ĐẠIăHỌCăĐÀăN NG
TRƯỜNGăĐẠIăHỌCăSƯăPHẠMă
⎯⎯⎯⎯⎯⎯⎯⎯⎯

HUỲNHăTHỊăBÍCHăTHU

NGHIÊNăCỨUăVỀăTÍNHăĐIỀUăKHIỂNăĐƯỢCăVÀă
QUANăSÁTăĐƯỢCăCỦAăHỆăMỌăTẢ

LUẬNăV NăTHẠCăSĨăTỐNăHỌC

ĐàăN ngă- N mă2019


ĐẠIăHỌCăĐÀăN NG
TRƯỜNGăĐẠIăHỌCăSƯăPHẠMă
⎯⎯⎯⎯⎯⎯⎯⎯⎯

HUỲNHăTHỊăBÍCHăTHU

NGHIÊNăCỨUăVỀăTÍNHăĐIỀUăKHIỂNăĐƯỢCăVÀă
QUANăSÁTăĐƯỢCăCỦAăHỆăMỌăTẢ

Chun ngành: Tốn giải tích
Mưăsố: 84.6.01.02

LUẬNăV NăTHẠCăSĨăTỐNăHỌC

Ng

iăh ngăd năkhoaăh c

TS.ăLÊăHẢIăTRUNG

ĐàăN ngă– N mă2019


LỜI CAM ĐOAN

Tôi xin cam đoan Luận văn là công trình nghiên cứu của riêng tơi
dưới sự hướng dẫn trực tiếp của TS. Lê Hải Trung.
Trong quá trình nghiên cứu, tôi đã kế thừa thành quả khoa học của
các nhà khoa học với sự trân trọng và biết ơn.

Đà Nẵng, tháng 05 năm 2019
Tác giả

Huỳnh Thị Bích Thu




▼ư❝ ❧ư❝
▼Ð ✣❺❯
✶ ❑✐➳♥ t❤ù❝ ❝ì sð

✹
✻

✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔

✾


✶✳✶ ▼❛ tr➟♥ ✈➔ ✤à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ❍↕t ♥❤➙♥ ✈➔ ↔♥❤ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỵ tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻
✽

✾
✶✺

✸ ❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔

✸✸

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✸

✸✳✶ ❑❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✷ ❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✸✸
✸✹


é

ỵ ỹ ồ t
ỵ tt ❦❤✐➸♥ ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ♥❣❤➔♥❤ ❦❤♦❛ ❤å❝
❦❤→❝ ữ t số ỵ tt tỷ
t ỵ t tr t ✤è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❝→❝ ✤è✐ t÷đ♥❣
tr➯♥ t❤÷í♥❣ ✤÷đ❝ ♠ỉ ♣❤ä♥❣ ❜➡♥❣ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè✱ ✈➻ t❤➳
✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❧➔ ♠ët ✈➜♥ ✤➲ ❝➛♥ t❤✐➳t ✈➔ ✤➣ ✤÷đ❝
♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠✳ ❈→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝
ù♥❣ ❞ư♥❣ ♥❣➔② ❝➔♥❣ ♥❤✐➲✉ ð ❝→❝ ỹ t tr t ỵ
✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ❝õ❛ ❤➺ ổ t ữủ t tr
t ỵ tt ✈➔ ù♥❣ ❞ư♥❣✳
❇➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ♠ët ❤➺ tố ữủ ồ tr
t ợ q t ♥❣❤✐➯♥ ❝ù✉✱ ❝â t❤➸ ❦➸ ✤➳♥ ♥❤÷ ✿ ❙✳P ❩✉❜♦✈❛✱ ❨✳❱✳P❛❦♦r♥✉✐✱
❊✳❱ ❘❛❡s❦❛②❛✱ ❆✳❆✐❧♦♥✱ ▲❡♥❛ ❙❝❤♦❧③ ✳✳✳ ✈➔ ✤➳♥ ♥❛② ✤➣ tr t ởt ữợ
ự ổ t t tr ỵ t❤✉②➳t ✤✐➲✉ ❦❤✐➸♥✳ ❚r♦♥❣ ❝→❝ ❝æ♥❣ tr➻♥❤
❝õ❛ ❝→❝ t→❝ ❣✐↔ ♥➯✉ tr➯♥✱ ❝→❝ ♠ỉ ❤➻♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ữợ
ởt ữỡ tr số
F (t, x, x,
˙ u) = 0, x(t0 ) = x0 ,
y − G(t, x) = 0.

❍➺ ♠ỉ t↔ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ t♦➔♥ ♣❤➛♥ ✭C ✲ ✤✐➲✉ ❦❤✐➸♥ ữủ
ợ t ý tr t x0 Rn ✈➔ tr↕♥❣ t❤→✐ ❝✉è✐ ❝ò♥❣ xf ∈ Rn
t❤➻ ❤➔♠ ✤✐➲✉ ❦❤✐➸♥ u(t) ❧➔ ❜✐➳♥ ✤ê✐ tø tr↕♥❣ t❤→✐ x0 ✤➳♥ tr↕♥❣ t❤→✐ xf tr♦♥❣
t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥ tf ≥ t0 ✭tù❝ ❧➔ ∃u, tf < ∞ s❛♦ ❝❤♦ x(tf , u, x0 ) = xf ✮✳
❱ỵ✐ ♠♦♥❣ ♠✉è♥ ♥❣❤✐➯♥ ự ỵ tt ừ ổ t ự
ử ừ õ ũ ợ sỹ ủ ỵ ữợ ồ tứ r



tæ✐ q✉②➳t ✤à♥❤ ❝❤å♥ ✤➲ t➔✐ ✿✧◆❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ q✉❛♥ s→t

✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧ ❝❤♦ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ♠➻♥❤ ✳

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉

✲ ❍➺ t❤è♥❣ ❧↕✐ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ❤➺ ♠ỉ t↔ ✳
✲ ◆❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔ ✳

✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
✸✳✶✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉

▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺
♠ỉ t↔ ❝â ❞↕♥❣✿
F (t, x, x,
˙ u) = 0, x(t0 ) = x0 ,
y − G(t, x) = 0.

✸✳✷✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

◆❣❤✐➯♥ ❝ù✉ ✈➲ ❤➺ ổ t ợ tỹ

Pữỡ ❝ù✉

❈→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ t❤✉ë❝ ❝→❝ ❧➽♥❤ ✈ü❝✿
✣↕✐ sè t✉②➳♥ t➼♥❤✱ ●✐↔✐ t➼❝❤✱ ỵ tt ữỡ tr ỵ tt
ữỡ tr s ỵ tt ữỡ tr s ỵ tt ờ
ữỡ tr s

ị ồ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t➔✐

✣➲ t➔✐ ❝â ❣✐→ trà t ỵ tt tỹ t õ t sỷ ❞ư♥❣ ❧✉➟♥ ✈➠♥

♥❤÷ ❧➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❞➔♥❤ ❝❤♦ s✐♥❤ ✈✐➯♥ ♥❣❤➔♥❤ t♦→♥ ✈➔ ❝→❝ ✤è✐ t÷đ♥❣
q✉❛♥ t➙♠ ✤➳♥ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✈➔ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔ ✳

✺


❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝ì sð
✶✳✶ ▼❛ tr➟♥ ✈➔ ✤à♥❤ t❤ù❝
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët ♠❛ tr➟♥ ❆ ❧♦↕✐ ✭❝➜♣✮ m × n tr➯♥ tr÷í♥❣ K ✭K ✕
❧➔ tr÷í♥❣ t❤ü❝ R✱ ❤♦➦❝ ự C ởt ỳ t ỗ m ì n ♣❤➛♥ tû

tr♦♥❣ K ✤÷đ❝ ✈✐➳t t❤➔♥❤ m ❞á♥❣ ✈➔ n ❝ët ♥❤÷ s❛✉✿






A=




a11 a12
a21 a22
a31 a32
✳✳
✳✳

✳
✳
am1 am2

a13
a23
a33
✳✳
✳

... a1n
... a2n
... a3n
✳
... ✳✳

am3 ... amn






.




❚r♦♥❣ ✤â aij ∈ K ❧➔ ♣❤➛♥ tû ð ✈à tr➼ ❞á♥❣ i✱ ❝ët j ❝õ❛ ❆✳ ✣ỉ✐ ❦❤✐ ❆
✤÷đ❝ ✈✐➳t ♥❣➢♥ ❣å♥ ❧➔ A = aij . tr tữớ ữủ ỵ ❇✱

❈ ✈➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ ❝ï m ì n tr trữớ K ữủ ỵ

Amìn (K).

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ A ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ AT ✱ ❧➔

♠❛ tr➟♥ ♥❤➟♥ ✤÷đ❝ tø A ❜➡♥❣ ❝→❝❤ ✤ê✐ ❤➔♥❣ t❤➔♥❤ ❝ët✳

❱➼ ❞ư ✶✳✶✳ ❈❤♦
A=

0 1 2 3
5 4 8 −6
✻


t❤➻



0 5

1 4
AT = 
2 8

3 −6






.



❚➼♥❤ ❝❤➜t ✶✳✶✳ P❤➨♣ ❝ë♥❣ ❤❛✐ ♠❛ tr➟♥ ❝ò♥❣ ❝ï✳ ❚ê♥❣ A + B ừ

tr ũ tữợ m ì n A B ữủ ởt tr ũ tữợ
ợ tû tr♦♥❣ ✈à tr➼ t÷ì♥❣ ù♥❣ ❜➡♥❣ tê♥❣ ❝õ❛ ❤❛✐ ♣❤➛♥ tû t÷ì♥❣ ù♥❣ ❝õ❛
♠é✐ ♠❛ tr➟♥✿
(A + B)i,j = Ai,j + Bi,j , 1 ≤ i ≤ m, 1 j n.

t ổ ữợ ởt số ợ tr cA ừ số
c ụ ữủ ồ ổ ữợ ợ tr A ữủ tỹ ❤✐➺♥ ❜➡♥❣ ❝→❝❤ ♥❤➙♥
♠é✐ ♣❤➛♥ tû ❝õ❛ A ✈ỵ✐ c✿ (cA)i,j = c.Ai,j ✳ P❤➨♣ t♦→♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔

ổ ữợ

t tr P ♠❛ tr➟♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❦❤✐

✈➔ ❝❤➾ ❦❤✐ sè ❝ët ❝õ❛ ♠❛ tr➟♥ ❜➯♥ tr→✐ ❜➡♥❣ sè ❤➔♥❣ ❝õ❛ ♠❛ tr➟♥ ❜➯♥ ♣❤↔✐✳
◆➳✉ A ❧➔ ♠ët ♠❛ tr➟♥ m × n ✈➔ B ❧➔ ♠ët ♠❛ tr➟♥ n × p✱ t❤➻ ♠❛ tr➟♥ t➼❝❤
AB ❧➔ ♠❛ tr➟♥ m × p ợ tỷ ữủ t t ổ ữợ ừ
tữỡ ự tr A ợ ởt tữỡ ự tr♦♥❣ B ✿
(AB)ij = Ai,1 B1,j + Ai,2 B2,j + ... + Ai,n Bn,j , 1 ≤ i ≤ m, 1 ≤ j ≤ n.

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈✉ỉ♥❣ A✱ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ det A,

❧➔ ♠ët ❣✐→ trà ❝❤ù❛ ✤ü♥❣ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ♥❤➜t ✤à♥❤ ❝õ❛ ♠❛ tr➟♥ ♥➔② A. ❱➔

✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿
det

a11 a12
a21 a22

= a11 .a22 − a12 .a21 .

❚r♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✱ ✤à♥❤ t❤ù❝ ❝õ❛ A ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿
n

n

ai,j det Ci,j

ai,j det Ci,j =

det A =

i=1

j=1

✈ỵ✐ det Ci,j = (−1)(i+j) det Mi,j , Mi,j ❧➔ ♠❛ tr➟♥ ♥❤➟♥ ✤÷đ❝ tø A ❜➡♥❣ ❝→❝❤
①â❛ ✤✐ ❤➔♥❣ i ✈➔ ❝ët j.
✼


✶✳✷


❍↕t ♥❤➙♥ ✈➔ ↔♥❤ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ◆➳✉ V ✈➔ W ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ ❝ị♥❣ ♠ët

tr÷í♥❣✱ ❝❤ó♥❣ t❛ ♥â✐ r➡♥❣ →♥❤ ①↕ f : V → W ❧➔ ♠ët ✭♣❤➨♣✮ ❜✐➳♥ ✤ê✐ t✉②➳♥
t➼♥❤ ♥➳✉ ❝❤♦ ❜➜t ❦ý ❤❛✐ ✈❡❝tì x ✈➔ y tr♦♥❣ V t ý ổ ữợ a tr K
ú t❛ ❝â✿
f (x ± y) = f (x) ± f (y) f (x ± y) = f (x) ± f (y) ✭t➼♥❤ ❦➳t ❤ñ♣✮
f (ax) = af (x) f (ax) = af (x) t t t
õ ỵ tữỡ ữỡ ợ f t tờ ủ t✉②➳♥
t➼♥❤✧✱ ❝â ♥❣❤➽❛ ❧➔✱ ❝❤♦ ❜➜t ❦ý ✈❡❝t♦r x1, ..., xm ổ ữợ a1, ..., am,
ú t õ
f (a1 x1 + · · · + am xm ) = a1 f (x1 ) + · · · + am f (xm ).

❚❤ỉ♥❣ t❤÷í♥❣✱ ❱ ✈➔ ❲ ❝â t❤➸ ①❡♠ ♥❤÷ ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ ❝→❝
tr÷í♥❣ ❦❤→❝ ♥❤❛✉✱ ✈➔ ❦❤✐ ✤â ✤✐➲✉ q✉❛♥ trå♥❣ ❧➔ ①→❝ ✤à♥❤ tr÷í♥❣ ♥➔♦ ✤÷đ❝
❞ị♥❣ ❝❤♦ ✤à♥❤ ♥❣❤➽❛ t✉②➳♥ t➼♥❤✳ ◆➳✉ ❱ ✈➔ ❲ t❤✉ë❝ ❦❤ỉ♥❣ ❣✐❛♥ tr➯♥ tr÷í♥❣
❑ ♥❤÷ ①→❝ ✤à♥❤ ð tr➯♥✱ ❝❤ó♥❣ t❛ ♥â✐ ✈➲ ❑✲→♥❤ ①↕ t✉②➳♥ t➼♥❤✳ ❱➼ ❞ư✱ ❧✐➯♥
❤đ♣ ❝õ❛ ♠ët sè ♣❤ù❝ ❧➔ ♠ët ❘✲→♥❤ ①↕ t✉②➳♥ t➼♥❤ ❈✱ ♥❤÷♥❣ ♥â ❦❤ỉ♥❣ ♣❤↔✐ ❧➔
❈✲t✉②➳♥ t➼♥❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ◆➳✉ f : V

❧➔ t✉②➳♥ t➼♥❤✱ t❛ ✤à♥❤ t
ừ ỵ r ừ ❢ ✈➔ ❤↕♥❣ ❝õ❛ ❢ ♥❤÷ s❛✉✿
→W

ker(f ) = { x ∈ V : f (x) = 0 }; im(f ) = { f (x) : x ∈ V },
ker(f ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ ❱ ✈➔ ✐♠✭❢✮ ổ


ổ tự s ữủ ỵ ✈➲ sè ❝❤✐➲✉✿

❝♦♥ ❝õ❛ ❲✳

dim(ker(f )) + dim(im(f )) = dim(V ) .
❙è dim(im(f )) ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❤↕♥❣ ừ f ỵ r r
ỏ số dim(ker(f )) ✤÷đ❝ ❣å✐ ❧➔ sè ✈ỉ ❦❤✉②➳t ✭♥✉❧❧✐t②✮ ❝õ❛ f ✈➔ ỵ
v(f ) V W ỳ ❤↕♥ ❝❤✐➲✉✱ ✈➔ f ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ♠❛ tr➟♥ A✱ t❤➻
❤↕♥❣ ✈➔ sè ✈ỉ ❤✐➺✉ ❝õ❛ f t÷ì♥❣ ù♥❣ ❜➡♥❣ ❤↕♥❣ ✈➔ sè ✈æ ❤✐➺✉ ❝õ❛ ♠❛ tr➟♥ A✳

✽


❈❤÷ì♥❣ ✷

❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛
❤➺ ♠ỉ t↔
◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ♥❤➡♠ ♠ư❝ ✤➼❝❤ tr➻♥❤ ❜➔② ✈➲ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥
✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✳❚➔✐ ❧✐➺✉ ❝❤➼♥❤ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣
❝❤÷ì♥❣ ♥➔② ❧➔ t➔✐

ỵ tt
t ①❡♠ ①➨t ❤➺ ♠æ t↔ ❝â ❞↕♥❣✿
F (t, x, x,
˙ u) = 0, x(t0 ) = x0 , y − G(t, x) = 0,

✭✷✳✶✮

tr♦♥❣ ✤â t ∈ [t0, tf ].


✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❍➺ ♠ỉ t↔ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❤♦➔♥ t♦➔♥ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥

♥➳✉ ✈ỵ✐ ❜➜t ❦ý tr↕♥❣ t❤→✐ ❜❛♥ ✤➛✉ ✤➣ ❝❤♦ x0 ∈ Rn
✈➔ tr↕♥❣ t❤→✐ ❝✉è✐ ũ xf Rn tỗ t ởt ✤➛✉ ✈➔♦ u ❜✐➳♥
✤ê✐ ❤➺ tø x0 t❤➔♥❤ xf tr♦♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥ t ∈ [t0, tf ].

✭❝â t❤➸

ố ợ ởt t ủ ữủ ồ õ t t ữủ tứ x0 ợ tt
xf Rx tỗ t ởt ❦✐➸♠ s♦→t ✤÷đ❝ u ❜✐➳♥ ✤ê✐ ❤➺ tø x0
t❤➔♥❤ xf tr tớ ỳ tự tỗ t u, tf < ∞ s❛♦ ❝❤♦
0

✾


x(tf , u, x0 ) = xf ∈ Rx0 ✳✮ Ð ✤➙②✿
Rx0 = {xf ∈ Rn |∃u, tf < ∞ : x(tf : u, x0 ) = xf } ⊆ Rn .
R = ∪x0 ∈Xct0 Rx0 ❜✐➸✉ t❤à t➟♣ ❤ñ♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ✭tr♦♥❣ ✤â Xct0 ⊆ Rn ❧➔
t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❣✐→ trà ❜❛♥ ✤➛✉ x0 t↕✐ t❤í✐ ✤✐➸♠ t0 ✮✳ ❍➺ ✭✷✳✶✮ ✤÷đ❝ ❣å✐
❧➔ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ tr♦♥❣ t➟♣ ❤đ♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ✭❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ ✕ ❘✮✱
♥➳✉ ❜➜t ❦ý tr↕♥❣ t❤→✐ ♥➔♦ tr♦♥❣ Rn ❝â t❤➸ ✤↕t ✤÷đ❝ tø ❜➜t ❦ý tr↕♥❣ t❤→✐
❜❛♥ ✤➛✉ x0 ∈ Rn tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❤ú✉ ❤↕♥ ✭tù❝ ❧➔ ✤è✐ ✈ỵ✐ ❜➜t ❦ý
x0 ∈ Rn , xf ∈ Rn , tf < ∞ s❛♦ ❝❤♦ x(tf ; u, x0 ) = tf
ú ỵ ✤✐➲✉ ❦❤✐➸♥ ❘ ✤ỉ✐ ❦❤✐ ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ♥➠♥❣ ✤✐➲✉

❦❤✐➸♥ ✤ë♥❣ ❤ú✉ ❤↕♥✳ ◆â✐ ❝❤✉♥❣✱ ❝→❝ ❤➺ ♠æ t↔ s➩ ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t ❧➔ ✤✐➲✉
❦❤✐➸♥ ✤÷đ❝ ❈✱ ✈➻ ❝→❝ ✤✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ s➩ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ tr➯♥ ♠ët ✤❛ t↕♣
♥❣❤✐➺♠ ♥❤➜t ✤à♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ E = In ✱ ❦❤↔ ♥➠♥❣ ✤✐➲✉ ❦❤✐➸♥ ❘ trị♥❣
✈ỵ✐ ❦❤↔ ♥➠♥❣ ✤✐➲✉ ❦❤✐➸♥ ❈✳

❱➼ ❞ư

✷✳✶✳

❳➨t ❤➺ ♠ỉ t↔
0 0
1 0

x˙ 1
x˙ 2

=

0 1
1 0

x1
x2

+

0
1

u.

❑❤✐ ✤â t➟♣ ❤đ♣ ❝â ✤↕t ✤÷đ❝ ❝❤♦ ❜ð✐ R = {(x1 , x2 ) ∈ R2 |x2 = 0} ✈➔ ❤➺ ❝â
t❤➸ ✤✐➲✉ ❦❤✐➸♥ ❘✳
❱➼ ❞ö


✷✳✷✳

❳➨t ❤➺ ♠ỉ t↔
0 1
0 0

x˙ 1
x˙ 2

=

1 0
0 1

x1
x2

+

−1
−1

u.

✈ỵ✐ ❤➔♠ ✤➛✉ ✈➔♦ u ✤÷đ❝ ❝❤♦ ❜ð✐✿
u(t) =

0, 0 ≤ t ≤ 1,
1, 1 ≤ t ≤ tf .


tù❝ ❧➔ ✉ ❝❤➾ ❧➔ ❧✐➯♥ tư❝ tø♥❣ ♣❤➛♥✳ ◆❣❤✐➺♠ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿
x1 (t) = u + u,
˙ x2 (t) = u.

❉♦ ✤â✱ ✤è✐ ✈ỵ✐ u ✤➣ ❝❤♦✱ ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♥➔♦ t❤❡♦ ờ tỗ t
ự tr t ❝õ❛ ❤➺ ❝â t❤➸ ✤÷đ❝ ♠ỉ t↔ ♥❤÷ tr♦♥❣ ❍➻♥❤
✶✳✶✳ ✈➔ x1 , x2 ❧➔ ♠ët ♥❣❤✐➺♠ t❤❡♦ ♥❣❤➽❛ ♣❤➙♥ ♣❤è✐✳
✶✵


❍➻♥❤ ✷✳✶✿ ❚r↕♥❣ t❤→✐ ①✉♥❣ ❧ü❝ ❝õ❛ ♥❣❤✐➺♠✳ ✭❳✉♥❣ tr♦♥❣ x1 ❧➔ ❞♦ tr↕♥❣ t❤→✐
✤➛✉ ✈➔♦✳✮

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

❜➜t ❦ý tr↕♥❣ t❤→✐ ❜❛♥ ✤➛✉ x0 ∈ Rn tỗ t ởt
✤÷đ❝ u✱ ❜✐➳♥ ✤ê✐ ❤➺ s❛♥❣ tr↕♥❣ t❤→✐ ①✉♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥
❤ú✉ ❤↕♥✳
❈â t❤➸ ❝❤➾ r❛ r➡♥❣ ❦❤↔ ♥➠♥❣ ✤✐➲✉ I tữỡ ữỡ ợ ừ
ọ tt ❝→❝ tr↕♥❣ t❤→✐ ①✉♥❣ ❜➡♥❣ ❝→❝❤ ❝❤å♥ ♠ët u ♣❤ò ❤đ♣✳ ✣✐➲✉ ♥➔② ❝â
t❤➸ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ❜➡♥❣ ✤✐➲✉ ❦❤✐➸♥ ỗ tr t ố ợ ồ
tr t x0 tỗ t ởt ỗ tr t s
ổ t ổ õ ♥❣❤✐➺♠ ①✉♥❣✮✳ ✣è✐ ✈ỵ✐ ❝→❝ ❤➺ ♠ỉ t↔ ❝â t❤í✐ ❣✐❛♥
❜➜t ❜✐➳♥ ✭▲❚■✮ ❞↕♥❣✿
✭✷✳✷✮

E x˙ = Ax + Bu, y = Cx, x(0) = x0 ,

✈ỵ✐ E, A ∈ Rn,n, B ∈ Rn,m, C ∈ Rp,n✳ ●✐↔ sû r = r(E) < n tỗ t

tr ổ s ❜✐➳♥ T, W ∈ Rn,n ✤➸ ❝❤♦ ❤➺ t❤è♥❣ ♥➔② 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 =

I nf 0
0 N

, W AT =

✶✶

J 0
0 I n∞

,


B1
B2

CT = (C1 C2 ), W B =

, T −1 x =


x1
x2

.

✈➔ ✤➦t v = ✐♥❞(E, A)✳ ❈❤ó♥❣ t❛ ❣å✐ ✭✷✳✸✮ ❧➔ ❤➺ t❤è♥❣ ❝♦♥ ❝❤➟♠ t❤ù ♥❣✉②➯♥
nf ✈➔ ✭✷✳✹✮ ❧➔ ❤➺ t❤è♥❣ ❝♦♥ ♥❤❛♥❤ ❝õ❛ t❤ù ♥❣✉②➯♥ n∞ ✳ ❚✐➳♣ t❤❡♦ t❛ ❜✐➳t r➡♥❣
♥❣❤✐➺♠ ✭❤➔♠ tr↕♥❣ t❤→✐✮ ❝õ❛ ✭✷✳✸✮ ❧➔✿
t

eJ(t−s) B1 u(s)ds, t > 0

Jt

x1 (t) = e x1 (0) +
0

v−1

N i B2 u(i) .

x2 (t) = −
i=0

❉♦ ✤â✱ ữủ ố ợ ờ ✤✐➸♥✮ ♣❤↔✐ t❤ä❛
♠➣♥ u ∈ Cpv−1(I, Rm) ✭tù❝ ❧➔ u(t), (v − 1) ✲ ❧➛♥ ✈✐ ♣❤➙♥ ❧✐➯♥ tö❝ t❤❡♦ tø♥❣
✤✐➸♠✮✳ ✣è✐ ✈ỵ✐ ❜➜t ❦ý t > 0 ❤➔♠ tr↕♥❣ t❤→✐ x(t) = T (xT1 xT2 )T ✤÷đ❝ ①→❝ ✤à♥❤
❞✉② ♥❤➜t ❜ð✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x1(0)✱ ✤➛✉ ✈➔♦ ✤✐➲✉ ❦❤✐➸♥ u(s)✱ 0 ≤ s ≤ t ✈➔
t❤í✐ ❣✐❛♥ t✳ ❈ư t❤➸✱ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x2(0) ♣❤↔✐ ✤÷đ❝ ①→❝ t

õ t ồ x1(0) tũ ỵ ❚r♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❜✐➸✉ t❤à R˜ 0 ❧➔
t➟♣ ❤đ♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❝õ❛ ✭✷✳✸✮ tø ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x1(0) = 0 ✭✈➔ x2(0)
❞✉② ♥❤➜t✮✳
❇ê ✤➲ ✷✳✶✳ ✣è✐ ✈ỵ✐ ❜➜t ❦ý ✤❛ t❤ù❝ p(t) = 0 ①➨t ♠❛ tr➟♥
t

W (p, t) =
0

T

p(s)eA1 s B1 B1T eA1 s p(s)ds,

✈ỵ✐ A1 ∈ Rn,n, B1 ∈ Rn,m✳ ❑❤✐ ✤â
Im(W (p, t)) = Im(B1 A1 B1 ...An−1
1 B1 )

✈ỵ✐ ❜➜t ❦ý t > 0✳
❈❤ù♥❣ ♠✐♥❤✳ ❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲ tữỡ ữỡ ợ
T i
T
ker(W (p, t)) = n1
i=0 ker(B1 (A1 ) ).

✣➸ x ∈ ker(W (p, t))✱ t❤➻
t
T

T


x p(s)e

x W (p, t)x =
0

A1 s

T
B1 B1T eA1 s p(s)xds

✶✷

t

=
0

T

B1T eA1 s p(s)x 22 ds = 0


T

✈➔ ❞♦ ✤â B1T eA1 s p(s)x = 0 ✈ỵ✐ 0 ≤ s ≤ t ✳ ✣❛ t❤ù❝ p(s) ❝â sè ♥❣❤✐➺♠ ❤ú✉
❤↕♥ tr♦♥❣ [0, t]✱ ✈➻ ✈➟② ❝❤ó♥❣ t❛ ❝â
T

B1T eA1 s x = 0, 0 ≤ s ≤ t.
T

T i
s tũ ỵ ú t õ x n1
i=0 ker(B1 (A1 ) ) t ỵ ✕

❍❛♠✐❧t♦♥✮ ✈➔ ✈➻ ker(W (p, t)) ⊆ ker(B1T (AT1 )i ). ố ợ x ker(B1T (AT1 )i )
ữủ q✉→ tr➻♥❤ ♥➔② ♠❛♥❣ ❧↕✐ x ∈ ker(W (p, t))✱✈ỵ✐

ker(B1T (AT1 )i ) ⊆ ker(W (p, t)).

❇ê ✤➲ ✷✳✷✳ ✣➸ xi ∈ Rn, i = 0, v − 1 ✈➔ t1 > 0. õ tỗ t ởt tự
p(t) ∈ Rn ❜➟❝ v − 1 s❛♦ ❝❤♦ p(i) (t1 ) = x1 ✈ỵ✐ i = 0, v − 1.
❇➡♥❣ ỹ p(t) ữợ

ự

p(t) = x0 + x1 (t − t1 ) + ... +

1
xv−1 (t − t1 )v−1
(v − 1)!

✈➔ sû ❞ö♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜❛♥ t ự ữủ sỹ tỗ t ừ
số x0 , x1 , ..., xv1 .

ỵ ✣➦t R˜0 ❧➔ t➟♣ ❤đ♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❝õ❛ ✭✷✳✷✮ tø ✤✐➲✉ ❦✐➺♥ ❜❛♥
✤➛✉ ❜➡♥❣ ❦❤æ♥❣ x1 (0) = 0 t❤➻

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

ú ỵ ⊕ ❝â ♥❣❤➽❛ ❧➔ t➼❝❤ ✣➲ ❝→❝✳

❈❤ù♥❣ ♠✐♥❤✳ ✣è✐ ✈ỵ✐ x1(0) = 0 ✈➔ t > 0 ❤➔♠ tr↕♥❣ t❤→✐ ❝õ❛ ✭✷✳✷✮ ✤÷đ❝
①→❝ ✤à♥❤ ❜ð✐

v−1

t

e

x1 (t) =

J(t−s)

N i B2 u(i) (t).

B1 u(s)ds, x2 (t) = −

0

i=0

❘ã r➔♥❣✱ x2 (t) ∈ Im[B2 N B2 ...N n∞ −1 B2 ] ✭✈➻ v n ỡ ỳ tỗ t
i (t) ∈ R, i = 0, v − 1 s❛♦ ❝❤♦

eJt = β0 (t)I + β1 (t)J + ... + βnf −1 (t)J nf −1 .
❱➻ ✈➟②✱
nf −1

t


e

x1 (t) =

J(t−s)

B1 u(s)ds =

0

J B1
i=0

✶✸

t
i

βi (t − s)u(s)ds
0


∈ Im[B1 JB1 ...J nf −1 B1 ]

✤è✐ ✈ỵ✐ t > 0✳ ❉♦ ✤â
x(t) =

x1 (t)
x2 (t)


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

❚r→✐ ❧↕✐✱ ✤➦t
x=

x1
x2

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

✈ỵ✐ x1 ∈ Im[B1 JB1 ...J n −1B1] ✈➔ x2
õ tỗ t i Rn , i = 0, v − 1 s❛♦ ❝❤♦
f

Im[B2 N B2 ...N n∞ −1 B2 ].❉♦

∞

v−1

N i B2 ωi .

x2 = −
i=0

❚ø ❇ê ✤➲ ✷✳✷ ❝❤♦ ❜➜t ❦ý t > 0 ❝è ✤à♥❤✱ ❦❤✐ õ tỗ t ởt tự p(s)
v 1 s❛♦ ❝❤♦ p(i) (t) = ωi . ❉♦ ✤â✱ sû ❞ư♥❣ ❤➔♠ ✤➛✉ ✈➔♦ u(t) = u1 (t) + p(t)
❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ ❤➔♠ tr↕♥❣ t❤→✐ ❝õ❛ ❤➺ ❧➔✿
t


t

e

x1 (t) =

J(t−s)

0

0

✈➔

t

x˜1 := x1 −

eJ(t−s) B1 p(s)ds

B1 u1 (s)ds +

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

0

✈ỵ✐ t > 0 ❝è ✤à♥❤✳ ❱ỵ✐ ♠å✐ t > 0 ❝è ✤à♥❤✱ ✤➦t q(s) = sv (s − t)v = 0. ứ ờ
ú t s r sỹ tỗ t ❝õ❛ z ∈ Rn s❛♦ ❝❤♦ W (q, t)z = x˜1. ✣➦t
u1 (s) = q(s)2 B1T eJ (t−s) z ❝❤♦ 0 ≤ s ≤ t ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ ❤➔♠ tr↕♥❣ t❤→✐
❝õ❛ ❤➺ ❧➔✿

f

T

t

e

x1 (t) =

J(t−s)

0
t

q(s)e

=
0

J(t−s)

B1 q(s)

2

T
B1T eJ (t−s) zds

T

B1 B1T eJ (t−s) q(s)dsz

t

eJ(t−s) B1 p(s)ds

+
0
t

eJ(t−s) B1 p(s)ds

+
0

= W (q, t)z + x1 − x1 = x1
✶✹


✈➔

v−1

v−1

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

(i)

i


N B2 u (t) = −

x2 (t) = −

i=0

i=0
v−1

N i B2 ωi = x2 .

=−
i=0

❱➼ ❞ö ✷✳✸✳ ✶✳ ❳➨t ❤➺
x˙ 1 =

1 1
0 1

x1 +

0
1

u, x1 (0) = x01 ,

0 = x2 + [−1 0]u,
✈ỵ✐ n = 4✱nf = 2✱n∞ = 2 ❝â ❞↕♥❣ ❝❤➼♥❤ t➢❝✳ ❚➟♣ ❤đ♣ ❝â t❤➸ ✤↕t ✤÷đ❝ tø


x1 (0) = 0 ✤÷đ❝ ✤÷❛ r❛ ❜ð✐
R˜0 = Im[B1 JB1 ] ⊕ [B2 N B2 ] = R2 ⊕ (R ⊕ {0}) = R3 ⊕ {0}.
✷✳ ❳➨t ❤➺

0 1
0 0

x˙2 = x2 +

−1
−1

u.

˜0 = R2 ✈➔ ❤➔♠ tr↕♥❣ t❤→✐ ❝õ❛ ❤➺
❚➟♣ ❤đ♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ✤÷đ❝ ❝❤♦ ❜ð✐ R
♥➔② ✤÷đ❝ ❝❤♦ ❜ð✐
v−1

u + u˙
u

N i B2 u(i) (t) =

x2 (t) = −
i=0

.


❉♦ ✤â✱ ✤è✐ ✈ỵ✐ ❜➜t ❦ý ω = [ω1 ω2 ]T ∈ R2 ✈➔ t1 > 0 ❝❤ó♥❣ t❛ ❝â t❤➸ ❧ü❛
❝❤å♥ u(t) s❛♦ ❝❤♦ u(t1 ) = ω2 , u(t
˙ 1 ) = ω1 − ω2 ✈➔ ❝❤ó♥❣ t❛ ❝â

x2 (t1 ) =

ω1
ω2

.

✷✳✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔
❇ê ✤➲ ✷✳✸✳ ✭❇ê ✤➲ ❍❛✉t✉s ✕ P♦♣♦✈✮✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✶✳ ❍➺ x˙ = Ax + Bu ❧➔ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ✳

✶✺


✷✳ rank(K) = rank[B AB ...An−1B] = n.
✸✳ ◆➳✉ z ❧➔ ✈➨❝tì ✤➦❝ tr÷♥❣ ❝õ❛ AT t❤➻ z T B = 0.
✹✳ rank[λI − A B] = n, ∀λ ∈ C.

ỵ tố ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ ❈ ❦❤✐ ✈➔ ❝❤➾
❦❤✐ rank[λE − A B] = n, ∀λ ∈ C.
✷✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ ❍➺ t❤è♥❣ ❝♦♥ ♥❤❛♥❤ ✭✷✳✹✮ ❧➔ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ✳
❜✮ rank[B2 N B2 ...N v−1B2] = n∞.
❝✮ rank[N B2] = n∞.
❞✮ rank[E B] = n.
❡✮ ✣è✐ ✈ỵ✐ ♠å✐ ♠❛ tr➟♥ ❦❤ỉ♥❣ s✉② ❜✐➳♥ Q1✱ P1 t❤ä❛ ♠➣♥ E = Q1


I 0
0 0

P1 ,

˜1
B
˜2
B

.
✈ỵ✐ QB =
✸✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ ❍➺ ✭✷✳✷✮ ❧➔ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ✳
❜✮ ❈→❝ ❤➺ t❤è♥❣ ❝♦♥ ❝❤➟♠ ✈➔ ♥❤❛♥❤ ✭✷✳✸✮ ✈➔ ✭✷✳✹✮ ✤➲✉ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥
C✳
❝✮ rank[B1 JB1 ...J n −1B1] = nf ✈➔ rank[B2 N B2 ...N v−1B2] = n∞.
❞✮ rank[λE − A B] = n ❤ú✉ ❤↕♥ ∀λ ∈ C ✈➔ rank[E B] = n.
❡✮ rank[αE − βA B] = n ✈ỵ✐ ∀(α, β) ∈ C2\{(0, 0)}.
f

❈❤ù♥❣ ♠✐♥❤✳

✶✳ ❍➺ t❤è♥❣ ❝♦♥ ❝❤➟♠ ❧➔ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✭❖❉❊✮✱ ❞♦
✤â✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝â ❦❤↔ ♥➠♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤è✐ ✈ỵ✐ ❝→❝ ❤➺ ❜➜t ❜✐➳♥ t❤❡♦ t❤í✐
❣✐❛♥ ✭▲❚■ ✮ t✐➯✉ ❝❤✉➞♥ ✤÷đ❝ →♣ ❞ư♥❣ ✈➔ ✭✷✳✸✮ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ❦❤✐ ✈➔ ❝❤➾
❦❤✐ ❤↕♥❣ [λI − J B1] = nf ❝❤♦ t➜t ❝↔ λ ∈ C ❤ú✉ ❤↕♥✳ ❍ì♥ t❤➳ ♥ú❛✱ ❝❤ó♥❣
t❛ ❝â✿
rank[λE−A B] = rank[λW ET −W AT W B] = rank


λI − J
0
B1
0
λN − I B2

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

.


✷✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ ❤➺ t❤è♥❣ ❝♦♥ ♥❤❛♥❤ ✭✷✳✹✮ ❧➔ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝
C ♥➳✉ t➟♣ ❤đ♣ ❝â t❤➸ ✤↕t ✤÷đ❝ ❧➔
Im[B2 N B2 ...N v−1 B2 ] = Rn∞ ⇐⇒ rank[B2 N B2 ...N v−1 B2 ] = n∞ .

❍➺ (N B2) ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ✭t❤❡♦ ❤➺ ▲❚■ t✐➯✉ ❝❤✉➞♥✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤↕♥❣
[λI − N , B2 ] = n∞ ❝❤♦ ♠å✐ λ ∈ C✳ ❱➻ ✈➟②✱ ✤➙② ❧➔ ✤✐➲✉ ❦✐➺♥ ❝❤ù❛ t➜t ❝↔
λ ∈ σ(N ) = {0} ✈➻ N ❧ô② t✐♥❤ ✈➔ ❞♦ ✤â rank[λI − N , B2 ] = n∞ ✈ỵ✐ ♠å✐
λ ∈ C ⇐⇒ rank[−N B2 ] = rank[N B2 ] = n∞ . ❈❤ó♥❣ t❛ ❝â
rank[E B] = rank[W ET W B] = rank

Inf 0 B1
0 N B2

= nf +rank[N B2 ].

❉♦ ✤â✱ rank[N B2] = n∞ ⇐⇒ rank[E B] = n.

ữỡ tỹ ữ trữợ
❤➺ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ✈➔ ✤➸ x1(0) = 0. ❙❛✉ ✤â ✈ỵ✐ ❜➜t ❦ý t1 > 0
✈➔ ω Rn tỗ t ✤÷đ❝ u ∈ Cpv−1 s❛♦ ❝❤♦
x(t1 ) = w✳ ◆❤÷ ✈➟②✿
˜ 0 = Im[B1 JB1 ...J nf −1 B1 ] ⊕ Im[B2 N B2 ...N v−1 B2 ] = Rn
R

✈➔ rank[B2 N B2 ...N v−1B2] = n∞.
▼➦t ❦❤→❝✱ ✤➸ ❣✐ú ❝→❝ ✤✐➲✉ ❦✐➺♥ t❤ù ❤↕♥❣ ✳ ❈❤ó♥❣ t❛ ❜✐➳t r➡♥❣
⇐⇒ rank[B1 JB1 ...J nf −1 B1 ] = nf

˜0 + {
Rx1 (0) = R

x1
x2

| x1 = eJt x1 (0) ∈ Rnf , x2 = 0 ∈ Rn∞ } = Rn

✈➔ ❞♦ ✈➟② ❤➺ ✭✷✳✷✮ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ✳
α
⇐⇒ rank[λE − A B] = rank[ E − A B] = rank[αE − βA B].
β
❱➼ ❞ö

✷✳✹✳

❳➨t ❤➺
x˙ =


1 1
0 1

x1 +

0
1

−1
0

u.

0 = x2 +
✶✼

u


❈❤ó♥❣ t❛ ❝â rank[B1
−1 0
0 0

0 1
1 1

JB1 ] = rank

= 2


✈➔ rank[B2

BN2 ] =

❉♦ ✤â✱ ❤➺ ❦❤æ♥❣ t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ✱ tr♦♥❣ ❦❤✐ ❤➺
t❤è♥❣ ❝♦♥ ❝❤➟♠ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ✳

rank

= 1 < 2.

●❤✐ ❝❤ó ✷✳✶✳ ✣è✐ ✈ỵ✐ ❝→❝ ❤➺ t❤ù ♥❣✉②➯♥ tr↕♥❣ t❤→✐ ❧ỵ♥ ❝→❝ t✐➯✉ ữủ

ữ r tr ỵ trữợ ổ ũ ủ ✈ỵ✐ ❝→❝ t➼♥❤ t♦→♥ sè✱ ✈➻ ♣❤➙♥ t→❝❤
❤➺ t❤è♥❣ t❤➔♥❤ ✭❲❈❋✮ ❤♦➦❝ ❝→❝ ❣✐→ trà r✐➯♥❣ ❧➔ ❝➛♥ t❤✐➳t✳ ▼ët ❤➺ ♠ỉ t↔
t✉②➳♥ t➼♥❤ t❤❡♦ ❜➜t ❜✐➳♥ t❤í✐ ❣✐❛♥ t❤ỉ♥❣ t❤÷í♥❣ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ R ✭♥❣❤➽❛
❧➔ ❝â t❤➸ ✤✐➲✉ tr t ủ õ t t ữủ ợ ❜➜t ❦ý tf > 0
♥➔♦ ✈➔ x1(0) ∈ R, w R, tỗ t ởt ♥❤➟♥ ✤÷đ❝
u ∈ Cpv−1 s❛♦ ❝❤♦ x(tf ) = ω.

✣à♥❤ ỵ s tữỡ ữỡ

❍➺ ✭✷✳✷✮ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ R.
✷✳ ❍➺ t❤è♥❣ ❝♦♥ ❝❤➟♠ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C.
✸✳ rank[λE − A B] = n ✈ỵ✐ ♠å✐ λ ∈ C ❤ú✉ ❤↕♥✳
✹✳ rank[B1 JB1 ...J n −1B1] = nf .
f

❈❤ù♥❣ ♠✐♥❤✳


✶⇐⇒✷✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❤➺ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ R ♥➳✉
˜ 0 = Im[B1 JB1 ...J nf −1 B1 ] ⊕ Im[B2 N B2 ...N v−1 B2 ]
R
= Rnf ⊕ Im[B2 N B2 ...N v−1 B2 ].

❉♦✱ Im[B1 JB1 ...J n −1B1] = Rn ⇐⇒ ❤➺ t❤è♥❣ ❝♦♥ ❝❤➟♠ ✭✷✳✸✮ ❝â t❤➸
✤✐➲✉ ❦❤✐➸♥ C.
✷⇐⇒✸ ✤÷đ❝ s✉② r❛ trü❝ t✐➳♣✳
✸⇐⇒✹✳ ❍✐➸♥ ♥❤✐➯♥✱ ✈➻ ❦❤↔ ♥➠♥❣ ✤✐➲✉ ❦❤✐➸♥ C ❜❛♦ ❤➔♠ ❦❤↔ ♥➠♥❣ ✤✐➲✉
❦❤✐➸♥ R✳
f

f

❱➼ ❞ư ✷✳✺✳ ✶✳ ❳➨t ❤➺ tr♦♥❣ ❱➼ ❞ư ✷✳✹ ✤÷đ❝ ❝❤♦ ❜ð✐
x˙ =

1 1
0 1

x1 +
✶✽

0
1

u


−1

0

0 = x2 +

u.

❍➺ t❤è♥❣ ❝♦♥ ❝❤➟♠ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C ♥❤÷ ❝❤ó♥❣ t❛ ✤➣ ♥❤➻♥ t❤➜② ✈➔ ❞♦ ✤â
❤➺ t❤è♥❣ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ R✳
✷✳❚❛ ❝â ♠❛ tr➟♥ N ❧➔ ❧ô② ❧✐♥❤ ✈➔ ①➨t ❤➺ N x˙ = x + Bu
ỗ tố ❝♦♥ ♥❤❛♥❤ ✈➔ ✤✐➲✉ ♥➔② ❧✉æ♥ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ ữủ R.

q t ợ tr λE − A✱ t❤➻ ❤➺ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ C

❦❤✐ ✈➔ ❦❤✐ ❤➺ ❝â t❤➸ ✤✐➲✉ ❦❤✐➸♥ R ✈➔ rank[E

B] = n✳

❈❤ù♥❣ ♠✐♥❤✳ ❍➔♠ tê♥❣ q✉→t ✈➔ ❝→❝ ♥❣❤✐➺♠ ♣❤➙♥ ♣❤è✐ ❝❤♦ ♣❤➨♣ ♥❣❤✐➺♠

❣✐→♥ ✤♦↕♥ t↕✐ ♠ët sè ✤✐➸♠ ❦❤→❝ tr♦♥❣ I ❑➼ ❤✐➺✉ Dn = C0∞(R, Rn) ❧➔ t➟♣
❤ñ♣ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈ỉ ❤↕♥ ✈ỵ✐ ❝→❝ ❣✐→ trà tr♦♥❣ Rn ✈➔ ❣✐→ tr♦♥❣ R✳ ❈→❝ ♣❤➛♥
tû ❝õ❛ Dn ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❤➔♠ ❦✐➸♠ tr❛✳
x(t)

✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ▼ët ❤➔♠ t✉②➳♥ t➼♥❤ f : Dn → Rn ✈ỵ✐

✈ỵ✐ ♠å✐ φ1, φ2 ∈ Dn, α1, α2 ∈ R
✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ tê♥❣ q✉→t ❤♦➦❝ ♣❤➙♥ ♣❤è✐ ♥➳✉ ♥â ❧✐➯♥ tö❝✱ ✤â ❧➔ f (φ) → 0
tr♦♥❣ Rn ❝❤♦ t➜t ❝↔ ❝→❝ ❝❤✉é✐ (φi)i∈N ✈ỵ✐ φi → 0 ∈ Dn ✳
❈❤✉é✐ (φi(t))i∈N ❤ë✐ tư ✈➲ ❦❤ỉ♥❣ ♥➳✉ t➜t ❝↔ ❝→❝ ❤➔♠ φi t✐➳♥ tỵ✐ ✈➔

(q)
(φi (t))i∈N ❤ë✐ tư ✤➲✉ ✤➳♥ ✵ ❝❤♦ t➜t ❝↔ q ∈ N0 ✳ ❈❤ó♥❣ t❛ ❜✐➸✉ t❤à ❦❤æ♥❣
❣✐❛♥ ❝õ❛ t➜t ❝↔ ❝→❝ ❤➔♠ s✉② rë♥❣ tr➯♥ Dn t❤❡♦ C n✳ ✣➸ sû ❞ö♥❣ ❝→❝ ❤➔♠ s✉②
rë♥❣ tr♦♥❣ ❦❤✉ỉ♥ ❦❤ê ❝õ❛ ❝→❝ ❤➺ ♠ỉ t↔✱ ❝❤ó♥❣ t❛ ❝➛♥ ❦❤→✐ ♥✐➺♠ ❝→❝ ✤↕♦ ❤➔♠
✈➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣✳ ✣↕♦ ❤➔♠ ❝➜♣ q ❝õ❛ f ✤÷đ❝ ✈✐➳t ❧➔
f (q) , q ∈ N0 ❝õ❛ ❤➔♠ s✉② rë♥❣ f ∈ C n ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐
f (q) (φ) = (−1)q f (φ(q) ) ✈ỵ✐ ♠å✐ φ ∈ Dn .
❍➔♠ f (q) ❧➔ t✉②➳♥ t➼♥❤ ✈➔ ❧✐➯♥ tö❝✱ ✈➻ ✈➟② ♠å✐ ❤➔♠ s✉② rë♥❣ ✤➲✉ ❝â ✤↕♦
❤➔♠ t❤❡♦ tự tỹ tũ ỵ tr C n ợ ởt ❤➔♠ s✉② rë♥❣ f ∈ C n ✈➔ ✈ỵ✐ ♠ët
❤➔♠ X ∈ C n ✈➔ t❤ä❛ ♠➣♥✿
f (α1 φ1 + α2 φ2 ) = α1 f (φ1 ) + α2 f (φ2 )

˙
X(φ)
= f (φ), φ ∈ Dn

✤÷đ❝ ❣å✐ ❧➔ ❣è❝ ❝õ❛ f, ❤❛② ♥â✐ ❝→❝❤ ❦❤→❝ X ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
X˙ = f. ❈❤♦ A ∈ C ∞ (R, Rm,n ) ✈➔ x ∈ C n ❦❤✐ ✤â ♣❤➨♣ ♥❤➙♥ ♠❛ tr➟♥ ✤÷đ❝
✤à♥❤ ♥❣❤➽❛ ❜ð✐✿
✶✾


Ax(φ) = x(AT φ)
✈ỵ✐ ♠å✐ φ ∈ Dn .

❱➼ ❞ư ✷✳✻✳ ❍➔♠ ♣❤➙♥ ❜è ❉❡❧t❛✕❞✐r❛❝ δα ∈ C n ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ δα(φ) =

φ(α) ✈ỵ✐ ♠å✐ φ ∈ Dn , α ∈ R✳ ❱➔ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿
δα (x) =


+∞, x = α,
0, x = α.

●❤✐ ❝❤ó ✷✳✷✳ ❱➻ ✈ỵ✐ φ ∈ Dn ✤➣ ❝❤♦ ✈➔ t > 0 ✤õ ❧ỵ♥✱ t❛ ❝â✿
t

φ(0) = −(φ(t) − φ(0)) = −φ(t)

|t0 =

−

∞

˙
φ(t)dt
=−

0

˙
φ(t)dt

0

˙
˙
H(t)φ(t)dt
=: H(φ),


=−
R

tr♦♥❣ ✤â

H(t) =

0, t < 0,
1, t ≥ 0.

❧➔ ữợ s ú t t ố q 0 = H˙ ✳ ❈❤ó♥❣ t❛ ❝ơ♥❣
❝â t❤➸ ①→❝ ✤à♥❤ sü ❞à❝❤ ❝❤✉②➸♥ ❝õ❛ H t❤❡♦ Hα (t) := H(t − α) ✈➔ δα = H˙ α ✳
❍❛✐ ❤➔♠ s✉② rë♥❣ f1 , f2 ∈ C n ❜➡♥❣ ♥❤❛✉ ♥➳✉ f1 (φ) = f2 (φ) ✈ỵ✐ ♠å✐ φ ∈ Dn ✳
❚r♦♥❣ ♣❤➛♥ s❛✉ ✤➙②✱♠ët ❤➔♠ x : I → Rn , I ⊆ R ✤÷đ❝ ❝♦✐ ❧➔ ♠ët ❤➔♠ ✤÷đ❝
①→❝ ✤à♥❤ tr➯♥ R ❜➡♥❣ ❝→❝❤ ✤➦t x(t) = 0 ✈ỵ✐ t ∈
/ I. ❍ì♥ ♥ú❛✱ tr↕♥❣ t❤→✐
❦❤ỉ♥❣ trì♥ ❝õ❛ ♥❣❤✐➺♠ ❜à ❤↕♥ ❝❤➳ ①↔② r❛ ð ♠ù❝ ❝â t❤➸ ✤➳♠ ✤÷đ❝ sè ✤✐➸♠

τ j ∈ T ⊆ R✳

✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ●✐↔ sû t➟♣ ❤ñ♣ T = {τj ∈ R | τj < τj+1, ∈ Z} ❦❤æ♥❣ ❝â
✤✐➸♠ ❣✐→♥ ✤♦↕♥✳ ❍➔♠ ♣❤➙♥ ♣❤è✐ x ∈ C n ✤÷đ❝ ❣å✐ ❧➔ trì♥ ①✉♥❣ õ õ t

ữủ t ữợ

x = x + ximp , xˆ =

xˆj ,

✭✷✳✻✮


j∈Z

tr♦♥❣ ✤â x
ˆj ∈ C ∞ ([τj , τj+1 ], Rn ) ✤è✐ ✈ỵ✐ ♠å✐ j ∈ Z ✈➔ ♣❤➛♥ ①✉♥❣ ximp ❝â
❞↕♥❣

qj

, cij ∈ Rn , qj ∈ N0 .
cij δτ(i)
j

ximp,j =
i=0

✷✵

✭✷✳✼✮


n
❚➟♣ ❤đ♣ ❝õ❛ t➜t ❝↔ ❝→❝ ❤➔♠ s✉② rë♥❣ trì♥ ①✉♥❣ ✤÷đ❝ ❜✐➸✉ t❤à ❜ð✐ Cimp
(T)✳
n
❇ê ✤➲ ✷✳✹✳ ✶✳ ▼ët ❤➔♠ ♣❤➙♥ ♣❤è✐ x ∈ Cimp
(T) ❞✉② ♥❤➜t ①→❝ ✤à♥❤ ♣❤➙♥
t➼❝❤ ✭✷✳✻✮
n
✷✳ ✣è✐ ✈ỵ✐ x ∈ Cimp

(T) ❝❤ó♥❣ t❛ ❝â t❤➸ ❣→♥ ❣✐→ trà ❤➔♠ x(t) ❝❤♦ ♠é✐ t ∈
R\T ❜ð✐ x(t) = x
ˆj (t) ✈ỵ✐ t ∈ (τj , τj+1 ) ✈➔ ❣✐ỵ✐ ❤↕♥ x(τj − ) = limt→τ xˆj−1 (t)
✈➔ x(τj +) = limt→τ xˆj (t) ✈ỵ✐ ♠å✐ τj ∈ T✳
n
n
✸✳ ❚➜t ❝↔ ❝→❝ ✤↕♦ ❤➔♠ ✈➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ x ∈ Cimp
(T) ❝â tr♦♥❣ Cimp
( T) ✳
n
✹✳ ❚➟♣ ❤đ♣ Cimp
(T) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ✈➔ ✤÷đ❝ ✤â♥❣ t❤❡♦ ♣❤➨♣ ♥❤➙♥
✈ỵ✐ ❝→❝ ❤➔♠ A ∈ C ∞(R, Rm,n)✳
n
✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ❚❤ù tü ①✉♥❣ ❝õ❛ x ∈ Cimp
(T) t↕✐ τj ∈ T ✤÷đ❝ ①→❝ ✤à♥❤
❧➔ iord(x) |τ := −q − 2 ♥➳✉ x ❝â t❤➸ ✤÷đ❝ ❧✐➯♥ ❦➳t ✈ỵ✐ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝ tr♦♥❣
[τj−1 , τj+1 ] ✈➔ q ✱ ✈ỵ✐ 0 ≤ q ≤ ∞ ❧➔ sè ♥❣✉②➯♥ ❧ỵ♥ ♥❤➜t s❛♦ ❝❤♦
j

j

−

+

j

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


◆❤÷ ✤à♥❤ ♥❣❤➽❛ iord(x) |τ := −1 ♥➳✉ x ❝â t❤➸ ❧✐➯♥ ❦➳t ✈ỵ✐ ❤➔♠ ❧✐➯♥ tö❝ tr♦♥❣
[τj−1 , τj+1 ] trø t = τj ✈➔ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔
j

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

▼➦t ❦❤→❝✱ t❤ù tü ①✉♥❣ ❝õ❛ x ✤÷đ❝ ①→❝ ✤à♥❤ ❧➔ iordx := maxτ ∈Tiord(x) |τ ✳
n
(T) ✈➔ A ∈ C ∞ (R, Rm,n )✳ ❚❤➻ iordAx ≤ iordx ❝â
❇ê ✤➲ ✷✳✺✳ ✣➸ x ∈ Cimp
✤➥♥❣ t❤ù❝ ✈ỵ✐ m = n ✈➔ A(τj ) ❦❤ỉ♥❣ t❤➸ ✤↔♦ ♥❣÷đ❝ ❝❤♦ ♠é✐ τj ∈ T✳
❱➼ ❞ư ✷✳✼✳ ❳➨t ♠ỉ ❤➻♥❤ ✈✐ ♠↕❝❤ ✭①❡♠ ❍➻♥❤ ✷✳✷✮ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ❉❆❊ s❛✉
✤➙②
j

x1 − x4 = u(t),
1
C(x˙ 1 − x˙ 2 ) + (x3 − x2 ) = 0,
R
x3 = A(x4 − x2 ),
x4 = 0.

❱ỵ✐ x4 = 0, x1 = u(t), x˙ 1 = u˙ ✈➔ x3 = −Ax2 ❝❤ó♥❣ t❛ ❝â
x˙ 2 = −

1
(A + 1)x2 + u.
˙
CR
✷✶


j


❍➻♥❤ ✷✳✷✿ ▼æ ❤➻♥❤ ✈✐ ♠↕❝❤

❉♦ sü ❝❤✉②➸♥ ❧➺♥❤ ð ✤✐➺♥ →♣ ✤➛✉ ✈➔♦✱ u ❦❤ỉ♥❣ ✈✐ ♣❤➙♥ ✤÷đ❝✳ ✣è✐ ợ u = H
A ữỡ tr ♠æ ❤➻♥❤ ❝â ❞↕♥❣
x1 − x4 = H,
1
(x3 − x2 ) = 0,
R
x2 = 0,

C(x˙ 1 − x˙ 2 ) +

x4 = 0,

✈ỵ✐ ♥❣❤✐➺♠ x1 = H, x2 = 0, x3 = −RCH = −RCδ0 ✈➔ x4 = 0✳ ❍➺ ❧➔ ♠ët
❉❆❊ ❝õ❛ sè ♠ơ v = 2 ✭❤♦➦❝ µ = 1✮ ✈➔ iord f = −1✳ ✣è✐ ✈ỵ✐ ♠ët tr
ỗ t ử x(1) = 0 ❝❤ó♥❣ t❛ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t x
✈ỵ✐ iord x = 0
ố ợ trữớ ủ t ừ ❜➜t ❜✐➳♥ t❤í✐ ❣✐❛♥ t❤ỉ♥❣ t❤÷í♥❣
n
❝â ❞↕♥❣ E x˙ = Ax + f ✈ỵ✐ f ∈ Cimp
(T) ❝õ❛ t❤ù tü ①✉♥❣ ✐♦r❞ f = q ∈
Z ∪ {−∞} ❝❤ó♥❣ t❛ ❝â t❤➸ t✐➳♥ ❤➔♥❤ ♥❤÷ s❛✉✳ ✣➛✉ t✐➯♥✱ ❝❤ó♥❣ t❛ ❝â t❤➸
❝❤✉②➸♥ ✤ê✐ ❝➦♣ ♠❛ tr➟♥ (E, A) t❤➔♥❤ (W CF )
(E, A) ∼ (W ET, W AT ) = (


Inf 0
0 N

❉♦ ✤â✱ ❝❤ó♥❣ t❛ ❝â
x˙ 1 = Jx1 + f1 ,
N x˙ 2 = Jx2 + f2 ,

✷✷

,

J 0
0 In∞

).

✭✷✳✽✮
✭✷✳✾✮