15ù
?5~???§|)??{
N￠?4?[
wg??
06/01/2022
N￠?4?[ (wg??) 15ù 06/01/2022 1 / 92
81
1 ?5???§|??n?
? Vg
3??5?n
àg?5???§|
?àg?5???§|
2 ~Xê?5???§|
Y
?ê
?)Y
O?
3 A^ˉKT~
N￠?4?[ (wg??) 15ù 06/01/2022 2 / 92
81
1 ?5???§|??n?
? Vg
3??5?n
àg?5???§|
?àg?5???§|
2 ~Xê?5???§|
Y
?ê
?)Y
O?
3 A^ˉKT~
N￠?4?[ (wg??) 15ù 06/01/2022 3 / 92
? Vg
??n??ê?tx(t)?
x(t) =
26
6666
6666
6664
x1(t)
x2(t):
::
xn(t)
37
7777
7777
7775; ?xT(t) = [x1(t);x2(t); ;xn(t)]; (1.1)
ùpxT(t)L?x(t)=?,ù￥z???txi(t)(i = 1;2; ;n)??
ma6t6bt?Y?ê.
??n n ?êY
A(t)?
A(t) =
26
6666
6666
6664
a11(t) a12(t) a1n(t)
a21(t) a22(t) a2n(t):
:: ::: :::
an1(t) an2(t) ann(t)
37
7777
7777
7775; (1.2)
§z???aij(t)(i;j = 1;2; ;n)??ma6t6bt?Y?ê.
N￠?4?[ (wg??) 15ù 06/01/2022 4 / 92
??
?Y: ?tx(t)??Y
A(t)z????3?ma6t6bt?Y,K?
T?t??Y
3?ma6t6bt′?Y.
??: ?tx(t)??Y
A(t)z????3?ma6t6bt??,K?
T?t??Y
3?ma6t6bt′??,?k
x0(t) =
26
6666
6666
6666
664
x01(t)
x02(t)
:::
x0n(t)
37
7777
7777
7777
775
; (1.3)
A0(t) =
26
6666
6666
6666
664
a011(t) a012(t) a01n(t)
a021(t) a022(t) a02n(t)
::: ::: :::
a0n1(t) a0n2(t) a0nn(t)
37
7777
7777
7777
775
:
N￠?4?[ (wg??) 15ù 06/01/2022 5 / 92
e??ó Y
A(t);B(t)9ó??tx(t);y(t)t3?ma6t6bt??,
Kk
[A(t) + B(t)]0 = A0(t) + B0(t);
[x(t) + y(t)]0 = x0(t) + y0(t);
[A(t)B(t)]0 = A0(t)B(t) + A(t)B0(t);
[A(t)x(t)]0 = A0(t)x(t) + A(t)x0(t):
??
?è: ?tx(t)??Y
A(t)z????3?ma6t6bt?è,K?
T?t??Y
3?ma6t6bt′?è,?k
N￠?4?[ (wg??) 15ù 06/01/2022 6 / 92
Z b
a
A(t)dt =
26
6666
6666
6666
6666
6666
6666
6666
4
R b
a a11(t)dt
R b
a a12(t)dt
R b
a a1n(t)dt
R b
a a21(t)dt
R b
a a22(t)dt
R b
a a2n(t)dt
::: ::: :::
R b
a an1(t)dt
R b
a an2(t)dt
R b
a ann(t)dt
37
7777
7777
7777
7777
7777
7777
7777
5
;
Z b
a
x(t)dt =
26
6666
6666
6666
6666
6666
6666
6664
R b
a x1(t)dt
R b
a x2(t)dt
:::
R b
a xn(t)dt
37
7777
7777
7777
7777
7777
7777
7775
:
N￠?4?[ (wg??) 15ù 06/01/2022 7 / 92
??
n??tx = (x1;x2; ;xn)T?ê?
kxk=
nX
i=1
jxij;
n n Y
A?ê?
kAk=
nX
i=1
jaijj:
??n n Y
A =fai;j;i;j = 1;2; ;ng;B =fbi;j;i;j = 1;2; ;ngún?
?tx =fxi;i = 1;2; ;ng;y =fyi;i = 1;2; ;ng,±9?ma6t6bt
?è?êY
ú?ê?tA(t);x(t),Kk
N￠?4?[ (wg??) 15ù 06/01/2022 8 / 92
1.kxk>0?kxk= 0??^?′xi = 0(i = 1;2; ;n);
kAk>0?kAk= 0??^?′aij = 0(i;j = 1;2; ;n);
2.é??~êk,k
kkxk=jkjkxk; kkAk=jkjkAk;
3.kx + yk6kxk+kyk; kA + Bk6kAk+kBk;
4.kAxk6kAkkxk;kABk6kAkkBk;
5.
R b
a x(t)dt
6
R b
a kx(t)kdt;
R b
a A(t)dt
6
R b
a kA(t)kdt; a6b:
N￠?4?[ (wg??) 15ù 06/01/2022 9 / 92
??
e?tSfxkg(xk = (x1k;x2k; ;xnk)T)éz??k(k = 1;2; ;n)?′?
?,K??tSfxkg′???.
??
e?ê?tSfxk(t)g(xk(t) = (x1k(t);x2k(t); ;xnk(t))T)éz??k(k = 1;
2; ;n)3?ma6t6bt?′??(????),K??ê?tS
fxk(t)g′???(????).
??
e?ê?t?êP1k=1 xk(t)ü?ú¤¤?ê?tS3?
ma6t6bt′??(????),K?P1k=1 xk(t)3?ma6t6bt′
??(????).
N￠?4?[ (wg??) 15ù 06/01/2022 10 / 92
?Xe? ???§|
8>>
>>>>
<>
>>>>
>:
x01(t) = a11(t)x1(t) + a12(t)x2(t) + + a1n(t)xn(t) + f1(t);
x02(t) = a21(t)x1(t) + a22(t)x2(t) + + a2n(t)xn(t) + f2(t);
x0n(t) = an1(t)x1(t) + an2(t)x2(t) + + ann(t)xn(t) + fn(t);
(1.4)
??
?§|(1.4)''ux1;x2; ;xn9x01;x02; ;x0n′?5,
??§|(1.4)
?? ?5?????§|. e?§|￥??êê′ 9 ±
t,K?ù?p ?????§|.
N￠?4?[ (wg??) 15ù 06/01/2022 11 / 92
e?ê?tu(t)êu0(t)3?m[ ; ] [a;b]t?Y,?÷v
u0(t) = A(t)u(t) + f(t); t2[ ; ] (1.5)
K?u(t)??§|(1.4)3?m[ ; ] [a;b]t??).
XJ??D?^?
x(t0) = ; (1.6)
K÷v^?(1.6)?§|(1.4)|)ˉK??D?ˉK??)ˉK.3
?1t0?m[ ; ] [a;b]S,)u(t)÷v
u(t0) = :
N￠?4?[ (wg??) 15ù 06/01/2022 12 / 92
~
y?ê?t
u(t) =
" sint
cost
#
′D?ˉK
x0(t) =
" 0 1
1 0
#
x(t); x(0) =
" 0
1
#
3?m( 1;+1)t).
): du sint;cost3?m( 1;+1)t?Y?,?\?§
u0(t) =
" sint
cost
#0
=
" cost
sint
#
=
" 0 1
1 0
#" sint
cost
#
=
" 0 1
1 0
#
u(t):
?k
u(0) =
" sin0
cos0
#
=
" 0
1
#
;
?d, u(t)??§D?ˉK).
N￠?4?[ (wg??) 15ù 06/01/2022 13 / 92
??n ?5???§D?ˉK(
x(n) + a1(t)x(n 1) + + an 1(t)x0+ an(t)x = f(t);
x(t0) = 1;x0(t0) = 2; ;x(n 1)(t0) = n; (1.7)
?LCtC?
x1(t) = x(t);x2(t) = x01(t); ;xn(t) = x0n 1(t)
òD?ˉK(1.7)z?Xe? ?5???§|D?ˉK
8>>
>>>>
>>>>
>>><
>>>>
>>>>
>>>>
>:
x0(t) =
26
6666
6666
6666
6666
6666
4
0 1 0 0
0 0 1 0
::: ::: ::: :::
0 0 0 1
an(t) an 1(t) an 2(t) a1(t)
37
7777
7777
7777
7777
7777
5
x(t) +
26
6666
6666
6666
6666
6666
4
0
0
:::
0
f(t)
37
7777
7777
7777
7777
7777
5
;
x(t0) = ;
(1.8)
N￠?4?[ (wg??) 15ù 06/01/2022 14 / 92
ù￥
x(t) =
26
6666
6666
6666
664
x1(t)
x2(t)
:::
xn(t)
37
7777
7777
7777
775
=
26
6666
6666
6666
664
x(t)
x0(t)
:::
x(n 1)(t)
37
7777
7777
7777
775
; =
26
6666
6666
6666
664
1
2
:::
n
37
7777
7777
7777
775
:
5:ˉ￠t, n ?5???§D?ˉK(5-8)?? ?5???§|
D?ˉK(5-9)3ù??e′ d. =??ù￥??D?ˉK),
?E,??D?ˉK).
~
òD?ˉK
x000+ tx00 3t2x0+ 7x = et; x(0) = 1; x0(0) = 0; x00(0) = 1
z??? d? ?5???§|D?ˉK.
N￠?4?[ (wg??) 15ù 06/01/2022 15 / 92
):-
x1(t) = x; x2(t) = x01(t) = x0; x3(t) = x02(t) = x00;
Kk
x =
26
6666
6664
x1(t)
x2(t)
x3(t)
37
7777
7775; x0 =
26
6666
6664
x01(t)
x02(t)
x03(t)
37
7777
7775; =
26
6666
6664
1
0
1
37
7777
7775:
q??
x01(t) = x0 = x2(t);
x02(t) = x00 = x3(t);
x03(t) = x000 = tx00+ 3t2x0 7x + et
= tx3(t) + 3t2x2(t) 7x1(t) + et;
u′
x0 =
26
6666
6664
0 1 0
0 0 1
7 3t2 t
37
7777
7775x +
26
6666
6664
0
0
et
37
7777
7775;
N￠?4?[ (wg??) 15ù 06/01/2022 16 / 92
?k
x(0) =
26
6666
6664
x1(0)
x2(0)
x3(0)
37
7777
7775 = :
I???′: z??n ?5???§??z?n?? ?5???§
¤¤?§|,?L5%?¤á. ~X,?§|
x0 =
" 1 0
0 1
#
x; x =
" x
1
x2
#
;
?Uz??? ?5???§.
N￠?4?[ (wg??) 15ù 06/01/2022 17 / 92
3??5?n
?n
XJ?êY
A(t)ú?ê?tf(t)3?m[a;b]t?Y,K? ?5???
§|D?ˉK (
x0(t) = A(t)x(t) + f(t);
x(t0) = ; t0 2[a;b]; (1.9)
3?m[a;b]S3??)x = x(t).
ún
? ?5???§|D?ˉK(1.9)3?m[a;b]t?Y) duè?
?§|
x(t) = +
Z t
t0
[A(s)x(s) + f(s)]ds (1.10)
3?m[a;b]t?Y).
N￠?4?[ (wg??) 15ù 06/01/2022 18 / 92
E?kS??ê?tS
8>>
<>
>:
’0(t) = ;
’k(t) = + R tt
0
[A(s)’k 1(s) + f(s)]ds; t2[a;b];k = 1;2; : (1.11)
ùp,?kS??ê?t’k(t)??? ?5???§|D?ˉ
K(1.9)11kgCq).
ún
éu¤kêk,?kS??ê?tSf’k(t)g3?m[a;b]tk??
??Y.
ún
?kS??ê?tSf’k(t)g3?m[a;b]t′????.
N￠?4?[ (wg??) 15ù 06/01/2022 19 / 92
y2: du
’k(t) = ’0(t) +
kX
j=1
[’j(t) ’j 1(t)]; (1.12)
?d?y2?kS??ê?tSf’k(t)g3?m[a;b]t????,?Iy
2a(1.12)mà?ê?ê3?m[a;b]t????=?.
??A(t), f(t)34?m[a;b]t?Y,¤±kA(t)k,kf(t)kt3[a;b]tk.,=
3~êLúK,|
kA(t)k6L; kf(t)k6K; t2[a;b]:
u′k
k’1(t) ’0(t)k=
Z t
t0
[A(s)’0(s) + f(s)]ds
6
Z t
t0
kA(s)’0(s) + f(s)kds
6M(t t0); t06t6b;
ù￥M = Lk k+ K.
N￠?4?[ (wg??) 15ù 06/01/2022 20 / 92
ók
k’2(t) ’1(t)k=
Z t
t0
kA(s)[’1(s) ’0(s)]kds
6L
Z t
t0
k’1(s) ’0(s)kds
6 ML2 (t t0)2; t06t6b:
dê?8B{
k’k(t) ’k 1(t)k6 ML
k 1
k! (t t0)
k6 MLk 1
k! (b t0)
k: (1.13)
d?dA.dO{?,
1P
k=1
MLk 1
k! (b t0)
k3?m[t0;b]t????,
Kk?ê?ê kP
j=1
[’j(t) ’j 1(t)]3?m[t0;b]t????,l
?kS
??ê?tSf’k(t)g3[t0;b]t????.
N￠?4?[ (wg??) 15ù 06/01/2022 21 / 92
ún
’(t)′è??§|(1.10)??3?m[a;b]t?Y),?d,’(t)?′D?
ˉK(1.9)??3?m[a;b]t).
y2:-lim
k!1
’k(t) = ’(t),Kéa(1.11)￥1aüàó?4??
’(t) = lim
k!1
’k(t) = lim
k!1
"
+
Z t
t0
(A(s)’k 1(s) + f(s))ds
#
= +
Z t
t0
lim
k!1
[A(s)’k 1(s) + f(s)]ds
= +
Z t
t0
[A(s)’(s) + f(s)]ds:
(1.14)
N￠?4?[ (wg??) 15ù 06/01/2022 22 / 92
ún
e (t)′è??§|(1.10)??3?m[a;b]t,??),K7
k (t) ’(t), t2[a;b],=è??§|(1.10))′??.
y2: du
(t) = +
Z b
t0
[A(s) (s) + f(s)]ds; (1.15)
a(1.14)?a(1.15)?~?
k’(t) (t)k=
Z t
t0
[A(s)(’(s) (s))]ds
6
Z b
t0
kA(s)kk’(s) (s)kds
6L
Z t
t0
k’(s) (s)kds:
N￠?4?[ (wg??) 15ù 06/01/2022 23 / 92
-
g(t) =k’(t) (t)k; u(t) =
Z t
t0
g(s)ds;
K
g(t)6L
Z t
t0
g(s)ds = Lu(t):
u0(t) = g(t),¤±
u0(t)6Lu(t);
u′
[u0(t) Lu(t)]e Lt60:
üàè?? Z
t
t0
u0(s)e Lsds L
Z t
t0
u(s)e Lsds
= u(s)e Ls
t
t0
= u(t)e Lt u(t0)e Lt0
60;
N￠?4?[ (wg??) 15ù 06/01/2022 24 / 92
Kk
u(t)6u(t0);
= Z t
t0
g(s)ds =
Z t
t0
k’(t) (t)kds60:
?d,k’(t) (t)k= 0,’(t) (t).
d±túnB??ny2.
5:3?y2L§￥,·?uy? ?5???§|D?ˉK(1.9)
)3?m?XêY
A(t)ú?àg?f(t)?Y?m[a;b].ù?:?
12ù￥''u? ???§D?ˉK)3??5?n9)
3?m?ó.??3E?k?ú%C?êS￥=IXêY
A(t)ú
?àg?f(t)?Y=?,¤±éu? ?5???§|D?ˉK(1.9)5
`,?3)ò?.
N￠?4?[ (wg??) 15ù 06/01/2022 25 / 92
í?
XJai(t)(i = 1;2; ;n), f(t)3?m[a;b]t?Y,Ké??t0 2[a;b]9
??~ê i(i = 1;2; ;n),?§
x(n) + a1(t)x(n 1) + + an 1(t)x0+ an(t)x = f(t)
3??3?m[a;b]t??)w(t),?÷vD?^?
w(t0) = 1;w0(t0) = 2; ;w(n 1)(t0) = n:
N￠?4?[ (wg??) 15ù 06/01/2022 26 / 92
àg?5???§|
!·?k5??àg?5???§|
dx(t)
dt = A(t)x(t) (1.16)
?)(,ù￥x(t);A(t)
?n
(U\n) ex1(t);x2(t)′àg?5???§|(1.16)),K§??
5|ü x(t) = c
1x1(t) + c2x2(t) (1.17)
E,′àg?5???§|(1.16)),ù￥c1;c2???~ê.
N￠?4?[ (wg??) 15ù 06/01/2022 27 / 92
y2: du
dx(t)
dt =
d
dt(c1x1(t) + c2x2(t))
= c1dx1(t)dt + c2dx2(t)dt
= c1A(t)x1(t) + c2A(t)x2(t)
= A(t)[c1x1(t) + c2x2(t)]
= A(t)x(t);
y.
-V =fàg?5???§|(1.16)3?m[a;b]t??)g,K?n`2
8üV′???5?m.
ún
8üV′??n??5?m.
N￠?4?[ (wg??) 15ù 06/01/2022 28 / 92
y2:-t0 2[a;b],d)3??5?n?,éu??~ê?
tx0 2Rn,3V￥3???x(t),?÷vx(t0) = x0. l
3N
H : Rn !V;(=òx0Nx(t)):
duéu??x(t)2V,kx(t0)2Rn,?
H(x(t0)) = x(t);
?dNH′÷N. q??éu??x1;x2 2Rn,-
x1(t) = H(x1); x2(t) = H(x2);
d)??5?,
x1(t) , x2(t); t2[a;b]; ?=x1 , x2;
¤±NHq′??N,?dU\nú)??5
H(c1x1 + c2x2) = c1H(x1) + c2H(x2):
?d,NH′?5,?′??lRnVóN.
N￠?4?[ (wg??) 15ù 06/01/2022 29 / 92
ù??XRn?Vk?ó?5(,¤±V′??n??5?m,=àg?
5???§|(1.16))¤?5?m?ê??§| ê?ó.
?n
(?)() eàg?5???§|(1.16)3?m[a;b]tkn??5?''
)
x1(t);x2(t); ;xn(t); (1.18)
K§?)?L??
x(t) = c1x1(t) + c2x2(t) + + cnxn(t);
ù￥ci(i = 1;2; ;n)???~ê.
)|(1.18))¤
)?5?mV,?~?ù?àg?5???§
|(1.16)??? )|.?d,|àg?5???§|(1.16)?)?I
|?§??? )|=?.
N￠?4?[ (wg??) 15ù 06/01/2022 30 / 92
@o·?I?)?ü?ˉK:
(1)|?àg?5???§|(1.16)n?)x1(t);x2(t); ;xn(t);
(2)y2)|x1(t);x2(t); ;xn(t)?5?''.
??
)|xi(t)(i = 1;2; ;n)?t/a?
xi(t) =
26
6666
6666
6666
664
x1i(t)
x2i(t)
:::
xni(t)
37
7777
7777
7777
775
; i = 1;2; ;n;
K§Kd?1a?
W(t) =
x11(t) x12(t) x1n(t)
x21(t) x22(t) x2n(t)
::: ::: :::
xn1(t) xn2(t) xnn(t)
:
N￠?4?[ (wg??) 15ù 06/01/2022 31 / 92
ún
)|(1.18)Kd?1a÷v4?úa
W(t) = W(t0)e
R t
t0 tr(A(t))dt; x2[a;b]; (1.19)
ù￥, t0 2[a;b]; tr(A(t))L?Y
A(t),,=tr(A(t)) = nP
i=1
aii(t).
y2:
dW(t)
dt =
nX
i=1
x11(t) x12(t) x1n(t)
x21(t) x22(t) x2n(t)
::: ::: :::
dxi1(t)
dt
dxi2(t)
dt
dxin(t)
dt:
:: ::: :::
xn1(t) xn2(t) xnn(t)
N￠?4?[ (wg??) 15ù 06/01/2022 32 / 92
=
nX
i=1
x11(t) x12(t) x1n(t)
x21(t) x22(t) x2n(t)
::: ::: :::
nX
j=1
aijxj1
nX
j=1
aijxj2
nX
j=1
aijxjn
::: ::: :::
xn1(t) xn2(t) xnn(t)
=
nX
i=1
aii W(t) = tr(A(t)) W(t);
üàè?Ba(1.19).
t?ún`2:)|(1.18)Kd?1aW(t)e3,:t0??",K3
??mt??";e3,:t0???",K3??mt???".
=W(t)?oe?",?oe??".
N￠?4?[ (wg??) 15ù 06/01/2022 33 / 92
?n
àg?5???§|(1.16))|(1.18)?5?''??^?′
W(t) , 0; t2[a;b]:
y2:e)|x1(t);x2(t); ;xn(t)?5?'',=?§|
c1x1(t) + c2x2(t) + + cnxn(t)
=
26
6666
6666
6666
664
x11(t) x12(t) x1n(t)
x21(t) x22(t) x2n(t)
::: ::: :::
xn1(t) xn2(t) xnn(t)
37
7777
7777
7777
775
26
6666
6666
6666
664
c1
c2
:::
cn
37
7777
7777
7777
775
= 0
?k"),¤±XêY
1a÷?,=W(t) , 0;t2[a;b].
N￠?4?[ (wg??) 15ù 06/01/2022 34 / 92
??,eW(t) , 0,dún?,éu,:t0 2[a;b],kW(t0) , 0,=
x1(t0);x2(t0); ;xn(t0)
3Rn￥?5?'',u′k
H(0) = H(c1x1(t0) + c2x2(t0) + + cnxn(t0))
= c1x1(t) + c2x2(t) + + cnxn(t) = 0;
í?
)|x1(t);x2(t); ;xn(t)?5?''.)|(1.18)?5?''??^?
′
W(t) 0; t2[a;b]:
N￠?4?[ (wg??) 15ù 06/01/2022 35 / 92
~
y x
1
x2
!
= c1
et
0
!
+ c2
tet
et
!
′???§| x0
1x0
2
!
=
1 1
0 1
! x
1
x2
!
(1.20)
?),ù￥c1;c2???~ê.
)μ-
’1(t) =
et
0
!
; ’2(t) =
tet
et
!
:
??
’01(t) =
et
0
!
=
1 1
0 1
! et
0
!
=
1 1
0 1
!
’1(t);
¤±’1(t)′?§|(1.20)).
N￠?4?[ (wg??) 15ù 06/01/2022 36 / 92
q??
’02(t) =
tet
et
!0
=
et + tet
et
!
=
1 1
0 1
! tet
et
!
=
1 1
0 1
!
’2(t)
¤±’2(t)?′?§|(1.20)).
?Det(’1(t);’2(t)) , 0,¤±)’1(t);’2(t)?5?'', c1’1(t) + c2’2(t)?
?§|?).
éu)|x1(t);x2(t); ;xn(t),·???±ò§?¤Y
/a
(t) =
26
6666
6666
6666
664
x11(t) x12(t) x1n(t)
x21(t) x22(t) x2n(t)
::: ::: :::
xn1(t) xn2(t) xnn(t)
37
7777
7777
7777
775
;
(t)??àg?5???§|(1.16)))YY
.
N￠?4?[ (wg??) 15ù 06/01/2022 37 / 92
XJ)|x1(t);x2(t); ;xn(t)?5?'',=′??? )|?,·?
? (t)?àg?5???§|(1.16)?))YY
.AO/, (t) = E(E?
ó ü?Y
)?,?ù?IO?))YY
.
ò)Y
(t)?\àg?5???§|(1.16)?
0(t) = (x0ij(t))n n =
0B
BBBB
B@
nX
k=1
aik(t)xkj(t)
1C
CCCC
CA
n n
= (aij(t))n n(xij(t))n n = A(t) (t);
¤±)Y
(t)?′àg?5???§|(1.16)Y
/a),=Y
).
??,e (t)′àg?5???§|(1.16)Y
),K§z??′
àg?5???§|(1.16)).
N￠?4?[ (wg??) 15ù 06/01/2022 38 / 92
?n
XJ (t)?àg?5???§|(1.16)?)Y
,@oàg?5???
§|(1.16)??)’(t)?L??
’(t) = (t)c;
ùp, c?(?n?~ê?t.
?n
àg?5???§|(1.16))Y
(t)′?)Y
??^?′
(1)1aDet( (t)) , 0; t2[a;b];?
(2)3t0 2[a;b];Det( (t0)) , 0,l
Det( (t)) , 0.
?n
XJ (t)′àg?5???§|(1.16)3?m[a;b]t?)Y
,
C′n n ??é~êY
,@o (t)C?′àg?5???§|(1.16)
3?m[a;b]t?)Y
.
N￠?4?[ (wg??) 15ù 06/01/2022 39 / 92
y2:??
( (t)C)0 = 0(t)C = A(t) (t)C = A(t)( (t)C);
¤± (t)C′àg?5???§|(1.16))Y
. q?? (t)′?)Y
, C??é,Kk
Det( (t)C) = Det( (t))Det(C) , 0;
¤±, (t)C′?)Y
.
dt??n?,ü??)Y
?m? ????é~êY
.
í?
e (t); (t)′àg?5???§|(1.16)3?m[a;b]tü??)Y
,
K??3??n n ??é~êY
C,|3?m[a;b]t
(t) = (t)C:
N￠?4?[ (wg??) 15ù 06/01/2022 40 / 92
y2:?? (t); (t)′?m[a;b]t?)Y
,¤± 1(t)3,?
U(t) = 1(t) (t)
E,?n n Y
,Kk
(t) = (t)U(t);
u′
0(t) = 0(t)U(t) + (t)U0(t)
= A(t) (t)U(t) + (t)U0(t)
= A(t) (t) + (t)U0(t):
du (t)÷vàg?5???§|(1.16),¤±
(t)U0(t) = 0:
u′kU0(t) = 0,U(t) = C?n n ~êY
.
N￠?4?[ (wg??) 15ù 06/01/2022 41 / 92
?àg?5???§|
??§|
x0(t) = A(t)x(t) + f(t); (1.21)
T?§|?éAuàg?5???§|(1.16)?àg?5???§|.
5?
e’(t)′?àg?5???§|(1.21)), (t)′?àg?5???§
|(1.21)éAàg?5???§|(1.16)),K’(t) + (t)′?àg?
5???§|(1.21)).
y2: du
’0(t) = A(t)’(t) + f(t);
0(t) = A(t) (t);
òüa?\
(’(t) + (t))0 = A(t)(’(t) + (t)) + f(t);
’(t) + (t)′?àg?5???§|(1.21)).
N￠?4?[ (wg??) 15ù 06/01/2022 42 / 92
5?
e’(t);’(t)′?àg?5???§|(1.21)ü?),K’(t) ’(t)′éA
àg?5???§|(1.16)).
y2: du
’0(t) = A(t)’(t) + f(t);
’0(t) = A(t)’(t) + f(t);
üa?~,
(’(t) ’(t))0 = ’0(t) ’0(t) = A(t)(’(t) ’(t));
’(t) ’(t)′àg?5???§|(1.16)).
?n
e (t)′àg?5???§|(1.16)???)Y
, (t)′?àg?5
???§|(1.21)??A),K?àg?5???§|(1.21)?)?
L??
’(t) = (t)c + (t); (1.22)
ù￥, c?(?~ê?t.
N￠?4?[ (wg??) 15ù 06/01/2022 43 / 92
y2:??
’0(t) = 0(t)c + 0(t)
= A(t) (t)c + A(t) (t)
= A(t)( (t)c + (t))
= A(t)’(t);
¤±’(t)′?àg?5???§|(1.21)).
u′d5??,’(t) (t)′àg?5???§|(1.16)).2d?n
?,3~ê?tc,|
’(t) (t) = (t)c;
u′?n(?.
N￠?4?[ (wg??) 15ù 06/01/2022 44 / 92
e?éu? ?àg?5???§||),·?a''12ù? ?àg
?5???§|)§E,?±|^~êC′{,=òéAàg?5?
??§|(1.16)) (t)c￥~ê?tcC?c(t).
b?
’(t) = (t)c(t) (1.23)
??àg?5???§|(1.21)),òù?\?§|
0(t)c(t) + (t)c0(t) = A(t) (t)c(t) + f(t):
du (t)′éAàg?5???§|(1.16)?)Y
,?dta?z
{?
(t)c0(t) = f(t):
3t0ttè??
c(t) = c(t0) +
Z t
t0
1(s)f(s)ds:
da(1.23)?
c(t0) = 1(t0)’(t0);
N￠?4?[ (wg??) 15ù 06/01/2022 45 / 92
u′k
c(t) = 1(t0)’(t0) +
Z t
t0
1(s)f(s)ds:
¤±|
’(t) = (t) 1(t0)’(t0) + (t)
Z t
t0
1(s)f(s)ds: (1.24)
,???,éa(1.24)üà|?
’0(t) = 0(t) 1(t0)’(t0) + 0(t)
Z t
t0
1(s)f(s)ds + (t) 1(t)f(t)
= A(t) (t)[ 1(t0)’(t0) +
Z t
t0
1(s)f(s)ds] + f(t)
= A(t)’(t) + f(t):
?d,?àg?5???§|(1.21)÷vD?^?’(t0) = )?
’(t) = (t) 1(t0) + (t)
Z t
t0
1(s)f(s)ds: (1.25)
N￠?4?[ (wg??) 15ù 06/01/2022 46 / 92
~
|D?ˉK
x0(t) =
" 1 1
0 1
#
x(t) +
" e t
0
#
; x(0) =
" 1
1
#
).
): dc~?
(t) =
" et tet
0 et
#
??§éAàg?5???§|?)Y
,K|^~êC′{ò
(t)c(t) =
" et tet
0 et
#
c(t)
?\?§,?
(t)c0(t) =
" et tet
0 et
#
c0(t) =
" e t
0
#
= f(t):
N￠?4?[ (wg??) 15ù 06/01/2022 47 / 92
du
1(t) =
" et tet
0 et
# 1
= e 2t
" et tet
0 et
#
= e t
" 1 t
0 1
#
;
K
c0(t) = e t
" 1 t
0 1
#" e t
0
#
=
" 1 t
0 1
#" e 2t
0
#
=
" e 2t
0
#
:
üàè?
c(t) = 1(0)
" 1
1
#
+
Z t
0
" e 2s
0
#
ds =
" 1
1
#
+
26
6666
64
1
2e
2t
0
37
7777
75 =
26
6666
64 1
1
2e
2t
1
37
7777
75;
¤±,?§3D?^?e)?
’(t) = (t)c(t) =
" et tet
0 et
#2666
6664 1
1
2e
2t
1
37
7777
75 =
26
6666
64 (1 t)e
t 1
2e
t
et
37
7777
75:
N￠?4?[ (wg??) 15ù 06/01/2022 48 / 92
í?
eai(t)(i = 1;2; ;n), f(t)t3?m[a;b]t?Y, xi(t)(i = 1;2; ;n)??
m[a;b]tàg?5???§
x(n) + a1(t)x(n 1) + + an(t)x = 0
? )|,K?àg?5???§
x(n) + a1(t)x(n 1) + + an(t)x = f(t) (1.26)
÷vD?^?
’(t0) = 0;’0(t0) = 0; ;’(n 1)(t0) = 0; t0 2[a;b]
)?L??
’(t) =
nX
k=1
xk(t)
Z t
t0
Wk[x1(s);x2(s); ;xn(s)]
W[x1(s);x2(s); ;xn(s)] f(s)ds; (1.27)
N￠?4?[ (wg??) 15ù 06/01/2022 49 / 92
í?
ù￥, W[x1(s);x2(s); ;xn(s)]′? )|x1(s);x2(s); ;xn(s)Kd?1
a, Wk[x1(s), x2(s); ;xn(s)]′òù￥1k?±(0;0; ;0;1)TO?
1a. @o,?àg?5???§(1.26)??)L??
u(t) = c1x1(t) + c2x2(t) + + cnxn(t) +’(t);
ùpci(i = 1;2; ;n)?(?~ê.
a(1.27)???àg?5???§(1.26)~êC′úa. d?,?àg
?5???§(1.26)?)?
x(t) = c1x1(t) + c2x2(t) + + cnxn(t) +’(t);
ùpci(i = 1;2; ;n)???~ê.
~
??§x00+ x = 0??? )|?x1(t) = cost;x2(t) = sint,|éA
?àg?5???§x00+ x = 2tant??).
N￠?4?[ (wg??) 15ù 06/01/2022 50 / 92
):·??±?|^a(1.27)?1|)
W[x1(t);x2(t)] =
cost sint sint cost
= 1;
t0 = 0,Kk
’(t) =
Z t
0
(sintcoss costsins) 2tansds
= 2sint
Z t
0
sinsds 2cost
Z t
0
sinstansds
= 2sint 2costlnjsect + tantj:
|’(t)′ˉK￥?àg?5???§??).q?
?sint′x00+ x = 0??),¤±
2costlnjsect + tantj
?′x00+ x = 2tant??).
N￠?4?[ (wg??) 15ù 06/01/2022 51 / 92
81
1 ?5???§|??n?
? Vg
3??5?n
àg?5???§|
?àg?5???§|
2 ~Xê?5???§|
Y
?ê
?)Y
O?
3 A^ˉKT~
N￠?4?[ (wg??) 15ù 06/01/2022 52 / 92
Y
?êexpA
!·?k5??XêY
?~êY
àg?5???§|
x0(t) = Ax(t) (2.1)
? )||),ùpA = (aij)n n?~êY
.
??
A?n n ~êY
,Y
?ê
1X
k=0
Ak
k! = E + A +
A2
2! + +
Am
m! + = expA
ú??Y
?êexpA,?P?eA,ù￥, E?n ü?Y
.
Am?Y
Amg?,·?5?A0 = E; 0! = 1.A?"Y
?,=A = O,
kexpO = E.
N￠?4?[ (wg??) 15ù 06/01/2022 53 / 92
ún
?ê1P
k=0
Ak
k!éu¤kA?′??,?dexpA′??(?Y
.
ún
?ê
expAt =
1X
k=0
Aktk
k! (2.2)
3t??k??mt′????.
N￠?4?[ (wg??) 15ù 06/01/2022 54 / 92
5?
eY
A;B??,=AB = BA,K
exp(A + B) = expAexpB:
y2: duAB = BA,Kd?a?n9yé???ê|ê?n?
exp(A + B) =
1X
k=0
(A + B)k
k! =
1X
k=0
0B
BBBB
B@
1X
l=0
AlBk l
l!(k l)!
1C
CCCC
CA
=
1X
l=0
Al
l!
0B
BBBB
B@
1X
k=0
Bk
k!
1C
CCCC
CA = expAexpB:
5?
é??Y
A,Y
?ê_=(expA) 13,?
(expA) 1 = exp( A):
y2:-B = A,B?y.
N￠?4?[ (wg??) 15ù 06/01/2022 55 / 92
5?
eT′??éY
,K
exp(T 1AT) = T 1(expA)T:
y2:
exp(T 1AT) = E +
1X
k=1
(T 1AT)k
k!
= E +
1X
k=1
T 1AkT
k!
= E + T 1
0B
BBBB
B@
1X
k=1
Ak
k!
1C
CCCC
CAT
= T 1(expA)T:
N￠?4?[ (wg??) 15ù 06/01/2022 56 / 92
?n
Y
(t) = expAt
′~Xêàg?5???§|(1.16)?)Y
,? (0) = E.
y2:??
0(t) = (expAt)0
= A + A
2t
1! +
A3t2
2! + +
Aktk 1
(k 1)! +
= AexpAt
= A (t);
¤± (t)′~Xêàg?5???§|(2.1))Y
. q??
(0) = expO = E;
?
Det( (0)) = Det(E) = 1 , 0;
¤± (t)′~Xêàg?5???§|(2.1)?)Y
.
N￠?4?[ (wg??) 15ù 06/01/2022 57 / 92
?n
Y
(t) = expAt
′~Xêàg?5???§|(1.16)?)Y
,? (0) = E.
y2:??
0(t) = (expAt)0
= A + A
2t
1! +
A3t2
2! + +
Aktk 1
(k 1)! +
= AexpAt = A (t);
¤± (t)′~Xêàg?5???§|(2.1))Y
. q??
(0) = expO = E;
?
Det( (0)) = Det(E) = 1 , 0;
¤± (t)′~Xêàg?5???§|(2.1)?)Y
.
N￠?4?[ (wg??) 15ù 06/01/2022 58 / 92
u′,d?n5.5?±~Xêàg?5???§|(2.1)??)?L
??
’(t) = (expAt)c;
c?~ê?t.’(0) = ?,’(t) = (expAt) .
~
A′éY
?,=
A =
26
6666
6666
6666
664
1
2
:::
n
37
7777
7777
7777
775
;
á|?x0(t) = Ax(t)?)Y
.
N￠?4?[ (wg??) 15ù 06/01/2022 59 / 92
): dún?
expAt = E +
26
6666
6666
6666
664
1
2
:::
n
37
7777
7777
7777
775
t
1! +
26
6666
6666
6666
664
21
22
:::
2n
37
7777
7777
7777
775
t2
2! + +
26
6666
6666
6666
664
k1
k2
:::
kn
37
7777
7777
7777
775
tk
k! + =
26
6666
6666
6666
664
e 1t
e 2t
:::
e nt
37
7777
7777
7777
775
:
N￠?4?[ (wg??) 15ù 06/01/2022 60 / 92
~
|?§|x0(t) =
" 1 1
0 1
#
x(t)?)Y
.
): du
A =
" 1 1
0 1
#
=
" 1 0
0 1
#
+
" 0 1
0 0
#
;
ü?Y
′??,K
expAt = exp
" 1 0
0 1
#
t exp
" 0 1
0 0
#
t
=
" et 0
0 et
#
0B
BBBB
@E +
" 0 1
0 0
#
t +
" 0 1
0 0
#2 t2
2! +
1C
CCCC
A:
?)Y
?
expAt = et
" 1 t
0 1
#
:
N￠?4?[ (wg??) 15ù 06/01/2022 61 / 92
?)Y
O?
??
j E Aj= 0; ? Det( E A) = 0: (2.3)
?§(2.3)??~Xêàg?5???§|(2.1)A?§, ??A
?,
?tv??? éAA?t.
du?§(2.3)¤A??1?ó,?ò?)Y
|)??±e
ü??1?1??. (1)A?t?ü?(=Y
AkA
? 1; 2; ; n????ó)?,dt????±e?n.
?n
eY
Akn?p??óA? 1; 2; ; n,@où
A?éAXn
??5?''A?tv1;v2; ;vn,K~Xêàg?5???§
|(2.1)?)Y
?L??
(t) = [e 1tv1;e 2tv2; ;e ntvn]; 1< t < +1:
N￠?4?[ (wg??) 15ù 06/01/2022 62 / 92
y2: dc???·??2, e itvi(i = 1;2; ;n)?~Xêàg?
5???§|(2.1)),?
Det( (0)) = Det([v1;v2; ;vn]) , 0;
d?n?, (t)′~Xêàg?5???§|(2.1)???)Y
.
~
á|?§|
x0(t) = Ax(t);ù￥A =
" 1 2
2 1
#
???)Y
(t),?|)expAt.
N￠?4?[ (wg??) 15ù 06/01/2022 63 / 92
):A?§?
j E Aj=
1 22 1
= ( 1)2 + 4 = 0;
)A? 1 = 1 + 2i; 2 = 1 2i.
1 = 1 + 2i?,d?§|
( 1E A)v1 = 0
)A?t
v1 = c1
" 1
i
#
; c1???~ê:
1 = 1 2i?,d?§|
( 2E A)v2 = 0
)A?t
v2 = c2
" i
1
#
; c2???~ê:
N￠?4?[ (wg??) 15ù 06/01/2022 64 / 92
d?n?,Y
(t) =
" e(1+2i)t ie(1 2i)t
ie(1+2i)t e(1 2i)t
#
??§|??? )|.
q??
(0) =
" 1 i
i 1
#
;
1(0) = 12
" 1 i
i 1
#
;
¤±
expAt = (t) 1(0) = et
" cos2t sin2t
sin2t cos2t
#
:
5:3?n(5.10)￥, (t)???ò′expAt. d?n(5.7)?,ü??m?
????é~êY
C.
N￠?4?[ (wg??) 15ù 06/01/2022 65 / 92
(2)A? 1; 2; ; k-ê?O?n1;n2; ;nk
?n1 +n2 + +nk = n(ek = n,Káuü??1(1))?,éAuz??nj-
A? j,?5?ê?§|
(A jE)njv = 0 (2.4)
) N¤n??Ap?mU??nj?f?mUj(j = 1;2; ;k),
ù?n??Ap?m?L??U1;U2; ;Uk?ú.@o,éu??
?tu2U,3???t|uj 2Uj(j = 1;2; ;k),|
u = u1 + u2 + + uk:
k = 1?,Y
A?k??A?,?7én??Ap?m?1?);
k = n?, uj = cjvj(j = 1;2; ;n),
vjB′A? j¤éAA?
t, cj?,
~ê.
N￠?4?[ (wg??) 15ù 06/01/2022 66 / 92
????5,·???16k6n?1.??D?^?’(0) = ,K
= v1 + v2 + + vk; (2.5)
ù￥vj 2Uj(j = 1;2; ;k),duf?mUj′d?§|(2.4)),@
ovj??′?§|(2.4)),=k
(A jE)lvj = 0; l>nj; j = 1;2; ;k: (2.6)
u′?
e jtexp( jEt) = e jt
26
6666
6666
6666
664
e jt
e jt
:::
e jt
37
7777
7777
7777
775
= E; (2.7)
(üa(2.6)úa(2.7)
(expAt)vj = (expAt)e jt[exp( jEt)]vj
= e jt[exp(A jE)t]vj
= e jt[E + t(A jE) + t
2
2!(A jE)
2 + + tnj 1
(nj 1)!(A jE)
nj 1]vj:
N￠?4?[ (wg??) 15ù 06/01/2022 67 / 92
da(2.5),éta|ú?
’(t) = (expAt) = (expAt)
kX
j=1
vj =
kX
j=1
(expAt)vj
=
kX
j=1
e jt[E + t(A jE) + t
2
2!(A jE)
2 + + tnj 1
(nj 1)!(A jE)
nj 1]vj:
?d,~Xêàg?5???§|(2.1)÷v’(0) = )?
’(t) =
kX
j=1
e jt
0B
BBBB
BB@
nj 1X
i=0
ti
i!(A jE)
i
1C
CCCC
CCAvj: (2.8)
AO/,A?k??A??,?7
)D??t ,Ké???tu?
k
(A E)nu = 0;
=(A E)n?"Y
. u′
expAt = e texp(A E)t = e t
n 1X
i=0
(A E)i t
i
i!: (2.9)
N￠?4?[ (wg??) 15ù 06/01/2022 68 / 92
￠St,·???±lúa(2.8)￥?|?expAt,??
expAt = (expAt)E = [(expAt)e1;(expAt)e2; ;(expAt)en];
ù￥
e1 =
26
6666
6666
6666
6666
6666
4
1
0
:::
0
0
37
7777
7777
7777
7777
7777
5
;e2 =
26
6666
6666
6666
6666
6666
4
0
1
:::
0
0
37
7777
7777
7777
7777
7777
5
; ;en =
26
6666
6666
6666
6666
6666
4
0
0
:::
0
1
37
7777
7777
7777
7777
7777
5
′ü??t. d’(t) = (expAt) ?,O??)Y
expAt￠St?I?ò
D??t ?O?
= e1; = e2; ; = en;
|n?)??,B?E?Y
expAt.
N￠?4?[ (wg??) 15ù 06/01/2022 69 / 92
~
á|?§|
x0(t) = Ax(t); ù￥A =
" 2 1
1 2
#
???)Y
(t),?|)expAt.
):A?§?
j E Aj=
2 1 1 2
= ( 1)( 3) = 0;
)A? 1 = 1; 2 = 3.
1 = 1?,d?§|
( 1E A)v1 = 0
?A?t
v1 = c1
" 1
1
#
; c1???"??~ê:
N￠?4?[ (wg??) 15ù 06/01/2022 70 / 92
2 = 3?,d?§|
( 2E A)v2 = 0
?A?t
v2 = c2
" 1
1
#
; c2???~ê:
d?n?,?§??? )|?
(t) =
" et e3t
et e3t
#
:
q??
(0) =
" 1 1
1 1
#
;
1(0) = 12
" 1 1
1 1
#
;
¤±
expAt = (t) 1(0) = 12
" et + e3t e3t et
e3t et et + e3t
#
:
N￠?4?[ (wg??) 15ù 06/01/2022 71 / 92
~
|)D?ˉKx0(t) = Ax(t);’(0) = ,ù￥A =
" 2 1
1 4
#
,?|expAt.
): dA?§
j E Aj=
2 11 4
= 2 6 + 9 = 0
) = 3?A-A?,A?t
v = c1
" 1
1
#
; c1???~ê?c1 , 0:
??Y
A?k??A?,=?k??f?mU1,P =
"
1
2
#
;Kk
’(t) = e3t[E + t(A 3E)]
= e3t
"
1 + t( 1 + 2)
2 + t( 1 + 2)
#
:
N￠?4?[ (wg??) 15ù 06/01/2022 72 / 92
dúa(2.8)?
expAt = e3t[E + t(A 3E)]
= e3t
" 1 0
0 1
#
+ t
" 1 1
1 1
#!
= e3t
" 1 t t
t 1 + t
#
:
~
e
A =
26
6666
6666
6666
6666
664
3 1
3 1
3
3
3
37
7777
7777
7777
7777
775
;
á|?expAt.
N￠?4?[ (wg??) 15ù 06/01/2022 73 / 92
): dA?§j E Aj= 0?) = 3?Y
Aê-A?,?
(A + 3E)3 = 0:
?d,dúa(2.8)?
expAt = e 3t
E + t(A + 3E) + t
2
2!(A + 3E)
2
!
= e 3t
26
6666
6666
6666
6666
6666
664
1 t t
2
2! 0 00 1 t 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
37
7777
7777
7777
7777
7777
775
:
N￠?4?[ (wg??) 15ù 06/01/2022 74 / 92
~
á|?§|
x0(t) = Ax(t); ù￥A =
26
6666
6664
0 1 1
1 1 1
0 1 1
37
7777
7775
÷vD?^?’(0) = )’(t),9?)Y
expAt.
): dA?§
j E Aj=
1 1
1 + 1 1
0 1 + 1
= ( + 1)2 = 0
)ü? 1 = 0ú-? 2 = 1.
1 = 0?,A?t?
u =
26
6666
6664
2
1
1
37
7777
7775; ???"??~ê:
N￠?4?[ (wg??) 15ù 06/01/2022 75 / 92
2 = 1?,A?t?
v =
26
6666
6664
37
7777
7775; ; ?? ?"??~ê:
?d,f?mU1d?tuü¤,f?mU2d?tvü¤.éD??t ?1
),-z1 2U1;z2 2U2; = [ 1; 2; 3]T,@ok
z1 + z2 =
26
6666
6664
2
1
1
37
7777
7775 +
26
6666
6664
37
7777
7775 =
26
6666
6664
2 +
+
+
37
7777
7775 =
26
6666
6664
1
2
3
37
7777
7775:
)
= 1 3; = 2 3 1; = 2 1 + 3:
u′k
z1 =
26
6666
6664
2( 1 3)
1 3
1 3
37
7777
7775; z2 =
26
6666
6664
2 3 1
2 1 + 3
2 3 1
37
7777
7775:
N￠?4?[ (wg??) 15ù 06/01/2022 76 / 92
dúa(2.8)?,÷vD?^?’(0) = )?
’(t) = e 1tEz1 + e 2t(E + t(A + E))z2
= z1 + e t(E + t(A + E))z2
=
26
6666
6664
2( 1 3)
1 3
1 3
37
7777
7775 + e t
26
6666
6664
2 3 1 + t( 1 + 2 + 3)
2 1 + 3
2 3 1 + t( 2 1 + 3)
37
7777
7775:
?O = [1;0;0]T;[0;1;0]T;[0;0;1]T,?3??5?'')
’1(t) =
26
6666
6664
2 e t(1 + t)
1 e t
1 e t(1 + t)
37
7777
7775;’2(t) =
26
6666
6664
te t
e t
te t
37
7777
7775;’3(t) =
26
6666
6664
2 + (2 + t)e t
1 + e t
1 + (2 + t)e t
37
7777
7775;
?d
expAt =
26
6666
6664
2 (1 + t)e t te t 2 + (2 + t)e t
1 e t e t 1 + e t
1 (1 + t)e t te t 1 + (2 + t)e t
37
7777
7775:
N￠?4?[ (wg??) 15ù 06/01/2022 77 / 92
~
|?§| 8
>>><
>>>:
x01 = 2x1 + 2x2 2x3;
x02 = 2x1 + 5x2 4x3;
x03 = 2x1 4x2 + 5x3;
÷vD?^?’(0) = )’(t),?|expAt.
): XêY
A =
26
6666
6664
2 2 2
2 5 4
2 4 5
37
7777
7775;
dA?§)A? 1 = 10?ü?, 2 = 1?-?.
1 = 10?,éAA?t
u = c1
26
6666
6664
1
2
2
37
7777
7775; c1???~ê:
N￠?4?[ (wg??) 15ù 06/01/2022 78 / 92
2 = 1?,éAA?t
v =
26
6666
6664
2c2 + 2c3
c2
c3
37
7777
7775; c2;c3???~ê:
?d,f?mU1′d?tu¤ü¤,f?mU2′d?tv¤ü¤.
-z1 2U1;z2 2U2; = [ 1; 2; 3]T,ó~(5.12)?)
z1 = 19
26
6666
6664
1 + 2 2 2 3
2( 1 + 2 2 2 3)
2( 1 + 2 2 2 3)
37
7777
7775; z2 = 19
26
6666
6664
8 1 2 2 + 2 3
2 1 + 5 2 + 4 3
2 1 + 4 2 + 5 3
37
7777
7775:
dd?÷vD?^?’(0) = )?
’(t) = e10tEz1 + et(E + t(A E))z2
= 19e10t
26
6666
6664
1 + 2 2 2 3
2( 1 + 2 2 2 3)
2( 1 + 2 2 2 3)
37
7777
7775 + 19et
26
6666
6664
8 1 2 2 + 2 3
2 1 + 5 2 + 4 3
2 1 + 4 2 + 5 3
37
7777
7775:
N￠?4?[ (wg??) 15ù 06/01/2022 79 / 92
?O = [1;0;0]T;[0;1;0]T;[0;0;1]T,?\ta3??5?'')
’1(t) = 19
26
6666
6664
e10t + 8et
2e10t 2et
2e10t + 2et
37
7777
7775;’2(t) = 19
26
6666
6664
2e10t 2et
4e10t + 5et
4e10t + 4et
37
7777
7775;’3(t) = 19
26
6666
6664
2e10t + 2et
4e10t + 4et
4e10t + 5et
37
7777
7775:
u′
expAt = 19
26
6666
6664
e10t + 8et 2e10t 2et 2e10t + 2et
2e10t 2et 4e10t + 5et 4e10t + 4et
2e10t + 2et 4e10t + 4et 4e10t + 5et
37
7777
7775:
5:éu?)Y
){,?k, ü?~??{.??′òùz?e
(Jordan)IO/., ??′|^M??-p4(Hamilton-Cayley)?n.
N￠?4?[ (wg??) 15ù 06/01/2022 80 / 92
éu?àg?5???§|
x0 = Ax + f(t) (2.10)
d5.1!￥a(1.25)?±??)’(t)O?úa.?? (t) = expAt,¤±
1(s) = exp( As);
1(t0) = exp( At0):
’(t0) = ?,òt?üa?\a(1.25),?
’(t) = exp[A(t t0)] +
Z t
t0
exp[A(t s)]f(s)ds: (2.11)
a(2.11)??
|)~Xê?àg?5???§|?{.
N￠?4?[ (wg??) 15ù 06/01/2022 81 / 92
~
|?§|
x0 = Ax + f(t); A =
" 1 2
2 1
#
; f(t) =
" et
0
#
;
÷vD?^?
’(0) =
" 0
1
#
)’(t).
): d~5.8?
expAt = et
" cos2t sin2t
sin2t cos2t
#
;
N￠?4?[ (wg??) 15ù 06/01/2022 82 / 92
): d~5.8?
expAt = et
" cos2t sin2t
sin2t cos2t
#
;
?\a(2.11)
’(t) = et
" cos2t sin2t
sin2t cos2t
#" 0
1
#
+
Z t
0
et s
" cos2(t s) sin2(t s)
sin2(t s) cos2(t s)
#" es
0
#
ds
= et
" sin2t
cos2t
#
+ et
Z t
0
" cos2(t s)
sin2(t s)
#
ds
= et
" sin2t
cos2t
#
12et
" sin2t
1 cos2t
#
=
26
6666
6666
664
3
2e
t sin2t
1
2e
t(3cos2t 1)
37
7777
7777
775:
N￠?4?[ (wg??) 15ù 06/01/2022 83 / 92
?,·?0 X?|^.ê.dC?{|?)Y
.
???t?ê.ê.dC??
￡[f(t)] =
Z +1
0
e stf(t)dt; (2.12)
ù￥, f(t)′n??t?ê,§z???t?3.ê.dC?.A^.
ê.dC??±ò|)?5???§|ˉKz?|)?5?ê?§|
ˉK.
~
|^.ê.dC?(2.12)|)~14.
): ò?§|??t/a
( x0
1 = x1 + 2x2 + e
t;
x02 = 2x1 + x2: (2.13)
D?^??x1(0) = 0;x2(0) = 1.
N￠?4?[ (wg??) 15ù 06/01/2022 84 / 92
-
X1(s) = ￡[x1(t)]; X2(s) = ￡[x2(t)];
é?§|(2.13)?1.ê.dC?8
>>><
>>>: (s 1)X1(s) 2X2(s) =
1
s 1;2X
1(s) + (s 1)X2(s) = 1:
u′)
8>>
>>>>
>>><
>>>>
>>>>
>:
X1(s) = 3(s 1)2 + 22 = 32 2(s 1)2 + 22;
X2(s) =
s 1 2s 1
(s 1)2 + 22 =
3
2(s 1)
(s 1)2 + 22
1
2
s 1:
dL4-1 (.ê.dC?L)B?
x1(t) = 32et sin2t; x2(t) = 32et cos2t 12et:
ù?~14(J??.
N￠?4?[ (wg??) 15ù 06/01/2022 85 / 92
~
|^.ê.dC?{|?§|
( x0
1 = x1 + 2x2;x0
2 = x1 + x2;
÷vD?^?’1(0) = 0;’2(0) = 1)’1(t);’2(t),?|?§?)Y
.
):-
X1(s) = ￡[’1(t)]; X2(s) = ￡[’2(t)]:
b
x1 = ’1(t); x2 = ’2(t)
÷v?§|,é?§|?1.ê.dC??
( (s 1)X
1(s) 2X2(s) = ’1(0) = 0;
X1(s) + (s 1)X2(s) = ’2(0) = 1;
N￠?4?[ (wg??) 15ù 06/01/2022 86 / 92
= 8
>>>>
>><
>>>>
>>:
X1(s) = 2(s 1)2 + 2;
X2(s) = s 1(s 1)2 + 2:
dL4-1 (.ê.dC?L)?
( ’
1(t) =
p2et sinp2t;
’2(t) = et cosp2t:
?
E?)Y
,·?I?2|?÷vD?^?
1(0) = 1; 2(0) = 0
)( 1(t); 2(t)). aqt?C?L§?
( (s 1)X
1(s) 2X2(s) = 1(0) = 1;
X1(s) + (s 1)X2(s) = 2(0) = 0;
N￠?4?[ (wg??) 15ù 06/01/2022 87 / 92
) 8
>>>>
>><
>>>>
>>:
X1(s) = s 1(s 1)2 + 2;
X2(s) = 1(s 1)2 + 2:
L4-1 8
>>><
>>>:
1(t) = et cosp2t;
2(t) = 1p2et sinp2t:
?d,?§|?)Y
?
(t) =
" ’
1(t) 1(t)
’2(t) 2(t)
#
=
26
6666
6664
p2et sinp2t et cosp2t
et cosp2t 1p2et sinp2t
37
7777
7775:
N￠?4?[ (wg??) 15ù 06/01/2022 88 / 92
81
1 ?5???§|??n?
? Vg
3??5?n
àg?5???§|
?àg?5???§|
2 ~Xê?5???§|
Y
?ê
?)Y
O?
3 A^ˉKT~
N￠?4?[ (wg??) 15ù 06/01/2022 89 / 92
(0k?u?) 0k?′#?? ;?§?￥
?§l
?U??NS¤k0§é?úk?￥1kL?0. u?
0k?~^?{′?:0Ftu?(GTT)§B?\?Jt?:
0,,3?ó?mu??:0?Y(G)ú-?à?Y(H)§à§?Cz
?1???. 20-V60c?,{IAckermanJ?
{′u? .. b
8>>
>>><
>>>>
>:
dG
dt = F1(G;H) + J(t);
dH
dt = F2(G;H);
ù￥, J(t)?ú?é0?YO\ ü??.m???2?G
( F
1(G0;H0) = 0;
F2(G0;H0) = 0:
?C?
g = G G0;h = H H0
N￠?4?[ (wg??) 15ù 06/01/2022 90 / 92
?ù? ?§
8>>
>>><
>>>>
>:
dg
dt = F1(G0 + g;H0 + h) + J(t);
dh
dt = F2(G0 + g;H0 + h):
òmà?êDm{z?? ?"1;"2,?P
m1 = @F1(G0;H0)@G ; m2 = @F1(G0;H0)@H ;
m4 = @F2(G0;H0)@G ; m3 = @F2(G0;H0)@H ;
K?§|?±z?
8>>
>>><
>>>>
>:
dg
dt = m1g m2h + J(t);
dh
dt = m3h + m4g;
(3.1)
ù￥, mit?~ê.ùB′0k?u? ..
N￠?4?[ (wg??) 15ù 06/01/2022 91 / 92
(?ì? .) d>′ú?C?|C¤[^N?ì .′?
?~???§|.???C>
>?E(t)°?X???U?ì,l
|
(ì? u)?.U?ì ?t′[^|Sgd$?.Cz
>6?L??,?3[^N^?ú>6)^??p?^e
?1$?.PU?ì=zXê?T, f?U?ì?(ì?p?^?,
R?U?ì>{, L?U?ìaAXê.?t?m(ì?′??
?k{Z?Xú?.{ZXê?C,?5Xê?k,u′k''
Xa
e = Tdxdt; f = Ti:
ù￥, e??üà>?ü, x???￡, x0???Y.dú?1
??9￡′>???,????§|
8>>
>>>>
<>
>>>>
>:
md
2x
dt2 + c
dx
dt + kx = Ti;
Tdxdt + Ldidt + Ri = E(t):
(3.2)
N￠?4?[ (wg??) 15ù 06/01/2022 92 / 92
|
|
