Bài toán điều khiển H – vô cùng cho một lớp hệ phương trình vi phân không ôtônôm

▼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
∞

❝❤♦ ♠ét ❧í♣ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥
❦❤➠♥❣ ➠t➠♥➠♠ ✹✵
✸✳✶✳ ➜✐Ò✉ ❦❤✐Ó♥ H
∞
❜Ò♥ ✈÷♥❣ ❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ
❤➺♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✷✳ ➜✐Ò✉ ❦❤✐Ó♥ H
∞
❜Ò♥ ✈÷♥❣ ❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ
❜✐Õ♥ t❤✐➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✷
✸✳✸✳ ➜✐Ò✉ ❦❤✐Ó♥ H
∞
❜Ò♥ ✈÷♥❣ ❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ
❤ç♥ ❤î♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
❑Õt ❧✉❐♥ ✻✷
❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✻✸
✸
▼ét sè ❦Ý ❤✐Ö✉ sö ❞ô♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥
• R
+
❧➭ t❐♣ ❝➳❝ sè t❤ù❝ ❦❤➠♥❣ ➞♠✳
• R
n
❧➭ ❦❤➠♥❣ ❣✐❛♥ ❊✉❝❧✐❞ n ❝❤✐Ò✉ ✈í✐ ❝❤✉➮♥ . ✈➭ tÝ❝❤ ✈➠ ❤➢í♥❣ ., .✳
• R
n×m
❧➭ t❐♣ ❝➳❝ ♠❛ tr❐♥ ❝✃♣ n × m✳
• L
2

([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,∞)✳

• C([a, b], R
n
) ❧➭ t❐♣ ❝➳❝ ❤➭♠ ❧✐➟♥ tô❝ tr➟♥ [a, b] ✈➭ ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ R
n
✳
✹
ờ ó
ý tết ề ể t ọ ột tr ữ ĩ ự t ọ
ứ ụ q trọ ớ ợ t ệ t trể tr t ỉ
ụ í ủ ý tết ề ể t ọ ữ ì
t ọ ợ ứ ụ ể qết ữ ề ị tí ủ
ệ tố ề ể t ề t tự tễ tr ọ ệ
tế ợ t ở trì t ọ ề ể t tý
ế ữ ụ t ọ t ệ ể tì ờ r tự tễ
ề t ề ề ĩ tt ề ể tờ q ế ệ
ộ ự t ở trì PP t ọ ớ tờ
tụ rờ r
x(t) = f(t, x(t), u(t))
x(k + l) = f(k, x(k), u(k)), k = 0, 1, 2, ...
tr ó x(.) ế tr t t ố tợ r u(.) ế ề ể
t ố tợ ủ ệ tố ột ệ tố ề ể
ột ì t ọ ợ t ở trì t ọ ể tị sự
ệ r
ột tr ữ ụ í í ủ t ề ể ệ tố tì
ề ể s ệ tố r ó ữ tí t t
ố ứ ữ ụ í ụ tể ủ ệ tố r ờ t
t ề ể t ề ể ợ t
ổ ị ổ ị t ề ể tố ệ ý tết ề
ể t ọ ợ t trể t ớ ý tết ứ ụ
ợ ề t ọ tr ớ q t ứ ó ề

♣❤➢➡♥❣ ♣❤➳♣ ➤➢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✐

✈í✐ ❤Ö P❚❱P ❝❤❐♠✳ ❚✐Õ♣ ➤Õ♥ tr×♥❤ ❜➭② ❝➳❝ ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝✱ ❜➭✐ t♦➳♥
æ♥ ➤Þ♥❤ ❤♦➳ ✈➭ ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
✳ P❤➬♥ ❝✉è✐ ❈❤➢➡♥❣ ✶ ➤Ò ❝❐♣ ➤Õ♥ ♠ét sè
❜æ ➤Ò ➤➢î❝ sö ❞ô♥❣ ♥❤✐Ò✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✳
❚r♦♥❣ ❈❤➢➡♥❣ ✷✱ ❧✉❐♥ ✈➝♥ ❣✐í✐ t❤✐Ö✉ ♠ét sè ❦Õt q✉➯ ➤➲ ❝ã ✈Ò ➤✐Ò✉ ❦✐Ö♥ ❣✐➯✐
➤➢î❝ ❝ñ❛ ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝ã trÔ
tr♦♥❣ ❬✾❪ ❞ù❛ tr➟♥ ♠è✐ q✉❛♥ ❤Ö ❣✐÷❛ ➤✐Ò✉ ❦❤✐Ó♥ ➤Ò✉ ❤♦➭♥ t♦➭♥ ❤♦➷❝ ➤✐Ò✉ ❦❤✐Ó♥
➤➢î❝ ✈Ò 0 ❝ñ❛ ❤Ö ➤✐Ò✉ ❦❤✐Ó♥ ✈➭ sù tå♥ t➵✐ ♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ❘✐❝❝❛t✐ ✈✐
♣❤➞♥ ✭❘❉❊✮✳ ❈✉è✐ ❝❤➢➡♥❣✱ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ➤✐Ò✉ ❦✐Ö♥ ❝ã ❧ê✐ ❣✐➯✐ ❝ñ❛ ❜➭✐ t♦➳♥
➤✐Ò✉ ❦❤✐Ó♥ H
∞
❜Ò♥ ✈÷♥❣ ❝❤♦ ❧í♣ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ
✻
ồ tờ ở ỗ ết q ề r í ụ
ết q ứ ớ ủ ợ trì tr
ứ ề ệ ủ t ề ể H

ề ữ ột ớ
ệ PP t ó trễ trễ ế t ỗ ợ ự
ề ể ợ ổ ị ự tr ệ ủ trì t
r sốt q trì ọ t ợ sự ú
ỡ t tì sự ỉ tú ủ t ớ ũ
ọ Pt ỉ tr tứ ĩ tết ò trề t
ữ ọ ổ í ứ ọ
tỏ ò ết s s t tớ t r ể t
ũ ợ sự ộ í ệ ủ t tr tổ ộ t
tí trờ ọ ọ tự ọ ố

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

0
) = x
0
, x
0
∈ R
n
, t
0
≥ 0
✭✶✳✶✮
tr♦♥❣ ➤ã
f(t, x) : I × D → R
n
, D = {x ∈ R
n
: x − x
0
 ≤ a}.
◆❣❤✐Ö♠ x(t) ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ✭✶✳✶✮ ❧➭ ❤➭♠ sè x(t) ❦❤➯ ✈✐ ❧✐➟♥ tô❝
t❤♦➯ ♠➲♥✿
✐✮ (t, x(t)) ∈ I × D,
✽
✐✐✮ x(t) t❤♦➯ ♠➲♥ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ✭✶✳✶✮✳
●✐➯ sö ❤➭♠ f(t, x(t)) ❧✐➟♥ tô❝ tr➟♥ I × D✱ ❦❤✐ ➤ã ♥❣❤✐Ö♠ x(t) ❝❤♦ ❜ë✐ ❞➵♥❣ tÝ❝❤
♣❤➞♥ s❛✉
x(t) = x
0
+


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❝

❦Ý ❤✐Ö✉ ❧➭ ❘❋❉❊✭f✮✱ φ(t) ∈ C✳
❱Ý ❞ô ♠ét sè ❞➵♥❣ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝ã trÔ ➤➢î❝ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❧✉❐♥
✈➝♥ ♥❤➢✿
◦ P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ rê✐ r➵❝
˙x(t) = A(t)x(t) + A
1
(t)x(t − h), t ≥ 0,
x(t) = φ(t), t ∈ [−h, 0],
tr♦♥❣ ➤ã h ≥ 0; x(t) ∈ R
n
❀ A(t), A
1
(t) ∈ R
n×n
❧➭ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ ❧✐➟♥ tô❝
❝❤♦ tr➢í❝ tr➟♥ R
+
✱ φ ∈ C([−h, 0], R
n
) ❧➭ ❤➭♠ ❜❛♥ ➤➬✉ ✈í✐ ❝❤✉➮♥
φ = sup
t∈[−h,0]
φ(t).
✾
P trì tế tí t ó trễ ố
x(t) = A(t)x(t) + A
1
(t)

t

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

n
A(t), A
1
(t), A
2
(t) R
nìn
tr
tụ trớ tr R
+
C([ max(h, k), 0], R
n
) ớ

= sup
t[ max(h,k),0]
(t).
í ổ ị ủ ệ trì
ét ệ trì ó ớ tết f(t, 0) 0 tứ
ệ ó ệ tự t ổ ị ủ ệ trì
tờ t ó ị ĩ s
ị ĩ
ệ ủ ệ ợ ọ ổ ị ế ớ ọ số > 0, t
0
0,
tồ t số = (, t
0
) > 0 s t ì ệ x(t
0
, )(t) ủ ệ t

< 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
.

P
ử ụ ố ớ trì
tờ ú t ó tể ét ợ tí ổ ị ủ ệ f
ị ĩ ét ệ f tụ V : R
+
ì C R
ợ ọ ủ ệ ế tồ t số
1
,
2
,
3
> 0
t

1
x(t)
2
V (t, x
t
)
2
x
t

2
,


V (t, x

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

m
)
ét ệ ề ể tế tí ớ trị x(0) = x
0
trớ
ó ứ ớ ỗ ề ể ợ u(t) t ủ ệ
trì tế tí ó ệ x(t, x
0
, u) t tờ ể t
ợ ở
x(t, x
0
, u) = U(t, 0)x
0
+

t
0
U(t, s)B(s)u(s)ds, t 0
tr ó U(t, s) tr ệ ủ ệ tế tí t t
x(t) = A(t)x(t), t 0.
ị ĩ tr t x
0
, x
1
R
n
(x
0
, x

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

1
ó tồ t T > 0 s

T
0
U(T, s)B(s)B
T
(s)U
T
(T, s)ds > 0.
rớ t ú t ét ết q sở t ề tí ề ể ợ ủ
ệ tế tí ừ
x(t) = Ax(t) + Bu(t), t 0,

tr♦♥❣ ➤ã x(t) ∈ R
n
, u(t) ∈ R
m
, A, B ❧➭ ❝➳❝ ♠❛ tr❐♥ ❤➺♥❣ ❝ã sè ❝❤✐Ò✉ t➢➡♥❣
ø♥❣✳
➜Þ♥❤ ❧ý ✶✳✷✳✸✿ ✭❚✐➟✉ ❝❤✉➮♥ ❤➵♥❣ ❑❛❧♠❛♥✮
❍Ö t✉②Õ♥ tÝ♥❤ ❞õ♥❣ ✭✶✳✹✮ ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❤♦➭♥ t♦➭♥ ✈Ò 0 ❦❤✐ ✈➭ ❝❤Ø ❦❤✐
rank[B, AB, ..., A
n−1
B] = n.
◆❤➢ ✈❐② ➤Ó ①Ðt tÝ♥❤ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❝ñ❛ ♠ét ❤Ö t✉②Õ♥ tÝ♥❤ ❞õ♥❣ ✭✶✳✹✮✱ t❛
❝❤Ø ❝➬♥ ①➳❝ ❧❐♣ ♠❛ tr❐♥ [B, AB, ..., A
n−1
B]−(n×m)✱ s❛✉ ➤ã ❦✐Ó♠ tr❛ ❤➵♥❣ ❝ñ❛
♥ã ❧➭ ➤ñ✳ ▼❛ tr❐♥ ♥➭② ➤➢î❝ ❣ä✐ ❧➭ ♠❛ tr❐♥ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝✱ ❦Ý ❤✐Ö✉ ❧➭ [A/B]✳

❱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 =



1
0


.
❱×
rank[A/B] = rank


1 −2
0 1


= 2
♥➟♥ ❤Ö ➤➲ ❝❤♦ ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❤♦➭♥ t♦➭♥ ✈Ò 0✳
❇➟♥ ❝➵♥❤ ➤ã ❝❤ó♥❣ t❛ ❝ã t❤Ó ❦✐Ó♠ tr❛ ➤➢î❝ tÝ♥❤ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❤♦➭♥ t♦➭♥
❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ❞õ♥❣ ❞➢í✐ ❞➵♥❣ ➤✐Ò✉ ❦✐Ö♥ ❑❛❧♠❛♥✳
➜Þ♥❤ ❧ý ✶✳✷✳✺✿ ❬✷❪ ●✐➯ sö ❝➳❝ ♠❛ tr❐♥ A(t), B(t) ❧➭ ❝➳❝ ❤➭♠ ❣✐➯✐ tÝ❝❤ tr➟♥
[t
0
,∞)✳ ❍Ö ✭✶✳✸✮ ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❤♦➭♥ t♦➭♥ ✈Ò 0 ❦❤✐ ✈➭ ❝❤Ø ❦❤✐
∃t
1
∈ [t
0
,∞) : rank[M
0
(t
1

), 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) =



1
2
cos t 0
0
−1
2
sin t


,
B(t) =


e
− sin t
0
0 e
− cos t


.
❚❛ ❝ã
M
0
(t) = B(t) =


e
− sin t
0

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

(t), M
1
(t)] ❝ã ❤➵♥❣ ❜➺♥❣ ✷ ✈í✐ ♠ä✐ t > t
0
= 0 ♥➟♥ t❤❡♦ ➤Þ♥❤ ❧ý
✶✳✷✳✺✱ ❤Ö [A(t), B(t)] ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ❤♦➭♥ t♦➭♥ ✈Ò 0✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✼✿ ❬✾❪ ❍Ö [A(t), B(t)] ❣ä✐ ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ➤Ò✉ ❤♦➭♥ t♦➭♥ ♥Õ✉
tå♥ t➵✐ N > 0, c
1
, c
2
, c
3
, c
3
, c
4
> 0 s❛♦ ❝❤♦ ✈í✐ ♠ä✐ t ∈ R
+
✿
✐✮ c
1
I ≤ W (t, t + N) ≤ c
2
I
✐✐✮ c
3
I ≤ U(t, t + N)W (t, t + N)U(t, t + N) ≤ c
4
I

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

s❛♦ ❝❤♦ ✈í✐ ❤➭♠ ➤✐Ò✉ ❦❤✐Ó♥ ♥➭② ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥
x(t) = f(t, x(t), h(x(t))), t ≥ 0,
❧➭ æ♥ ➤Þ♥❤ t✐Ö♠ ❝❐♥✳ ❍➭♠ h(x) t❤➢ê♥❣ ❣ä✐ ❧➭ ❤➭♠ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝✳
❚r➢ê♥❣ ❤î♣ ❤Ö ✭✶✳✺✮ ❧➭ ❤Ö t✉②Õ♥ tÝ♥❤
˙x = Ax + Bu
t❤× ❤Ö ❧➭ æ♥ ➤Þ♥❤ ❤♦➳ ➤➢î❝ ♥Õ✉ tå♥ t➵✐ ♠❛ tr❐♥ K s❛♦ ❝❤♦ ♠❛ tr❐♥ (A + BK) ❧➭
æ♥ ➤Þ♥❤✳
➜Þ♥❤ ❧ý ✶✳✸✳✷✿ ❍Ö t✉②Õ♥ tÝ♥❤ ✭✶✳✺✮ ❧➭ æ♥ ➤Þ♥❤ ❤♦➳ ➤➢î❝ ♥Õ✉ ♥ã ❧➭ ➤✐Ò✉ ❦❤✐Ó♥
➤➢î❝ ❤♦➭♥ t♦➭♥ ✈Ò 0✳
❱Ý ❞ô ✶✳✸✳✸✿ ❳Ðt ❤Ö ➤✐Ò✉ ❦❤✐Ó♥ t✉②Õ♥ tÝ♥❤ ✭✶✳✺✮ tr♦♥❣ ➤ã
A =


−1 0
0 0


, B =


1
0


.
❚❛ ❝ã ❤Ö ˙x = Ax ❧➭ æ♥ ➤Þ♥❤✱ ❞♦ ➤ã ❤Ö ➤➲ ❝❤♦ ❧➭ æ♥ ➤Þ♥❤ ❤♦➳ ➤➢î❝ ✈í✐ K = 0✳
❚✉② ♥❤✐➟♥ ❤Ö ❦❤➠♥❣ ❧➭ ●◆❈ ✈×
rank[A/B] = 1 < 2.
❱Ý ❞ô tr➟♥ ❝❤Ø r❛ r➺♥❣ ♥Õ✉ ❤Ö ❧➭ æ♥ ➤Þ♥❤ ❤♦➳ ➤➢î❝ t❤× ❤Ö ➤ã ❝❤➢❛ ❝❤➽❝ ➤➲
❧➭ ●◆❈✳ ❉♦ ➤ã ♣❤➬♥ ➤➯♦ ❝ñ❛ ➤Þ♥❤ ❧ý ✶✳✸✳✷ ❦❤➠♥❣ ➤ó♥❣✳

✶✺
❚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

3
1
˙x
2
= −x
1
− x
3
2
− u
3
2
❳Ðt ❝➳❝ ❤➭♠
V (x
1
, x
2
) = x
2
1
+ x
2
2
, a(t) = γ(t) = t
2
,
b(t) = 2t
2
, u = h(x) = x
✈í✐ u = (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

2
. ˙x
2
= −4x
4
1
− 4x
4
2
≤ −x
2
1
− x
2
2
= −γ((x
1
, x
2
), ∀(x
1
, x
2
) ∈ R
2
.
❉♦ ➤ã ❤Ö ➤➲ ❝❤♦ ❧➭ æ♥ ➤Þ♥❤ ❤♦➳ ➤➢î❝ ✈í✐ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝
u(t) = h(x(t)) = x(t).
✶✻
✶✳✹ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H

∞
❳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

, C(t) ∈ R
l×n
, D(t) ∈ R
l×m
❧➭ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ ❧✐➟♥ tô❝ ❝❤♦ tr➢í❝ tr➟♥
R
+
✳ ❍➭♠ ♥❤✐Ô✉ ω(t) ❧➭ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ♥Õ✉ ω ∈ L
2
([0,∞), R
r
)✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✹✳✶✿ ❬✾❪ ❈❤♦ γ > 0✳ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö ✭✶✳✻✮ ❧➭ ❜➭✐
t♦➳♥ t×♠ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝ u(t) = K(t)x(t) t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉✿
✭✐✮ ❱í✐ ω = 0✱ ♠ä✐ ♥❣❤✐Ö♠ ❝ñ❛ ❤Ö ➤ã♥❣
˙x(t) = [A(t) + B(t)K(t)]x(t) ✭✶✳✼✮
æ♥ ➤Þ♥❤ t✐Ö♠ ❝❐♥ ▲②❛♣✉♥♦✈❀
✭✐✐✮ ❚å♥ t➵✐ c
0
> 0 s❛♦ ❝❤♦
sup

∞
0
z(t)
2
dt
c

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

1
(t)ω(t) ✭✶✳✾✮
✶✼
t❤✉é❝ L
2
([0,∞), R
n
) ✈í✐ ♠ä✐ ♥❤✐Ô✉ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ω(t)❀
✭✐✐✮ ❚å♥ t➵✐ c
0
> 0 s❛♦ ❝❤♦
sup

∞
0
z(t)
2
dt
c
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
)✳
✶✳✺ ▼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



ν
0
ω(s)ds

T
W


ν
0
ω(s)ds

≤ ν

ν
0
ω
T
(s)W ω(s)ds.
❇æ ➤Ò ✶✳✺✳✸✿ ❬✼❪ ❱í✐ ❜✃t ❦× ♠❛ tr❐♥ A(t) ❜Þ ❝❤➷♥ tr➟♥ R
+
✱ tå♥ t➵✐ Q ∈
BM
+
(0,∞) t❤♦➯ ♠➲♥ Q(t) − A(t) ≥ 0.
❑Õt ❤î♣ ✈í✐ ❤Ö ➤✐Ò✉ ❦❤✐Ó♥ ✭✶✳✸✮✱ ①Ðt ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❘✐❝❝❛t✐
˙
P (t) + A
T

(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

+
.
✶✾
❈❤➢➡♥❣ ✷
▼ét sè ❦Õt q✉➯ ✈Ò ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ✈í✐ ❣✐➯
t❤✐Õt ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝
P❤➬♥ ➤➬✉ ❝❤➢➡♥❣ ✷✱ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❦Õt q✉➯ ❣✐➯✐ ➤➢î❝ ❝ñ❛ ❜➭✐ t♦➳♥ ➤✐Ò✉
❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝ã trÔ ❞ù❛ tr➟♥
♠è✐ q✉❛♥ ❤Ö ❣✐÷❛ tÝ♥❤ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ➤Ò✉ ❤♦➭♥ t♦➭♥ ✈➭ sù tå♥ t➵✐ ♥❣❤✐Ö♠ ❝ñ❛
♣❤➢➡♥❣ tr×♥❤ ❘✐❝❝❛t✐ ✈✐ ♣❤➞♥✳ ❚✐Õ♣ ➤ã ➤➢❛ r❛ ♠ét sè ❦Õt q✉➯ ♠ë ré♥❣ tr♦♥❣ ❬✼❪
✈Ò ❜➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❜Ò♥ ✈÷♥❣ ❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ
❤➺♥❣ tr➟♥ ❜✐Õ♥ tr➵♥❣ t❤➳✐ ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt ✈Ò ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ♥❤Ñ ❤➡♥✳
✷✳✶ ❚Ý♥❤ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ✈➭ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö t✉②Õ♥ tÝ♥❤
❧✐➟♥ tô❝ ❦❤➠♥❣ ➠t➠♥➠♠
❳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
, C(t) ∈ R
l×n
, D(t) ∈ R
l×m
❧➭ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ ❧✐➟♥ tô❝ ❝❤♦ tr➢í❝ tr➟♥ R
+
✳
❍➭♠ ♥❤✐Ô✉ ω(t) ❧➭ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ♥Õ✉ ω ∈ L

2
([0,∞), R
r
)✳
❳Ðt ❤Ö ✭✷✳✶✮ ✈í✐ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ B
1
(t), C(t) ❧✐➟♥ tô❝ ❜Þ ❝❤➷♥ tr➟♥ [0,∞)
✈➭ ❣✐➯ t❤✐Õt
D
T
(t)[C(t), D(t)] = [0, I], ∀t ≥ 0 ✭✷✳✷✮
➤Ó ❣✐➯♠ sù ♣❤ø❝ t➵♣ ❦❤✐ ➤➳♥❤ ❣✐➳ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥✳ ❚❛ ❝ã ➤Þ♥❤ ❧ý s❛✉✿
➜Þ♥❤ ❧ý ✷✳✶✳✶✿ ●✐➯ sö ❤Ö [A(t), B(t)] ❧➭ ➤✐Ò✉ ❦❤✐Ó♥ ➤➢î❝ ➤Ò✉ ❤♦➭♥ t♦➭♥✳ ❇➭✐
t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö ✭✷✳✶✮ ❝ã ❧ê✐ ❣✐➯✐ ♥Õ✉ tå♥ t➵✐ P ∈ M(R
n
+
) t❤♦➯ ♠➲♥
♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❘✐❝❝❛t✐ ✭❘❉❊✮
˙
P (t) + A
T
(t)P (t) + P (t)A(t)
−P (t)

B(t)B
T
(t) −
1

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)P (t)]x(t)
①Ðt ❤➭♠ ▲②❛♣✉♥♦✈ ❞➵♥❣ s❛✉
V (t, x) = P (t)x, x.
✷✶
❚❛ ❝ã
β
1
x(t) ≤ V (t, x(t)) ≤ β
2
x(t).
▲✃② ➤➵♦ ❤➭♠ ❝ñ❛ V (.) ❞ä❝ t❤❡♦ ♥❣❤✐Ö♠ ❝ñ❛ ❤Ö ➤ã♥❣✱ ✈í✐ ω = 0✱ t❛ ❝ã
˙
V (t, x(t)) = 
˙
P (t)x(t), x(t) + 2P (t)x(t), ˙x(t)
= −x(t)
2
− P (t)B(t)B
T
(t)P (t)x(t), x(t)
−
1
γ
P (t)B
1
(t)B
T
1
(t)P (t)x(t), x(t) − C

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)

−
1
γ
P (t)B
T
1
(t)B
1
(t)P (t)x(t), x(t) − C
T
(t)C(t)x(t), x(t)
+2P (t)B
1
(t)ω(t), x(t)
❍➡♥ ♥÷❛✱ ✈í✐ u(t) = −B
T
(t)P (t)x(t) ✈➭ ➤✐Ò✉ ❦✐Ö♥ ✭✷✳✷✮ ❝ã
z(t)
2
= C
T
(t)C(t)x(t), x(t) + P (t)B(t)B
T
(t)P (t)x(t), x(t).
❉♦ ➤ã

∞
0

z(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


− x(t)
2
−
1
γ
P (t)B
T
1
(t)B
1
(t)P (t)x(t), x(t)
+2P (t)B
1
(t)ω(t), x(t) − γω(t)
2

dt + P (0)x
0
, x
0
.
❙ö ❞ô♥❣ ❜æ ➤Ò ✭✶✳✺✳✶✮ t❛ ❝ã
2P (t)B
1
(t)ω(t), x(t) − γω(t)
2
≤
1
γ
P (t)B

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

z(t)
2
dt
c
0
x
0

2
+

∞
0
ω(t)
2
dt
≤ γ
✈í✐ c
0
=
P (0)
γ
> 0 ✈× P (0) > β
1
> 0✱ s✉♣r❡♠✉♠ ❧✃② tr➟♥ x
0
∈ R
n
✈➭ ❤➭♠
♥❤✐Ô✉ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ω ∈ L

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

(t) =


√
γ
8
e
− sin
2
t−4
0
0
√
γ
8
e
− cos
2
t−5


,
C(t) =





0 0
√

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

(t)C(t) = 0, D
T
(t)D(t) = I ✈➭ ♠❛ tr❐♥ U(t, s) ➤➢î❝ ❝❤♦ ❜ë✐
U(t, s) =


e
cos
2
s−cos
2
t
0
0 e
s−t


.
✷✸
❱í✐ x = (x
1
, x
2
) ∈ R
2
t❛ ❝ã

t+N
t
B

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

♥➟♥ ❝❤ä♥ N = 1 t❛ ❝ã
e
−2
x ≤

t+N
t
B
T
(s)U
T
(N, s)x
2
ds ≤ x
t❤♦➯ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ✭✐✮ ❝ñ❛ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✼ ✈í✐ c
1
= e
−2
, c
2
= 1.
▼➷t ❦❤➳❝
U(t, s)
2
= e
2(cos
2
s−cos
2
t)

+ 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×♥❤ ✈✐ ♣❤➞♥








˙p
1
(t) + 2 sin 2tp
1
(t) −

e
−2 cos
2
t
−
e
−2 sin
2
t−8
64

p
2
1
(t) +
11
4
sin
2
t = −1
˙p

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

T
(t)[C(t), D(t)] = [0, I], ∀t ≥ 0. ✭✷✳✹✮
✷✹
t❛ ❝ã ❜æ ➤Ò s❛✉✿
❇æ ➤Ò ✷✳✷✳✶✿ ❇➭✐ t♦➳♥ ➤✐Ò✉ ❦❤✐Ó♥ H
∞
❝❤♦ ❤Ö ✭✷✳✶✮ ❝ã ❧ê✐ ❣✐➯✐ ♥Õ✉ tå♥ t➵✐ ♠❛ tr❐♥
X ∈ BMU
+
(0,∞), R ∈ BMU
+
(0,∞) s❛♦ ❝❤♦ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❘✐❝❝❛t✐
s❛✉ t❤♦➯ ♠➲♥
˙
X + A
T
X + XA − X[BB
T
−
1
γ
B
1
B
T
1
]X + C
T
C + R = 0, t ≥ 0. ✭✷✳✺✮
❍➭♠ ➤✐Ò✉ ❦❤✐Ó♥ ♥❣➢î❝ ❧➭

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
➜➵♦ ❤➭♠
˙

V (.) ❞ä❝ t❤❡♦ ♥❣❤✐Ö♠ ❝ñ❛ ❤Ö ➤ã♥❣ t❛ ❝ã
˙
V (t, x) = 
˙
X(t)x(t), x(t) + 2X(t)x(t), ˙x(t)
= (
˙
X(t) + A
T
(t)X(t) + X(t)A(t)x(t), x(t)
−2X(t)B(t)B
T
(t)X(t)x(t), x(t)
= −X(t)B(t)B
T
(t)X(t)x(t), x(t) − C
T
(t)C(t)x(t), x(t)
−
1
γ
X(t)B
1
(t)B
T
1
(t)X(t)x(t), x(t) − R(t)x(t), x(t)
≤ −R(t)x(t), x(t)
✷✺
❜ë✐ ✈×

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

3
λ
2
V (t, x).
❚õ ➤ã t❛ ❝ã
V (t, x(t)) ≤ V (0, x
0
)e
−
λ
3
λ
2
t
, ∀t ≥ 0
♠➭
λ
1
x(t)
2
≤ V (t, x(t))
♥➟♥
x(t) ≤

V (0, x
0
)
λ
1
e

−
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.
✷✻