1oÙ CXê�5XÚ

November 17, 2011

() November 17, 2011 1 / 32

CXê�5XÚ

CXê�5XÚ'å~Xê�5XÚ`5,�¹�E,�õ.

1�Ü©¥cn!?ØÙ�~Xê�5XÚ�aq�5�,=�55�, ±ÏXêXÚÚ�êìC­½.

14!0�'u)�O�Gronwall-Bellman Ø�ª.

1�Ü©¹5-8!.Ì�l��,*ÿ(�þ?1?Ø, Ñ\ÑÑ­½�XÚ")ìC­½�'X.

() November 17, 2011 2 / 32

CXê�5XÚ�A�

CXê�5XÚx = A(t)x (4.1.1)

Ù¥A(t)´©ãëY¼êÝA(t) : J→ Rn×n, x ∈ Rn.

x = 0´Ù�²ï �.

XÚ(4.1.1)��Ü)|¤�n��5�mS,

S = {x(t)∣x(t) = A(t)x(t)} (4.1.2)

C�x(t0) = x0 → x(t) (4.1.3)

ïá��dx0 ∈ Rn�S��_�5C�.

() November 17, 2011 3 / 32

CXê�5XÚ�A�

∀� ∈ J. Pfj(t, �)�÷vЩ^�

xj(�) = ej

�)§Ù¥e1, e2, ⋅ ⋅ ⋅ , en�g,Ä."Ä�)Ý:

F (t, �) = (f1(t, �), ⋅ ⋅ ⋅ , fn(t, �)) (4.1.4)

(1) dF (t,�)dt = A(t)F (t, �),=F (t, �)´(4.1.1)�Ý);

(2)Ä�)ÝF (t, �)äkeã+(�:

(∀�1, �2 ∈ J) : F (t, �2) = F (t, �1)F (�1, �2)

(∀� ∈ J) : F (�, �) = I

(∀t, � ∈ J) : F (t, �) = F (�, t)−1

�NF (t, �)|¤��C�+,�´Ø����.

() November 17, 2011 4 / 32

CXê�5XÚ�A�

(3)duF (t, �)F (�, t) = I

KkdF (t, �)

dtF (�, t) + F (t, �)

dF (�, t)

dt= 0

½�d/�¤

A(t) + F (t, �)dF (�, t)

dt= 0

dd��F (�, t)÷v�§

dF (�, t)

dt= −F (�, t)A(t)

ePF ∗(t, �)´(4.1.1)��ÝXÚ

x = −AT (t)x

�Ä�)Ý,KF ∗(t, �) = F (�, t)T

() November 17, 2011 5 / 32

CXê�5XÚ�A�

~Xê�5XÚx = Ax (4.1.5)

Ä�)Ý�eA(t−�),ù�C�+´���+.

CXê�5XÚ�(J,Ì�3uvk���{,U3�½ÝA(t)�ÒUá=�äÙéAXÚ�­½5.

XÚ(4.1.5)�")ìC­½⇔

�(A) ⊂oC− (4.1.6)

éuCXê�5XÚ(4.1.1),=¦k

(∀t ∈ J) : �(A(t)) ⊂oC− (4.1.7)

�Ã{��(4.1.1)�")´ÄìC­½.

() November 17, 2011 6 / 32

CXê�5XÚ�A�

~~~4.1.1 �ÄCXê�5XÚ

x =

[−� M [1− '(t)]

M'(t) −�

]x = A(t)x

Ù¥

'(t) =1

2[1 + (−1)i], i ≤ t < i+ 1,

A(t) =

[−� 0M −�

]= A1, 2k ≤ t < 2k + 1,

A(t) =

[−� M0 −�

]= A2, 2k + 1 ≤ t < 2k + 2.

eA1(t−�) = e−�(t−�)[

1 0M(t− �) 1

]eA2(t−�) = e−�(t−�)

[1 M(t− �)0 1

]() November 17, 2011 7 / 32

CXê�5XÚ�A�

�ÄÐ�X(0) = I2�Ý),��X(t).K

X(1) = e−�[

1 0M 1

]

X(2) = e−2�[1 M0 1

] [1 0M 1

]= e−2�

[M2 + 1 MM 1

]dde�´y

X(2n) = e−2n�[M2n + ∗ ∗∗ ∗

]y�� > 0®�½.eÀM¦Me−� > 1,KokX(2n)Ã..u´XÚ")�½Ø­½.�é?¿� > 0§k

(∀t ∈ J) : �(A(t)) = {−�,−�} ⊂oC−

() November 17, 2011 8 / 32

CXê�5XÚ�A�

~~~4.1.2 ��XÚ

� = �(t)�, �(t) =

{−4 2k ≤ t < 2k + 12 2k + 1 ≤ t < 2k + 2

XÚ(´ì?­½�§��(t) ⊈oC−.

Lyapunov1��{�ä~Xê�5XÚìC­½: 3�½W =W T�½e, Ý�§

V A+ATV = −W

´Ä�3�½)V = V T .

() November 17, 2011 9 / 32

CXê�5XÚ�A�

CXê�g.V (t, x) = xTV (t)x

��XÚ�Lyapunov¼ê,K

V ∣(4.1.1) = xT [V (t) + V (t)A(t) +AT (t)V (t)]x

éA�Lyapunov�§C�

V (t) + V (t)A(t) +AT (t)V (t) = −W (t) (4.1.8)

=3�½�½ÝW (t)�, �©�§(4.1.8)´Ä�3�½k.Ý(ÄKV (t, x)�7äá�þ.)V (t).

�±���~XêXÚ3nØþ�éA. ��ý¦)(4.1.8)Ù(J´����.

() November 17, 2011 10 / 32

CXê�5XÚ�A�

�ÄXÚx = A(t)x+B(t)u

F (t, �)´éAgdXÚ(1.1)�Ä�)Ý,K

x(t; t0, x0) = F (t, t0)x0 +

∫ t

t0

F (t, �)B(�)u(�)d� (4.1.9)

3"Ð^�eXÚ�Ñ\�G��m´���5C�

x(t) =

∫ t

t0

F (t, �)B(�)u(�)d�

ù�'X^5ïáCXê�5XÚBIBO­½�gdXÚ��ìC­½�m�'X. ��5!����5!�*ÿ5����*ÿ5,,,§¤å���^�~Xê�5XÚ¥eA(t−�)��^�aq.

() November 17, 2011 11 / 32

CXê�5XÚ�A�

éuXÚ(4.1.1),ÏÙ?Û)�~ê�þE�).Ï XÚ(4.1.1)��

Ü)�A5þdÐ�u)3�:�����oBr= {x ∣∥ x ∥< r}S)

�5�LyÑ5,Ï éuCXê�5XÚ`5,eã(Ø3ïÄ?Ø�k¿Â.

10XÚ�")´­½��duXÚ��Ü)þk..

20XÚ")��ÛáÚ�du")´ÛÜáÚ�.

30XÚ")��Û��ìC­½�du")��ìC­½.

() November 17, 2011 12 / 32

LyapunovC��±Ï�5XÚ

½½½ÂÂÂ4.2.1 CXê�5C�

x = T (t)z (4.2.1)

¡�´�LyapunovC�,X�10 (∀t ∈ J) : T (t) ∈ C1(J);20 (∃M1,M2 > 0)(∀t ∈ J) : ∥T (t)∥ ≤M1, ∥T (t)∥ ≤M2;30 (∃N > 0)(∀t ∈ J) : ∥T−1(t)∥ ≤ N.e

(∃� > 0)(∀t ∈ J) : T (t) = T (t+ �)

KT (t)´±��±Ï�±Ï�5C�;�T (t)ét ∈ [0, � ]þ�_q��,KT (t)´�LyapunovC�.

�ÛÉ~Xê�5C�Ñ´LyapunovC�.

() November 17, 2011 13 / 32

LyapunovC��±Ï�5XÚ

½½½ÂÂÂ4.2.2 CXê�5XÚ

x = A(t)x (4.2.2)

¡�´Lyapunov¿Âe�z,½�zXÚ,X��3LyapunovC�(4.2.1)ò(4.2.2)z¤~Xê�5XÚ

z = Az, A ∈ Rn×n (4.2.3)

LyapunovC�¿Ø��, Ïd^ØÓLyapunovC�¤��~Xê�5XÚÒ�U�Oé�, ØK�­½5.

LyapunovC�9Ù_þ���k.�5C�, ÏdC�c�XÚ�­½5�ØC"

() November 17, 2011 14 / 32

LyapunovC��±Ï�5XÚ

~~~4.2.1 ïÄXÚ

x =

[0 1−1 0

]x (4.2.4)

C�

x =

[cos t sin t− sin t cos t

]z

´LyapunovC�. TC�ò(4.2.4)C�

z = 0

ØÓLyapunovC��ò�XÚC�*d¿Ø�q�~Xê�5XÚ.�üXÚ�:þ­½�Øì?­½.

() November 17, 2011 15 / 32

LyapunovC��±Ï�5XÚ

ÚÚÚnnn4.2.1 �(4.2.1)��LyapunovC�, KÙ_½�LyapunovC�.yyy²²² x = T (t)z�LyapunovC�,KT−1�3�k..-S(t) = T−1(t),duT (t)S(t) = I,K

d

dtT (t)S(t) = T (t)S(t) + T (t)S(t) = 0

u´S(t) = −T−1(t)T (t)S(t)7k.. =z = S(t)x´LyapunovC�.

½½½nnn4.2.1 eXÚ(4.2.2)´�z5XÚ,K

(I) ")­½=⇒")��­½.

(II) ")ìC­½=⇒")��ìC­½.

() November 17, 2011 16 / 32

LyapunovC��±Ï�5XÚ

x = A(t)x, V = xTV (t)x (4.2.5)

LyapunovC�x = T (t)z,

z = A(t)z, A = (S + SA)T, S = T−1 (4.2.6)

V = xTV (t)x = zTT TV Tz = zTWz

KW +WA+ ATW = T T (V + V A+ATV )T

=d

dt(zTW (t)z)∣(4.2.6) =

d

dt(xTV (t)x)∣(4.2.5)

() November 17, 2011 17 / 32

LyapunovC��±Ï�5XÚ

�ıÏXê�5XÚµ

x = A(t)x (4.2.7)

�F (t, �)´(4.2.7)�Ä�)Ý,KF (t+ T, �)E,´(4.2.7)�),u´�3���ÝG1(�),¦

F (t+ T, �) = F (t, �)G1(�)

⇒F (t+ T, t) = G1(t)

d��5�F (t, �) = F (t+ T, � + T )

⇒ G1(t+ T ) = G1(t), ¿�´�_�.

() November 17, 2011 18 / 32

LyapunovC��±Ï�5XÚ

�ÝXÚz = −AT (t)z

�´±ÏXê�"F T (�, t)�ÙÄ�)Ý. Ó��3ÝG2(t),¦

F (t+ T, �) = G2(t)F (t, �)

¿�G2(t+ T ) = G2(t). 2-� = t,K

G2(t) = F (t+ T, t) = G1(t)

½½½nnn4.2.2 eF (t, �)´±ÏXê�5XÚ(4.2.7)�Ä�)Ý, K�3����_±Ï¼êÝG(t),¦k

F (t+ T, �) = F (t, �)G(�) = G(t)F (t, �)

�é?Ût��þkG(t)�G(�)�q.

() November 17, 2011 19 / 32

LyapunovC��±Ï�5XÚ

½½½nnn4.2.3 ±ÏXê�5XÚ(4.2.7)´�zXÚ, ��3±Ï�5C�òÙC�¤~Xê�5XÚ,éØÓëê� ó, éAXÚ�m´�q�.

y²µduG(t)´�ÛÉ�,K�3B(t)¦k

G(t) = eB(t)T (4.2.8)

-J(t, �) = F (t, �)e−B(�)t

KJ(t+ T, �) = F (t+ T, �)e−B(�)(t+T )

= F (t, �)G(�)G−1(�)e−B(�)t = J(t, �)

() November 17, 2011 20 / 32

LyapunovC��±Ï�5XÚ

��5C�x = J(t, �)z (4.2.9)

duJ(t, �)´t����ä±ÏT��ÛÉÝ,Ï (4.2.9)��LyapunovC�.

þªü>ét¦�, K

z = J−1(t, �)[A(t)J(t, �)− J(t, �)]z = J−1(t, �)[A(t)J(t, �)

−A(t)F (t, �)e−B(�)t + F (t, �)e−B(�)tB(�)]z = B(�)z

duG(�)3ØÓ��*d�q,K��B(�)3ØÓ��½*d�q££4.2.8¤¤.

() November 17, 2011 21 / 32

LyapunovC��±Ï�5XÚ

5551µ~4.2.1¥

F (t, �) = e

⎛⎝ 0 1−1 0

⎞⎠(t−�)

G(t) = F (t+ T, t) = e

⎛⎝ 0 1−1 0

⎞⎠T

B(�) =

(0 1−1 0

)

J(t, �) = e

⎛⎝ 0 1−1 0

⎞⎠(t−�)e−B(�)t = e

−

⎛⎝ 0 1−1 0

⎞⎠�òXÚz¤~Xê�5XÚ�C��±´õ«/ª�,½n4.2.3�´Ù¥��«, 3ù�a¥éAØÓëê�¤C�¤�XÚ�mâ�½´�q�.

() November 17, 2011 22 / 32

LyapunovC��±Ï�5XÚ

(4.2.8)⇒Λ(G(�)) ⊂

oS1⇌ Λ(B(�)) ⊂

oC−

½½½nnn4.2.4 ±ÏXê�5XÚ(4.2.7)�")ìC­½��=�

Λ(G(�)) ⊂oS1= {z∣ ∣z∣ < 1} (4.2.10)

")­½��=�

Λ(G(�)) ⊂ S1 = {z∣ ∣z∣ ≤ 1} (4.2.11)

�éA∂S1 = {z∣ ∣z∣ = 1}þG(�)�A���éA�gÐ�Ïf.

() November 17, 2011 23 / 32

LyapunovC��±Ï�5XÚ

XÚ")Ø­½��=�

Λ(G(�)) ∩o

Sc1 ∕= � (4.2.12)

½�k(4.2.11),�éA∂S1þG(�)�A����k��éA��g

Ð�Ïf.±þSc1 = {z∣ ∣z∣ ≥ 1},o

Sc1´ÙmØ.

éXÚ(4.2.7),ek(4.2.12),K¡éAXÚ")Ø­½���Ø­½.

() November 17, 2011 24 / 32

LyapunovC��±Ï�5XÚ

ü�¯¢:

±ÏXê�5XÚ�²LyapunovC�C�¤~Xê�5XÚ.

���5XÚ�gCq�~Xê�5XÚ�,�gCqXÚ�ìC­½�y��5XÚìC­½. �gCqXÚ��Ø­½Kí���5XÚØ­½.

() November 17, 2011 25 / 32

LyapunovC��±Ï�5XÚ

½½½nnn4.2.5 é±ÏXêXÚ

x = f(t, x), f(t, x) ≡ f(t+ T, x) (4.2.13)

Ù�gCqXÚ�

x = A(t)x, A(t) ≡ A(t+ T ) (4.2.14)

Ù¥A(t) = ∂f∂x ∣x=0. K

(I) (4.2.14)")ìC­½�y(4.2.13)")ÛÜ�êìC­½.(II) (4.2.14)")��Ø­½�í�(4.2.13)")Ø­½.

() November 17, 2011 26 / 32

LyapunovC��±Ï�5XÚ

5552µµµXJæ^~Xê½±ÏXê�g.��Lyapunov¼ê5ïÄ�5±ÏXÚ�ìC­½, éuV (x) = xTV x,

V (x)∣(4.2.7) = −xTW (t)x (A(t)TV + V A(t) = −W (t))

�K½^�£J±��¤�±~f.

duV��½Ý,K2ÂA��¯K£V −1W�A��¤

�V x =Wx

k�����2ÂA���1(t)��2(t),¿k

(∀(t, x) ∈ J×Rn, x ∕= 0) : �1(t) ≥xTW (t)x

xTV x≥ �2(t)

u´dV (x)

dt∣(4.2.7) ≤ −�2(t)V (x)

() November 17, 2011 27 / 32

LyapunovC��±Ï�5XÚ

duA(t) = A(t+ T ),Ï �2(t) = �2(t+ T ). -∫ V (T )

V (0)

dV (x)

V (x)= lnV (x(T ))− lnV (x(0))

∫ T

0−�2(�)d� = −�

K

lnV (x(0))

V (x(T ))≥ �

�� > 0,KkV (x(T ))

V (x(0))< e−� < 1

dd U�äXÚ")�ìC­½, ÃI�¦

(∀t ∈ J) : �2(t) ≥ � > 0

ù�¦xTW (t)x�½�^�.

() November 17, 2011 28 / 32

LyapunovC��±Ï�5XÚ

~~~4.2.2 �ÄXÚ{�1 = (−1

2 + "� cos 2t)�1 + (1− "� sin 2t)�2�2 = (−1− "� sin 2t)�1 + (−1

2 − "� cos 2t)�2(4.2.15)

éA" = 0�XÚÝ

C(1) =

[−1

2 1−1 −1

2

]-

V = �21 + �22

9V ∣(4.2.15) = −[�21 + �22 − 2"�[(�21 − �22) cos 2t− 2�1�2 sin 2t]]

() November 17, 2011 29 / 32

LyapunovC��±Ï�5XÚ

u´

W (t) =

[1− 2 cos 2t −2 sin 2t−2 sin 2t 1 + 2 cos 2t

], = "�

l �2(t) = 1− 2 §� < 1/2½" < 1/2��§éAXÚ")´ìC­½�.

() November 17, 2011 30 / 32

LyapunovC��±Ï�5XÚ

½½½nnn4.2.6 éuXÚ(4.2.7),Ù")ìC­½=I,�3V = V T ∈ Rn×n�½,¦

W (t) = −(V P (t) + P T (t)V )

�V|¤��KÝå(V,W (t))���2ÂA��

� = minx ∕=0

xTW (t)x

xTV x

k ∫ T

0�(t)dt = � > 0

() November 17, 2011 31 / 32

LyapunovC��±Ï�5XÚ

aq,��Ä

�(t) = maxx ∕=0

xTW (t)x

xTV x

KXÚ")��Ø­½=I∫ T

0�(t)dt = � < 0

() November 17, 2011 32 / 32