CC BY
3Central Asian Journal of
Education and Innovation
TURLI EKSTREMAL MASALALAR. BAZI QADIMIY EKSTREMAL MASALALAR
Latipova Shahnoza Salim qizi
Osiyo Xalqaro Universiteti "Umumtexnik fanlar" kafedrasi o'qituvchisi
[email protected] https://doi.org/10.5281/zenodo.10686649
ARTICLE INFO
ABSTRACT
Qabul qilindi: 10-February 2024 yil Ma'qullandi: 15- February 2024 yil Nashr qilindi: 21- February 2024 yil
KEY WORDS
Funksiya, ekstremum, maksimum, minimum, grafik
"Hosila"atamasi derivee degan fransuzcha so'zning rus tiliga aynan tarjimasidir, bu atamani 1797-yilda
V f V f
J.Lagranj xozirgi - '■ - '1 belgilashlarni ham kiritgan. Bunday atash tushuncha mazmunini aks
ettiradfunksiya / CO / 1 funksiyadan
kelib chiqadi, u^ f ' 1 ' funksiyaning hosilasidir. I.Nyuton hosilaviy funksiyani fyuksiya, funksiyaning o'zini esa flyuenta deb atagan.G.Leybnis differensial
d/d/
nisbat haqida gapiradi va hosilani dxdx kabi belgilaydi.
Braxistoxrona haqidagi masala. 1696 yilda I.Bernulli tomonidan qo'yilgan bu masalada bir vertical to'g'ri chiziqda yotmagan ikkita Ava B nuqtalarni tutashtiruvchi shunday chiziqni topish talab qilinadiki, material nuqta o'zog'irlik kuchi ta'siri ostida shu chiziq bo'ylab harakat qilib, A nuqtadan nuqtaga eng qisqa vaqtda yetib kelsin.
Masalaning nomi grekcha "braxistos" -eng qisqa, "xronos" -vaqt so'zlaridan kelib chiqqan. Braxistoxrona haqidagi masalani hozirgi zamon matematikasi tilida ifodalash uchun to'g'ri
burchakli OxyOxy k0oordinatalar sistemasini 1-chizmada kcfrsatilganidek,ya'ni o'qni pastga yo'naltirib, qaraymiz.A nuqtani koordinatalar boshiga joylashtiramiz. B nuqtaning koordinatalari(x1,y1)bo"lsin.
ixtiyoriy y = y(x)y = y€*)siUiq chiziq
bilan tutashtiramiz.
Shu chiziq bo'ylab og'irlik kuchi ta'sirida harakatlanuvchi material nuqtaningmassasi mga,
w
tvaqtmomentidagitezligi gatengbcflsin.U holda, tvaqtda harakatdagi nuqtaning kinetik
jç fflii2^, mv2
energiyasi 2 2 potensial energiyasi
ladi, bu yerda
g«9.8 m/c2 -erkin tushish tezlanishi o'zgarmasi. Fizikadan yaxshi ma'lum bo'lgan energiyaningsaqlanishqonunigako'ra,
mv
-mgy H—— = 0
tenglikniolamiz.Buyerdan
v = fl-gyv = -J^iy
.Endi
v = —, ds = tJcLx1 + dy2,dy = y dx v = —, ds = yjdx2 + dy2,dy = y dx
ekanligini hisobga olsak,
dt=^ =
V
l^dsdt = ^ =
1+y
•2
ds
bcfladi.
Demak, ^ }'(.x)chiziq bcTylab A nuqtadan B nuqtagakcTchish uchun
sarflangan^ — ~~ ^^^vaqt uchun
T[y] =
f11HZ
1 yj 2 gy
dx
(1.1)
ifodaga ega bcTlamiz. (1-1) kcfrinishdagi y = yOO * G [O^Jy = y(x) x G [0,x-l]
i
T = T(y)T = T Cv)
miqdor
uzluksiz differensiallanuvchi funksiyalar
fazosida aniqlanganbcTlib,braxistoxronahaqidagi masalaesa,^-^^^ ^funksionalning
y (0) = 0, y (Xi) = yL y ( 0 ) = 0, y (.Vj ) = yx shartlarni qanoatlantiruvchi uzluksiz differensiallanuvchi funksiyalar to'plamida minimumini topish masalasidan iboratdir. Bu masala I.Bernulli,I.Nyuton,G.Leybnislar tomonidan yechilgan bo"lib,eng tez o"tish(sirpanish)chizig"isikloidadebataluvchi chiziqdan iborat bo'lar ekan.
Geodezik chiziqlar haqidagi masala.Masala quyidagicha qo'yiladi: Berilgan sirtda
j «
yotuvchi va sirtning °J °vanuqtalarini tutashtiruvchi eng qisqa uzunlikka ega bcTlgan chiziq topilsin(2-chizma).
Bunday eng qisqa uzunlikka ega chiziqlar geodezik chiziqlar deb ataladi.Masalaning matematik modelini tuzish uchun, ^ ^ sirt — 0 (p(x,),z) — 0 ^engiama
bilan berilgan,'^o-^o Va ^î-^i nuqtalarning koordinatalari, mos ravishda ^oiYo' zo)
(y y 7 ^ ^ 1J Z V 7 1
v. 0'. 0' oJ va l'^i' is- i>. i> u bcflsin, deb faraz qilamiz.Qaralayotgan nuqtalarni
y = yOOy = y00 z = z(x)z = z(.y)
xQ < x < xx
tutashtiruvchi ixtiyoriy
, silliq chiziqni qaraymiz. Bu chiziqning uzunligi
(1.2)
x0 < X < xx
r*i ,_
L = L[y; z] = y/1 + y2 + z2 dx
j v-
Formula orqali topiladi.Masalaning qo'yilishiga ko'ra,
yU'o) = yo z(x0) = z0 y(x1) = y1 z(Xj) = Zj <p{x,y(x),z(x)) = 0,
Xn < X < Xi Xn < X < Xi
(1.3)
Munosabatlarga ega bo'lamiz.Shunday qilib, geodezik chiziqlar haqidagi masala (1.2) kcTrinishdagi [}'>Z]L — [} >z] cfzgaruvchi miqdorni (1.3) shartlarni
qanoatlantiruvchi uzluksiz differensiallanuvchi, — ■' _—_'..
y = y(x), z = z(x),
Y T 'Y y < Y Y
□ — — ■ i vo — -v — • i funksiyalar tcTplamida minimallashtirish masalasidan iborat.Bu masala, 1698-yilda Ya. Bernulli tomonidan yechilgan.
Klassik izoperimetrik masala. Bu masala berilgan l uzunlikka ega bo'lgan barcha yopiq
SS
chiziqlar ichida maksimal yuzani chegaralovchi chiziqni toppish talab qilinadi. Bunday chiziqning aylanadan iborat ekanligi qadimgi Yunonistonda ma'lum edi. Izoperimetrik masala deb ataluvchi bu masalani hozirgi zamon matematikasi tilida ifodalash
uchun, yopiq chiziq * = V = KO î E [t^tj* = x(t) V = v(t) t<E[tQ,t±]
parametrik tenglamalar bilan berilgan, deb faraz qilamiz. U holda, shu yopiq chiziq bilan chegaralangan yuza,
f'* . S[x,y] = xyt dt
Jto
(1.4)
Formula orqali topiladi. Chiziq uzunligi 1 ga tengligi va chiziqning yopiqligini hisobga olsak, ti
i
X'
+ y2 dt = 1
(1.5)
x(_t0) = x(_t1), y(t0) = y(t1) (1.6)
(1.6)
Izoperemetrik shaklga va x(t0) = xCti), y(t0) = y (tj) Chegaraviy shartlarga ega bo'lamiz.Shunday qilib, qaralayotgan masala (1.4) ko'rinishdagi o'zgaruvchi miqdorning, (1.5) (1.6) shartlarni qanoatlantiruvchi X = x(t) V = y(t) t0<t <t1
Uzluksiz differensiallanuvchi funksiyalar to'plamida, minimumini topishdan ibotdir. Yuqorida keltirilgan masalalarda (1.1),(1.2) va(1.4) ko'rinishdagi o'zgaruvchi miqdorlarga ega bo'ldik.Ular funksional tipidagi o'zgaruvchi miqdorlarga misol bo'la oladi. Funksionallar esa funksional analiz kursining asosiy tushunchalaridan biri bo'lib, berilgan W funksional fazo V to'plamning har bir u elementiga biror J(u) haqiqiy sonni mos qo'yuvchi akslantirishni bildiradi. Funksionallar odatda cheksiz o'lchovli funksional fazolarda berilgan bo'ladi.Ularning eng katta (maksimal) va eng kichik(minimal)qiymatlarini topish haqidagi masalalar cheksiz o'lchovli ekstremal masalar bo'lib hisoblanadi.
XVII asrning oxiridan XX asr o'rtalarigacha bo'lgan davr klassik variatsion hisobning paydo bo'lishi va rivojlanishini o'z ichiga oladi.Bu davrda dastlabki fundamental tadqiqotlar L.Eyler va J.Lagranj tomonidan bajarildi. XVIIIasrning oxirlarida Eyler,Lagranj va Lejandrlarning ilmiy tadqiqotlari natijasida variatsion hisob birinchi variatsiyani tekshirish qismi bo'yicha tugallangan shaklga ega bo'ldi.XIX asrda esa, avval ma'lum bo'lgan variatsion masalalarni umumlashtirish boshlandi va variatsion hisobning tadbiqlari bo'yicha natijalar olindi(M.I.Ostrogradskiy tomonidan 1834-yilda karrali integrally variatsion masalalar uchun zaruriy shartlar olindi, variatsion hisobning mexanikaga tadbig'I qaraldi). XIXasrning ikkinchi
yarmida funksionallar ekstremumlarining
yetarli shartlari olindi(K.Veyershtrasstomonidan,1879yilda).
XX asrda variatsion hisobning to'g'ri usullari yuzaga keldi. Ular variatsion masalalarni taqribiy yechish uchun,hamda ularda yechimning mavjudligini isbotlash uchun juda muhimdir. XX asrning boshlarida matematikada yangi yo'nalish - funksional analiz yuzaanalizning tarkibiy qismiga aylandi. XXasrnig ikkinchi yarmiga kelib optimal boshqaruvning matematik nazariyasiga asos solinishi va uning jadal rivojlanishi variatsion hisob taraqqiyotida yangi davrni boshlab berdi. Bu yangi yo'nalishda Sobiq Ittifoq akademiki L.S.Pontryaginning «maksimum prinsipi», amerikalik R.Bellmanning dinamik pogrammalashtirish usuli asosiy natijalar hisoblanadi.
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