▼ô❝ ❧ô❝
▼ét sè ❦Ý ❤✐Ö✉ sö ❞ô♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥ ✹
▲ê✐ ♥ã✐ ➤➬✉ ✺
❈❤➢➡♥❣ ✶✿ ❈➡ së t♦➳♥ ❤ä❝ ✽
✶✳✶✳ P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝❤❐♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✶✳✷✳ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✸✳ ❇➭✐ t♦➳♥ æ♥ ➤Þ♥❤ ❤♦➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✶✳✹✳ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✶✳✺✳ ▼ét sè ❜æ ➤Ò ❜æ trî ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
❈❤➢➡♥❣ ✷✿ ●✐í✐ t❤✐Ö✉ ♠ét sè ❦Õt q✉➯ ✈Ò ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö
❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝ã trÔ ✈➭ ❝ã trÔ ✈í✐ ❣✐➯ t❤✐Õt ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ✷✵
✷✳✶✳ ❚Ý♥❤ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ✈➭ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tô❝
❦❤➠♥❣ ➠t➠♥➠♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✷✳ ▼è✐ ❧✐➟♥ ❤Ö ❣✐÷❛ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
✈➭ tÝ♥❤ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❝ñ❛ ❤Ö
t✉②Õ♥ tÝ♥❤ ❧✐➟♥ tô❝ ❦❤➠♥❣ ➠t➠♥➠♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷✳✸✳ ❇➭✐ t♦➳♥ æ♥ ➤Þ♥❤ tr♦♥❣ L
2
✈➭ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❜Ò♥ ✈÷♥❣ ❝❤♦ ❤Ö
t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
❈❤➢➡♥❣ ✸✿ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞

([t, s], R
n
) ❧➭ t❐♣ ❝➳❝ ❤➭♠ ▲
2
✲❦❤➯ tÝ❝❤ tr➟♥ [s, t]✳
• A
T
❧➭ ♠❛ tr❐♥ ❝❤✉②Ó♥ ✈Þ ❝ñ❛ ♠❛ tr❐♥ A✳
• Q ≥ 0 (Q > 0)✱ ❦Ý ❤✐Ö✉ ♠❛ tr❐♥ Q ①➳❝ ➤Þ♥❤ ❦❤➠♥❣ ➞♠ ✭t➢➡♥❣ ø♥❣ ①➳❝
➤Þ♥❤ ❞➢➡♥❣✮✱ tø❝ ❧➭
Qx, x ≥ 0 (Qx, x > 0).
• M(R
n
+
) ❧➭ t❐♣ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ ➤è✐ ①ø♥❣✱ ①➳❝ ➤Þ♥❤ ❦❤➠♥❣ ➞♠ tr♦♥❣ R
n
✱
❧✐➟♥ tô❝ tr➟♥ t ∈ [0,∞)✳
• BM
+
(0,∞) ❧➭ t❐♣ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ ❜Þ ❝❤➷♥✱ ➤è✐ ①ø♥❣✱ ①➳❝ ➤Þ♥❤ ❦❤➠♥❣
➞♠ tr♦♥❣ R
n
✱ ❧✐➟♥ tô❝ tr➟♥ t ∈ [0,∞)✳
• BMU
+
(0,∞) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ ❜Þ ❝❤➷♥✱ ➤è✐ ①ø♥❣✱ ①➳❝ ➤Þ♥❤
❞➢➡♥❣ ➤Ò✉ tr♦♥❣ R
n
✱ ❧✐➟♥ tô❝ tr➟♥ t ∈ [0,∞)✳


♣❤➢➡♥❣ ♣❤➳♣ ➤➢î❝ sö ❞ô♥❣ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ò✉ ❦❤✐Ó♥ ♥❤➢✿ ➤✐Ò✉ ❦❤✐Ó♥ t➢➡♥❣
t❤Ý❝❤ ✭❛❞❛♣t✐✈❡ ❝♦♥tr♦❧✮✱ ➤✐Ò✉ ❦❤✐Ó♥ ❜Ò♥ ✈÷♥❣✱ ➤✐Ò✉ ❦❤✐Ó♥ tè✐ ➢✉✱✳✳✳
❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ sö ❞ô♥❣ ♣❤➢➡♥❣ ♣❤➳♣ H
∞
✭❜➭✐ t♦➳♥ ➤✐Ò✉
❦❤✐Ó♥ H
∞
✮ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ò✉ ❦❤✐Ó♥ ➤Ó ➤➵t ➤➢î❝ q✉➳ tr×♥❤ ➤✐Ò✉ ❦❤✐Ó♥ æ♥ ➤Þ♥❤
❜Ò♥ ✈÷♥❣✳ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❧➭ sù ❦Õt ❤î♣ ❝ñ❛ ❜➭✐ t♦➳♥ æ♥ ➤Þ♥❤ ❤♦➳ ✈➭
❜➭✐ t♦➳♥ tè✐ ➢✉ ❤♦➳✳ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❧➭ t×♠ ❤➭♠ ➤✐Ò✉ ❦❤✐Ó♥ ➤Ó ❤Ö ➤➲
❝❤♦ ❧➭ æ♥ ➤Þ♥❤ ✈➭ t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ♠ø❝ ❝❤♦ tr➢í❝✳ ❇➭✐ t♦➳♥ ➤✐Ò✉
❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ➠t➠♥➠♠✱ ♣❤➢➡♥❣ ♣❤➳♣ ♣❤æ ❞ô♥❣ ❧➭ sö ❞ô♥❣ ❤➭♠
▲②❛♣✉♥♦✈✲❑r❛s♦✈s❦✐✐ ✈➭ ➤✐Ò✉ ❦✐Ö♥ æ♥ ➤Þ♥❤ ➤➵t ➤➢î❝ ❞ù❛ tr➟♥ ✈✐Ö❝ ❣✐➯✐ ♥❣❤✐Ö♠
❝ñ❛ ❜✃t ➤➻♥❣ t❤ø❝ ♠❛ tr❐♥ t✉②Õ♥ tÝ♥❤ ❤♦➷❝ ♣❤➢➡♥❣ tr×♥❤ ❘✐❝❝❛t✐ ➤➵✐ sè✳ ➜è✐ ✈í✐
❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ t❤× ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤➢î❝ ❞ù❛ tr➟♥ ♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣
tr×♥❤ ❘✐❝❝❛t✐ ✈✐ ♣❤➞♥✳ ❇➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ➤ã✱ tr♦♥❣ ❬✾✱ ✶✵❪ ❝➳❝ t➳❝ ❣✐➯ ➤➲ ➤➢❛
r❛ ➤✐Ò✉ ❦✐Ö♥ ➤ñ ➤Ó ❣✐➯✐ ➤➢î❝ ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣
➠t➠♥➠♠ ❦❤➠♥❣ ❝ã trÔ ✈í✐ ❣✐➯ t❤✐Õt ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❝ñ❛ ❤Ö ➤✐Ò✉ ❦❤✐Ó♥✳
▲✉❐♥ ✈➝♥ ❣å♠ ✸ ❝❤➢➡♥❣✿
❈❤➢➡♥❣ ✶ tr×♥❤ ❜➭② ♥❤÷♥❣ ❦✐Õ♥ t❤ø❝ ❝➡ së ✈Ò ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t❤➢ê♥❣✱
♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝ã ❝❤❐♠✱ tÝ♥❤ æ♥ ➤Þ♥❤ ✈➭ ♣❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ▲②❛♣✉♥♦✈ ➤è✐

ộ ù ớ sự q t t ề ệ ủ trờ ọ tự
ò tố ề ể ệ ọ rt ề ữ ó
ữ ồ ộ ự ớ ể ó ộ ợ ọ t tr ổ
ứ ử ờ t t tớ t
ị ó tr
ì tờ ự t ó tể
tr ỏ tế sót ế rt ợ sự ó ý ủ t


❈❤➢➡♥❣ ✶
❈➡ së t♦➳♥ ❤ä❝
❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ❝ñ❛ ♣❤➢➡♥❣
tr×♥❤ ✈✐ ♣❤➞♥ ❝ã ❝❤❐♠✱ tÝ♥❤ æ♥ ➤Þ♥❤ ✈➭ ♣❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ▲②❛♣✉♥♦✈ ➤è✐ ✈í✐ ❤Ö
♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝ã ❝❤❐♠✱ s❛✉ ➤ã ➤Þ♥❤ ♥❣❤Ü❛ ✈➭ ♥➟✉ ❝➳❝ ❦Õt q✉➯ ❧✐➟♥ q✉❛♥
➤Õ♥ ❝➳❝ ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝✱ ❜➭✐ t♦➳♥ æ♥ ➤Þ♥❤ ❤♦➳ ✈➭ ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥
H
∞
♠➭ ❧✉❐♥ ✈➝♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ sö ❞ô♥❣✳
✶✳✶ P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝❤❐♠
✶✳✶✳✶ P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t❤➢ê♥❣
❳Ðt ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥



˙x = f(t, x), t ∈ I = [t
0
, t
0
+ b]
x(t

t
t
0
f(s, x(s))ds
✶✳✶✳✷ P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝❤❐♠
●✐➯ sö h > 0✳ ❑Ý ❤✐Ö✉ C = C([−h, 0], R
n
) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝➳❝ ❤➭♠ ❧✐➟♥ tô❝
tõ [−h, 0] ✈➭♦ R
n
✈í✐ ❝❤✉➮♥ ➤➢î❝ ①➳❝ ➤Þ♥❤ ❜ë✐ φ = sup
−h≤θ≤0
φ(θ). ❱í✐
❜✃t ❦× t ≥ 0✱ ➤➷t x
t
(θ) = x(t + θ),−h ≤ θ ≤ 0 ❧➭ ➤♦➵♥ q✉ü ➤➵♦ ❝ñ❛ x(t) ✈í✐
❝❤✉➮♥ x
t
 = sup
s∈[−h,0]
x(t + s). P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝❤❐♠ ✭❝ã trÔ✮ ❞➵♥❣
˙x(t) = f(t, x
t
), t ≥ 0, ✭✶✳✷✮
x(t) = φ(t), t ∈ [−h, 0],
tr♦♥❣ ➤ã f : R
+
× C → R
n
❧➭ ❤➭♠ ❝❤♦ tr➢í❝✳ P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝ã trÔ ➤➢î❝

th
x(s)ds, t 0,
x(t) = (t), t [h, 0],
tr ó h 0; x(t) R
n
A(t), A
1
(t) R
nìn
tr tụ
trớ tr R
+
C([h, 0], R
n
) ớ
= sup
t[h,0]
(t).
P trì tế tí t ó trễ ỗ ợ
x(t) = A(t)x(t) + A
1
(t)x(t h) + A
2
(t)

t
tk
x(s)ds, t 0,
x(t) = (t), t [ max(h, k), 0],
tr ó h, k 0; x(t) R

< tì
x(t
0
, )(t) < , t t
0
.

ệ ủ ệ ợ ọ ổ ị tệ ế ó ổ ị
ữ ớ ỗ t
0
0 tồ t = (t
0
) > 0 s ớ ọ C t
< t ó
lim
t
x(t
0
, )(t) = 0.
ệ ủ ệ ợ ọ ổ ị ũ ế tồ t số M >
0, > 0 s ọ ệ ủ ệ t
x(t
0
, )(t) Me
(tt
0
)
, t t
0
.

t
)
3
x(t)
2
ớ ọ ệ x(t) ủ ệ
ị ý ế ệ f tồ t tì ệ ổ
ị tệ
t ề ể ợ
ét ột ệ tố ề ể t ở trì tế tí í
ệ [A(t), B(t)]
x(t) = A(t)x(t) + B(t)u(t), t 0,

tr ó x(t) R
n
t tr t u(t) R
m
t ề ể n
m; A(t) R
nìn
, B(t) R
nìm
, t 0 tr tụ tr R ột
ét u(t) ị tr [0,) tí ị trị tr
R
m
sẽ ợ ọ ề ể ợ ủ ệ ớ ề
ể ợ t tờ tr L
p
([0,), R

1
) ợ ọ ề
ể ợ s tờ t
1
> 0 ế tồ t ề ể ợ u(t) s
ệ x(t, x
0
, u) ủ ệ t ề ệ
x(0, x
0
, u) = x
0
, x(t
1
, x
0
, u) = x
1
.
ị ĩ ệ [A(t), B(t)] ọ ề ể ợ t ề 0 ế
ớ t ì tr t x
0
R
n
tồ t ột tờ t
1
> 0 s (x
0
, 0)
ề ể ợ s tờ t

❱Ý ❞ô ✶✳✷✳✹✿ ❳Ðt tÝ♥❤ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❝ñ❛ ❤Ö





˙x
1
= −2x
1
+ 2x
2
+ u
˙x
2
= x
1
− x
2
.
❚❛ ❝ã
A =


−2 2
1 −1


B =


), M
1
(t
1
), ..., M
n
(t
1
)] = n,
tr♦♥❣ ➤ã
M
0
(t) = B(t),
✶✸
M
k+1
(t) = −A(t)M
k
(t) +
d
dt
M
k
(t), k = 0, 1, ..., n − 1.
❈❤ó ý r➺♥❣ ♥Õ✉ ❤Ö ❧➭ ❞õ♥❣✱ tø❝ ❧➭ ❝➳❝ ♠❛ tr❐♥ A(.), B(.) ❧➭ ❤➺♥❣ sè✱ t❤×
❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ❑❛❧♠❛♥ tr♦♥❣ ❤❛✐ ➤Þ♥❤ ❧ý ✶✳✷✳✸ ✈➭ ✶✳✷✳✺ ❧➭ ➤å♥❣ ♥❤✃t✳
❱Ý ❞ô ✶✳✷✳✻✿ ❳Ðt ❤Ö ✭✶✳✸✮ tr♦♥❣ ➤ã
A(t) =



0 e
− cos t


,
M
1
(t) = −A(t)B(t) +
d
dt
M
0
(t) =


−3
2
cos te
− sin t
0
0
3
2
sin te
− cos t


.
❱× ♠❛ tr❐♥ [M
0

tr♦♥❣ ➤ã
W (t, t + N) =

t+N
t
U(N, s)B(s)B
T
(s)U
T
(N, s)ds.
✶✹
✶✳✸ ❇➭✐ t♦➳♥ æ♥ ➤Þ♥❤ ❤♦➳
❳Ðt ❤Ö ➤✐Ò✉ ❦❤✐Ó♥ ♠➠ t➯ ❜ë✐ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥





˙x(t) = f(t, x(t), u(t)), t ≥ 0
x(t) ∈ R
n
, u(t) ∈ R
m
.
✭✶✳✺✮
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✶✿ ❍Ö ✭✶✳✺✮ ❣ä✐ ❧➭ æ♥ ➤Þ♥❤ ❤♦➳ ➤➢î❝ ♥Õ✉ tå♥ t➵✐ ❤➭♠
h(x) : R
n
→ R
m

✶✺
❚r➢ê♥❣ ❤î♣ ❤Ö ✭✶✳✺✮ ❧➭ ❤Ö ♣❤✐ t✉②Õ♥✱ t❛ ❝ã ➤Þ♥❤ ❧ý s❛✉✿
➜Þ♥❤ ❧ý ✶✳✸✳✹✿ ❳Ðt ❤Ö ➤✐Ò✉ ❦❤✐Ó♥ ♣❤✐ t✉②Õ♥ ✭✶✳✺✮✳ ●✐➯ sö tå♥ t➵✐ ❤➭♠ V (t, x) ✈➭
❤➭♠ ✈Ð❝t➡ h(x) : R
n
→ R
m
s❛♦ ❝❤♦
✐✮ V (t, x) ①➳❝ ➤Þ♥❤ ❞➢➡♥❣
✐✐✮ ❚å♥ t➵✐ γ(.) ∈ K :
∂V
∂x
f(x, h(x)) ≤ −γ(x), ∀x ∈ R
n
\ 0.
❑❤✐ ➤ã ❤Ö ❧➭ æ♥ ➤Þ♥❤ ❤♦➳ ➤➢î❝ ✈í✐ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝ u(t) = h(x(t))✳
❱Ý ❞ô ✶✳✸✳✺✿ ❳Ðt tÝ♥❤ æ♥ ➤Þ♥❤ ❤♦➳ ➤➢î❝ ❝ñ❛ ❤Ö ♣❤✐ t✉②Õ♥





˙x
1
= x
2
− x
3
1
− u

1
, u
2
), x = (x
1
, x
2
). ❚❛ ❝ã
V (0, 0) = 0,
a((x
1
, x
2
) ≤ V (x
1
, x
2
) ≤ b((x
1
, x
2
))
✈➭
∂V
∂x
f(x, h(x)) = 2x
1
. ˙x
1
+ 2x

∞
❳Ðt ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠
˙x(t) = A(t)x(t) + B(t)u(t) + B
1
(t)ω(t), t ≥ 0,
z(t) = C(t)x(t) + D(t)u(t), t ≥ 0, ✭✶✳✻✮
x(0) = x
0
, x
0
∈ R
n
,
tr♦♥❣ ➤ã x(t) ∈ R
n
❧➭ ✈❡❝t➡ tr➵♥❣ t❤➳✐✱ u(t) ∈ R
m
❧➭ ❤➭♠ ➤✐Ò✉ ❦❤✐Ó♥✱ ω(t) ∈ R
r
❧➭ ❜✐Õ♥ ♥❤✐Ô✉✱ z(t) ∈ R
l
❧➭ ❤➭♠ q✉❛♥ s➳t✱ A(t) ∈ R
n×n
, B(t) ∈ R
n×m
, B
1
(t) ∈
R
n×r

0
x
0

2
+

∞
0
ω(t)
2
dt
≤ γ ✭✶✳✽✮
✈í✐ s✉♣r❡♠✉♠ tr➟♥ ♠ä✐ ❣✐➳ trÞ ❜❛♥ ➤➬✉ x
0
∈ R
n
✈➭ ♠ä✐ ❤➭♠ ♥❤✐Ô✉ ❦❤➳❝ ❦❤➠♥❣
ω ∈ L
2
([0,∞), R
r
)✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✹✳✷✿ ❬✼❪ ❈❤♦ γ > 0✳ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❜Ò♥ ✈÷♥❣ ❝❤♦ ❤Ö
✭✶✳✻✮ ❧➭ ❜➭✐ t♦➳♥ t×♠ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝ u(t) = K(t)x(t) t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥
s❛✉✿
✭✐✮ ▼ä✐ ♥❣❤✐Ö♠ ❝ñ❛ ❤Ö ➤ã♥❣
˙x(t) = [A(t) + B(t)K(t)]x(t) + B

dt
≤ γ ✭✶✳✶✵✮
✈í✐ s✉♣r❡♠✉♠ tr➟♥ ♠ä✐ ❣✐➳ trÞ ❜❛♥ ➤➬✉ x
0
∈ R
n
✈➭ ♠ä✐ ❤➭♠ ♥❤✐Ô✉ ❦❤➳❝ ❦❤➠♥❣
ω ∈ L
2
([0,∞), R
r
)✳
✶✳✺ ▼ét sè ❜æ ➤Ò ❜æ trî
❇æ ➤Ò ✶✳✺✳✶✿ ❬✼❪ ✭❇✃t ➤➻♥❣ t❤ø❝ ♠❛ tr❐♥ ❈❛✉❝❤②✮ ❈❤♦ ◗✱ ❙ ❧➭ ❤❛✐ ♠❛ tr❐♥ ➤è✐
①ø♥❣ ✈➭ S > 0✱ ❦❤✐ ➤ã
2Qy, x − Sy, y ≤ QS
−1
Q
T
x, x, ∀x, y ∈ R
n
.
❇æ ➤Ò ✶✳✺✳✷✿ ❱í✐ ♠ä✐ ♠❛ tr❐♥ ➤è✐ ①ø♥❣ ①➳❝ ➤Þ♥❤ ❞➢➡♥❣ W ∈ R
n×n
✱ ✈➠ ❤➢í♥❣
ν ≥ 0 ✈➭ ❤➭♠ ✈Ð❝t➡ ω : [0, ν] → R
n
s❛♦ ❝❤♦ ❝➳❝ tÝ❝❤ ♣❤➞♥ ❝ã ❧✐➟♥ q✉❛♥ ➤Ò✉ ①➳❝
➤Þ♥❤✱ t❛ ❝ã


(t)P (t) + P (t)A(t) − P (t)B(t)B
T
(t)P (t) + Q(t) = 0 ✭✶✳✶✶✮
t❛ ❝ã ♠ét sè ❜æ ➤Ò s❛✉✿
❇æ ➤Ò ✶✳✺✳✹✿ ❬✼❪ ●✐➯ sö A(t), B(t) ❜Þ ❝❤➷♥ tr➟♥ R
+
✳ ◆Õ✉ ❤Ö [A(t), B(t)] ❧➭
➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❤♦➭♥ t♦➭♥ ✈Ò 0 t❤× ✈í✐ ❜✃t ❦× ♠❛ tr❐♥ Q ∈ BM
+
(0,∞)✱ ♣❤➢➡♥❣
tr×♥❤ ✈✐ ♣❤➞♥ ❘✐❝❝❛t✐ ✭✶✳✶✶✮ ❝ã ♥❣❤✐Ö♠ P ∈ BM
+
(0,∞)✳
✶✽
❇æ ➤Ò ✶✳✺✳✺✿ ❬✾❪ ◆Õ✉ ❤Ö [A(t), B(t)] ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ➤Ò✉ ❤♦➭♥ t♦➭♥ t❤× ❦❤➻♥❣
➤Þ♥❤ s❛✉ ❧✉➠♥ ➤ó♥❣✿
P❤➢➡♥❣ tr×♥❤ ❘✐❝❝❛t✐ ✈✐ ♣❤➞♥ ✭✶✳✶✶✮✱ tr♦♥❣ ➤ã Q(t) = I✱ ❝ã ♥❣❤✐Ö♠ P ∈
M(R
n
+
) ❜Þ ❝❤➷♥ ➤Ò✉ tr➟♥ ✈➭ ❞➢í✐✱ tø❝ ❧➭ tå♥ t➵✐ β
1
, β
2
≥ 0 t❤♦➯ ♠➲♥
β
1
≤ P (t) ≤ β
2
, ∀t ∈ R

0
∈ R
n
,
✷✵
tr♦♥❣ ➤ã x(t) ∈ R
n
❧➭ ✈❡❝t➡ tr➵♥❣ t❤➳✐✱ u(t) ∈ R
m
❧➭ ❤➭♠ ➤✐Ò✉ ❦❤✐Ó♥✱ ω(t) ∈ R
r
❧➭ ❜✐Õ♥ ♥❤✐Ô✉✱ z(t) ∈ R
l
❧➭ ❤➭♠ q✉❛♥ s➳t✱ A(t) ∈ R
n×n
, B(t) ∈ R
n×m
, B
1
(t) ∈
R
n×r
, C(t) ∈ R
l×n
, D(t) ∈ R
l×m
❧➭ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ ❧✐➟♥ tô❝ ❝❤♦ tr➢í❝ tr➟♥ R
+
✳
❍➭♠ ♥❤✐Ô✉ ω(t) ❧➭ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ♥Õ✉ ω ∈ L

γ
B
1
(t)B
T
1
(t)

P (t) + I = 0, ✭✷✳✸✮
✈➭ ❤➭♠ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝ ❧➭
u(t) = −B
T
(t)P (t)x(t), t ≥ 0.
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ❤Ö [A(t), B(t)] ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ➤Ò✉ ❤♦➭♥ t♦➭♥✱ t❤❡♦
❜æ ➤Ò ✶✳✺✳✺✱ ♣❤➢➡♥❣ tr×♥❤ ❘❉❊ ✭✷✳✸✮ ❝ã ♥❣❤✐Ö♠ P (t) ∈ M(R
n
+
) t❤♦➯ ♠➲♥ ➤✐Ò✉
❦✐Ö♥
β
1
≤ P (t) ≤ β
2
, ∀t ∈ R
+
.
❱í✐ ❤➭♠ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝ u(t) = −B
T
(t)P (t)x(t) ✈➭ ❤Ö ➤ã♥❣ ✈í✐ ω = 0✿
˙x(t) = [A(t) − B(t)B

T
(t)C(t)x(t), x(t)
≤ −x(t)
2
❜ë✐ ✈×
P (t)B(t)B
T
(t)P (t)x(t), x(t) ≥ 0
P (t)B
1
(t)B
T
1
(t)P (t)x(t), x(t) ≥ 0
C
T
(t)C(t)x(t), x(t) ≥ 0, ∀t ≥ 0.
❱❐② t❤❡♦ ➜Þ♥❤ ❧ý ✶✳✶✳✸✱ ❤Ö ➤ã♥❣ ✈í✐ ω = 0 ❧➭ æ♥ ➤Þ♥❤ t✐Ö♠ ❝❐♥✳
❚✐Õ♣ t❤❡♦ ❝❤ó♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ò✉ ❦✐Ö♥ ✭✶✳✽✮ ❝ñ❛ ✈í✐ ♠ä✐ ❣✐➳ trÞ ❜❛♥ ➤➬✉
x
0
∈ R
n
✈➭ ❤➭♠ ♥❤✐Ô✉ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ω(t)✳ ❚❛ ❝ã
˙
V (t, x(t)) = −x(t)
2
− P (t)B
T
(t)B(t)P (t)x(t), x(t)

2
− γω(t)
2

dt
≤

∞
0

z(t)
2
− γω(t)
2
+
˙
V (t, x(t))

dt −

∞
0
˙
V (t, x(t))dt
✷✷
≤

∞
0


1
(t)B
T
1
(t)P (t)x(t), x(t).
❑❤✐ ➤ã

∞
0

z(t)
2
− γω(t)
2

dt ≤ P(0)x
0
, x
0

≤ P (0)x
0

2
.
❱❐②
sup

∞
0

2
([0,∞), R
r
)✳ ❉♦ ➤ã t❤❡♦ ➤Þ♥❤ ❧ý ✶✳✹✳✶✱ ❜➭✐ t♦➳♥
➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö ✭✷✳✶✮ ❝ã ❧ê✐ ❣✐➯✐✳ 
❱Ý ❞ô ✷✳✶✳✷✿ ❈❤♦ γ > 0✳ ❳Ðt ❤Ö ✭✷✳✶✮ tr♦♥❣ ➤ã
A(t) =


sin 2t 0
0 −1


, B(t) =


e
− cos
2
t
0
0 e
−t


,
B
1

11
4
sin t
√
11
4
cos t
0 0





, D(t) =





1 0
0 0
0 1





.
❚❛ ❝ã D
T

T
(s)U
T
(N, s)x
2
ds = e
−2 cos
2
N
x
2
1
N + e
−2N
x
2
2
N.
❱×
e
−2 cos
2
N
≥ e
−2N
, e
−2 cos
2
N
≤ 1, ∀N ≥ 1

+ e
2(s−t)
≤ e
2
+ 1
✈í✐ s < t ♥➟♥ ➤✐Ò✉ ❦✐➟♥ ✭✐✐✮ ❝ñ❛ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✼ t❤♦➯ ♠➲♥✳ ❑❤✐ ➤ã t❛ ❝ã ❤Ö
[A(t), B(t)] ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ➤Ò✉ ❤♦➭♥ t♦➭♥✳
❱❐② ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
✭✷✳✶✮ ❝ã ❧ê✐ ❣✐➯✐ ✈í✐ u(t) ➤➢î❝ ①➳❝ ➤Þ♥❤ ❜ë✐
u(t) = −B
T
(t)P (t)x(t)
tr♦♥❣ ➤ã ♥❣❤✐Ö♠
P (t) =


p
1
(t) 0
0 p
2
(t)


❝ñ❛ ❘❉❊ ✭✷✳✸✮ ➤➢î❝ ➤Þ♥❤ ♥❣❤Ü❛ ❜ë✐ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥





2
(t) + 2p
2
(t) −

e
−2t
−
1
64
e
−2 cos
2
t−10

p
2
2
(t) +
1
4
cos
2
t + 1 = 0.
✷✳✷ ▼è✐ ❧✐➟♥ ❤Ö ❣✐÷❛ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
✈➭ tÝ♥❤ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝
❝ñ❛ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠
❳Ðt ❤Ö ✭✷✳✶✮ ✈í✐ ❣✐➯ t❤✐Õt
D

u(t) = −B
T
(t)X(t)x(t), t ≥ 0.
❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❤➭♠ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝ u(t) = −B
T
(t)X(t)x(t) ✈➭ ❤Ö ➤ã♥❣
✈í✐ ω(t) = 0
˙x(t) = [A(t) − B(t)B
T
(t)X(t)]x(t), ✭✷✳✻✮
①Ðt ❤➭♠ ▲②❛♣✉♥♦✈ ❞➵♥❣✿
V (t, x) = X(t)x, x.
❉♦ X ∈ BMU
+
(0,∞) ♥➟♥ tå♥ t➵✐ λ
1
, λ
2
> 0 s❛♦ ❝❤♦
λ
1
x
2
≤ V (t, x) ≤ λ
2
x
2
, ∀t ≥ 0
➜➵♦ ❤➭♠
˙

XBB
T
Xx(t), x(t) ≥ 0,
XB
1
B
T
1
X(t), x(t) ≥ 0,
C
T
Cx(t), x(t) ≥ 0.
▼➷t ❦❤➳❝ R ∈ BMU
+
(0,∞) ♥➟♥ tå♥ t➵✐ λ
3
> 0 s❛♦ ❝❤♦
Rx(t), x(t) ≥ λ
3
x
2
.
❉♦ ➤ã
˙
V (t, x) ≤ −λ
3
x
2
≤ −
λ

−
λ
3
2λ
2
t
, ∀t ≥ 0.
❱❐② ❤Ö ➤ã♥❣ ✭✷✳✻✮ æ♥ ➤Þ♥❤ ♠ò ♥➟♥ æ♥ ➤Þ♥❤ t✐Ö♠ ❝❐♥✳
➜Ó ❤♦➭♥ t❤➭♥❤ ❝❤ø♥❣ ♠✐♥❤✱ ❝❤ó♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ò✉ ❦✐Ö♥ ✭✶✳✽✮ ✈í✐ ♠ä✐
❣✐➳ trÞ ❜❛♥ ➤➬✉ x
0
∈ R
n
✈➭ ❤➭♠ ♥❤✐Ô✉ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ω(t)✳
❱í✐ u(t) = −B
T
(t)X(t)x(t) ✈➭ ➤✐Ò✉ ❦✐Ö♥ ✭✷✳✹✮ t❛ ❝ã
z
2
= C
T
Cx, x + X(t)B(t)B
T
(t)X(t)x, x.
✷✻