Lý Thuyết ổn định là một trong những tính chất định tính tiêu biểu của lý thuyết phương trình vi phân và tích phân. Được bắt đầu nghiên cứu từ những năm cuối thế kỷ 19 bởi nhà toán học người Nga A.M.Lyapunov, đến nay lý thuyết ổn định đã có những bước phát triển mạnh mẽ và thu được nhiều thành tựu rực rỡ.
▼ơ❝ ❧ơ❝ ▼ét sè ❦Ý ❤✐Ư✉ ❞ï♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥ ✷ ▼ë ➤➬✉ ✸ ✶ ❈➡ së t♦➳♥ ❤ä❝ ✻ ✶✳✶ ❇➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤ ✈➭ ỉ♥ ➤Þ♥❤ ❤♦➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✶ ❇➭✐ t♦➳♥ æ♥ ➤Þ♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✷ P❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ▲②❛♣✉♥♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✸ ❇➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤ ❤♦➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷ ❇➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤✱ ỉ♥ ➤Þ♥❤ ❤♦➳ ❤Ư ❝ã trÔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✶ ❇➭✐ t♦➳♥ æ♥ ➤Þ♥❤ ❤Ư ❝ã trƠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t ổ ị ệ trì ➤✐Ị✉ ❦❤✐Ĩ♥ ❝ã trƠ ✳ ✶✷ ✶✳✸ ▼ét sè ❜ỉ ➤Ị ❜ỉ trỵ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❚Ý♥❤ æ♥ ị ổ ị ệ trì ❤➭♠ ✷✳✶ ❚Ý♥❤ ỉ♥ ➤Þ♥❤ ❤Ư ❝ã ♥❤✐Ơ✉ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ ✶✺ ✳ ✳ ✳ ✶✺ ✷✳✷ ❚Ý♥❤ ỉ♥ ➤Þ♥❤ ❤♦➳ ❤Ư ➤✐Ị✉ ❦❤✐Ĩ♥ ❝ã ♥❤✐Ơ✉ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ ❚Ý♥❤ ỉ♥ ➤Þ♥❤ ✈➭ ỉ♥ ➤Þ♥❤ ❤♦➳ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝❤➽❝ ❝❤➽♥ ❝ã trÔ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ❚Ý♥❤ æ♥ ➤Þ♥❤ ✈➭ ỉ♥ ➤Þ♥❤ ❤♦➳ ❤Ư ❝ã ♥❤✐Ơ✉ ♣❤✐ t✉②Õ♥ ó trễ ỗ ợ í ổ ị ệ ó ễ tế t ó trễ ỗ ❤ỵ♣ ✸✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ❚Ý♥❤ ỉ♥ ➤Þ♥❤ ❤♦➳ ❝➳❝ ❤Ư ❝ã ♥❤✐Ơ✉ tế t ó trễ ỗ ợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ❑Õt ❧✉❐♥ ✼✶ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✶ ✷ ▼ét sè ❦Ý ❤✐Ö✉ ❞ï♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥ ◦ R+ ✿ t❐♣ ❝➳❝ sè t❤ù❝ ❦❤➠♥❣ ➞♠❀ ◦ Rn ✿ ❦❤➠♥❣ ❣✐❛♥ ✈Ð❝ t➡ n − ❝❤✐Ị✉ ✈í✐ ❦Ý ❤✐Ư✉ tÝ❝❤ ✈➠ ❤➢í♥❣ ❧➭ , ✈➭ ❝❤✉➮♥ ✈Ð❝ t➡ ❧➭ ❀ ◦ Rn×r ✿ ❦❤➠♥❣ ❣✐❛♥ ❝➳❝ ♠❛ tr❐♥ (n × r) − ❝❤✐Ị✉❀ ◦ C([a, b], Rn )✿ t❐♣ t✃t ❝➯ ❝➳❝ ❤➭♠ ❧✐➟♥ tơ❝ tr➟♥ [a, b] ✈➭ ♥❤❐♥ ❣✐➳ trÞ tr➟♥ Rn ❀ ◦ A❚ ✿ ♠❛ tr❐♥ ❝❤✉②Ĩ♥ ✈Þ ❝đ❛ ♠❛ tr❐♥ A❀ ◦ I ✿ ♠❛ tr❐♥ ➤➡♥ ✈Þ❀ ◦ λ(A)✿ t❐♣ t✃t ❝➯ ❝➳❝ ❣✐➳ trÞ r✐➟♥❣ ❝đ❛ A❀ ◦ λ♠❛① (A) := ♠❛①{Reλ : λ ∈ λ(A)}❀ ◦ A ≥ 0✿ ♠❛ tr❐♥ A ①➳❝ ➤Þ♥❤ ❦❤➠♥❣ ➞♠❀ ◦ A > 0✿ ♠❛ tr❐♥ A ①➳❝ ➤Þ♥❤ ❞➢➡♥❣❀ ◦ BM + (0, ∞)✿ t❐♣ ❝➳❝ ❤➭♠ ♠❛ tr❐♥ ➤è✐ ①ø♥❣✱ ①➳❝ ➤Þ♥❤ ❦❤➠♥❣ ➞♠ ✈➭ ❜Þ ❝❤➷♥ tr➟♥ (0, ∞); ◦ µ(A) := 12 λ♠❛① (A + A❚ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é ➤♦ ❝đ❛ ♠❛ tr❐♥ A❀ ◦ η(A) := λ♠❛① (AA❚ ) ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✉➮♥ t❤❡♦ ♣❤ỉ ❝đ❛ ♠❛ tr❐♥ A ✸ ▼ë ý tết ổ ị ột tr ữ tí t ị tí t ể ủ ý tết trì tí ó ột ì tợ ột ệ tố ợ ọ ổ ị t ột tr➵♥❣ t❤➳✐ ❝➞♥ ❜➺♥❣ ♥➭♦ ➤ã ♥Õ✉ ❝➳❝ ♥❤✐Ô✉ ❜Ð tr♦♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❜❛♥ ➤➬✉ ❤♦➷❝ tr♦♥❣ ❝✃✉ tró❝ ❝đ❛ ❤Ư t❤è♥❣ ❦❤➠♥❣ ❧➭♠ ❝❤♦ ❤Ư t❤è♥❣ ➤ã ❜Þ t❤❛② ➤ỉ✐ q✉➳ ♥❤✐Ị✉ s♦ ✈í✐ tr➵♥❣ t❤➳✐ ❝➞♥ ❜➺♥❣ ó ợ t ứ từ ữ ố ❝ñ❛ t❤Õ ❦û ✶✾ ❜ë✐ ♥❤➭ t♦➳♥ ❤ä❝ ♥❣➢ê✐ ◆❣❛ ❆✳ ▼✳ ▲②❛♣✉♥♦✈✱ ➤Õ♥ ♥❛② ❧ý t❤✉②Õt ỉ♥ ➤Þ♥❤ ➤➲ ó ữ t trể ẽ t ợ ♥❤✐Ị✉ t❤➭♥❤ tù✉ rù❝ rì✳ ➜Õ♥ ♥❤÷♥❣ ♥➝♠ ❝đ❛ t❤❐♣ ❦û ✻✵✱ ❝ï♥❣ ✈í✐ sù ♣❤➳t tr✐Ĩ♥ ❝đ❛ ❧ý t❤✉②Õt ➤✐Ị✉ ❦❤✐Ĩ♥✱ ♥❣➢ê✐ t❛ ❝ị♥❣ ❜➽t ➤➬✉ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ổ ị ủ ệ ề ể ò ọ ❧➭ tÝ♥❤ ỉ♥ ➤Þ♥❤ ❤♦➳ ❝đ❛ ❝➳❝ ❤Ư ➤✐Ị✉ ❦❤✐Ĩ♥✱ ❞♦ ➤ã ❧ý t❤✉②Õt ỉ♥ ➤Þ♥❤ ♠➭ ▲②❛♣✉♥♦✈ ➤Ị ①➢í♥❣ tr➢í❝ t✐➟♥ ❝➭♥❣ t❤Ĩ ❤✐Ư♥ ➤➢ỵ❝ t➬♠ q✉❛♥ trä♥❣ ❝đ❛ ♠×♥❤ tr♦♥❣ sù ♣❤➳t tr✐Ĩ♥ ❧✐➟♥ tơ❝ ❝đ❛ t♦➳♥ ❤ä❝✳ ì ữ ý tí tr ế tí ổ ị ợ ứ ✈➭ ♣❤➳t tr✐Ó♥ ♥❤➢ ♠ét ❧ý t❤✉②Õt t♦➳♥ ❤ä❝ ➤é❝ ❧❐♣ ❝ã r✃t ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ ❤÷✉ ❤✐Ư✉ tr♦♥❣ t✃t ❝➯ ❝➳❝ ❧Ü♥❤ ✈ù❝ tõ ❦✐♥❤ tÕ ➤Õ♥ ❦❤♦❛ ❤ä❝ ❦ü t❤✉❐t✳ ◆❤➢ ❝❤ó♥❣ t❛ ➤➲ ❜✐Õt✱ ❝ã ♥❤✐Ị✉ ♣❤➢➡♥❣ ể ứ tí ổ ị ệ trì ✈✐ ♣❤➞♥✳ ❈❤➻♥❣ ❤➵♥ ♥❤➢ ✿ ♣❤➢➡♥❣ ♣❤➳♣ t❤ø ♥❤✃t ▲②❛♣✉♥♦✈ ✭❤❛② ❝ß♥ ❣ä✐ ❧➭ ♣❤➢➡♥❣ ♣❤➳♣ sè ♠ị ➤➷❝ tr➢♥❣✮✱ ♣❤➢➡♥❣ ♣❤➳♣ t❤ø ❤❛✐ ▲②❛♣✉♥♦✈ ✭❤❛② ❝ß♥ ❣ä✐ ❧➭ ♣❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ▲②❛♣✉♥♦✈✮✱ ♣❤➢➡♥❣ ♣❤➳♣ ①✃♣ ①Ø✱ ♣❤➢➡♥❣ ♣❤➳♣ s s à à à ỗ ề ó ➢✉ ➤✐Ĩ♠✱ ♥❤➢ỵ❝ ➤✐Ĩ♠ r✐➟♥❣✳ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ỉ♥ ➤Þ♥❤ ✈➭ ỉ♥ ➤Þ♥❤ ❤♦➳ ❝đ❛ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ♣❤➳♣ ì ó ột ữ ệ ể ứ trì ữ tế ➤➞② ❝ị♥❣ ❧➭ ♠ét ❝➠♥❣ ❝ơ q✉❛♥ trä♥❣ tr♦♥❣ ❧ý t❤✉②Õt ➤Þ♥❤ tÝ♥❤ ❝➳❝ ❤Ư ➤✐Ị✉ ❦❤✐Ĩ♥✱ ❝➳❝ ❤Ư ➤é♥❣ ❧ù❝✳ ◆❣♦➭✐ r❛✱ ✈× ❤➬✉ ❤Õt tr♦♥❣ ❝➳❝ q✉➳ tr×♥❤ ✈❐t ❧ý✱ ❤♦➳ ❤ä❝✱ s✐♥❤ ❤ä❝✱ ❦✐♥❤ tÕ✱ ❦ü t❤✉❐t✱ ✳✳✳ t❤➢ê♥❣ ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤é trÔ t❤ê✐ ❣✐❛♥ ✹ ♥➟♥ ♠ét ❝➳❝❤ t✃t ♥❤✐➟♥✱ ❧í♣ ❤Ư ❝ã trƠ ➤➲ t❤✉ ❤ót ➤➢ỵ❝ ♥❤✐Ị✉ sù q✉❛♥ t➞♠ ❝đ❛ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝✳ ➜Ĩ ❝ã t❤Ĩ ø♥❣ ❞ơ♥❣ ♥❤✐Ị✉ ❤➡♥ tr♦♥❣ t❤ù❝ t✐Ơ♥✱ ♥❣➢ê✐ t❛ ❦❤➠♥❣ ❝❤Ø q✉❛♥ t➞♠ ➤Õ♥ ✈✐Ư❝ ➤➢❛ r❛ ❝➳❝ t✐➟✉ ❝❤✉➮♥ ỉ♥ ➤Þ♥❤ ❝❤♦ ♠ét ❧í♣ ❤Ư ❝ã trƠ ♠➭ ♣❤➯✐ ➤➳♥❤ ❣✐➳ ➤➢ỵ❝ ✧➤é✧ ỉ♥ ị ủ ệ ó trễ ó ột tr ữ ➤➳♥❤ ❣✐➳ ➤é ỉ♥ ➤Þ♥❤ ❝đ❛ ♠ét ❤Ư ❝ã trƠ ❧➭ ➤➳♥❤ ❣✐➳ ❜➺♥❣ ❤➭♠ ♠ị✳ ❱× ✈❐②✱ tÝ♥❤ ỉ♥ ị ũ ổ ị ợ ũ ủ ❝➳❝ ❤Ư ❝ã trƠ ➤➲ ➤➢ỵ❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ề tr ữ trì ♠ét ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈Ị tÝ♥❤ ỉ♥ ➤Þ♥❤ ♠ị ✈➭ ổ ị ợ ũ ột số ệ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❤➭♠✳ ❇è ❝ơ❝ ❝đ❛ ❧✉❐♥ ✈➝♥ ❣å♠ ♣❤➬♥ ♠ë ➤➬✉✱ ❜❛ ❝❤➢➡♥❣ ✈➭ ♣❤➬♥ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦✳ ❈❤➢➡♥❣ ✶ ❧➭ ❝➡ së t♦➳♥ ❤ä❝✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ❜➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤ t ổ ị ệ trì ✈✐ ♣❤➞♥ t❤➢ê♥❣✱ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝ã trƠ ✈➭ ♠ét sè ❜ỉ ➤Ị sÏ ➤➢ỵ❝ sư ❞ơ♥❣ tr♦♥❣ ♥❤÷♥❣ ❝❤➢➡♥❣ s❛✉ ❝đ❛ ❧✉❐♥ ✈➝♥✳ ❚r♦♥❣ ❝❤➢➡♥❣ ✷ ❝❤ó♥❣ t trì ột số ết q ợ ❝ø✉ tr♦♥❣ ♠✃② ♥➝♠ ❣➬♥ ➤➞② ✈Ị sù ỉ♥ ➤Þ♥❤ ũ ổ ị ợ ũ ủ ột sè ❧í♣ ❤Ư ❝ã ♥❤✐Ơ✉ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trƠ rê✐ r➵❝ ✈➭ t✐➟✉ ❝❤✉➮♥ ✈Ị sù ỉ♥ ➤Þ♥❤ ũ ề ữ ổ ị ợ ũ ❜Ị♥ ✈÷♥❣ ❝đ❛ ❧í♣ ❤Ư t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝❤➽❝ ❝❤➽♥ ❝ã trÔ rê✐ r➵❝✳ ❈❤➢➡♥❣ ✸ ❧➭ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ ❝ñ❛ ❧✉❐♥ ✈➝♥✳ ❈❤➢➡♥❣ ♥➭② ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ỉ♥ ➤Þ♥❤ ✈➭ ỉ♥ ➤Þ♥❤ ❤♦➳ ❝❤♦ ♠ét ❧í♣ ❤Ư ❝ã trƠ ♠ë ré♥❣ ❤➡♥✳ ❈ơ t❤Ĩ✱ ❝❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤✐Ị✉ ❦✐Ư♥ ➤đ ♠í✐ ❝❤♦ tÝ♥❤ ổ ị ổ ị ủ ệ trì ✈✐ ♣❤➞♥ ❝ã ♥❤✐Ô✉ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trÔ ỗ ợ ó trễ rờ r ố ❦Õt q✉➯ tr➟♥ ➤➢ỵ❝ ➳♣ ❞ơ♥❣ ➤Ĩ ➤➢❛ r❛ ♠ét sè ➤✐Ị✉ ❦✐Ư♥ ➤đ ♠í✐ ❝❤♦ tÝ♥❤ ỉ♥ ➤Þ♥❤ ♠ị ề ữ ổ ị ợ ũ ề ✈÷♥❣ ❝❤♦ ❧í♣ ❤Ư t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝❤➽❝ ó trễ ỗ ợ r ỗ ệ ➤➢ỵ❝ ①Ðt ➤Õ♥ ë ➤➞② ➤Ị✉ ❝ã ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❦❤➞♠ ♣❤ơ❝ ✈➭ ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ●❙✳ ❚❙❑❍ ❱ị ✺ ◆❣ä❝ P❤➳t ✲ ◆❣➢ê✐ ❚❤➬② ➤➲ t❐♥ t×♥❤ ❝❤Ø ❜➯♦ t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❧➭♠ ❧✉❐♥ ✈➝♥ ✈➭ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❝❤ó♥❣ t➠✐ ❤ä❝ ❝❤✉②➟♥ ♥❣➭♥❤ ✧ ❧ý t❤✉②Õt tè✐ ➢✉ ✈➭ ➤✐Ị✉ ❦❤✐Ĩ♥✧ t➵✐ ❱✐Ư♥ t♦➳♥ ❤ä❝✳ ➜å♥❣ t❤ê✐✱ t➠✐ ❝ị♥❣ ❜➭② tá sù ❦Ý♥❤ ♣❤ơ❝ ✈➭ ❜✐Õt ➡♥ tí✐ ♥❤÷♥❣ t❤➬②✱ ❝➠ ë ❱✐Ư♥ t♦➳♥ ❤ä❝✳ ◆❤÷♥❣ ♥❣➢ê✐ ➤➲ tr✉②Ị♥ ❝❤♦ t➠✐ ❧ß♥❣ ➤❛♠ ♠➟ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❞➭♥❤ t➷♥❣ ❧✉❐♥ ✈➝♥ ì t ữ ộ í ệ ỗ ự t t ữ ❝❤♦ t➠✐ tr♦♥❣ ❝✉é❝ sè♥❣✱ tr♦♥❣ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉✳ ▼➷❝ ❞ï ➤➲ ❝è ❣➽♥❣ r✃t ♥❤✐Ị✉ ♥❤➢♥❣ ✈× tờ trì ộ ò ế ✈➝♥ ♥➭② ❦❤➠♥❣ t❤Ĩ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ s❛✐ ❧➬♠✱ t❤✐Õ✉ ①ãt✳ ❚➠✐ r✃t ♠♦♥❣ ♥❤❐♥ ➤➢ỵ❝ sù ❝❤Ø ❜➯♦ ✈➭ ♥❤÷♥❣ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝đ❛ q✉Ý t❤➬② ❝➠ ✈➭ ❝➳❝ ❜➵♥✳ ❈❤➢➡♥❣ ✶ ❈➡ së t♦➳♥ ❤ä❝ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ị tÝ♥❤ ỉ♥ ➤Þ♥❤ ✈➭ tÝ♥❤ ỉ♥ ➤Þ♥❤ ❤♦➳ ợ ủ ệ trì tờ ❧í♣ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝ã trƠ✳ ❈❤ó♥❣ t➠✐ ❝ị♥❣ ♥❤➽❝ ❧➵✐ ♠ét sè ❦Õt q✉➯ ❦✐♥❤ ➤✐Ĩ♥ ✈➭ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❝➡ ❜➯♥✳ ✶✳✶ ❇➭✐ t♦➳♥ æ♥ ➤Þ♥❤ ✈➭ ỉ♥ ➤Þ♥❤ ❤♦➳ ✶✳✶✳✶ ❇➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤ ❳Ðt ♠ét ❤Ư t❤è♥❣ ➤➢ỵ❝ ♠➠ t➯ ❜ë✐ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ x(t) ˙ = f (t, x(t)), tr♦♥❣ ➤ã x(t) ∈ Rn ❧➭ ✈Ð❝ t➡ tr➵♥❣ t❤➳✐✱ f : R+ × Rn → Rn ❧➭ ❤➭♠ ✈Ð❝ t➡ ❝❤♦ tr➢í❝✳ ❳✉②➟t s✉èt ❧✉❐♥ ✈➝♥ ♥➭② t❛ ❣✐➯ t❤✐Õt r➺♥❣ ❤➭♠ ❦✐Ư♥ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ➤✐Ĩ♠ ✭✶✳✶✮ t ≥ 0, f (.) t❤♦➯ ♠➲♥ ➤✐Ò✉ (t0 , x0 ) ∈ R+ × Rn ❤Ư (1.1) ❝ã ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ➤✐ q✉❛ (t0 , x0 ) ✈➭ ♥❣❤✐Ö♠ ❦Ð♦ ❞➭✐ ➤➢ỵ❝ ✈í✐ ♠ä✐ t ≥ t0 ❑❤✐ ➤ã ♥❣❤✐Ư♠ ♥➭② ➤➢ỵ❝ ❦ý ❤✐Ư✉ ❧➭ ❱í✐ x(t, t0 , x0 ) z(t) ❧➭ ♠ét ♥❣❤✐Ư♠ ❝đ❛ ❤Ư (1.1), ❜➺♥❣ ♣❤Ð♣ ➤ỉ✐ ❜✐Õ♥ y(t) = x(t) − z(t), t❤× ❤Ư (1.1) sÏ ➤➢ỵ❝ ➤➢❛ ✈Ị ❞➵♥❣ y(t) ˙ = f (t, y(t) + z(t)) − f (t, z(t)) ✻ ✭✶✳✷✮ ✼ ➜➷t F (t, y(t)) = f (t, y(t) + z(t)) − f (t, z(t)) t❤× F (t, 0) = ✈➭ ♥❣❤✐Ư♠ y(t) ≡ ❝đ❛ ❤Ư (1.2) sÏ t➢➡♥❣ ø♥❣ ✈í✐ ♥❣❤✐Ư♠ z(t) ❝đ❛ ❤Ư (1.1) ❱× ✈❐②✱ t ì ứ tí ổ ị ủ ệ ứ tí ổ ị ủ ệ z(t) ủ ệ (1.1) tì t❛ ♥❣❤✐➟♥ y(t) ≡ ❝đ❛ ❤Ư (1.2) ❈❤Ý♥❤ ✈× ❧ý ❞♦ ♥➭② ♥➟♥ ❦❤➠♥❣ ♠✃t tÝ♥❤ tæ♥❣ q✉➳t t❛ ❝ã t❤Ĩ ❣✐➯ sư f (t, 0) = 0, tø❝ ❧➭ ❣✐➯ sư ❤Ư (1.1) ❧✉➠♥ ❝ã ♥❣❤✐Ư♠ ❦❤➠♥❣ ✭♥❣❤✐Ư♠ ➤å♥❣ ♥❤✃t ❜➺♥❣ ✵✮✳ ❑❤✐ ➤ã✱ t❛ ❝ã ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ s❛✉✿ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶ ❬✸❪✳ ◆❣❤✐Ư♠ ❦❤➠♥❣ ❝đ❛ ❤Ư (1.1) ợ ọ ổ ị ế t ❦ú sè > 0, t0 > 0, tå♥ t➵✐ sè δ( , t0 ) > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ♥❣❤✐Ư♠ x(t, t0 , x0 ) ❝đ❛ ❤Ư ✈í✐ ◦ x0 < δ, t❤× t❛ ❝ã ◆❣❤✐Ư♠ ❦❤➠♥❣ ❝đ❛ ❤Ư (1.1) ợ ọ ổ ị tệ ế ó ❧➭ ỉ♥ ➤Þ♥❤ ✈➭ tå♥ t➵✐ sè δ0 ✈í✐ ◦ x(t, t0 , x0 ) < , ∀t ≥ t0 > ✭♣❤ô t❤✉é❝ ✈➭♦ t0 ✮ s❛♦ ❝❤♦ ♠ä✐ ♥❣❤✐Ö♠ x(t, t0 , x0 ) x0 < δ t❤× ❧✐♠t→+∞ x(t, t0 , x0 ) = ◆❣❤✐Ư♠ ủ ệ (1.1) ợ ọ ổ ị ũ ♥Õ✉ tå♥ t➵✐ sè N > ✈➭ sè α > s❛♦ ❝❤♦ x(t, t0 , x0 ) ≤ N e−α(t−t0 ) ❑❤✐ ➤ã ➤Þ♥❤✳ ❱➭ x0 , ∀t ≥ t0 ✭✶✳✸✮ N ➤➢ỵ❝ ❣ä✐ ❧➭ ❤Ư sè ổ ị ợ ọ số ũ ổ α, N ➤➢ỵ❝ ❣ä✐ ❝❤✉♥❣ ❧➭ ❝➳❝ ❝❤Ø sè ỉ♥ ị ể ọ t ì ó ệ ❝đ❛ ❤Ư ➤Þ♥❤ t✐Ư♠ ❝❐♥✱ ỉ♥ ➤Þ♥❤ ♠ị✮ t❛ sÏ ♥ã✐ ❤Ư (1.1) ❧➭ ỉ♥ ➤Þ♥❤ ✭ỉ♥ (1.1) ❧➭ ỉ♥ ➤Þ♥❤ ✭ỉ♥ ➤Þ♥❤ t✐Ư♠ ❝❐♥✱ ỉ♥ ➤Þ♥❤ ♠ị✮✳ ◆❣❛② tõ ữ trì t r ột t✐➟✉ ❝❤✉➮♥ q✉❛♥ trä♥❣ ❝❤♦ tÝ♥❤ ỉ♥ ➤Þ♥❤ ♠ị ❝đ❛ ❤Ö t✉②Õ♥ tÝ♥❤ ➠t➠♥➠♠ x(t) ˙ = Ax(t), t ≥ 0, ✭✶✳✹✮ ✽ ❞ù❛ ✈➭♦ ❝➳❝ ❣✐➳ trÞ r✐➟♥❣ ❝đ❛ ♠❛ tr❐♥ A ❈ơ t❤Ĩ ❧➭ ❤Ư (1.4) ỉ♥ ➤Þ♥❤ ♠ị ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♣❤➬♥ t❤ù❝ ❝đ❛ t✃t ❝➯ ❝➳❝ ❣✐➳ trÞ r✐➟♥❣ ❝đ❛ ♠❛ tr❐♥ ❝❤✉➮♥ ❝ỉ ➤✐Ĩ♥ ❦❤➳❝ ❧➭ ❤Ö tr❐♥ A ❧➭ ➞♠✳ ▼ét t✐➟✉ (1.4) ỉ♥ ➤Þ♥❤ ♠ị ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ❜✃t ❦ú ột Q ố ứ ị trì ▲②❛♣✉♥♦✈ A❚ P + P A = −Q ❝ã ♥❣❤✐Ö♠ P ➤è✐ ①ø♥❣✱ ①➳❝ ➤Þ♥❤ ❞➢➡♥❣✳ ❍❛✐ ❦Õt q✉➯ q✉❛♥ trä♥❣ ♥➭② t✐➟✉ ❜✐Ó✉ ❝❤♦ ❤❛✐ ♣❤➢➡♥❣ ♣❤➳♣ ❝➡ ❜➯♥ ứ tí ổ ị ủ ột ệ trì ✈✐ ♣❤➞♥✳ ➜ã ❧➭ ♣❤➢➡♥❣ ♣❤➳♣ ♣❤æ ✈➭ ♣❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ▲②❛♣✉♥♦✈✳ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ sÏ sư ❞ơ♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ▲②❛♣✉♥♦✈ ❧➭ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤Ý♥❤ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ ❜➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤ ✈➭ ỉ♥ ➤Þ♥❤ ❤♦➳✳ ✶✳✶✳✷ P❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ▲②❛♣✉♥♦✈ ❚❛ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ❤➭♠ ▲②❛♣✉♥♦✈ ❝❤♦ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t ≥ 0, x(t) ˙ = f (t, x(t)), x(t) ∈ Rn ❧➭ ✈Ð❝ t➡ tr➵♥❣ t❤➳✐ ❝đ❛ ❤Ư✱ f : R+ × Rn → Rn ❧➭ ❤➭♠ tr♦♥❣ ➤ã ✈Ð❝ t➡ ❝❤♦ tr➢í❝ ✈➭ ❣✐➯ t❤✐Õt ❧✐➟♥ tơ❝ t➝♥❣ ❝❤➷t ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳ f (t, 0) = 0, ∀t ≥ ❑ý ❤✐Ö✉ K ❧➭ t❐♣ ❝➳❝ ❤➭♠ a(.) : R+ → R+ , a(0) = ❍➭♠ V (t, x) : R+ × Rn → R, V (t, 0) = 0, ∀t ≥ 0, ❦❤➯ ✈✐ ❧✐➟♥ tơ❝ ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ▲②❛♣✉♥♦✈ ❝đ❛ ❤Ư ✐✮ ✭✶✳✺✮ (1.5) ♥Õ✉✿ V (t, x) ❧➭ ❤➭♠ ①➳❝ ➤Þ♥❤ ❞➢➡♥❣ t❤❡♦ ♥❣❤Ü❛ ∃a(.) ∈ K : V (t, x) ≥ a( x ), ✐✐✮ V˙ (t, x(t)) := ∂V ∂t + ∂V ∂x f (t, x(t)) ∀(t, x) ∈ R+ × Rn , ≤ 0, ✈í✐ ♠ä✐ ♥❣❤✐Ư♠ x(t) ❝đ❛ ❤Ư (1.5) ◆Õ✉ ❤➭♠ V (t, x) t❤♦➯ ♠➲♥ t❤➟♠ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ∀(t, x) ∈ R+ × Rn , ✐✐✐✮ ∃b(.) ∈ K : ✐✈✮ ∃c(.) ∈ K : V˙ (t, x(t)) ≤ −c( x ), ✈í✐ ♠ä✐ ♥❣❤✐Ư♠ x(t) ❝đ❛ ❤Ư V (t, x) ≤ b( x ), (1.5) t❤× t❛ ❣ä✐ ❤➭♠ V (t, x) ❧➭ ❤➭♠ ▲②❛♣✉♥♦✈ ❝❤➷t ❝ñ❛ ❤Ư (1.5) ❙❛✉ ➤➞② ❧➭ ❤❛✐ ➤Þ♥❤ ❧ý ỉ♥ ➤Þ♥❤ ủ ợ tr [16] ị ❧ý ✶✳✶✳ ◆Õ✉ ❤Ư (1.5) ❝ã ❤➭♠ ▲②❛♣✉♥♦✈ t❤× ❤Ư ổ ị ữ ế t tì ệ ổ ị tệ ị ý ◆Õ✉ ❤Ö (1.5) ❝ã ❤➭♠ ▲②❛♣✉♥♦✈ t❤♦➯ ∃λ1 , λ2 > : λ1 x ≤ V (t, x) ≤ λ2 x , ∀(t, x) ∈ R+ × Rn , ✐✐✮ ∃α > : V˙ (t, x(t)) ≤ −2αV (t, x(t)) ✈í✐ ♠ä✐ ♥❣❤✐Ư♠ x(t) ❝đ❛ ❤Ư ✐✮ (1.5), tì ệ (1.5) ổ ị ũ ❧➭ ❝➳❝ ❝❤Ø sè ỉ♥ ➤Þ♥❤ α, N = ▲②❛♣✉♥♦✈✳ ✶✳✶✳✸ ❇➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤ ❤♦➳ ❳Ðt ♠ét ❤Ư ➤✐Ị✉ ể ợ t ệ trì x(t) ˙ = f (t, x(t), u(t)), tr♦♥❣ ➤ã t ≥ 0, ✭✶✳✻✮ x(t) ∈ Rn ❧➭ ✈Ð❝ t➡ tr➵♥❣ t❤➳✐✱ u(t) ∈ Rm ❧➭ ✈Ð❝ t➡ ➤✐Ị✉ ❦❤✐Ĩ♥✳ ❍➭♠ ➤✐Ị✉ ❦❤✐Ĩ♥ u(.) t❤✉é❝ ❧í♣ ❤➭♠ ❦❤➯ tÝ❝❤ ❜❐❝ ❤❛✐ tr➟♥ ❝➳❝ ➤♦➵♥ ❤÷✉ ❤➵♥ [0, s], ∀s > ✈➭ ❧✃② ❣✐➳ trÞ tr♦♥❣ Rm ❍➭♠ f : R+ × Rn × Rm → Rn ❧➭ ❤➭♠ ✈Ð❝ t➡ ❝❤♦ tr➢í❝ ➤➢ỵ❝ ❣✐➯ t❤✐Õt t❤♦➯ f (t, 0, 0) = 0, ∀t ≥ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸ ❬✸❪✳ ệ ề ể (1.6) ợ ọ ổ ị ➤➢ỵ❝ ♥Õ✉ ♥❤➢ tå♥ t➵✐ ❤➭♠ g : Rn → Rm s❛♦ ❝❤♦ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ s❛✉ ✭t❤➢ê♥❣ ❣ä✐ ❧➭ ❤Ö ➤ã♥❣✱ ❝❧♦s❡❞ ✲ ❧♦♦♣ s②st❡♠✮ x(t) ˙ = f t, x(t), g(x(t)) , ❧➭ ỉ♥ ➤Þ♥❤ t✐Ư♠ ❝❐♥✳ ❍➭♠ t ≥ 0, ✭✶✳✼✮ u(t) = g(x(t)) ➤➢ỵ❝ ọ ề ể ợ ủ ệ ị ĩ ✶✳✹✳ ❍Ư ➤✐Ị✉ ❦❤✐Ĩ♥ ♥Õ✉ ♥❤➢ tå♥ t➵✐ ❤➭♠ ỉ♥ ị ũ (1.6) ợ ọ ổ ị ợ ❞➵♥❣ ♠ị g : Rn → Rm s❛♦ ❝❤♦ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ (1.7) ❧➭ ✶✵ ◆Õ✉ ♠ét ❤Ư ổ ị ũ ổ ị ợ ũ ✈í✐ tè❝ ➤é ❤é✐ tơ ♠ị α ❝❤♦ tr➢í❝ t❤× ệ ó ợ ọ ệ ổ ị ổ ị ợ t♦➳♥ ỉ♥ ➤Þ♥❤✱ ỉ♥ ➤Þ♥❤ ❤♦➳ ❤Ư ❝ã trƠ ❇➭✐ t♦➳♥ ỉ♥ ➤Þ♥❤ ❤Ư ❝ã trƠ ❈❤ó♥❣ t❛ ♥❤❐♥ t❤✃② r ệ trì tờ q ệ ữ ❜✐Õ♥ t❤ê✐ ❣✐❛♥ ➤ỉ✐ ❝đ❛ tr➵♥❣ t❤➳✐ (1.1) ♠➠ t➯ ♠è✐ t, tr➵♥❣ t❤➳✐ ❝đ❛ ❤Ư t❤è♥❣ x(t) ✈➭ ✈❐♥ tè❝ t❤❛② x(t) t➵✐ ❝ï♥❣ ♠ét t❤ê✐ ➤✐Ó♠ t ❙♦♥❣ tr➟♥ t❤ù❝ tÕ✱ ❝➳❝ q✉➳ tr×♥❤ ①➯② r❛ tr♦♥❣ tù ♥❤✐➟♥ t❤➢ê♥❣ ❝ã sù ❧✐➟♥ q✉❛♥ tí✐ q✉➳ ❦❤ø✱ ➤Ị✉ ♠❛♥❣ Ýt ♥❤✐Ị✉ tÝ♥❤ ❞✐ tr✉②Ị♥✳ ❱× ✈❐② ❦❤✐ ♠➠ t q trì ú ợ ể ễ ❜➺♥❣ ❝➳❝ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝ã trƠ✳ ●✐➯ sư ♠ét ❤Ư t❤è♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ q✉➳ ❦❤ø ✈í✐ ➤é trƠ (0 ≤ h < +∞) ❱í✐ x(.) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ tr➟♥ R+ , ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ Rn , ❝❤ó♥❣ t❛ ①➞② ❞ù♥❣ ❤➭♠ − h, , Rn xt ∈ C := C ◆❤➢ ✈❐②✱ ♥❤➢ s❛✉ xt (s) = x(t + s), ∀s ∈ [−h, 0] xt ❧➭ ♠ét ➤♦➵♥ q✉ü ➤➵♦ tr➟♥ [t − h, t] ❝đ❛ ❤➭♠ x(.) ✈í✐ ❝❤✉➮♥ tr♦♥❣ C ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐ xt = s✉♣s∈[−h,0] x(t + s) ❑❤✐ ➤ã ❤Ư ♣❤➢➡♥❣ tr×♥❤ ❝ã trƠ ♠➠ t➯ sù ♣❤ơ t❤✉é❝ ❝đ❛ ✈❐♥ tè❝ t❤❛② ➤ỉ✐ t➵✐ t❤ê✐ ➤✐Ĩ♠ t❤➳✐ ❝đ❛ ❤Ư t❤è♥❣ tr♦♥❣ ❦❤♦➯♥❣ t❤ê✐ ❣✐❛♥ tr➢í❝ ➤ã t ✈➭♦ tr➵♥❣ [t − h, t] ➤➢ỵ❝ ❝❤♦ ❞➢í✐ ❞➵♥❣ x(t) ˙ = f (t, xt ), tr♦♥❣ ➤ã t ≥ 0, ✭✶✳✽✮ f : R+ × C → Rn ▼ét ♥❣❤✐Ư♠ x(.) ❝đ❛ ❤Ư (1.8) ➤✐ q✉❛ ể (t0 , ) R+ ì C ợ ý ❤✐Ư✉ x(t, φ) ❑❤✐ ➤ã✱ ❤➭♠ ❣✐➳ trÞ ❜❛♥ ➤➬✉ ❝đ❛ ♥❣❤✐Ư♠ ♥➭② tr♦♥❣ ❦❤♦➯♥❣ x(t0 + s) = φ(s), [t0 − h, t0 ] ❝❤Ý♥❤ ❧➭ ❤➭♠ φ, tø❝ ❧➭ xt0 (t0 , φ)(s) = ∀s ∈ [−h, 0] ❚❛ ❝ị♥❣ ❣✐➯ t❤✐Õt r➺♥❣ ❤➭♠ f (.) t❤♦➯ ➤✐Ị✉ ệ s ỗ ể (t0 , ) R+ × C ❤Ư (1.8) ❝ã ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ➤✐ q✉❛ ➤✐Ĩ♠ ♥➭② ✈➭ ♥❣❤✐Ư♠ ❦Ð♦ ❞➭✐ ➤➢ỵ❝ ✈í✐ ♠ä✐ t ≥ ❚➢➡♥❣ tù ♥❤➢ ❝➳❝ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t❤➢ê♥❣✱ t❛ ❝ò♥❣ ❣✐➯ t❤✐Õt ✺✾ ❈❤♦ ❝➳❝ sè ❞➢➡♥❣ α, σ, , , , , , h, k, a, b, c, d, t❛ ➤➷t 3b2 2c2 k 2αk γ= + dθ) + −2αh + e , σe (1 − δ) µ(A) = s✉♣t∈R+ µ(A(t)), η(A1 ) = s✉♣t∈R+ η(A1 (t)), −1 (a p = s✉♣t∈R+ P (t) , B = s✉♣t∈R+ B(t) , θ = (p + β) B , ❚ R1 (t) = R(t) + B(t)B (t), Q1 (t) = Q(t) + β η (B)I, η(A2 ) = s✉♣t∈R+ η(A2 (t)), M = p + β + h + 2h2 e2αh + k , A(t) = A(t) + tr♦♥❣ ➤ã −1 N= M , β βke2αk A2 (t)A❚2 (t) + (α + )I, R(t), Q(t) ợ ị tr ị ❧ý ✸✳✶✱ tø❝ ❧➭ R(t) = − A1 (t)A❚1 (t) − −2αh σe (1 − δ) −1 ke2αk A2 (t)A❚2 (t) − γI, 3β η (A1 ) Q(t) = 2αβ + β γ + + 2à(A) + 2h e (1 ) +2 ị ❧ý ✸✳✷✳ ❑❤✐ ➤ã ❤Ö −1 β ke2αk η (A2 ) I + Q(t) + he2αh W (t) + kH(t) ❈❤♦ tr➢í❝ α > ✈➭ ❣✐➯ sư r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ (3.8) ➤➢ỵ❝ t❤♦➯ ♠➲♥✳ (3.7) ❧➭ α ổ ị ợ ế tồ t số > 0, ♠❛ tr❐♥ ①➳❝ ➤Þ♥❤ ❦❤➠♥❣ ➞♠ ✈➭ ❜Þ ❝❤➷♥ ➤Ị✉ ❝❤➷♥ ➤Ị✉ Q(t), H(t), ❝ơ t❤Ĩ t❛ ❝ã Q(t) ≤ Q(t)x(t), x(t) ≥ σ tr♦♥❣ ➤ã W (t), ❝➳❝ ♠❛ tr❐♥ ①➳❝ ➤Þ♥❤ ❞➢➡♥❣ ➤Ị✉ ✈➭ ❜Þ , , , σ, 1, W (t) ≤ x(t) 2, H(t) ≤ , H(t)x(t), x(t) ≥ ❧➭ ♥❤÷♥❣ sè ❞➢➡♥❣ ✈➭ 3, x(t) P (t) ∈ BM + (0, ∞) , s❛♦ ❝❤♦ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❘✐❝❝❛t✐ s❛✉ ➤➢ỵ❝ t❤♦➯ ♠➲♥ P˙ (t) + A❚ (t)P (t) + P (t)A(t) − P (t)R1 (t)P (t) + Q1 (t) = (RDE4) ✻✵ x(t, φ) ❝ñ❛ ❤Ư t❤♦➯ ♠➲♥ ❍➡♥ ♥÷❛✱ ♥❣❤✐Ư♠ x(t, φ) ≤ N e−αt ❈❤ø♥❣ ♠✐♥❤✳ ❣✐➯ t❤✐Õt ❈❤♦ φ u(t) = K(t)x(t), tr♦♥❣ ➤ã K(t) sÏ ➤➢ỵ❝ ❝❤ä♥ s❛✉ ✈➭ t❛ K(t) ≤ θ ❑❤✐ ➤ã ❤Ư (3.7) sÏ ➤➢ỵ❝ ✈✐Õt ❧➵✐ ♥❤➢ s❛✉ t x(t) ˙ = A(t)x(t) + A1 (t)x(t − h(t)) + A2 (t) x(s) ds t−k(t) t + f˜ t, x(t), x(t − h(t)), ✭✸✳✶✵✮ x(s) ds , t−k(t) h = ♠❛①{h, k}, t ∈ [−h, 0], x(t) = φ(t), tr♦♥❣ ➤ã A(t) = A(t) + B(t)K(t), t f˜ t, x(t), x(t − h(t)), x(s) ds t−k(t) t = f t, x(t), x(t − h(t)), x(s) ds, u(t) , t−k(t) ✈➭ f˜(.) = f (.) t ≤a x(t) x(t − h(t)) +b +c x(s) ds +d u(t) x(s) ds +d K(t) x(s) ds t−k(t) t ≤a x(t) x(t − h(t)) +b +c x(t) t−k(t) t = (a + dθ) x(t) +b x(t − h(t)) +c t−k(t) ✭✸✳✶✶✮ ➜è✐ ✈í✐ ❤Ư ➤ã♥❣ (3.10), t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✸✳✶✱ t❛ ❝ị♥❣ ❧✃② ❤➭♠ V = V1 + V2 + V3 + V4 + V5 , tr♦♥❣ ➤ã ❝➳❝ ❤➭♠ V1 , V2 , V3 , V4 , V5 ợ ị tr ị ý ❉♦ ➤ã t❛ ❝ã ➤➳♥❤ ❣✐➳ s❛✉ ✻✶ V˙ + 2αV ≤ ❚ P˙ (t) + A (t) + −1 + P (t) A(t) + + P (t) +2 −1 −1 βke2αk A2 (t)A❚2 (t) + (α + βγ)I P (t) βke2αk A2 (t)A❚2 (t) + (α + βγ)I x(t), x(t) A1 (t)A❚1 (t) −2αh σe (1 − δ) ke2αk A2 (t)A❚2 (t) + γI P (t)x(t), x(t) + Q(t)x(t), x(t) + he2αh W (t)x(t), x(t) + k H(t)x(t), x(t) ❚ A(t) + A (t) x(t), x(t) 3β 2 + 2αβ + β γ + + −2αh η (A1 ) σe (1 − δ) +β +2 −1 β ke2αk η (A2 ) x(t) ❚r♦♥❣ ➤ã A(t) = A(t) + B(t)K(t), ❚ A(t) + A (t) = A(t) + B(t)K(t) + A(t) + B(t)K(t) ❚ = A(t) + A❚ (t) + B(t)K(t) + K ❚ (t)B ❚ (t) ❚õ ➤ã s✉② r❛ V˙ + 2αV ≤ P˙ (t) + A❚ (t) + + P (t) A(t) + −1 −1 βke2αk A2 (t)A❚2 (t) + (α + βγ)I P (t) βke2αk A2 (t)A❚2 (t) + (α + βγ)I x(t), x(t) + P (t)R(t)P (t)x(t), x(t) + Q(t)x(t), x(t) + he2αh W (t)x(t), x(t) + k H(t)x(t), x(t) + β A(t) + A❚ (t) x(t), x(t) 3β 2 + 2αβ + β γ + + −2αh η (A1 ) + −1 β ke2αk η (A2 ) σe (1 − δ) + K ❚ (t)B ❚ (t)P (t) + P (t)B(t)K(t) x(t), x(t) + β B(t)K(t) + K ❚ (t)B ❚ (t) x(t), x(t) ❍❛② ❧➭ x(t) ✻✷ V˙ (t, xt ) + 2αV (t, xt ) ≤ + P˙ (t) + A❚ (t)P (t) + P (t)A(t) − P (t)R(t)P (t) + Q(t) x(t), x(t) K ❚ (t)B ❚ (t)P (t) + P (t)B(t)K(t) x(t), x(t) + β B(t)K(t) + K ❚ (t)B ❚ (t) x(t), x(t) ❇➞② ❣✐ê✱ t❛ ❧✃② K(t) = − 12 B ❚ (t)P (t) + 21 βB ❚ (t) t❤× 1 K ❚ (t) = − P (t)B(t) + βB(t) 2 ❉♦ ➤ã K ❚ (t)B ❚ (t)P (t) + P (t)B(t)K(t) = 1 − P (t)B(t) + βB(t) B ❚ (t)P (t) 2 1 + P (t)B(t) − B ❚ (t)P (t) + βB ❚ (t) 2 1 = − P (t)B(t)B ❚ (t)P (t) + βB(t)B ❚ (t)P (t) − P (t)B(t)B ❚ (t)P (t) 2 + βP (t)B(t)B ❚ (t) ❙✉② r❛ K ❚ (t)B ❚ (t)P (t) + P (t)B(t)K(t) x(t), x(t) = − P (t)B(t)B ❚ (t)P (t)x(t), x(t) + β P (t)B(t)B ❚ (t)x(t), x(t) ❚❛ ❝ã B(t)K(t) + K ❚ (t)B ❚ (t) 1 1 = B(t) − B ❚ (t)P (t) + βB ❚ (t) + − P (t)B(t) + βB(t) B ❚ (t) 2 2 1 = − B(t)B ❚ (t)P (t) + βB(t)B ❚ (t) − P (t)B(t)B ❚ (t) 2 ✻✸ ❚õ ➤ã s✉② r❛ β B(t)K(t) + K ❚ (t)B ❚ (t) x(t), x(t) = β B(t)B ❚ (t)x(t), x(t) − β P (t)B(t)B ❚ (t)x(t), x(t) ❙✉② r❛ V˙ (t, xt ) + 2αV (t, xt ) P˙ (t) + A❚ (t)P (t) + P (t)A(t) − P (t)R(t)P (t) + Q(t) x(t), x(t) ≤ − P (t)B(t)B ❚ (t)P (t)x(t), x(t) + β B(t)B ❚ (t)x(t), x(t) ❚❛ ❧➵✐ ❝ã B(t)B ❚ (t)x(t), x(t) ≤ η (B) x(t) ❉♦ ➤ã V˙ (t, xt ) + 2αV (t, xt ) P˙ (t) + A❚ (t)P (t) + P (t)A(t) − P (t) R(t) + B(t)B ❚ (t) P (t) ≤ + Q(t) + β η (B) x(t), x(t) ❍❛② ❧➭ V˙ (t, xt ) + 2αV (t, xt ) ≤ P˙ (t) + A❚ (t)P (t) + P (t)A(t) − P (t)R1 (t)P (t) + Q1 (t) x(t), x(t) , tr♦♥❣ ➤ã ❱× R1 (t) = R(t) + B(t)B ❚ (t), Q1 (t) = Q(t) + β η (B) P (t) ❧➭ ♥❣❤✐Ö♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ (RDE4) ♥➟♥ V˙ (t, xt ) + 2αV (t, xt ) ≤ 0, ∀t ≥ ❉♦ ➤ã V˙ (t, xt ) ≤ −2αV (t, xt ), ▲✃② tÝ❝❤ ♣❤➞♥ ✷ ✈Õ ❝ñ❛ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② tõ ∀t ≥ 0 tí✐ t✱ t❛ ❝ã V (t, xt ) ≤ V (0, x0 )e−2αt , ∀t ≥ ✻✹ ▼➷t ❦❤➳❝✱ tõ ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ❤➭♠ β x(t) V (t, xt ) t❛ ❝ã ≤ V (t, xt ), ∀t ≥ 0, ✈➭ ❞♦ ➤ã x(t, φ) ≤ V (0, x0 ) −αt e , β ∀t ≥ ❚❛ ❧➵✐ ❝ã V (0, x0 ) = P (0)x(0), x(0) + β x(0) e2αs Q(s)x(s), x(s) ds + −h(0) 0 e2α(s+h) W (s)x(s), x(s) ds dτ + −h τ −h(τ ) 2ατ −k s e + H(τ )x(τ ), x(τ ) dτ ds ≤ (p + β + h + 2h2 e2αh + k ) φ ❱× ✈❐② x(t, φ) ≤ N e−αt φ ➜ã ❝ị♥❣ ❧➭ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❱Ý ❞ơ ✸✳✸✳ ❳Ðt ❤Ư (3.7), ✈í✐ ❤➭♠ ➤✐Ị✉ ❦✐Ư♥ ❜❛♥ ➤➬✉ ❜✃t ❦ú φ(t) ❝➳❝ ❤➭♠ ❝ã trÔ ˙ = sin t, h(t) = sin2 t =⇒ h(t) 5 k(t) = cos3 t, ✈➭ A(t) = √ A2 (t) = ∈ C([− 12 , 0], R2 ), a(t) −1 , A1 (t) = b(t) 2sint √ 0 3cost sin2 t + , cos2 t + , B(t) = t f (.) = sint , cost f1 (.) − f2 (.) + √12 t−k(t) x1 (s) ds , u2 (t)cos[tx1 (t)] ✻✺ tr♦♥❣ ➤ã 1 f1 (.) = x1 (t)sin[x2 (t − h(t))], f2 (.) = x1 (t − h(t))sin[tx1 (t)], 39 15 23 e 419 a(t) = −2sin2 t− + e−2t (sin4 t− sin2 t− )− e2t +3e− sin2 t , 16 16 16 39 11 25 e 499 b(t) = −3cos2 t− + e−2t (cos4 t− cos2 t+ )− e2t + e− sin2 t 16 16 16 2 ❚❛ ❝ã f (.) ≤ + x(t) t +√ x(t−h(t)) x(s) ds + t−k(t) u(t) ❉♦ ➤ã √ 1 1 a = , b = , c = √ , d = , µ(A) = 2, η(A1 ) = 1, η(A2 ) = 3, η(B) = 4 ❈❤♦ α = β = 1, ✈➭ e + sin2 t , W (t) = + sin2 t Q(t) = − sin2 t , − sin2 t e + 2cos2 t + 2cos2 t H(t) = ❙✉② r❛ σ = e, = e, R1 (t) = = 4, sin4 t − = 5, 15 sin t − = 6, 419 16 ◆❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ = 1, θ = 3, γ = cos t − 11 cos t 499 16 − 2e sin2 t 23 16 Q1 (t) = + 3e − 2e sin2 t 0 + + 25 16 39 , 16 , 2e (RDE4) ❧➭ P (t) = e−2t e−2t ❉Ô t❤✃② t✃t ❝➯ ề ệ tr ị ý ợ t ó ệ ổ ị ợ ữ ề ể ợ ổ ị ợ ❝❤♦ ❜ë✐ u(t) = (1 − e−2t )(sin2 t + 1) (1 −2t −e x(t) )(cos2 t + 2) ✻✻ ➜Þ♥❤ ❧ý ✸✳✷ ❝❤♦ t❛ ♠ét ❤Ư q✉➯ ✈Ị tÝ♥❤ ỉ♥ ➤Þ♥❤ ❤♦➳ ợ ề ữ ủ ệ ề ể tế tí t ó trễ ỗ ợ ét ệ ➤✐Ị✉ ❦❤✐Ĩ♥ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝❤➽❝ ❝❤➽♥ ❝ã trễ ỗ ợ x(t) = [A(t) + A(t)]x(t) + [A1 (t) + ∆A1 (t)]x(t − h(t)) t + [A2 (t) + ∆A2 (t)] x(s) ds + [B(t) + ∆B(t)]u(t), ✭✸✳✶✷✮ t−k(t) x(t) = φ(t), tr♦♥❣ ➤ã t ∈ [−h, 0], h = ♠❛①{h, k}, x(t) ∈ Rn ❧➭ ✈Ð❝ t➡ tr➵♥❣ t❤➳✐ ❝đ❛ ❤Ư✱ u(t) ∈ Rm ❧➭ ❤➭♠ ➤✐Ị✉ ❦❤✐Ĩ♥✱ A(t), A1 (t), A2 (t) ❧➭ ❝➳❝ ♠❛ tr❐♥ ❤➭♠ ❧✐➟♥ tô❝ tr➟♥ R+ , B(t) ❧➭ ♠❛ tr❐♥ ❤➭♠ ❧✐➟♥ tơ❝ ✈➭ ❜Þ ❝❤➷♥ tr➟♥ R+ , ❝ß♥ h(t), k(t) ❧➭ ❝➳❝ ❤➭♠ ❝ã trƠ ❦❤➯ ✈✐ ❧✐➟♥ tơ❝ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ≤ h(t) ≤ h, ˙ h(t) ≤ δ < 1, ≤ k(t) ≤ k, ❈ß♥ ❝➳❝ ♠❛ tr❐♥ ♥❤✐Ơ✉ ∀t ≥ 0, ∀t ≥ ∆A(t), ∆A1 (t), ∆A2 (t), ∆B(t) ❝ã ❞➵♥❣ ∆A(t) = EF (t)H, ∆A1 (t) = E1 F (t)H1 , ∆A2 (t) = E2 F (t)H2 , ∆B(t) = E3 F (t)H3 ❚r♦♥❣ ➤ã✱ t❤Ý❝❤ ❤ỵ♣✱ E, E1 , E2 , H, H1 , H2 ❧➭ ❝➳❝ ♠❛ tr❐♥ ❤➺♥❣ ❝❤♦ tr➢í❝ ❝ã ❝❤✐Ị✉ F (t) ❧➭ ♠❛ tr❐♥ ❦❤➠♥❣ ①➳❝ ➤Þ♥❤ ♥❤➢♥❣ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❣✐í✐ ♥é✐ F (t) ≤ 1, ∀t ∈ R+ ❇➺♥❣ ❝➳❝❤ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✸✳✷✱ t❛ ❝ã ➤✐Ị✉ ệ ủ tí ổ ị ợ ề ✈÷♥❣ ❝❤♦ ❤Ư ➤✐Ị✉ ❦❤✐Ĩ♥ (3.12) ✻✼ ❚❤❐t ✈❐②✱ ❤Ư (3.12) ➤➢ỵ❝ ❜✐Õ♥ ➤ỉ✐ t❤➭♥❤ t x(t) ˙ = A(t)x(t) + A1 (t)x(t − h(t)) + A2 (t) x(s) ds + B(t)u(t) t−k(t) t + ∆A(t)x(t) + ∆A1 (t)x(t − h(t)) + ∆A2 (t) x(s) ds t−k(t) + ∆B(t)u(t), h = ♠❛①{h, k} t ∈ [−h, 0], x(t) = φ(t), ✭✸✳✶✸✮ ◆Õ✉ t❛ ➤➷t t f t, x(t), x(t − h(t)), x(s) ds, u(t) t−k(t) t = ∆A(t)x(t) + ∆A1 (t)x(t − h(t)) + ∆A2 (t) x(s) ds + ∆B(t)u(t) t−k(t) t❤× t f t, x(t), x(t − h(t)), x(s) ds, u(t) t−k(t) ≤ E H x(t) + E1 H1 x(s) ds + ✭✸✳✶✹✮ x(t − h(t)) t + E2 H2 E3 H3 u(t) t−k(t) ❱× ✈❐②✱ t❛ ❝ã ❦Õt q✉➯ s❛✉ ❈❤♦ ❝➳❝ sè ❞➢➡♥❣ α, σ, , , , , , h, k, t❛ ➤➷t 3b2 2c2 k 2αk γ= + dθ) + −2αh + e , σe (1 − δ) µ(A) = s✉♣t∈R+ µ(A(t)), η(A1 ) = s✉♣t∈R+ η(A1 (t)), −1 (a η(A2 ) = s✉♣t∈R+ η(A2 (t)), p = s✉♣t∈R+ P (t) , B = s✉♣t∈R+ B(t) , θ = (p + β) A1 (t)A❚1 (t) − −2αh σe (1 − δ) − γI + B(t)B ❚ (t), R2 (t) = − −1 B , ke2αk A2 (t)A❚2 (t) ✻✽ 3β Q2 (t) = 2αβ + β γ + + 2βµ(A) + −2αh η (A1 ) σe (1 − δ) +2 −1 β ke2αk η (A2 ) I + Q(t) + he2αh W (t) + kH(t) + β η (B), M = p + β + h + 2h2 e2αh + k , A(t) = A(t) + −1 M , β N= βke2αk A2 (t)A❚2 (t) + (α + βγ)I, tr♦♥❣ ➤ã a= E H , b = E1 ❍Ö q✉➯ ✸✳✸✳ β > 0, ❍Ö H1 , c = E2 H2 , d = E3 (3.12) ❧➭ ỉ♥ ➤Þ♥❤ ❤♦➳ ợ ũ ề ữ ế tồ t số tr❐♥ ①➳❝ ➤Þ♥❤ ❦❤➠♥❣ ➞♠ ✈➭ ❜Þ ❝❤➷♥ ➤Ị✉ ➤Þ♥❤ ❞➢➡♥❣ ➤Ị✉ ✈➭ ❜Þ ❝❤➷♥ ➤Ị✉ Q(t) ≤ Q(t)x(t), x(t) ≥ σ tr♦♥❣ ➤ã H3 , , , σ, 1, W (t), ❝➳❝ ♠❛ tr❐♥ ①➳❝ Q(t), H(t), ❝ơ t❤Ĩ t❛ ❝ã W (t) ≤ x(t) 2, H(t) ≤ , H(t)x(t), x(t) ≥ ❧➭ ♥❤÷♥❣ sè ❞➢➡♥❣ ✈➭ 3, x(t) P (t) ∈ BM + (0, ∞) , s❛♦ ❝❤♦ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❘✐❝❝❛t✐ s❛✉ ➤➢ỵ❝ t❤♦➯ ♠➲♥ P˙ (t) + A❚ (t)P (t) + P (t)A(t) − P (t)R2 (t)P (t) + Q2 (t) = (RDE5) ❍➡♥ ♥÷❛✱ ♥❣❤✐Ư♠ x(t, φ) ❝đ❛ ❤Ö t❤♦➯ ♠➲♥ x(t, φ) ≤ N e−αt φ ❙❛✉ ➤➞②✱ ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ✈Ý ❞ơ ➳♣ ❞ơ♥❣ ❍Ư q✉➯ ✸✳✸ ♥➭②✳ ❱Ý ❞ơ ✸✳✹✳ ❳Ðt ❤Ư ➤✐Ị✉ ❦❤✐Ĩ♥ (3.12), ✈í✐ ❤➭♠ ➤✐Ị✉ ❦✐Ư♥ ❜❛♥ ➤➬✉ ❜✃t ❦ú φ(t) ∈ C([− 21 , 0], R2 ), ❝➳❝ ❤➭♠ ❝ã trÔ 1 ˙ h(t) = sin2 t =⇒ h(t) = sin2t, 2 1 k(t) = sin2 t, ✻✾ ✈➭ A(t) = √ A2 (t) = a(t) , −1 b(t) 3sint √ , 5cost A1 (t) = B(t) = sint , cost sin2 t + , cos t + √ √0 √0 √0 , E3 = , E2 = , E= , E1 = 0 √ √ √ √0 √0 2 , H1 = , H2 = , H3 = H= 3 0 ❚r♦♥❣ ➤ã a(t) = −289 − 6sin2 t − + e−2t (sin4 t − 14sin2 t − 285) √ 7e e − e2t 821 + 60 14 + − sin t , e−2t b(t) = −289 − 10cos t + (cos4 t − 14cos2 t − 273) √ 3e e 1649 − e2t + 60 14 + − cos t 2 2 ❈❤♦ α = β = 1, ✈➭ Q(t) = 2e + sin2 t , W (t) = + cos2 t H(t) = ❙✉② r❛ 2e − sin2 t , − cos2 t + 2cos2 t + 2sin2 t √ √ √ E = 2, E1 = 3, E2 = 5, E3 = 2, √ √ H = 7, H1 = 3, H2 = 3, H3 = 3, √ µ(A) = 2, η(A1 ) = 1, η(A2 ) = 5, η(B) = 5, 1 σ = e, ε = e, = 3, = 3, = 5, 2 √ √ √ a = 7, b = 3, c = 5, d = 2, θ = 5, ✼✵ ❞♦ ➤ã γ = 289, sin4 t − 14sin2 t − 285 , cos t − 14cos2 t − 273 √ e 821 + 60 14 + 7e − sin t √ 3e e 1649 + 60 14 + − cos t R2 (t) = Q2 (t) = ❉♦ ➤ã ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ (RDE5) ❧➭ P (t) = e−2t e−2t ❉Ơ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❍Ư q✉➯ ✸✳✸ ➤➢ỵ❝ t❤♦➯ ♠➲♥✳ ❉♦ ➤ã ❤Ư (3.12) ổ ị ợ ũ ề ữ ữ ề ể ợ ổ ị u(t) = ◆❤❐♥ ①Ðt✳ (1 − e−2t )(sin2 t + 2) (1 −2t −e x(t) )(cos2 t + 4) ❈➳❝ ✈Ý ❞ô ♠✐♥❤ ❤♦➵ tr ị ý ệ q ợ r ♠ét ❝➳❝❤ ✧❧➽♣ ❣❤Ð♣✧ ✈í✐ ♠ơ❝ ➤Ý❝❤ ❦✐Ĩ♠ tr❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❝➳❝ ➤Þ♥❤ ❧ý ✈➭ ❝➳❝ ❤Ư q✉➯✳ ❱✐Ư❝ ❣✐➯✐ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❘✐❝❝❛t✐ ❝❤♦ ♥❣❤✐Ư♠ ❝❤Ý♥❤ ①➳❝ ❝ß♥ ❧➭ ✈✃♥ ➤Ị ❦❤ã✱ t✉② ♥❤✐➟♥ ♠ét sè ♣❤➢➡♥❣ ♣❤➳♣ ①✃♣ ①Ø ✈➭ ♣❤➢➡♥❣ ♣❤➳♣ sè t×♠ ♥❣❤✐Ư♠ ủ trì t ợ trì tr [4, 15] ✼✶ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ♠ét sè ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈Ị tÝ♥❤ ỉ♥ ➤Þ♥❤ ✈➭ ổ ị ợ ột số ệ ó ♥❤✐Ơ✉ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trƠ ✈➭ ❤Ư t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝❤➽❝ ❝❤➽♥ ❝ã trÔ✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ ✿ ✶✳ ●✐í✐ t❤✐Ư✉ ♠ét sè ❦Õt q✉➯ ❣➬♥ ➤➞② ✈Ị tÝ♥❤ ỉ♥ ➤Þ♥❤ ♠ị ổ ị ợ ũ ột số ❧í♣ ❤Ư ❝ã ♥❤✐Ơ✉ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ➠t➠♥➠♠ ❝ã trƠ rê✐ r➵❝ ✈➭ ❤Ö t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ ➠t➠♥➠♠ ❦❤➠♥❣ ❝❤➽❝ ❝❤➽♥ ❝ã trƠ rê✐ r➵❝ ✭❝❤➢➡♥❣ ✷✮✳ ✷✳ ❇➺♥❣ ✈✐Ư❝ ❝➯✐ t✐Õ♥ ❤➭♠ ▲②❛♣✉♥♦✈ ✈➭ ❜➺♥❣ ♠ét sè ❦ü t❤✉❐t ❝❤ø♥❣ ♠✐♥❤ ♣❤ï ❤ỵ♣✱ ❝❤ó♥❣ t➠✐ ➤➲ ➤➢❛ r❛ ♠ét sè ➤✐Ị✉ ❦✐Ư♥ ➤đ ♠í✐ ❝❤♦ tÝ♥❤ ỉ♥ ➤Þ♥❤ ♠ị ổ ị ợ ũ ệ ó ễ tế t ó trễ ỗ ợ ❈❤ó♥❣ t➠✐ ❝ị♥❣ ➤➲ ➳♣ ❞ơ♥❣ ❦Õt q✉➯ ♥➭② ➤Ĩ ➤➢❛ r❛ ♠ét sè ➤✐Ị✉ ❦✐Ư♥ ➤đ ♠í✐ ❝❤♦ tÝ♥❤ ổ ị ũ ề ữ ổ ị ợ ❞➵♥❣ ♠ị ❜Ị♥ ✈÷♥❣ ❝❤♦ ❧í♣ ❤Ư t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ t ó trễ ỗ ợ ết q✉➯ ♥➭② ➤Ị✉ ➤➢ỵ❝ ♠✐♥❤ ❤♦➵ ❜➺♥❣ ❝➳❝ ✈Ý ❞ơ ❦✐Ĩ♠ tr❛ ❝ơ t❤Ĩ ✭❝❤➢➡♥❣ ✸✮✳ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ◆❣✉②Ơ♥ ❚❤Õ ❍♦➭♥✱ P❤➵♠ P❤✉✱ ❈➡ së ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ✈➭ ❧ý t❤✉②Õt ỉ♥ ➤Þ♥❤✱ ◆❳❇ ●✐➳♦ ❞ơ❝✱ ✷✵✵✵✳ ❬✷❪ P❤❛♥ ❚❤❛♥❤ ◆❛♠✱ ❚Ý♥❤ ỉ♥ ➤Þ♥❤ ♠ị ✈➭ ổ ị ợ ũ ủ ệ trì ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❝ã ❝❤❐♠✱ ▲✉❐♥ ➳♥ t✐Õ♥ sÜ t♦➳♥ ❤ä❝✱ ✷✵✵✽✳ ❬✸❪ ❱ò ◆❣ä❝ P❤➳t✱ ◆❤❐♣ ♠➠♥ ❧ý t❤✉②Õt ➤✐Ị✉ ❦❤✐Ĩ♥ t♦➳♥ ❤ä❝✱ ◆❳❇ ➜➵✐ ❤ä❝ ◗✉è❝ ●✐❛ ❍➭ ◆é✐✱ ✷✵✵✶✳ ❬✹❪ ❆❜♦✉✲❑❛♥❞✐❧✱ ❍✳✱ ❋r❡✐❧✐♥❣✱ ●✳✱ ▲♦♥❡s❝✉✱ ❱✳✱ ❏❛♥❦✱ ●✳✱ ▼❛tr✐① ❘✐❝❝❛t✐ ❊q✉❛t✐♦♥s ✐♥ ❈♦♥tr♦❧ ❛♥❞ ❙②st❡♠s ❚❤❡♦r②✱ ăsr s ❋❡r♦♥ ❛♥❞ ❱✳ ❇❛❧❛❦r✐s❤♥❛♥✱ ▲✐♥❡❛r ▼❛tr✐① ■♥✲ ❡q✉❛❧✐t✐❡s ✐♥ ❙②st❡♠ ❛♥❞ ❈♦♥tr♦❧ ❚❤❡♦r②✱ ❙■❆▼ ❙t✉❞✐❡s ✐♥ ❆♣♣❧✳ ▼❛t❤✳✱ ❙■❆▼ P❆✱ ✈♦❧✳✶✺✱ ✶✾✾✹✳ ❬✻❪ ■❦❡❞❛ ▼✳✱ ▼❛❡❞❛ ❍✳ ❛♥❞ ❑♦❞❛♠❛ ❙✳✱ ❙t❛❜✐❧✐③❛t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧✱ ✱ ♣♣✳ ✼✶✻ ✲ ✼✷✾✱ ✶✾✼✷✳ ✶✵ ❬✼❪ ❍❛❧❡ ❏✳ ❛♥❞ ❱❡r❞✉②♥ ▲✉♥❡❧ ❙✳ ▼✳✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✉♥❝t✐♦♥❛❧ ❉✐❢❢❡r❡♥✲ t✐❛❧ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✶✾✾✸✳ ❬✽❪ ▲❡ ❱❛♥ ❍✐❡♥ ❛♥❞ ❱✉ ◆❣♦❝ P❤❛t✱ ❘♦❜✉st st❛❜✐❧✐③❛t✐♦♥ ♦❢ ❧✐♥❡❛r ♣♦❧②t♦♣✐❝ s②st❡♠s ✇✐t❤ ♠✐①❡❞ ❞❡❧❛②s ✐♥ st❛t❡ ❛♥❞ ❝♦♥tr♦❧✱ ✱ ♣♣✳ ✷✽✷ ✲ ✷✾✵✱ ✷✵✵✾✳ ✶✽✶ ✼✷ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳✱ ✼✸ ❬✾❪ P❛r❦ ❏✳ ❍✳ ❛♥❞ ❑✇♦♥ ❖✳ ▼✳✱ ❘♦❜✉st st❛❜✐❧✐③❛t✐♦♥ ❝r✐t❡r✐♦♥ ❢♦r ✉♥❝❡rt❛✐♥ s②st❡♠s ✇✐t❤ ❞❡❧❛② ✐♥ ❝♦♥tr♦❧ ✐♥♣✉t✱ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳✱ ✶✼✷✱ ♣♣✳ ✶✵✻✼ ✲ ✶✵✼✼✱ ✷✵✵✻✳ ❬✶✵❪ P✳ ◆✐❛♠s✉♣✱ ❑✳ ▼✉❦❞❛s❛✐ ❛♥❞ ❱✳ ◆✳ P❤❛t✱ ■♠♣r♦✈❡❞ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❢♦r t✐♠❡✲✈❛r②✐♥❣ s②st❡♠s ✇✐t❤ ♥♦♥❧✐♥❡❛r ❞❡❧❛②❡❞ ♣❡rt✉r❜❛t✐♦♥s✱ ▼❛t❤✳ ❈♦♠♣✉t✳✱ ❆♣♣❧✳ ✱ ♣♣✳ ✹✾✵✲✹✾✺✱ ✷✵✵✽✳ ✷✵✹ ❬✶✶❪ P✳ ◆✐❛♠s✉♣✱ ❑❛♥✐t ▼✉❦❞❛s❛✐ ❛♥❞ ❱✳ ◆✳ P❤❛t✱ ▲✐♥❡❛r ✉♥❝❡rt❛✐♥ ♥♦♥✲ ❛✉t♦♥♦♠♦✉s t✐♠❡✲❞❡❧❛② s②st❡♠s ✿ st❛❜✐❧✐③❛❜✐❧✐t② ✈✐❛ r✐❝❝❛t✐ ❡q✉❛t✐♦♥s✱ ❊❧❡❝tr♦♥✐❝ ❏♦✉r♥❛❧ ♦❢ ❉✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❬✶✷❪ ❘❡♥ ❋✳ ❛♥❞ ❈❛♦ ❏✳✱ ◆♦✈❡❧ t✐♠❡ ❞❡❧❛②s✱ α − st❛❜✐❧✐t② ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳✱ ✱ ♣♣✳ ✶ ✲ ✶✵✱ ✷✵✵✽✳ ✷✻ ♦❢ ❧✐♥❡❛r s②st❡♠s ✇✐t❤ ♠✉❧t✐♣❧❡ ✱ ♣♣✳ ✷✽✷ ✲ ✷✾✵✱ ✷✵✵✻✳ ✶✽✶ ❬✶✸❪ ❱✳ ◆✳ P❤❛t ❛♥❞ P✳ ◆✐❛♠s✉♣✱ ❙t❛❜✐❧✐t② ❛♥❛❧②s✐s ❢♦r ❛ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥❛❧ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ t♦ ❛♣♣❡❛r ✭✷✵✵✾✮ ✳ ❬✶✹❪ ❱✳ ◆✳ P❤❛t ❛♥❞ ❱✳ ❚✳ ❚❛✐✱ ●❧♦❜❛❧ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐③❛t✐♦♥ ♦❢ ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✈✐❛ ❘✐❝❝❛t✐ ❡q✉❛t✐♦♥s✱ t✐♦♥❛❧✱ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝✲ ✱ ♣♣✳ ✼✺✷ ✲ ✼✻✽✱ ✷✵✵✽✳ ✶✸ ❬✶✺❪ ❲✐❧❧✐❛♠ ❚❤♦♠❛s✱ ❘✳✱ ❘✐❝❝❛t✐ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❆❝❞❡♠✐❝Pr❡ss✱ ❙❛♥ ❉✐❡❣♦✱ ✶✾✼✷✳ ❬✶✻❪ ❨♦s❤✐③❛✇❛ ❚✳✱ ❙t❛❜✐❧✐t② ❚❤❡♦r② ❜② ▲②❛♣✉♥♦✈ ❙❡❝♦♥❞ ▼❡t❤♦❞✱ P✉❜❧✐❝❛✲ t✐♦♥ ♦❢ t❤❡ ▼❛t❤✳ ❙♦❝✐✳ ♦❢ ❏❛♣❛♥✱ ◆♦✳✾✱ ❚♦❦②♦✱ ✶✾✻✻✳