U teoremi 2.2 ta vysnovku 2.1 dovedeno, wo pobudovani relejni keruvannq
zabezpeçugt\ zbiΩnist\ do nulq kincevoho stanu wT
v Hs
0 × Hs
0
1−
lyße za umo-
vy s < – 1 / 2. PokaΩemo na prykladi, wo cq umova [ istotnog.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
952 L. V. FARDYHOLA, K. S. XALINA
Pryklad 2.1. Nexaj T ∈ ( 0, π ], w0
0 = 1
2
na ( 0, π ), w x1
0( ) ∈ Hs
0
1 0− ( π), , n ∈
∈ N, uα ∈ BN
( 0, T ), α = 0, π. Z formul (1.2), (1.9) ta lemy 1.3 oderΩu[mo
c w x u x u xl
s2
0
0
0 0
Ω( )( ) − ( ) − (π − )π ≤
≤
T
2
0
0
0
1
0
0
π
∈
π
π
∑ ( )
( ( ))
( ) − ( ) − (π − )
( ) − ( ) − (π − ) ′
k
k l
s
w x u x u x
w x u x u xZ
Ω
Ω Ξ
≤
≤
E( − ) ≤ π + ≤ π +x T W C wl
s T
l
s
l
T s
, 4 6 4 62 2 2
0
Ω .
OtΩe, zvidsy na pidstavi lemy 2.1 ma[mo
w
c
P C
w x u x u xT s l
s l
s
0
2
2 2 0
0
0 08 12
≥
π +
( ) − ( ) − (π − )π .
Dlq s = 0
w x u x u x0
0
0 0
0
2
( ) − ( ) − (π − ) ≥ π
π ,
tomu
w
c
P C
T l
s l
0
0 2
2 22 8 12
≥ π
π +
= ε0
. (2.14)
Takym çynom, dlq zadanyx T, w0
u vypadku s = 0 ≥ – 1 / 2 isnu[ ε0 > 0 take, wo
dlq bud\-qkoho N ∈ N ta bud\-qkyx uα ∈ B N ( 0, T ), α = 0, π, kincevyj stan wT
kerovano] systemy (0.1), (0.2), (1.1) zadovol\nq[ ocinku (2.14).
1. Lasiecka I., Triggiani R. Control theory for partial differential equations: continuous and
approximation theories. 2. Abstract hyperbolic-like systems over a finite time horizon. –
Cambridge Univ. Press, 2000.
2. Krabs W., Leugering G. On boundary controllability of one-dimension vibrating systems by
W p
0
1, -controls for p ∈ [0, ∞) // Math. Meth. Appl. Sci. – 1994. – 17. – P. 77 – 93.
3. Gugat M., Leugering G. Solutions of L
p
-norm-minimal control problems for the wave equation //
Comput. Appl. Math. – 2002. – 21, # 1. – P. 227 – 244.
4. Negreanu M., Zuazua E. Convergence of multigrid method for the controllability of a 1-d wave
equation // C. r. math. Acad. sci. – 2004. – 338, # 5. – P. 413 – 418.
5. Gugat M. Analytic solution of L
∞
-optimal control problems for the wave equation // J. Optimiz.
Theory and Appl. – 2002. – 114. – P. 151 – 192.
6. Fattorini H. O. Infinite dimensional optimization and control theory. – Cambridge Univ. Press,
1999.
7. Schwartz L. Théorie des distributions, I, II. – Paris: Hermann, 1950 – 1951.
8. Volevyç L. R., Hyndykyn S. H. Obobwenn¥e funkcyy y uravnenyq v svertkax. – M.: Fyzmat-
hyz, 1994. – 336 s.
9. Sklyar G. M., Fardigola L. V. The Markov power moment problem in problems of controllability
and frequency extinguishing for the wave equation on a half-axis // J. Math. Anal. and Appl. – 2002.
– 276, # 1. – P. 109 – 134.
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OderΩano 24.01.2005,
pislq doopracgvannq — 02.02.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
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| id | umjimathkievua-article-3358 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:04Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b5/7a6c560c660bd1f2716e25888cec0bb5.pdf |
| spelling | umjimathkievua-article-33582020-03-18T19:52:14Z Controllability problems for the string equation Проблеми керованості для рівняння струни Fardigola, L. V. Khalina, K. S. Фардигола, Л. В. Халіна, К. С. For the string equation controlled by boundary conditions, we establish necessary and sufficient conditions for 0-and ε-controllability. The controls that solve such problems are found in explicit form. Moreover, using the Markov trigonometric moment problem, we construct bangbang controls that solve the problem of ε-controllability. Получены необходимые и достаточные условия 0- и &epsilon;-управляемости для уравнения струны, управляемого краевыми условиями. Управления, решающие эти задачи, найдены в явном виде. Более того, с помощью тригонометрической проблемы моментов Маркова построены релейные управления, решающие задачу &epsilon;-управляемости. Institute of Mathematics, NAS of Ukraine 2007-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3358 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 7 (2007); 939–952 Український математичний журнал; Том 59 № 7 (2007); 939–952 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3358/3460 https://umj.imath.kiev.ua/index.php/umj/article/view/3358/3461 Copyright (c) 2007 Fardigola L. V.; Khalina K. S. |
| spellingShingle | Fardigola, L. V. Khalina, K. S. Фардигола, Л. В. Халіна, К. С. Controllability problems for the string equation |
| title | Controllability problems for the string equation |
| title_alt | Проблеми керованості для рівняння струни |
| title_full | Controllability problems for the string equation |
| title_fullStr | Controllability problems for the string equation |
| title_full_unstemmed | Controllability problems for the string equation |
| title_short | Controllability problems for the string equation |
| title_sort | controllability problems for the string equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3358 |
| work_keys_str_mv | AT fardigolalv controllabilityproblemsforthestringequation AT khalinaks controllabilityproblemsforthestringequation AT fardigolalv controllabilityproblemsforthestringequation AT halínaks controllabilityproblemsforthestringequation AT fardigolalv problemikerovanostídlârívnânnâstruni AT khalinaks problemikerovanostídlârívnânnâstruni AT fardigolalv problemikerovanostídlârívnânnâstruni AT halínaks problemikerovanostídlârívnânnâstruni |