to the solution of a bilinear optimal control problem with state constrains by the...
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![Page 1: To the Solution of a Bilinear Optimal Control Problem with State Constrains by the Doubled-Variations Method E.A. Rovenskaya Lomonosov Moscow State University,](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d3e5503460f94a165c8/html5/thumbnails/1.jpg)
To the Solution of a Bilinear To the Solution of a Bilinear Optimal Control Problem with Optimal Control Problem with
State Constrains State Constrains by the Doubled-Variations by the Doubled-Variations
MethodMethodE.A. RovenskayaE.A. Rovenskaya
Lomonosov Moscow State Lomonosov Moscow State University, University,
Moscow, RussiaMoscow, Russia
and International Institute for and International Institute for Applied Systems Analysis, Applied Systems Analysis,
Laxenburg, AustriaLaxenburg, Austria
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Problem Formulation Problem Formulation
Assumptions:
(A1)
(A2) and are commuting matrices;
(A3) is a continuous function increasing in x for all t
and measurable in t
(A4) for all is a continuously-differentiable function,
(A5)
)0(
0)(),(
)0(),()()()(
max))(,()]([
0
0
t
txtc
xxtutBxtAxtx
dttxtfuJT
nRtx )(1],[)( Ruutu
00 x
A Bnn RRTxtfxt ],0[:),(),(
nRTxcp ],0[:)(:00),0( 0 xc
0J0U
optimal utility value
the set of optimal controls
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Equivalent formulation Equivalent formulation
t
tBpAt dssutpxetx0
0)( )()(,)(
0)0(),()()()( xxtutBxtAxtx
],0[ Tt
T
dttxtfJ0
))(,(
TT
tBuAt dttptdtxetfJ00
0)( ))(,(),(
:)( pp is a continuous increasing function
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0),()( 0)( xetctv tBpAt
0)(),( txtc
)()(
)()()(
),()()()(
*2
*1
0)(
21
tcBt
tcAtct
xettuttv tBpAt
0),0()0( 00 xcvvAssumption (A5)
Equivalent formulation Equivalent formulation
0),0( 0 xc
![Page 5: To the Solution of a Bilinear Optimal Control Problem with State Constrains by the Doubled-Variations Method E.A. Rovenskaya Lomonosov Moscow State University,](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d3e5503460f94a165c8/html5/thumbnails/5.jpg)
max))(,()]([0
T
dttptuJ
0)0(,),()()()( 00)(
21 vvxettuttv tBpAt
t
dssutp0
)()(
]),0[( Tt
],[)( uutu
0)( tv
),()(
),()(
utvtv
utptp
0J
0U
optimal utility value
optimal controls set
Equivalent formulation Equivalent formulation
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Discrete approximation Discrete approximation
:],0[ T ,iti ),...,1,0( mi Ttm
},{)( uuutu i),[ 1 ii ttt
mm uuuuuu },{),...,,( 110
),( utp 0)(),( 0 uputp
1
0
)(),(i
jji uuputp
),[ 10 ttt
),[ 1 ii ttt
))(),...,(),((),( 110 upupuputp m
Uniform grid on
control approximation)1,...,1,0( mi
:)(tu
approximation of :)(tp
)(tu
)1,...,1,0( mi
Property:
*
*
),(),(
),(),(
uutputp
uutputp
*2),(),( uutputp
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],0[ t),(
2*
*
*),(),(*
)(
)(
)(1)(),( utpautpautpa e
tE
tE
tEetuaeauutv
)(2*
*
*)()(*
)(
)(
)(1),( upaupa
i
uipa ii etE
tE
tEeuaeauutv
),( utvApproximation of :)(tv
),[ 1 ii ttt)1,...,1,0( mi
0),0( vuv
)(uI
,))(()),(()(1
00
m
iii
t uphdtutpeuI
1i
i
t
t
ti dteh
),...,1,0( mi
Аппроксимация показателя качества :)(uJ
где
Discrete approximation Discrete approximation
![Page 8: To the Solution of a Bilinear Optimal Control Problem with State Constrains by the Doubled-Variations Method E.A. Rovenskaya Lomonosov Moscow State University,](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d3e5503460f94a165c8/html5/thumbnails/8.jpg)
),( utv i
max))(()(1
0
m
iii uphuI
mm uuuuu },0{),...,,( *110
0I
0U
Optimal utility value
Optimal controls set
),( ProblemProblem
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kkkk IUuu 0000 ,)( ),( kk
Let
.0,0 kk
,0,0 kk ,00 JIk 00 Uuk ].,0[2 TL
be an optimal control in problem,
Then
(i)
(ii)
so that
weakly in
LemmaLemma 1 1
),( Problem Problem
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Solution to Solution to ),(
)(2*
*
*)()(*
)(
)(
)(1)( upaupa
i
uipa ii etE
tE
tEeuaeautv
*0)(
0)(
0
,)(
0,)(),(
uuеслиet
uеслиetutv
i
uipa
i
uipa
Problem Problem
),[ 1 ii ttt )1,...,1,0( mi
1
1
)(
)(
i
i
i
i
t
t
i
t
t
i
dtt
dtt
0)(
)(
)()(
0)(
)()(
2*
*
*
*
2*
**
tE
tE
tE
aut
tE
tEaut
Условия (У2) и (У3)
*
0)(
0)(
001
,
0,),(),(
0
0
uuеслиe
uеслиeutvutv
iupai
iupai
ii
i
i
Рассмотрим 00 Uu
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Решение ),(
0u ),( 0utv i
задачи
- допустимый элемент, то ),...,1,0( mi
Определение.
mj tt - критическая точка, если замена 0ju
*u приводит к
),( 01 utv j или ,),( )(0
0
upajj
jeutv
Так как
на
mj tt - регулярная точка в противном случае.
Утверждение.
.00 jumj tt Пусть - критическая точка. Тогда
Лемма 2.
.*0 uu j 2 mj ttПусть - регулярная точка. Тогда
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Решение ),(
),...,( ** uuU }),(:2,...,1{ UtvmiL i
задачи
Рассмотрим
L
.min* iiL
:0K )},0{],,0[(),( * muutKutv
Лемма 4.
Пусть
L не пусто.
Тогда ).3,...,0(*0 miuu j
Лемма 3.
пусто.
Тогда (i)
(ii)
),2,...,0( **0 iiuu j
]).,[(2),( 20
* mitttKutv
и
Пусть
Пусть
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Решение исходной задачи
tесли
tEtaE
tE
tE
u
tеслиu
tu ,
)(1)(
)(
)(1
,
)(
*
2*
*
*
*
*
0
taueEtE*
0* )(
Теорема.
Управление
оптимально в исходной задаче.
где -единственный корень уравнения
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Решение исходной задачи
tеслиeE
E
tE
tET
tеслиeTtT
twtau
tau
,)(
)(
)(
)(
,)(
),()(*
*
*
*
0
)(0
0
*
*
tеслиeE
EEC
tеслиeCtC
twtau
t
,)(
)(
,)(
),()(*
**00
00
*
tеслиtE
tеслиeEtE
tau
),(
,)(
*
00
*
Оптимальные траектории
t
dssE
tw
)(
),(*
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Моделирование
ConstEtE ** )(
tеслиeT
tеслиeTtT
thau
tau
,
,)(
))((0
00
*
*
tеслиeeh
hTC
tеслиCtC
thauau ,11
,)(
)(00
00
**
tеслиE
tеслиeEtE
tau
,
,)(
*
00
*
0
*
*ln
1
E
E
au
*
1E
h
0
0
)(ln dttYeJ t
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Калибровка модели
0.4a
год2050
006.0*u
годГтCE /0.13*
USDтрилT .6.60
USDтрилC .3.00 годГтC /7.0
год2000
годГтСE /97.60 103.0 год
Начальный момент времени
Конечный момент времени
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Моделирование
год2026
)(tY)(tC
Время переключения
)(tE
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Введение
Измененияклимата
Влияние на окружающую среду
и жизнь человека
Эмиссии и концентрации
парниковых газов
Экономическийрост
Повышениетемпературы
• повышение уровня Мирового океана• дальнейшее потепление• лесные пожары• …
• сельское хозяйство• запасы пищи и пресной воды• здоровье людей• …
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Модель: экономическая часть
)(tC
)(tY валовый выпуск продукции
капитализация «производственных» технологий
капитализация «очищающих» технологий
)(tT
)(tY
0
))(( dttYfeJ t
)())(( * tYtuu
Показатель качества max
развитие технологий)(* tYu
)()( tYtu
развитие «производственных» технологий
развитие «очищающих» технологий
)()( taTtY
),()()()( tTtTtautT
),()())(()( * tCtTtuuatC
0)0( TT
0)0( CC
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)(0 * tE
)(tE)(
)(
)(
)()( 0 tC
tT
tC
tYtE
)()( * tEtE
Модель: экологическая часть
промышленные выбросы парниковых газов
(У1)
(У2) монотонно убывает на
(У3)
)(* tEt
],0[ t
)(|)(|02*
** tE
autE
для всех
функция ],0[ t
],0[ tдля всех
(У4) ,*00 EE ),0(**0 EE
0
00 C
TE где
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Постановка задачи
)()( taTtY
)()())(()( * tCtTtuuatC )()()()( tTtTtautT
max))((0
dttYfeJ t
)()(
)( * tEtC
tT
0)0( TT 0)0( CC
],0[ t
Uu )(
U
],0[)( *utu
Множество допустимых управлений
-- множество таких измеримых функций
:)(u ]),0[( t
монотонно возрастает на
)(yfy функция
],0[ t
Дополнительное условие:
Будем считать, что множество допустимых элементов не пусто
непрерывна,