最优控制变分法

1、1 baF (x, y, y dx1696J. Bernoulli , A , B ,A , B C , 0A BA , BA , B y =y (x , y (0=0, y (x 1 =y 1. v =ds 1+(y 21+(y 21+(y 2dt 2=mg cos =mg dy dt ,2ds dt 2=2g dy dt =2g dy dt 2=2gy +C. 0, C =0, v =1+(y 221 baF (x, y, y dx3dx F y dx = b addx F y dx.( =0(0=0,b aF y (x, y 0, y 0 d(Lagrange 17594dx F y (

2、x, y 0, y 0 =0. (1.1(1.1J (y Euler Euler 1744Euler F y F xy F yy y F y y y =0. (1.2Euler Euler F =F (y, y , F x F xy =0, (1.2,F y F yy y F y y y =0.1 ba F (x, y, y dx5dx (F F y y =(F y y +F y y (F yy y +F y y y y F y y =(F y F yy y F y y y y =0, F F y y =C ( . (1.31.1F =yx . y . F x , F (1.3,C =dx =

3、 y .x = ydyC 1y y 2=2arccos C 12y 1+cot 2=C 1sin 2=C 16y =2C 1sin cos d2 +C 2=C 12(2sin 2 +C 2, y =C 12, x =R (sin , y =R (1cos .RJ (y 1, y 2, ···, y n =b a F (x, y 1, ···, y n , y 1, ···, y n dx, D (J (y 1, y 2, ···, y n (1(y 1, y 2, ·

4、;··, y n C 2a, b ;(2y i (a =i , y i (b =i (i =1, 2, ···, n .J (y 1, y 2, ···, y n (y 10, y 20, ···, y n 0(1, 2, ···, n =b a F (x, y 10+1,···, y n 0+n ,y 10+1, ···, y n 0+n dx,C 2a, b ,(a =(b =0. (1, 2

5、, ···, n 1=0, 2=0, ···, n =02=0, ···, i = b a (F y i +F y i dx = b a F y id dx F y i =0, (i =1, 2, ···, n .1 ba F (x, y, y dx72(x 1(t 2+(x 2(t 2+···+(x n (t 2. HEulerUdt mx i =0(i =1, 2, 3 , Newtonm d 2x i xi (i =1, 2, 3 . Ne

6、wtonH L (t, q, q S = t 2t 1L (t, q, q dt Hamilton L (t, q, q La-grange DescartesEuler81+(y 2dx 2.J (y = b a y dx 3.J (y, z = b a(y 2+(z 2+y z dx 4. J (y = b aF (x, y, y , y dx,D (J =y (x ; y (x C 4a, b , y (a =,y (b =,y (a =µ,y (b =, F J (y y 0y 0EulerF y (x, y, y , y ddx 2F y (x, y, y , y =0.5

7、.J (y = b a (1+(y 2 dx y (0=0, y (0=1, y (1=1, y (1=1291+fy 2dxdyD (J (1f C 1(D ;(2(t =f (t , (t .J (u =D F x, y, u, uy dxdy, F C 2, D (J u (1u C 2(D ;(2u |D=, DD(x, y C 2(D , |D=0. J (u u 0Lagange( =D F x, y, u 0+,u0x, u0ydxdy,( = D +F u x y dxdy. ( = D F u Fu x y dxdy.1.1,10xFu yx, q =fx (1+p 2+q

8、2 1/2fy (1+p 2+q 2 1/2fx22pq 2fy2=0.211x2+ux 2+ux 2uy2ux2+2ux2+ux2+ux2+2ux2dxdy =Duux2dxdy =Dux2+2u12x2+ux=cos ,rx=1y=1rrrrr2+1 2 drd.F =rururFu r2ru 21=2 r2ur+12. r2ur+12=0.Laplace213xF px2F r+2y2F t =0,p =uy, r =2uxy, t =2ux22+2uxy22uf (x, y dxdy,141+(y (x 2dx =lJ (y =baF (x, y, y dxD (J (1y C 2;(

9、2y (a =,y (b =,baG (x, y, y dx =l.3.1y 0(x J (y G y ddx(F y +Gy =0.(3.1(3.1Lagrange1(x , 2(x C 2a, b ,i (a =i (b =0(i =1, 2.(1, 2 =baG (x, y 0+11+22, y 0+11+22 dx l.2(1, 2 =ba(G y 2+G y 2 dx =baG y d315dxG y 1.12(x 2(0, 0 =0.(1, 2 =baF (x, y 0+11+22, y 0+11+22 dx.(1, 2 =0(0, 0Lagrangei=0, (i =1, 2 .

10、2(x i =11=ba (F y 1+F y 1 dx+b a(G y 1+G y 1 dx= b aF y d dx G y 1dx = b a(F y +Gyd dx(F y +Gy =0.3.1F +G=y +dxG y =ddxy1+y 2=0.16 y 1 + y2 (y 2 = y=± (x c = x c. (x c2 . 2 (x c2 dx = 2 (x c2 + d. 3.2 2 (x c2 A, B b (y d2 + (x c2 = 2 . l ¤ J(y = a (p(xy 2 + r(x(y 2 dx. p(x, r (x 2 (1 y C a

11、, b; a, b ¨ D(J ¤ b (2 y(a = y(b = 0, y 2 (xdx = 1. a G y b d G = 2y. dx y y 2 (xdx = 1, a ¤ F + G = p(xy 2 + r(x(y 2 + y 2 , Euler J(y y (x ¤ y (x 0 0 2p(xy + 2y d (2r(xy = 0. dx ¤ × d (r(xy (p(x + y = 0 dx y(a = 0, y(b = 0 Ly = d (r(xy p(xy, dx L 3 Ly = y. 17 (3.2 y &

12、#164; (3.2 Lagrange L ¤ ¤ ¤ x z(x = a G(x, y, y dx, × z (x = G(x, y, y , z(a = 0, z(b = l. J(y = intb F (x, y, y dx, a y(a = , y(b = ¤ b (x, y, y , z, z = G(x, y, y z (x = 0, z(a = 0, z(b = l J= a F (x, y1 , , yn , y1 , , yn dx × ¤ yj (a = j , yj (b = j , (j = 1, 2, , n Lagr