De bai quen mat ko viet la C khong rong va .

Bai toan nay dung voi khong gian Hlibert bat ki chu ko bat buoc la thuc. Khi do phai thay dieu kien bang .

Thuc ra day la 2 dinh nghia tuong duong cua phep chieu vuong goc trong khong gian Hilbert len mot tap con khac rong, loi va dong. To nghi la trong sach ve giai tich ham the nao cung co. Chung minh cung khong qua kho.

Bai toan cua ban dung la xu*o*ng that. Thu nhat la dieu kien chan cua u (pointwise upper and lower bounds). To da doc dau do 1 bai bao ve viec giai quyet dieu kien nay nhung ko ro co ap dung cho bai toan cua ban duoc ko. Xin loi la dot nay to hoi ban nen ko co thoi gian de doc lai. Ban thu tim doc bai bao nay nhe
Casas, Eduardo; Tröltzsch, Fredi
Second-order necessary and sufficient optimality conditions for optimization problems and applications to control theory. (English)
SIAM J. Optim. 13, No.2, 406-431 (2002)
Neu ko tim duoc thi de lai dia chi email to se gui cho. Bai bao nay viet cho truong hop tong quat nhung co dua ra 1 VD cho ODE. Nho no ban co the xac dinh duoc optimality condition (va tinh dao ham x theo u).

Thu 2 la ham muc tieu va dieu kien cho x(T). Khong hieu de bai ra la the hay day la bai toan ban tu dat ra de giai 1 van de thuc te? Lieu ta co the viet ham muc tieu duoi dang
J(u)= u1^4 dt + K(x1(T)-L)^2 + Kx2(T)^2 ---> min
voi K>0 du lon duoc ko? Nhu vay ta da buoc x1(T)=L va x2(T)=0 va ban co the quen di dieu kien bien tai T va xet bai toan ODE voi gia tri ban dau x(0)=0. Tuy nhien u1 co thuc su la min hay ko thi chiu. Van de o cho neu K nho thi dieu kien bien tai T ko duoc thoa man (xap xi chua du tot), con neu K lon thi u1^4 dt lai tro nen qua nho so voi J, va nhu vay ko the kiem soat duoc hoan toan do lon cua u1. Neu va^~n de^? bai toan duoi dang cu thi to khong biet giai the nao. Ma tai sao phai can u1^4 vay? To khoai ham bac 2 hon, do phuc tap.

Ma ban dinh dung gi de giai ODE va thuat toan nao de toi uu vay?

Nếu mình đoán không nhầm thì là cái này:
xét hệ động lực
x'(t)=f(x,u,t)
sao cho x(0)=a, x(1)=b. Hãy tìm u(t) sao cho hàm mục tiêu
J(x) đạt max hoặc min. Nhưng tài liệu thì I don't know

Bai toan KK neu hoi bi chuoi vi voi moi u(t) (lien tuc) cho truoc ban than su ton tai cua x(t) cung khong chac chan! Nhu vay cau hoi dau tien la tim tap hop u de x ton tai. To ko chac chan lam voi ODE thi the nao nhung voi PDE de buoc ham trang thai tai thoi diem T nao do bang mot ham cho truoc (tham chi chi can xap xi thoi) hoan toan ko de dang, chu dung noi la con phai la nghiem toi uu cua 1 ham muc tieu nao do.

Ban thu neu DN cu the duoc ko? To biet it nhieu ve optimal control nhung ko hieu bai toan cua ban la the nao ca.