Nghiên cứu về tính điều khiển được của hệ mô tả

✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

✖✖✖✖✖

▲➊ ❚❍➚ ❚❍❯ ❍➪❆
❇⑩❖ ❈⑩❖ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P

◆●❍■➊◆ ❈Ù❯ ❱➋ ❚➑◆❍ ✣■➋❯ ìẹ
ế
ữợ ❍↔✐ ❚r✉♥❣

✣➔ ◆➤♥❣✱ ✵✶✴✷✵✷✵


▼ư❝ ❧ư❝
▼ð ✤➛✉
✶ ❑✐➳♥ t❤ù❝ ❝ì sð
✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺
✶✳✻

✶✳✼
✶✳✽
✶✳✾
✶✳✶✵
✶✳✶✶

▼❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ✈✉æ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ✤ì♥ ✈à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳✶ ❙ü ❜➡♥❣ ♥❤❛✉ ❝õ❛ ❤❛✐ ♠❛ tr➟♥
✶✳✻✳✷ P❤➨♣ ❝ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳✸ ❚➼❝❤ ♠ët sè ✈ỵ✐ ♠ët ♠❛ tr➟♥ ✳
✶✳✻✳✹ ◆❤➙♥ ❤❛✐ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳✺ ❈❤✉②➸♥ ✈à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻✳✻ ▲ô② t❤ø❛ ❜➟❝ n ❝õ❛ ♠❛ tr➟♥ ✳ ✳
✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍↕t ♥❤➙♥ ✈➔ ↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
●✐→ trà r✐➯♥❣✱ ✈❡❝t♦ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✶✵
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✶✶
✶✶

✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔

✶✷

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✺

✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷

ỵ ỹ ồ t

ổ t ❧➔ ♠ët ù♥❣ ❞ö♥❣ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳
❚➻♠ ❤✐➸✉ ✈➲ ❤➺ ♠ỉ t↔✱ ❝❤♦ t❛ ❝→✐ ♥❤➻♥ tê♥❣ q✉❛♥ ✈➲ ❜↔♥ ❝❤➜t ✈➟♥ ❤➔♥❤ ❝õ❛ ❝→❝
tr↕♥❣ t❤→✐ ♥❣❤✐➺♠ ❝ơ♥❣ ♥❤÷ ♠è✐ ❧✐➯♥ ❤➺ ♠➟t t❤✐➳t ❣✐ú❛ ❝→❝ tr↕♥❣ t❤→✐ tr♦♥❣ t❤í✐
❣✐❛♥ ❜➜t ❜✐➳♥ ❤ú✉ ❤↕♥✳ ❍➺ ♠æ t↔ ✤â♥❣ ✈❛✐ trá ✤✐➲✉ ❦❤✐➸♥ ♠å✐ sü ❜✐➳♥ ✤ê✐ ❝õ❛
♥❣❤✐➺♠ tr♦♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥ ❜➜t ❜✐➳♥ ❤ú✉ ❤↕♥✳ ❈❤➼♥❤ ✈➻ ✈➟②✱ tæ✐ ✈➔ ❚✐➳♥ s➽ ▲➯ ❍↔✐
❚r✉♥❣ ✤➣ q✉②➳t ✤à♥❤ ❝❤å♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✧❚➼♥❤ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧
♥❤➡♠ ♠ư❝ ✤➼❝❤ ♥➢♠ rã ❝→❝ q✉② ❧✉➟t ✤✐➲✉ ❦❤✐➸♥ ✈➔ t➻♠ ❤✐➸✉ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ tø♥❣
tr↕♥❣ t❤→✐ ✤✐➲✉ ❦❤✐➸♥ ❦❤→❝ ♥❤❛✉✳

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉

◆❤➡♠ ❤✐➸✉ rã t❤➜✉ ✤→♦ ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✳ ◆❣♦➔✐ r❛ ❝❤ó♥❣
tỉ✐ ♠♦♥❣ ố ữ r ữ t q ợ ử ử ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✳

✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✳

❚➼♥❤ ❈✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ❘✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ■✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔✱
♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❜❛ tr↕♥❣ t❤→✐ ✤✐➲✉ ❦❤✐➸♥ tr➯♥ ❝õ❛ ❤➺ ♠æ t↔ tr♦♥❣ Rn.

✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

◆❣❤✐➯♥ ❝ù✉ ❤➺ ♠æ t↔ ❜✐➳♥ t❤ü❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Rn✱ ❜❛ tr↕♥❣ t❤→✐ ✤✐➲✉ ❦❤✐➸♥
C − R − I ❝õ❛ ❤➺ ♠ỉ t↔ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❜➜t ❜✐➳♥ ❤ú✉ ❤↕♥✳

✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉

❈❤ó♥❣ tỉ✐ sû ❞ư♥❣ ữỡ ự ỵ tt tr q tr tỹ
t rữợ t ú tổ t t ồ ừ ỳ t

trữợ q ✤➳♥ t➼♥❤ ✤✐➲✉ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✳ ❚ø ✤â ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣
t÷ì♥❣ tü ❤â❛✱ ❝❤ó♥❣ tỉ✐ ❦❤→✐ q✉→t ♥❤ú♥❣ ❦➳t q✉↔ ✤â✱ ❝❤ó♥❣ tỉ✐ s➩ ✤÷❛ r❛ ♥❤ú♥❣
❦➳t ❧✉➟♥ ♠ỵ✐ ❝❤♦ ✤➲ t➔✐✳

✸


❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝ì sð

✶✳✶ ▼❛ tr➟♥

▼❛ tr➟♥ tr➯♥ tr÷í♥❣ K ❧➔ ♠ët ♠↔♥❣ ❝❤ú ♥❤➟t ❧✐➺t ❦➯ ❝→❝ sè ♥❤÷✿ sè t❤ü❝✱ sè ♣❤ù❝
❤❛② ♠ët ❤➔♠ sè✱✳✳ ✤÷đ❝ ①➳♣ t ởt trt t t ỗ m n ❝ët✳ ❑➼
❤✐➺✉✿


a11

a12

... a1n


 a21
A=
 ...


a22

... a2n 






am1 am2 ... amn

m.n

❚r♦♥❣ ✤â✿
aij ✿ ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ ♠❛ tr➟♥ ð ❞á♥❣ t❤ù i ✈➔ ❝ët j
i✿❝❤➾ sè ❞á♥❣✳
j ✿❝❤➾ số ởt
m.n ồ tữợ tr tữớ ❝→❝ ❝❤ú ❝→✐ A, B, C... ✤➸ ❦➼ ❤✐➺✉ ❝→❝ ♠❛
tr➟♥✳
❉↕♥❣ t❤✉ ❣å♥ A = [aij ]mn

❱➼ ❞ö ✶✳✶✳ ❈❤♦ ♠❛ tr➟♥
A=

9 −13

1

20 5

❧➔ ♠❛ tr➟♥ ❝ï 2.3 ✭✷ ❤➔♥❣ ✈➔ ✸ ❝ët✮✳

✹

−6


✶✳✷ ▼❛ tr➟♥ ✈✉æ♥❣

▼❛ tr➟♥ ✈✉æ♥❣ ❧➔ ♠❛ tr➟♥ ❝â sè ❤➔♥❣ ✈➔ sè ❝ët ❜➡♥❣ ♥❤❛✉✳ ▼❛ tr➟♥ ❝➜♣ n.n ✤÷đ❝
❣å✐ ❧➔ ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ n✳ ❈→❝ ♣❤➛♥ tû aij , (i = j) t↕♦ t❤➔♥❤ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤
❝õ❛ ♠❛ tr➟♥ ✈✉ỉ♥❣✳
❱➼ ❞ư ✶✳✷✳ ▼❛ tr➟♥


1 1 −1



B = 2 3 −2
0 2

1

❧➔ ♠❛ tr➟♥ ✈✉è♥❣ ❝➜♣ 3 ✭✸ ❤➔♥❣ ✈➔ ✸ ❝ët✮✳ ❈→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❧➛♥
❧÷đt ❧➔ 1; 3; 1.

✶✳✸ ▼❛ tr➟♥ ✤ì♥ ✈à

▼❛ tr➟♥ ✤ì♥ ✈à ❝➜♣ n tr♦♥❣ ✈➔♥❤ ❱ ❧➔ ♠➔ tr➟♥ ✈✉æ♥❣ ❝➜♣ n tr♦♥❣ ✤â t➜t ❝↔ ❝→❝

♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❜➡♥❣ ✶✱ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❦❤→❝ ❜➡♥❣ ❦❤æ♥❣✳




1 0 . . 0



0 1 . . 0

En = 
. . . . .


0 0 . . 1

✶✳✹ ▼❛ tr➟♥ ❝♦♥

▼❛ tr➟♥ ❝♦♥ ❝õ❛ A ❧➔ ♠❛ tr➟♥ B ✤÷đ❝ t❤➔♥❤ ❧➟♣ tø ♠❛ tr➟♥ ❜❛♥ ✤➛✉ ❜➡♥❣ ❝→❝❤ ❜ä
✤✐ ♠ët sè ❞á♥❣✱ ✈➔ ♠ët sè ❝ët✳


1 1

❱➼ ❞ö ✶✳✸✳ ❈❤♦ ♠❛ tr➟♥ A = 2
❝♦♥ ❝õ❛ ♠❛ tr➟♥ A.

2



0 −1 .





❚❛ ❝â ♠❛ tr➟♥ B = 2

3 1 −1

✶✳✺ ▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤
❈❤♦

1 1



0



❧➔ ♠❛ tr➟♥

3 1

❧➔ ♠❛ tr➟♥ ✈✉è♥❣ tr➯♥ tr÷í♥❣ K✱ ♥➳✉ tỗ t ởt tr B s
A ì B = En ✳ ❑❤✐ ✤â A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤✱ B ❧➔ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦
A

✺


❝õ❛ ♠❛ tr➟♥ A. ✈➼ ❞ư ✈➔ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦

✶✳✻ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ ♠❛ tr➟♥
✶✳✻✳✶ ❙ü ❜➡♥❣ ♥❤❛✉ ❝õ❛ ❤❛✐ ♠❛ tr➟♥

❍❛✐ ♠❛ tr➟♥ ❜➡♥❣ ♥❤❛✉ ❦❤✐ ❝â ❝→❝ ♣❤➛♥ tû t÷ì♥❣ ù♥❣ ❜➡♥❣ ♥❤❛✉ tứ ổ ởt
õ ũ tữợ
A = (aij )m×n ✈➔ B = (bhk )p×q


 m=p

❑❤✐ ✤â A = B ⇔ 


n=q

aij = bhk

✶✳✻✳✷ P❤➨♣ ❝ë♥❣

✣à♥❤ ỵ tr A B õ ũ tữợ n ì m ờ A + B

ởt tr ũ tữợ tr õ tỷ tr♦♥❣ ✈à tr➼ t÷ì♥❣ ù♥❣ ❜➡♥❣ tê♥❣
❤❛✐ ♣❤➛♥ tû t÷ì♥❣ ù♥❣ ❝õ❛ ♠é✐ ♠❛ tr➟♥✿
(A + B)ij = Aij + Bij , 1 ≤ i ≤ m,

✈➔1 ≤ i ≤ n

❱➼ ❞ö ✶✳✹✳
1 3 1
1 0 0

+

0 0 5
7 5 0

=

1+0 3+0 1+5
1+7 0+5 0+0

=

1 3 6
8 5 0

❚➼♥❤ ❝❤➜t
✶✳ A + B = B + A
✷✳ (A + B) + C = A + (B + C)
✸✳ 0 + A = A + 0 = A
✹✳ A + (−A) = (−A) + A = 0

✶✳✻✳✸ ❚➼❝❤ ♠ët sè ✈ỵ✐ ởt tr

ỵ cA ừ ởt số c ✈ỵ✐ ♠❛ tr➟♥ A ❧➔ ♠ët ♠❛ tr➟♥ ❝â ữủ
ộ tỷ ừ A ợ c

(cA)ij = c.Aij

✻


❱➼ ❞ö ✶✳✺✳
2.

1

−3

8

4 −2

5

=

2.1

2.8

2(−3)

2.4 2(−2)

2.5

=

2 16 −6
8 −4 10

❚➼♥❤ ❝❤➜t
✶✳ a.(A + B) = a.A + a.B
✷✳ (a + b).A = a.A + b.A
✸✳ a.(b.A) = (ab).A
✹✳ 1.A = A

✶✳✻✳✹ tr

ỵ tr A = (aij )m×n ❝â ❝➜♣ m × n ✈➔ ♠❛ tr➟♥ B = (bij )n×p ❝â

❝➜♣ n × p✳ ❚➼❝❤ ❝õ❛ ❤❛✐ ♠❛ tr➟♥ A ✈➔ B ❧➔ ♠ët ♠❛ tr C = (cij )mìp m ì p ợ
n

ckh =

aki bih ,
j=1

✈ỵ✐ k = 1, m, h = 1, p.

ữủ tr A ợ tr B✿ ❙è ♣❤➛♥ tû tr➯♥ ❞á♥❣ ❝õ❛
♠❛ tr➟♥ A ♣❤↔✐ ❜➥♥❣ sè ♣❤➛♥ tû tr➯♥ ❝ët ❝õ❛ ♠❛ tr➟♥ B t÷ì♥❣ ù♥❣✳



❱➼ ❞ư ✶✳✻✳ ❈❤♦ A =

1 0

1

0 1 −1

✈➔ B = 0

1.1 + 0.0 + 1.1

0



1



1 −1

❑❤✐ ✤â
C = A.B =

1

1.0 + 0.1 + 1(−1)

0.1 + 1.0 + (−1).1 0.0 + 1.1 + (−1)(−1)

❚➼♥❤ ❝❤➜t
✶✳ (A + B).C = A.C + B.C
✷✳ A(B + C) = A.B + A.C
✸✳ (A.B).C = A.(B.C)
✹✳ E.A = A.E = A ✭✈ỵ✐ E ❧➔ ♠❛ tr➟♥ ✤ì♥ ✈à✮
✺✳ (AB)T = B T AT ✳
✻✳ P❤➨♣ ♥❤➙♥ ♠❛ tr➟♥ ❦❤æ♥❣ ❝â t➼♥❤ ❣✐❛♦ ❤♦→♥✳
✼

=

2

−1

−1

2


❱➼ ❞ö ✶✳✼✳ ❈❤♦ A =

1 2
3 4

✈➔ B =

AB =

✈➔
BA =

0 −1
6

7

1 2

0 −1

3 4

6

7

0 −1

1 2

6

3 4

7

=

=

12 13
24 25
1 4
27

40

AB = BA.



ỵ ởt tr õ tữợ m ì n ✈ỵ✐ ❝→❝ ❣✐→ trà aij t↕✐

❤➔♥❣ i✱ ❝ët j t❤➻ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à B = AT ❧➔ ♠❛ tr õ tữợ n ì m ợ
tr bij = aji✳

❱➼ ❞ö ✶✳✽✳
T

1

2

3



1

0



= 2 −6



0 −6 7

3



7

❚➼♥❤ ❝❤➜t
✶✳ (A + B)T = AT + B T
✷✳ (a.A)T = a.AT
✸✳ (AB)T = B T AT
✹✳ (AT )T = A

✶✳✻✳✻ ▲ô② t❤ø❛ ❜➟❝ n ❝õ❛ ♠❛ tr➟♥

❈❤♦ A ❧❛ ♠ët tr ổ tr trữớ K A ì A ữủ ỵ A2
ữỡ ừ tr A
ợ n số ữỡ tũ ỵ t t A ì A ì ... ì A ỵ An ữủ ồ

ụ tứ n ừ A
ữợ A0 = E

✽






0 1 0

❱➼ ❞ö ✶✳✾✳ ❈❤♦ A = 0

0 1



0 0 0





0 0 1



⇒ A2 = 0 0 0
0 0 0





0 0 0

✈➔ A3 = 0

0 0



0 0 0

✶✳✼ ✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥

✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ❧➔ tê♥❣ ✤↕✐ sè ❝õ❛ n! sè ❤↕♥❣✱ ♠é✐ sè ❤↕♥❣ ❧➔
t➼❝❤ ❝õ❛ n ♣❤➛♥ tû ❧➜② tr➯♥ ❝→❝ ❤➔♥❣ ✈➔ ❝→❝ ❝ët ❦❤→❝ ♥❤❛✉ ❝õ❛ ♠❛ tr➟♥ A✱ ộ t
ữủ ợ tỷ +1 −1 t❤❡♦ ♣❤➨♣ t❤➳ t↕♦ ❜ð✐ ❝→❝ ❝❤➾ sè ❤➔♥❣ ✈➔
❝❤➾ sè ❝ët ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ t➼❝❤✳ ●å✐ Sn ❧➔ ♥❤â♠ ❝→❝ ❤♦→♥ ✈à ❝õ❛ ❝õ❛ n ♣❤➛♥
tû 1, 2, ..., n✱ t❛ ❝â ✭❝æ♥❣ t❤ù❝ ▲❡✐❜♥✐③✮
n
ai,δ(i)
sng(δ)πi=1

det(A) =
δ∈Sn

⑩♣ ❞ư♥❣ ❝❤♦ ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ ✶✱✷✱✸ t❛ ❝â✿

det[a] = a
det

a11 a12
a21 a22



a11 a12 a13

= a11 a22 − a12 a21



det a21 a22 a23  = a11 a22 a33 + a12 a23 a31 + a13 a21 a32





a31 a32 a33
− a13 a22 a31 − a12 a21 a33 − a11 a23 a32





−2 2 −3

❱➼ ❞ö ✶✳✶✵✳ ❚➼♥❤ ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ A = −1

2

1

3



0 −1

det (A) = (−2).1(−1) + 2.3.2 + (−3).(−1).0 − 2.1.(−3) − 0.3.(−2) − (−1).(−1).2
= −2 + 12 + 0 + 6 + 0 − 2
= 18

✾


❚➼♥❤ ❝❤➜t
✶✳ det (AB) = det (A) det (B) = det (B) det (A).
✷✳ ▼❛ tr➟♥ A ❦❤↔ ♥❣❤à❝❤ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤à♥❤ t❤ù❝ ❝õ❛ A ❦❤→❝ 0✱ t❛ ❝â✿
det (A−1 ) = det (A)−1 .

✸✳ det (AT ) = det (A).✳

✶✳✽ ▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤

▼❛ tr➟♥ A ✈✉æ♥❣ ❝➜♣ n ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ♥❣❤à❝❤ ✭❦❤ỉ♥❣ s✉② ❜✐➳♥✮ tr V
tỗ t tr A ũ s❛♦ ❝❤♦ AA = A A = E ✳ ❑❤✐ ✤â ❆✬ ✤÷đ❝ ❣å✐ ❧➔ ♠❛
tr➟♥ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ❆✱ A1

ữợ t tr

✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ A✿
✲ ◆➳✉ det(A) = 0 t❤➻ A ❦❤æ♥❣ ❝â ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ A−1✳
✲ ◆➳✉ det(A) = 0 t❤➻ A ❝â ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ A−1✳
✷✳ ▲➟♣ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à AT ❝õ❛ A✳
✸✳ ▲➟♣ ♠❛ tr➟♥ ♣❤ư ❤đ♣ AT : A∗ = (ATij )nn✳
1
A∗ .
✹✳ ❚➼♥❤ ♠❛ tr➟♥ A−1 = det(A)

❱➼ ❞ö ✶✳✶✶✳ ❈❤♦ A =
✶✳ det(A) =
✷✳ AT =
✸✳ A∗ =

1 −2
3

1

2

1 −2
3

2

✳ ❚➼♥❤ A−1

=8

3

−2 2
−4 2
−6 1

✹✳ A−1 = 81

2

2

−3 1

=

−0, 5

0, 25

−0, 75 0, 125

✶✳✾ ❍↕t ♥❤➙♥ ✈➔ ↔♥❤

◆➳✉ V ✈➔ W ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡✈t♦ tr➯♥ ❝ị♥❣ ♠ët tr÷í♥❣✱ t❛ ♥â✐ r➡♥❣ →♥❤ ①↕
f : V → W ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ t✉②➳♥ t➼♥❤✳ ❑❤✐ ✤â ❤↕t ♥❤➙♥ ừ f ỵ ker f

ừ f ỵ im(f ) ữủ ♥❣❤➽❛ ♥❤÷ s❛✉✿
ker f = {x ∈ V : f (x) = 0}
imf = {f (x) : x ∈ V}

❚➼♥❤ ❝❤➜t
dim(ker(f )) + dim(im(f )) = dim(V )

✶✳✶✵ ❍↕♥❣ ❝õ❛ ♠❛ tr➟♥

❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ A ❧➔ ❝➜♣ ❝❛♦ ♥❤➜t ❝õ❛ ❝→❝ ✤à♥❤ t❤ù❝ ❝♦♥ ❦❤→❝ ✵ ❝â tr♦♥❣ A✳ ỵ
rank(A) r(A)




1 2 3 4

ử ❈❤♦ ♠❛ tr➟♥ A = 2

4 6 8



3 5 7 9

▼❛ tr➟♥ ❝♦♥ ❝➜♣ ❝❛♦ ♥❤➜t ❝õ❛ A ❧➔





2 3 4

A234
123 = 4 6 8





5 7 9

♥➯♥ rank(A) = 3.

✶✳✶✶ ●✐→ trà r✐➯♥❣✱ ✈❡❝t♦ r✐➯♥❣

❈❤♦ A ❧➔ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n tr➯♥ tr÷í♥❣ sè K (K = R, C)✳ ❙è λ ∈ K ✤÷đ❝ ❣å✐ ❧➔
❣✐→ trà r✐➯♥❣ ✭trà r ừ tr A tỗ t ởt t u = 0, u ∈ Kn s❛♦ ❝❤♦
Au = λu

❑❤✐ ✤â ✈❡❝t♦ u ✤÷đ❝ ❣å✐ ❧➔ ✈❡❝t♦ r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ A ù♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ λ

✶✶


❈❤÷ì♥❣ ✷

❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛
❤➺ ♠ỉ t↔

✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉
❚✐➳♥ ❤➔♥❤ ①❡♠ ①➨t ❤➺ ♠æ t↔ ❝â ❞↕♥❣✿

F (t, x, x,
˙ u) = 0,

x(t0 ) = x0 ,

✭✷✳✶✮

y − G(x, t) = 0.

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳
✶✳ ❍➺ ♠ỉ t↔ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❤♦➔♥ t♦➔♥ ✭✤✐➲✉ ❦❤✐➸♥ ❈✮ ♥➳✉ ✈ỵ✐
♠å✐ tr↕♥❣ t❤→✐ ❜❛♥ ✤➛✉ ❝❤♦ trữợ x0 Rn tr t t tú xf Rn
tỗ t ởt ❜✐➳♥ ✤ê✐ ❤➺ tø tr↕♥❣ t❤→✐ x0 ✤➳♥
tr↕♥❣ t❤→✐ xf tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥ tf ≥ t0 ✭∃u, tf < ∞ s❛♦ ❝❤♦
x(tf , u, x0 ) = xf ✮✳
✷✳ ❚r♦♥❣ ❤➺ ♠æ t↔ ✭✷✳✶✮✱ t❛ ❦➼ ❤✐➺✉ Rx t õ t t tợ ữủ tứ x0 ∈ Rn
♥➳✉ ✈ỵ✐ t➜t ❝↔ ♠å✐ xf ∈ Rx ✱ tỗ t ởt ữủ ởt
❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✤➸ ❞à❝❤ ❝❤✉②➸♥ ❤➺ tø tr↕♥❣ t❤→✐ x0 ✤➳♥ tr↕♥❣ t❤→✐
xf tr♦♥❣ t❤í✐ ❣✐❛♥ ❣✐ỵ✐ ❤↕♥ ✭∃u, tf < ∞ s❛♦ ❝❤♦ x(tf , u, x0 ) = xf ∈ Rx ✮✿
0

0

0

Rx0 := {xf ∈ Rn | ∃u, tf < ∞ : x(tf , u, x0 ) = xf } ⊆ Rn .

✶✷


✣➦t R := x ∈χ Rx ✈ỵ✐ χtc ⊆ Rn ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❣✐→ trà ✤➛✉ x0 t↕✐
t❤í✐ ✤✐➸♠ t0✳ ❍➺ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ tr♦♥❣ t➟♣ ❝â t❤➸
✤↕t ✤÷đ❝ ✭❘✲✤✐➲✉ ❦❤✐➸♥✮✱ ♥➳✉ ♠å✐ tr↕♥❣ t❤→✐ tr♦♥❣ R ✤➲✉ ❝â t❤➸ ✤↕t ✤÷đ❝
tø ♥❤✐➲✉ tr t trữợ x0 tr tớ ỳ ❤↕♥ ✭❝❤♦
x0 ∈ R, xf ∈ R, ∃u, tf < ∞ s❛♦ ❝❤♦ x(tf ; u, x0 ) = tf
0

t0
c

0

0

ú ỵ
õ t ữủ ổ ❦❤✐ ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ✤ë♥❣ ❧ü❝ ✤✐➲✉
❦❤✐➸♥ ❤ú✉ ❤↕♥✳
✷✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✱ ❤➺ ❝â t❤➸ ❧➔ ổ ữủ
t ợ ỳ r ở sè ❝è ✤à♥❤ ♥❣❤✐➺♠ tr➯♥ ♠ët ✤❛ t↕♣ ✤➣ ❜✐➳t✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ E = In✱ ❤➺ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ ữủ trũ ợ
t

ử t ❤➺ ♠æ t↔ s❛✉
0 0

x˙ 1

1 0

x˙ 2

=

0 1

x1

1 0

x2

+

0
1

u

❚➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❝❤♦ ❜ð✐ R = {(x1 , x2 ) ∈ R2 |x2 = 0} ✈➔ ❤➺ tr➯♥ ❧➔ ❤➺ ✤✐➲✉ ❦❤✐➸♥
❝â t❤➸ ✤↕t ✤÷đ❝✳

❚r♦♥❣ ❤➺ ♠ỉ t↔ ❤✐➺♥ t÷đ♥❣ ❦❤→❝ ♣❤→t s✐♥❤ ♥➳✉ ❤➔♠ ✤➛✉ ✈➔♦ ❧✐➯♥ tö❝ ❞✉② ♥❤➜t✳ ❱➻
♥❣❤✐➺♠ ❝â t❤➸ ♣❤ö t❤✉ë❝ ✈➔♦ ✤↕♦ ❤➔♠ ❝õ❛ ✤➛✉ ✈➔♦ ♥➯♥ ❝â t❤➸ ①↔② r❛ ✈✐➺❝ ổ
õ ờ tỗ t ♠ỉ t↔ ❝â t❤➸ ♥❤➟♥ ♠ët ♥❣❤✐➺♠ s✉② rë♥❣✳

❱➼ ❞ư ✷✳✷✳ ❳➨t ❤➺ ♠æ t↔ s❛✉

0 1

x˙ 1

0 0

x˙ 2

=

0 1

x1

1 0

x2

+

1
1

ợ u ữủ
u(t) =

0, 0 ≤ t ≤ 1,
1, 1 ≤ t ≤ tf

u ❧➔ ❤➔♠ ❧✐➯♥ tö❝ t❤❡♦ t✳ ◆❣❤✐➺♠ ❝õ❛ ❤➺ ✤➣ ❝❤♦ ✤÷đ❝ ❝❤♦ ❜ð✐✿

x1 (t) = u + u˙
x2 (t) = u.

✶✸

u


trữợ ổ õ ờ tỗ t tr t
ỗ ừ ❝â t❤➸ ✤÷đ❝ ♠ỉ t↔ ♥❤÷ tr♦♥❣ ❤➻♥❤ ✈➔ (x1, x2) ❧➔ ♠ët ♥❣❤✐➺♠ t❤❡♦
♥❣❤➽❛ s✉② rë♥❣✳

❍➻♥❤ ✷✳✶✿ ❚r↕♥❣ t❤→✐ ①✉♥❣ ❝õ❛ ♥❣❤✐➺♠✳

✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ▼ët ❤➺ ♠æ t↔ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ①✉♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✭✤✐➲✉

❦❤✐➸♥ ■✮ ♥➳✉ ợ ồ tr t trữợ x0 Rn tỗ t↕✐ ♠ët ✤➛✉ ✈➔♦ ✤✐➲✉ ❦❤✐➸♥
✤÷đ❝✱ ✤✐➲✉ ✤â ❧➔♠ ❜✐➳♥ ✤ê✐ ❤➺ t❤➔♥❤ ❝→❝ tr↕♥❣ t❤→✐ ①✉♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥✳
◆â ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ t➼♥❤ ✤✐➸✉ ữủ ừ ỹ tữỡ ữỡ
ợ õ ❜ä t➜t ❝↔ ❝→❝ tr↕♥❣ t❤→✐ ①✉♥❣ ❧ü❝ ❜➡♥❣ ❝→❝❤ ❝❤å♥ ✉ t❤➼❝❤ ❤đ♣✳
❱✐➺❝ ♥➔② ❝â t❤➸ ✤÷đ❝ t❤ü❝ ❤✐➺♥ tr t ữủ ợ ộ tr
t trữợ x0 tỗ t ởt tr t ♥❣÷đ❝ s❛♦ ❝❤♦ ❤➺ ❧➦♣ ❦❤➨♣ ❦➼♥
❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ①✉♥❣ ❧ü❝ ♥➔♦✮✳
❈❤♦ ❤➺ ♠ỉ t↔ t❤í✐ ❣✐❛♥ ❜➜t ❜✐➳♥ ❝❤➼♥❤ q✉② ❝â ❞↕♥❣
E x˙ = Ax + Bu,

x(0) = x0

y = Cx

✭✷✳✷✮

✈ỵ✐ E, A ∈ Rn,m, B ∈ Rn,m, C Rp,n tỗ t sỹ ổ t trữ sè ✤ì♥ t❤✉➛♥
❝õ❛ ❝→❝ ❦❤→✐ ♥✐➺♠ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❦❤→❝ ♥❤❛✉✳ ❑❤ỉ♥❣ ♠➜t ✤✐ t➼♥❤ tê♥❣ q✉→t✱ ❝❤ó♥❣
t❛ ❣✐↔ sû r r = rank(E) < n õ tỗ t ♠❛ tr➟♥ ❦❤æ♥❣ s✉② ❜✐➳♥ T, W ∈ Rn,n
s❛♦ ❝❤♦ tữỡ ữỡ ợ
x 1 = Jx1 + B1 u,

x1 (0) = x1,0

N x˙ 2 = x2 + B2 u,

x2 (0) = x2,0

y = C 1 x1 + C 2 x2 .

✶✹

✭✷✳✸❛✮
✭✷✳✸❜✮
✭✷✳✸❝✮


✈ỵ✐
W ET =

CT =

Inf

0

0

N

, W BT =

C1 C2

J

, W AT =
B1
B2

0

0 In∞
, T −1 x =

,
x1
x2

✈➔ ❝❤♦ v = ind(E, A)✳ ❚❛ ❣å✐ ✭✷✳✸❛✮ ❧➔ ❤➺ ❝♦♥ ❝❤➟♠ ❝õ❛ nf ✈➔ ✭✷✳✸❜✮ ❧➔ ❤➺ ❝♦♥ ♥❤❛♥❤
❝õ❛ n∞✳
❑❤✐ ✤â✱ t❛ ❜✐➳t tr t ỗ ừ
t

x1 (t) = eJt x1 (0) +

eJ(t−s) B1 u(s)ds

(t > 0)

0
v−1

N i B2 u(i) .

x2 (t) = −
i=0

❱➻ ✈➟②✱ ♠ët ❤➔♠ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✭❝❤♦ ♠ët ♥❣❤✐➺♠ ❝ê ✤✐➸♥✮ ♣❤↔✐ t❤ä❛ ♠➣♥ u ∈
T
Cp v−1 (I, Rm )✳ ❱ỵ✐ ♠å✐ t > 0✱ x(t) = T x1 T x2 T
✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t
x1(0) trữợ u(s) 0 ≤ s ≤ t✱ ✈➔ t❤í✐ ✤✐➸♠ t✳ ❚r♦♥❣
tr÷í♥❣ ủ t trữợ x2(0) ữủ ♥❤➜t q✉→♥ ✭x2(0) ✤÷đ❝
①→❝ ✤à♥❤ ❞✉② ♥❤➜t✮ ✈➔ ❝❤➾ ❞✉② t x1(0) ợ õ t ữủ ồ ởt tũ þ✳
❈❤ó♥❣ t❛ ❦➼ ❤✐➺✉ R˜0 ❧➔ t➟♣ ❝â t❤➸ ✤↕t ữủ ừ tứ
trữợ x1(0) = 0 ✭✈➔ x2(0) ♥❤➜t q✉→♥✮✳

❇ê ✤➲ ✷✳✶✳ ❱ỵ✐ ♠å✐ ✤❛ t❤ù❝ p(t) = 0✱ ①➨t ♠❛ tr➟♥
t
T

p(s)eA1 s B1 B1 T eA1 s p(s)ds,

W (p, t) =
0

✈ỵ✐ A1 ∈ Rn,n, B1 ∈ Rn,m✳ ❑❤✐ ✤â✿
Im(W (p, t)) = Im

A1 A1 B1 ... A1 n−1 B1

✈ỵ✐ ♠å✐ t > 0
❈❤ù♥❣

ờ tr tữỡ ữỡ ợ
n1

ker(B1T (AT1 )i )

ker(W (p, t)) =
i=0

✶✺


❈❤♦ x ∈ ker(W (p, t))✱ ❦❤✐ ✤â
t

xT W (p, t)x =
0

✈➻ ✈➟② B1T eA
t❛ ❝â

T
1s

t

T

T

B1T eA1 s p(s)x

xT p(s)eA1 s B1 B1T eA1 s p(s)xds =
0

p(s)x = 0

2

ds = 0,
2

✈ỵ✐ 0 ≤ s ≤ t✳ ✣❛ t❤ù❝ p(s) ❝â ❤ú✉ ❤↕♥ ♥❣❤✐➺♠ tr♦♥❣ [0, t]✱
T

B1T eA1 s x = 0, 0 ≤ s ≤ t.
T
T i ♥➯♥ ker(W (p, t)) ⊆ ker(B T (AT )i ✳ ❱ỵ✐
❱➻ s tị② þ✱ ♥➯♥ t❛ ❝â x ∈ n−1
1

1
i=0 ker B1 (A1 )
T
T
i
x ∈ ker(B1 (A1 ) )✱ t✐➳♥ ❤➔♥❤ ♥❣÷đ❝ ❧↕✐ q✉→ tr➻♥❤ tr➯♥ t❛ ✤÷đ❝
x ∈ ker(W (p, t))✱ ❝â ♥❣❤➽❛ ❧➔

ker(B1T (AT1 )i ) ⊆ ker(W (p, t))

❱➟②
ker(B1T (AT1 )i ) = ker(W (p, t))

◆❣❤➽❛ ❧➔
Im(W (p, t)) = Im

A1 A1 B1 ... A1 n−1 B1

❇ê ✤➲ ✷✳✷✳ ❈❤♦ xi ∈ Rn, i = 0, 1, ..., v − 1 t1 > 0 õ tỗ t ởt ✤❛ t❤ù❝
p(t) ∈ Rn

❜➟❝ v − 1 s❛♦ ❝❤♦ p(i)(t) = xi ✈ỵ✐ i = 0, 1, ..., v − 1

❈❤ù♥❣ ♠✐♥❤✳

❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ❝→❝❤ ✤➦t

p(t) = x0 + x1 (t − t1 ) +

1

1
x2 (t − t1 )2 ... +
xv−1 (t − t1 )v−2
2!
(v − 1)!

⇒ p(t1 ) = x0 , p (t1 ) = x1 , p (t2 ) = x2 , ...

ỵ R˜0 ❧➔ t➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❝õ❛ ✭✷✳✸✮ tø
trữợ x1(0) = 0 õ

R0 = Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N n∞ −1 B2 ] .

ú ỵ r ỵ õ tữỡ tỹ ữ t srts = ì




❈❤♦ x1(0) = 0, t > 0✱ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✸✮ ✤÷đ❝ ❝❤♦ ❜ð✐

❈❤ù♥❣ ♠✐♥❤✳

v−1

t
J(t−s)

x1 (t) =

e

N i B2 u(i) (t).

x2 (t) = −

B1 u(s)ds,

0

i=0

❍✐➸♥ ♥❤✐➯♥✱ x2(t) ∈ Im[ B2 N B2 ... N n −1B2] ✭✈➻ v ≤ n∞✮✳ ❍ì♥ ỳ tỗ t
t i(t) R, i = 0, ..., nf − 1 s❛♦ ❝❤♦✿
∞

eJt (t) = β0 (t)I + β1 (t)I + ... + βnf −1 (t)J nf −1

❱➟②✱
nf −1

t
J(t−s)

x1 (t) =

e

t

βi (t − s)u(s)ds ∈ Im[B1 JB1 ...J nf −1 B1 ]

J i B1

B1 u(s)ds =

0

0

i=0

✈ỵ✐ ♠å✐ t > 0✳ ❱➟②✱
x(t) =

x1 (t)

∈ Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N n∞ −1 B2 ]

x2 (t)

▼➦t ❦❤→❝✱ ✤➦t
xˆ =

xˆ1
xˆ2

∈ Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N n∞ −1 B2 ] ,

✈ỵ✐ xˆ1 ∈ Im[ B1 JB1 ... J n 1B1] x2
tỗ t wi Rn , i = 0, 1, ..., v − 1 s❛♦ ❝❤♦

f

∈ Im[ B2 N B2 ... N n∞ −1 B2 ] .

◆➯♥

∞

v−1

N i B2 wi .

xˆ2 = −
i=1

❚ø ❜ê ✤➲ ✭✷✳✷✮✱ ✈ỵ✐ ồ t > 0 ố tỗ t ởt t❤ù❝ p(s) ❜➟❝ v − 1 s❛♦ ❝❤♦
p(i) (t) = wi . ◆➯♥✱ sû ❞ö♥❣ ❤➔♠ ✤➛✉ ✈➔♦ u(t) = u1 (t) + p(t) t❛ ✤÷đ❝ ❤➺
t

t
J(t−s)

x1 (t) =

e
0

✈➔

eJ(t−s) B1 p(s)ds

B1 u(s)ds +
0

t

eJ(t−s) B1 p(s)ds ∈ Im[ B1 JB1 ... J nf −1 B1 ]

x˜1 := xˆ1 −
0

✈ỵ✐ t > 0 ❝è ✤à♥❤✳ ❱ỵ✐ ♠å✐ t > 0 ❝è ✤à♥❤✱ ✤➦t q(s) = sv (s − t)v = 0✳ ❚ø ờ t
s r sỹ tỗ t ừ z ∈ Rn s❛♦ ❝❤♦ W (q, t)z = x˜1✳ ✣➦t u1(s) = q(s)2B1T eJ (t−s)z
T

f

✶✼


ợ 0 s t t ữủ ỗ
t
J(ts)

x1 (t) =

e

B1 q(s)

2

t

T
B1T eJ (ts) zds +

eJ(ts) B1 p(s)ds

0

0
t
J(ts)

=

q(s)e

T
B1 B1T eJ (t−s) q(s)dsz

t

eJ(t−s) B1 p(s)ds

+
0

0

= W (q, t) + xˆ1 − x˜1 = xˆ1

✈➔

v−1

v−1
i

(i)

N i B2 (u1 (t) + p(i) (t))

N B2 u (t) = −

x2 (t) = −

i=0

i=0
v−1

N i B2 wi = xˆ2

=−
i=0

❱➟②✱ xˆ ∈ R˜0.

❱➼ ❞ö ✷✳✸✳

✶✳ ❳➨t ❤➺ s❛✉
x˙1 =

1 1
0 1

0 = x2 +

0

x1 +

1

−1 0

x1 (0) = x01 ,

u,
u,

✈ỵ✐ n = 4, nf = 2, n∞ = 2✱ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ð ❞↕♥❣ ❝❤➼♥❤ t➢❝✳ ❚➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ tø
x1 (0) = 0 ✤÷đ❝ ❝❤♦ ❜ð✐
R˜0 = Im[ B1 JB1 ] ⊕ [ B2 N B2 ] = R2 ⊕ (R ⊕ {0}) = R3 ⊕ {0}.

✷✳ ❳➨t ❤➺ s❛✉
0 1
1 0

−1

x˙2 = x2 +

−1

❚➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❝❤♦ ❜ð✐ R˜ = R2 tr t ỗ ừ ữủ ❝❤♦
❜ð✐
v−1
N i B2 u(i) (t) =

x2 (t) = −
i=0

u + u˙
u

❱➟② ♥➯♥✱ ✈ỵ✐ ♠å✐ w = [ w1 w2 ]T ∈ R2 ✈➔ t > 0 t❛ ❝â t❤➸ ❝❤å♥ tø u(t) s❛♦ ❝❤♦
u(t1 ) = w2 , u(t
˙ 1 ) = w1 − w2 ✈➔ t❛ ✤÷đ❝
x2 (t) =

✶✽

w1
w1

.

ữ ỵ r ởt ữỡ tự õ t ỏ ọ ởt ữủ
ỡ ợ ||w1 − w2|| ❝➔♥❣ ❧ỵ♥ t❤➻ u˙ ❝➔♥❣ ❧ỵ♥✳ ✣➸ sû ử t q t t
t ỵ t❤✉②➳t ✤✐➲✉ ❦❤✐➸♥ s❛✉✳

✷✳✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔
❇ê ✤➲ ✷✳✸✳ ✭❇ê ✤➲ ❍❛✉t✉s✲P♦♣♦✈✮ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
✶✳ ❍➺ x˙ = Ax + Bu ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳
✷✳ rank(K) = rank[ B AB ... An−1B ] = n
✸✳ ◆➳✉ z ❧➔ ✈❡❝t♦ r✐➯♥❣ ❝õ❛ AT ✱ t❤➻ zT B ∈ C
✹✳ rank[ λI − A B ] = n ✈ỵ✐ ♠å✐ λ ∈ C

✣à♥❤ ỵ
t ♥➳✉ ✈➔ ❝❤➾ ♥➳✉
rank[ λI − A B ] = n, ∀λ ∈ C, λ

❤ú✉ ❤↕♥

✷✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ ❍➺ ❝♦♥ ♥❤❛♥❤ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳
❜✮ rank(K) = rank[ B N B ... N n−1B ] = n∞
❝✮ rank[ N B2 ] = n∞
❞✮ rank[ E B ] = n
❡✮ ❱ỵ✐ ♠å✐ ♠❛ tr➟♥ ❦❤ỉ♥❣ s✉② ❜✐➳♥ Q1 ✈➔ P1 t❤ã❛ ♠➣♥✿
E = Q1

I 0
0 0

P1 , QB =

B˜1
B˜2

.

❑❤✐ ✤â✿ rank(B˜2) = n − rank(E).
✸✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ ❍➺ ✷✳✸ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳
❜✮ ❍➺ ❝♦♥ ❝❤➟♠ ✈➔ ♥❤❛♥❤ ✷✳✸❛ ✈➔ ✷✳✸❜ ✤➲✉ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉
❦❤✐➸♥✳
❝✮ rank[B1 JB1 ... J n −1B1] = nf ✈➔ rank[B2 N B2 ... N v−1B2] = n∞
f

✶✾


❞✮ rank[λE − A B] = n ✈ỵ✐ t➜t ❝↔ ❤ú✉ ❤↕♥ λ ∈ C ✈➔ rank[E B] = n.
❡✮ rank[ αE − βA B ] = n ✈ỵ✐ t➜t ❝↔ (α, β) ∈ C2 \ (0, 0).
❈❤ù♥❣ ♠✐♥❤✳

✶✳ ❍➺ ❝♦♥ ❝❤➟♠ ❧➔ ♠ët ❖❉❊✱ ✈➟② ♥➯♥ ✤✐➲✉ ❦✐➺♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺
❝❤✉➞♥ ▲❚■ ✈➔ ✷✳✸❛ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ rank[ λ(I) − J B1 ] =
nf ✈ỵ✐ t➜t ❝↔ ❤ú✉ ❤↕♥ λ ∈ C
❍ì♥ ♥ú❛✱ t❛ ❝â
λI − J

rank[ λE − A B ] = rank[ λW ET − W AT W T B ] = rank

0

0

B1

λN − I B2

▼❛ tr➟♥ λN − I ❧➔ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ✈ỵ✐ ♠å✐ ❤ú✉ ❤↕♥ λ ∈ ✈➔
rank[ λE − A B ] = n∞ + rank[ λI − J B1 ] = n

✈➻ ◆ ❧ô② ❧✐♥❤✱ ✈➻ ✈➟②
rank[ λI − N B2 ] = n∞

✈ỵ✐ ♠å✐ λ ∈ C ⇔ rank[ −N

B2 ] = rank[ N B2 ] = n∞ ,

✷✳ ✷❛ ⇐⇒ ✷❜ tø ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ ❤➺ ❝♦♥ ♥❤❛♥❤ ✷✳✸❜ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥
♥➳✉ t➟♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❧➔
Im[ B2 N B2 ... N v−1 B2 ] = Rn∞ ⇔ rank[ B2 N B2 ... N v−1 B2 ] = n∞

✷❜ ⇐⇒ ✷❝ ❍➺ (N, B2) ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥ ✭t÷ì♥❣ tü ❤➺ ❝❤✉➞♥ ▲❚■✮

♥➳✉ ✈➔ ❝❤➾ ♥➳✉ rank[ λI − N B2 ] = n∞ ✈ỵ✐ ♠å✐ λ ∈ C. ✣✐➲✉ ❦✐➺♥ ♥➔② ❝è
✤à♥❤ ✈ỵ✐ ♠å✐ λ ∈ δ(N ) = {0} ✈➻ ◆ ❧ô② ❧✐♥❤✱ ✈➻ ✈➟②
rank[ λI − N B2 ] = n∞
⇐⇒

✈ỵ✐ ♠å✐ λ ∈ C

rank[ −N B2 ] = rank[ N B2 ] = n∞ .

✷❝ ⇐⇒ ✷❞ ❚❛ ❝â
rank[ E B ] = rank[ W ET W B ] = rank

❱➟②✱ rank[ N B2 ] = n∞ ⇐⇒ rank[ E
✷❞ ⇐⇒ ✷❡ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳
✷✵

Inf
0

B ] = n.

0

B1

N B2

= nf +rank[ N B2 ].

.


✸✳ ✸❛ ⇐⇒ ✸❝ ❈❤♦ ❤➺ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥ ✈➔ ❝❤♦ x1(0) = 0✳ ❑❤✐ ✤â✱ ✈ỵ✐
♠å✐ t1 > 0 w Rn tỗ t ởt ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ u = Cpv − 1 s❛♦
❝❤♦ x(t1) = w. ❱➟②
R˜0 = Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N n∞ −1 B2 ] = Rn
⇐⇒ rank[ B1 JB1 ... J nf −1 B1 ] = nf

✈➔ rank[ B2 N B2 ... N n −1B2 ] = n∞.
▼➦t ❦❤→❝✱ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❤↕♥❣✱ ❦❤✐ ✤â t❛ ❜✐➳t
∞

Rx1 (0) = R˜0 +

x1
x2

|x1 = eJt x1 (0) ∈ Rnf , x2 = 0 ∈ Rn∞

= Rn

◆➯♥ ❤➺ ✭✷✳✸✮ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳
✸❜ ⇐⇒ ✸❝ ❚❤❡♦ ✶ ✈➔ ✷✳
✸❜ ⇐⇒ ✸❞ ❚❤❡♦ ✶ ✈➔ ✷✳

✸❞ ⇐⇒ ✸❡ rank[ λE − A

α
βE

B ] = rank

−A B

= rank[ αE − βA B ]

❱➼ ❞ö ✷✳✹✳ ❳➨t ❤➺ s❛✉

x˙ 1 =

1 1
0 1

0 = x2 +

x1 +
−1
0

0
1

u

u.

❚❛ ❝â
rank[ B1 JB1 ] = rank

0 1
1 1

✈➔rank[ B2

=2

BN2 ] = rank

−1 0
0

0

= 1 < 2.

❱➟② ❤➺ tr➯♥ ❦❤æ♥❣ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✱ tr♦♥❣ ❦❤✐ ✤â ❤➺ t


ú ỵ ợ tr t tữợ ợ t tr ỵ

trữợ ổ ũ ủ t t sè✱ tø ❦❤❛✐ tr✐➸♥ ❝õ❛ ❤➺ tr♦♥❣ ✭❲❈❋✮ ❤❛②
❣✐→ trà r✐➯♥❣ ❝➛♥ t❤✐➳t✳ ▼ët ❝→❝❤ tèt ❤ì♥ ❧➔ t❤ỉ♥❣ q✉❛ ❞↕♥❣ ❜➟❝ t❤❛♥❣✳
❍➺ ♠æ t↔ t✉②➳♥ t➼♥❤ ❜➜t ❜✐➳♥ ❧➔ ổ t õ t t ữủ ợ ồ tf > 0 ✈➔
x1 (0) ∈ R, w ∈ R, tỗ t ởt ữủ u Cpv1 s x(tf ) = w.

ỵ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣
✷✶


✶✳ ❍➺ ✭✷✳✸✮ ❧➔ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ❝â t❤➸ ✤↕t ✤÷đ❝ ✭❘✲❝♦♥tr♦❧❧❛❜❧❡✮
✷✳ ❍➺ ❝♦♥ ❝❤➟♠ ❧➔ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ❝â t❤➸ ✤↕t ✤÷đ❝ ✭❘✲❝♦♥tr♦❧❧❛❜❧❡✮
✸✳ rank[ λE − A B ] = n ✈ỵ✐ t➜t ❝↔ ❤ú✉ ❤↕♥ λ ∈ C
✹✳ rank[ B1 JB1 ... J n −1B1 ] = nf .
f

❈❤ù♥❣ ♠✐♥❤✳

✶ ⇐⇒ ✷ ❚ø ✤à♥❤ ♥❣❤➽❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ❝â t❤➸ ✤↕t ✤÷đ❝ ♥➳✉

R˜0 = Im[ B1 JB1 ... J nf −1 B1 ] ⊕ Im[ B2 N B2 ... N v−1 B2 ]

= Rnf ⊕ Im[ B2 N B2 ... N v−1 B2 ]

❱➟②✱ Im[ B1 JB1 ... J n −1B1 ] = Rn ⇐⇒❍➺ ❝♦♥ ❝❤➟♠ ✭✷✳✸❛✮ ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉
❦❤✐➸♥✳
✷ ⇐⇒ ✸ ❙✉② r❛ trü❝ t✐➳♣ tø ✷✳✷✳
❱➟②✱ ✤✐➲✉ ❦❤✐➸♥ ❤♦➔♥ t♦➔♥ ❜❛♦ ❤➔♠ ✤✐➲✉ ❦❤✐➸♥ ❝â t❤➸ ✤↕t ✤÷đ❝✳ ❈❤✐➲✉ ♥❣÷đ❝ ❧↕✐
❦❤ỉ♥❣ ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✳
f

f

❱➼ ❞ư ✷✳✺✳

✶✳ ❳➨t ❤➺ ð ✈➼ ❞ư ✭✷✳✹✮ ✤÷đ❝ ❝❤♦ ❜ð✐
x˙1 =

1 1
0 1

0 = x2 +

x1 +
−1
0

0
1

u

u.

❍➺ ❝♦♥ ❝❤➟♠ tr➯♥ ❧➔ ❈✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥❤÷ ✤➣ ❝â✱ ♥➯♥ ❤➺ tr➯♥ ❧➔ ❘✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳
✷✳ ❈❤♦ ◆ ❧ô② ❧✐♥❤ ✈➔ ①➨t ❤➺ N x˙ = x + Bu. tr ỗ ✈➔ ♥â
❧✉ỉ♥ ❧➔ ❘✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳

❍➺ q✉↔ ✷✳✶✳ ❳➨t ❤➺ ✭✷✳✷✮ ✈ỵ✐ ❝❤ị♠ ♠❛ tr➟♥ ❝❤➼♥❤ q✉② λE − A✳ ❑❤✐ ✤â✱ ❤➺ tr➯♥ ❈✲
✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❤➺ ❘✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ rank[ E

B ] = n.

ứ t q trữợ t t ❣✐→♥ ✤♦↕♥ tr♦♥❣ ♥❣❤✐➺♠ x(t) t↕✐
♠ët sè ✤✐➸♠ ♣❤➙♥ ❜✐➺t tr♦♥❣ I✳ ❚❤❡♦ ✤â✱ ✤➦t Dn = ∞(R, Rn) ❧➔ t➟♣ ✈ỉ ❤↕♥ ❝→❝
❤➔♠ ❦❤↔ ✈✐ ✈ỵ✐ ❣✐→ trà tr♦♥❣ Rn ✈➔ ❝♦♠♣❛❝t tr♦♥❣ R. ❈→❝ ♣❤➛♥ tû Dn ✤÷đ❝ ❣å✐ ❧➔
❝→❝ ❤➔♠ t✐➯✉ ❝❤✉➞♥✳
❈❤ù♥❣ ♠✐♥❤✳

✷✷


✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ▼ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ f : Dn → Rn ✈ỵ✐
f (α1 φ1 + α2 φ2 ) = α1 f (φ1 ) + α2 f (φ2 )

✈ỵ✐ ♠å✐ φ1, φ2 ∈ Dn, α1, α2 ∈ R

✣÷đ❝ ❣å✐ ❧➔ ❤➔♠ s✉② rë♥❣ ♥➳✉ ♥â ❧✐➯♥ tö❝✱ ✤â ❧➔ f (φ) → 0 tr♦♥❣ Rn ✈ỵ✐ ♠å✐ ❞➣②
(φi )i∈N ✈ỵ✐ φ1 → 0 tr♦♥❣ Dn ✳

❱➼ ❞ư ✷✳✻✳ P❤➙♥ ❜è ❉✐r❛❝ ❞❡❧t❛ δα ∈ C n ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ δα(φ) = φ(α) ✈ỵ✐ ♠å✐
φ ∈ Dn , α R



+

(x) =

0

x=



ú ỵ ∈ Dn ✈➔ tˆ > 0 ✤õ ❧ỵ♠ t❤➻ ❝è ✤à♥❤ r➡♥❣
tˆ

∞

ˆ
φ(t)dt

ˆ
φ(t)dt
=−

φ(0) = −(φ(tˆ) − φ(0)) = −φ(t)|t0 = −

0

0

ˆ
ˆ
H(t)φ(t)dt
=: H()

=
R

H(t) = 0 ợt < 0 ữợ ✤ì♥ ✈à✳ ❚❛ t➻♠ ✤÷đ❝ q✉❛♥ ❤➺ δ0 = H.˙ ❚❛
1 ✈ỵ✐t ≥ 0
❝ơ♥❣ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ sü ❞à❝❤ ❝❤✉②➸♥ ❝õ❛ H ❜➡♥❣ Hα(t) := H(t − α) ✈➔ δα = H˙ α
❍❛✐ ♣❤➙♥ ❜è f1, f2 ∈ C n ữ f1() = f2() ợ ồ φ ∈ Dn✳ ❚❤❡♦ ✤â✱
x : I → Rn , I R ữủ ỷ ỵ ữ ởt số ①→❝ ✤à♥❤ tr➯♥ R ❜ð✐ x(t) = 0 ✈ỵ✐ t ∈
/ I.
❚✉② ♥❤✐➯♥✱ ♥❣❤✐➺♠ ❜à ❤↕♥ ❝❤➳ t↕✐ ♠ët sè ✤➳♠ ✤÷đ❝ ❝→❝ ✤✐➸♠ τj ∈ T ⊆ R.

✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ●✐↔ sû t➟♣ T = {τj ∈ R|τj < τj+1 ∈ Z} ❦❤ỉ♥❣ ❝â ✤✐➸♠ tư ♥➔♦✳

▼ët ❤➔♠ s✉② rë♥❣ x ∈ C n ✤÷đ❝ ❣å✐ ❧➔ ✐♠♣✉❧s✐✈❡ s♠♦♦t❤ õ õ t ữủ t
ữợ
x = x + ximp ,
xˆ =
xˆj ,
✭✷✳✹✮
j∈Z

t↕✐ xˆj ∈ C ∞(|τj , τj+1|, Rn) ✈ỵ✐ ♠å✐ j ∈ Z ✈➔ ❜ë ♣❤➟♥ ①✉♥❣ ❧ü❝ ximp ❝â ❞↕♥❣

qj
(

ximp,j =

✭✷✳✺✮

cij ∈ Rn , qj ∈ N0 .

cij δτj i),
i=0

n (T).
❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♣❤➙♥ ❜è ①✉♥❣ ❧ü❝ trì♥ ✤÷đ❝ ❦➼ ❤✐➺✉ Cimp
n (C) ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❦❤❛✐ ❦❤✐➸♥ ✭✷✳✸✮✳
❇ê ✤➲ ✷✳✹✳ ✶✳ P❤➙♥ ❜è x ∈ Cimp

n (C) t❛ ❝â t❤➸ ❦➼ ❤✐➺✉ ❣✐→ trà ❤➔♠ sè x(t) ✈ỵ✐ ♠é✐ t ∈ R \ T ❜ð✐ x(t) = x
✷✳ ❱ỵ✐ x ∈ Cimp
ˆj
✈ỵ✐ t ∈ (τj , τj+1) ✈➔ lim x(τj−) = limt→τ xˆj−1(t) ✈➔ lim x(τj+) = limt→τ xˆj−1(t) ✈ỵ✐ ♠
−
j

+
j

✷✸

τj ∈ T.

n (T) ♥➡♠ tr♦♥❣ C n (T).
✸✳ ❚➜t ❝↔ ❝→❝ ✤↕♦ ❤➔♠ ✈➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ x ∈ Cimp
imp
n
✹✳ ❚➟♣ Cimp(T) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦ ❜à ✤â♥❣ trữợ ợ A

C (R, Rm,n ).
n (C) t↕✐ τ ∈ T ✤÷đ❝ ❦➼ ❤✐➺✉ iord(x)|
✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ❇➟❝ ①✉♥❣ ❝õ❛ x ∈ Cimp
τ
j

j

:=

♥➳✉ x ❝â t❤➸ ❧✐➯♥ ❦➳t ✈ỵ✐ ❤➔♠ ❧✐➯♥ tư❝ tr♦♥❣ [τj−1; τ j + 1] ✈➔ q✱ ✈ỵ✐ 0 ≤ q ≤ ∞
❧➔ ♠ët sè ♥❣✉②➯♥ ✤õ ❧ỵ♥

−q − 2

x|[τj−1 ;τ j+1] ∈ C q ([τj−1 , τj+1 ], Rn ).

◆â ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ iord(x)|τ ; = −1 ♥➳✉ ① ❝â t❤➸ ữủ t ợ ởt
tử tr [j1; j + 1] ♥❣♦➔✐ ✤✐➸♠ t = τj ✈➔ ♥â ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛
j

iord(x)|τj := max{i ∈ N0 |0 ≤ i ≤ qj , cij = 0}

▼➦t ❦❤→❝✱ ❜➟❝ ①✉♥❣ ❝õ❛ x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ iordx; = maxτ ∈Tiord(x)|
j

τj

.

n (T) ✈➔ A ∈ C ∞ (R, Rm,n ). ❑❤✐ ✤â iordAx ≤ iordx ✈ỵ✐ t➼♥❤
❇ê ✤➲ ✷✳✺✳ ❈❤♦ x ∈ Cimp

✤ì♥ ♥❤➜t m = n ✈➔ A(τj ) ❦❤↔ ♥❣❤à❝❤ ✈ỵ✐ ♠é✐ τj ∈ T.

❱➼ ❞ư ✷✳✼✳ ❳➨t ♠➝✉ sì ♠↕❝❤ ✤✐➺♥ ❝õ❛ ♠ët ♠↕❝❤ ✈✐ ♣❤➙♥ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè s❛✉✿

❍➻♥❤ ✷✳✷✿ ▼↕❝❤ ✤✐➺♥ ❝õ❛ ♠ët ♠↕❝❤ ✈✐ ♣❤➙♥✳
x1 − x4 = u(t),
C(x˙ 1 − x˙ 2 ) +

1
(x3 − x2 ) = 0,
R
x3 = A(x4 − x2 ),
x4 = 0,

✷✹


✈ỵ✐ ✤➛✉ ✈➔♦ ✤✐➺♥ →♣ u(t) = 0 ✈ỵ✐t < 0,

1 ✈ỵ✐t ≥ 0
❱ỵ✐ x4 = 0, x1 = u(t), x˙ 1 = u˙ ✈➔ x3 = −Ax2 t❛ ✤÷đ❝
x˙ 2 = −

1
(A + 1)x2 + u.
˙
CR

❱➻ sü t❤❛② ✤ê✐ tr♦♥❣ ✤✐➺♥ →♣ ✤➛✉ ✈➔♦✱ ✉ ❦❤æ♥❣ ❦❤↔ ✈✐✳ ❈❤♦ u = H ✈➔ A → ∞✱
♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣
x1 − x4 = H,
C(x˙ 1 − x˙ 2 ) +

1
(x3 − x2 ) = 0,
R
x2 = 0,
x4 = 0,

✈ỵ✐ ♥❣❤✐➺♠ x1 = H, x2 = 0, x3 = −RC H˙ = −RCδ0✈➔ x4 = 0✳ ❍➺ tr➯♥ ❧➔ ♠ët ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè ✈ỵ✐ ❝❤➾ sè ν = 2 ✭❤♦➦❝ µ = 1✮ ✈➔ iordf = −1✳ Ð ♠é✐ ❣✐→ trà ♥❤➜t
q✉→♥ ❜❛♥ ✤➛✉✱ ✈➼ ❞ö x(−1) = 0 t❛ ❝â ♠ët ♥❣❤✐➺♠ ✤ì♥ x ✈ỵ✐ iordx = 0.
✣è✐ ợ trữớ ủ t ừ ữỡ tr ✤↕✐ sè ❜➜t ❜✐➳♥ ✤➲✉ ❝õ❛ ❞↕♥❣
n (T)✱ jordf = q ∈ Z ∪ {−∞} t❛ ❝â t❤➸ t✐➳♥ ❤➔♥❤ ữ s
E x = Ax + f ợ f Cimp
❚❤ù ♥❤➜t✱ t❛ ❝â t❤➸ ❜✐➳♥ ✤ê✐ ❝➦♣ ♠❛ tr➟♥ (E, A) t❤➔♥❤ (W CF )
(E, A) ∼ (W ET, W AT ) =

Inf

0

0

N

,

J

0

0 In∞

.

❱➟②✱ t❛ ❝â
x˙ = Jx1 + f1 ,
N x˙ 2 = x2 + f2 ,

✭✷✳✻❛✮
✭✷✳✻❜✮

❚↕✐ x1 = T −1x✳ ✣è✐ ✈ỵ✐ ♣❤➙♥ ❜è ❖❉❊ ✭✷✳✻❛✮ t❛ ❝â t❤➸ ①➨t ♥❣❤✐➯♠ ❝ì ❜↔♥ ❝õ❛
x2
♠❛ tr➟♥ X(t) t❤ä❛ ♠➣♥
˙
X(t)
= JX(t),

X(t0 ) = I,

n (T) s♦❧✈❡s ✭✷✳✻❛✮ ♥➳✉
♥❣❤➽❛ ❧➔ X(t) = eJ(t−t ) ∈ C ∞(R, Rn ,n )✳ ❱➟②✱ ♣❤➙♥ ❜è x˜ ∈ Cimp
n (T) s♦❧✈❡s
✈➔ ❝❤➾ ♥➳✉ z = X −1x˜ ∈ Cimp
0

f

f

z˙ = g1 = X −1 f1 ,

✷✺