Учебное пособие 800365

F 3 M(x1, x2, p0, p1, p2)/ H 3 3 3 3 |u| 1# 4 - u (t) = 1& p2(t) > 0 u (t) = −1& p2(t) <

0#

R

M(x1, x2, p0, p1, p2) = p0 + p1x2 + |p2|.

H / ! 3 3 3 ? / 3 3

p0(t) = −1

H(x1, x2, p0, p1, p2, u ) =

= p0(t) + p1(t)x2(t) + p2(t)u (t) = p1(0)x2(t) + |p2(0) − tp1(0)| − 1 = 0.

? G 3 u (t) = ±1/ / 4,B6/ 4 6

x2(t) = ±t + x2(0), x1(t) = ±t2/2 + x2(0)t + x1(0),

x2(t) = (t − tF ), x1(t) = (t − tF )2/2.

H 3 3 5

3 / 3 p2(t) = 0# ? 3

p2(0) − t p1(0) 3 / 3 3 5

3 [0, tF ] # A 5

E 3 3 / 3 ! 3 3 3 ? 5

3 3 ! &

#

( ) " 3 /

OY / 3

3 (a, H), a > 0, H > 0 4 # +#,6# H 5

/ 3 3 3 #

2 3 3 / / 5

/

0 u1, u2, u21 + u22 = 1

&6/ ! / 3 5

@ u1, u2/ @ 3 37

" / p0 = −1, p1(t) = p1(0)# F !/ u21 + u22 = = 1/ 3 3 3 H u1, u2 4 * 5A

6

M(x, y, p0, p1, p2) = p0 + 2gy p21 + p22. (21)

? G 3 3 p1, p2 3 3

F !/ 4&'6/ 4&,6 ! 3 3 3 ? / 5

3 3 3 x (t), y (t), u1(t), u2(t), pi (t), i = 0, 1, 2,

H = M = 2gy p12 + p22 − 1 = 0

yy¨ − y˙2 = −gy.

3 y˙ =

y(0) = 0/ 3 θ = 0# R

3 3 ! 4,'6 4 3# @

" 3 ! 3 3 T,O/

,BU#

A = (A\B) (A ∩ B).

! "# X = {a, b, c} $ % &' ( a, b c. *& & * τ = { , X, {a}, {b}} + * ,

- " # + ( C[a, b] L2[a, b]

x(t) = t2, a = 1, b = 1.

f (x, y) + # * F (x, y) = 0

F (x, y) = x2 + y3 − 1.

x2/3(y )2dx, y(0) = 0, y(1) = 1.

(A\B) ∩ (A ∩ B) = .

x1 x2 x3 x4 x1 x2 x1 x1 x3 x4 x1 x3

x2 x3 x4 x3 x1 x4

! "# X = {a, b, c} $ % &' ( a, b c. *& & * τ = { , X, {a}, {b}, {a, b}} + * ,

- " # + ( C[a, b] L2[a, b] x(t) = cos t, a = −π2 , b = π2 .

. / # # f (x, y) + # * F (x, y) = 0

(A\B) ∩ B = .

G= [{a, b, c}, {(a, a), (b, b), (c, c), (a, c), (b, c), (c, b), (b, a)}].

! "# X = {a, b, c} $ % &' ( a, b c. *& & * τ = { , X, {a}, {b}, {c}} + * ,

- " # + ( C[a, b] L2[a, b]

x(t) = t2 + t, a = −1, b = 1.

f (x, y) + # * F (x, y) = 0

F (x, y) = x3 − y4 − 1.

(y + (y )2)dx, y(0) = 0, y(1) = 1.

A\B = A\(A ∩ B).

G = [{x1 , x2 , x3 }, {(x1 , x2 ), (x1 , x1 ), (x3 , x2 ), (x1 , x3 ),

(x2 , x3 ), (x3 , x3 )}].

! "# X = {a, b, c} $ % &' ( a, b c. $ *& & * τ = { , X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}} + * ,

- " # + ( C[a, b] L2[a, b]

x(t) = sin 2t, a = 0, b = π/4.

. / # # f (x, y) + # * F (x, y) = 0

A ∩ (B C) = (A ∩ B) (A ∩ C).

G= [{a, b, c, d}, {(a, d), (b, b), (a, c), (a, b), (b, c), (c, d), (d, a)}].

! "# X = {a, b, c} $ % &' ( a, b c. *& & * τ = { , X, {a}} + * ,

- " # + ( C[a, b] L2[a, b]

x(t) = −2t3 + 2t, a = 0, b = 2.

. / # # f (x, y) + # * F (x, y) = 0

A (B ∩ C) = (A B) ∩ (A C).

G= [{d, e, f , g}, {(e, f ), (e, e), (g, e), (d, f ), (d, g), (g, g), (f , g)}].

! "# X = {a, b, c} $ % &' ( a, b c. *& & * τ = { , X, {a, b}, {b, c}} + * ,

- " # + ( C[a, b] L2[a, b]

x(t) = −t3 + t, a = −1, b = 1.

. / # # f (x, y) + # * F (x, y) = 0

(A\B)\C = A\(B C).

G= [{a, b, c}, {(a, b), (b, b), (c, a), (a, c), (c, c), (c, b), (b, a)}].

! "# X = {a, b, c} $ % &' ( a, b c. *& & * τ = { , X, {a, c}, {b, c}, {c}} + * ,

- " # + ( C[a, b] L2[a, b]

x(t) = −t2 − 1, a = −1, b = 1.

. / # # f (x, y) + # * F (x, y) = 0