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❯ ❑❍■➎◆ ✣×Đ❈ ❈Õ❆ ❍➏ ▼➷ 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✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✹ ✹ ✺ ✺ ✺ ✺ ✻ ✻ ✻ ✻ ✼ ✽ ✽ ✾ ✶✵ ✶✵ ✶✶ ✶✶ ✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔ ✶✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✺ ✷✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ỹ t➔✐ ❍➺ ♠ỉ t↔ ❧➔ ♠ët ù♥❣ ❞ư♥❣ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❚➻♠ ❤✐➸✉ ✈➲ ❤➺ ♠æ t↔✱ ❝❤♦ t❛ ❝→✐ ♥❤➻♥ tê♥❣ q✉❛♥ ✈➲ ❜↔♥ ❝❤➜t ✈➟♥ ❤➔♥❤ ❝õ❛ ❝→❝ tr↕♥❣ t❤→✐ ♥❣❤✐➺♠ ❝ơ♥❣ ♥❤÷ ♠è✐ ❧✐➯♥ ❤➺ ♠➟t t❤✐➳t ❣✐ú❛ ❝→❝ tr↕♥❣ t❤→✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❜➜t ❜✐➳♥ ❤ú✉ ❤↕♥✳ ❍➺ ♠ỉ 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✐ ♠♦♥❣ ♠✉è♥ ✤÷❛ r❛ ♥❤÷♥❣ ❦➳t q✉↔ ♠ỵ✐ ♣❤ư❝ ✈ư ❝❤♦ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✳ ✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✳ ❚➼♥❤ ❈✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ❘✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ■✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ 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✐ sỷ ữỡ ự ỵ tt tr q tr tỹ t rữợ t ú tổ t t❤➟♣ ❝→❝ ❜➔✐ ❜→♦ ❦❤♦❛ ❤å❝ ❝õ❛ ♥❤ú♥❣ t→❝ ❣✐↔ trữợ q t ữủ ♠ỉ t↔✳ ❚ø ✤â ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ t÷ì♥❣ tü ❤â❛✱ ❝❤ó♥❣ tỉ✐ ❦❤→✐ q✉→t ♥❤ú♥❣ ❦➳t q✉↔ ✤â✱ ❝❤ó♥❣ tỉ✐ s ữ r ỳ t ợ 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❤❡♦ ♠ët tr➟t t ♥❤➜t ỗ m n ởt  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 ✿❝❤➾ sè ❝ët✳ m.n tữợ tr tữớ ❝❤ú ❝→✐ A, B, C ✤➸ ❦➼ ❤✐➺✉ ❝→❝ ♠❛ tr➟♥✳ ❉↕♥❣ t❤✉ ❣å♥ A = [aij ]mn ❱➼ ❞ö ✶✳✶✳ ❈❤♦ ♠❛ tr➟♥ A= −13 20 ❧➔ ♠❛ tr➟♥ ❝ï 2.3 ✭✷ ❤➔♥❣ ✈➔ ✸ ❝ët✮✳ ✹ −6 ✶✳✷ ▼❛ tr➟♥ ✈✉æ♥❣ ▼❛ tr➟♥ ✈✉æ♥❣ ❧➔ ♠❛ tr➟♥ ❝â sè ❤➔♥❣ ✈➔ sè ❝ët ❜➡♥❣ ♥❤❛✉✳ ▼❛ tr➟♥ ❝➜♣ n.n ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n✳ ❈→❝ ♣❤➛♥ tû aij , (i = j) t↕♦ t❤➔♥❤ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❝õ❛ ♠❛ tr➟♥ ✈✉ỉ♥❣✳ ❱➼ ❞ö ✶✳✷✳ ▼❛ tr➟♥   1 −1   B = 2 −2 ❧➔ ♠❛ tr➟♥ ✈✉è♥❣ ❝➜♣ ✭✸ ❤➔♥❣ ✈➔ ✸ ❝ët✮✳ ❈→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❧➛♥ ❧÷đt ❧➔ 1; 3; ✶✳✸ ▼❛ tr➟♥ ✤ì♥ ✈à ▼❛ tr➟♥ ✤ì♥ ✈à ❝➜♣ n tr♦♥❣ ✈➔♥❤ ❱ ❧➔ ♠➔ tr➟♥ ✈✉æ♥❣ ❝➜♣ n tr♦♥❣ ✤â t➜t ❝↔ ❝→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❜➡♥❣ ✶✱ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❦❤→❝ ❜➡♥❣ ❦❤æ♥❣✳     0 0  En =   .   0 ✶✳✹ ▼❛ tr➟♥ ❝♦♥ ▼❛ tr➟♥ ❝♦♥ ❝õ❛ A ❧➔ ♠❛ tr➟♥ B ✤÷đ❝ t❤➔♥❤ ❧➟♣ tø ♠❛ tr➟♥ ❜❛♥ ✤➛✉ ❜➡♥❣ ❝→❝❤ ❜ä ✤✐ ♠ët sè ❞á♥❣✱ ✈➔ ♠ët sè ❝ët✳  1 ❱➼ ❞ö ✶✳✸✳ ❈❤♦ ♠❛ tr➟♥ A = 2 ❝♦♥ ❝õ❛ ♠❛ tr➟♥ A  −1   ❚❛ ❝â ♠❛ tr➟♥ B = 2 −1 ✶✳✺ ▼❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ❈❤♦ 1  0  ❧➔ ♠❛ tr➟♥ ❧➔ ♠❛ tr➟♥ ố tr trữớ K tỗ t ởt tr B s❛♦ ❝❤♦✿ A × 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♥❣ ỵ tr A B õ ũ tữợ n ì m A + B ởt tr ũ tữợ tr õ ♣❤➛♥ tû tr♦♥❣ ✈à tr➼ t÷ì♥❣ ù♥❣ ❜➡♥❣ tê♥❣ ❤❛✐ ♣❤➛♥ tû t÷ì♥❣ ù♥❣ ❝õ❛ ♠é✐ ♠❛ tr➟♥✿ (A + B)ij = Aij + Bij , ≤ i ≤ m, ✈➔1 ≤ i ≤ n ❱➼ ❞ö ✶✳✹✳ 1 0 + 0 = 1+0 3+0 1+5 1+7 0+5 0+0 = ❚➼♥❤ ❝❤➜t ✶✳ A + B = B + A ✷✳ (A + B) + C = A + (B + C) ✸✳ + A = A + = A ✹✳ A + (−A) = (−A) + A = ✶✳✻✳✸ ❚➼❝❤ ởt số ợ ởt tr ỵ cA ❝õ❛ ♠ët sè c ✈ỵ✐ ♠❛ tr➟♥ A ❧➔ ♠ët ♠❛ tr➟♥ ❝â ✤÷đ❝ ❜➡♥❣ ❝→❝❤ ♥❤➙♥ ♠é✐ ♣❤➛♥ tû ❝õ❛ A ✈ỵ✐ c (cA)ij = c.Aij ✻ ❱➼ ❞ö ✶✳✺✳ −3 −2 = 2.1 2.8 2(−3) 2.4 2(−2) 2.5 = 16 −6 −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 )m×p ❝➜♣ m × p✱ ✈ỵ✐ n ckh = aki bih , j=1 ✈ỵ✐ k = 1, m, h = 1, p ✣✐➲✉ ❦✐➺♥ ữủ tr A ợ tr B ố ♣❤➛♥ tû tr➯♥ ❞á♥❣ ❝õ❛ ♠❛ tr➟♥ A ♣❤↔✐ ❜➥♥❣ sè ♣❤➛♥ tû tr➯♥ ❝ët ❝õ❛ ♠❛ tr➟♥ B t÷ì♥❣ ù♥❣✳  ❱➼ ❞ö ✶✳✻✳ ❈❤♦ A = 1 −1 ✈➔ B = 0 1.1 + 0.0 + 1.1  1  −1 ❑❤✐ ✤â C = A.B = 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➼♥❤ ❣✐❛♦ ❤♦→♥✳ ✼ = −1 −1 ❱➼ ❞ö ✶✳✼✳ ❈❤♦ A = ✈➔ B = AB = ✈➔ BA = −1 −1 −1 = = 12 13 24 25 −1 −4 27 40 ♥➯♥ AB = BA ỵ ởt tr õ tữợ m ì n ợ ❝→❝ ❣✐→ trà aij t↕✐ ❤➔♥❣ i✱ ❝ët j t❤➻ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à B = AT ❧➔ ♠❛ tr➟♥ õ tữợ n ì m ợ tr bij = aji✳ ❱➼ ❞ö ✶✳✽✳ T   = 2 −6  −6  ❚➼♥❤ ❝❤➜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➟♥ ✈✉ỉ♥❣ tr➯♥ tr÷í♥❣ K✳ ❚➼❝❤ A ì A ữủ ỵ A2 ữỡ tr A ợ n số ữỡ tũ þ✱ t❤➻ t➼❝❤✿ A × A × × A ỵ An ữủ ụ tứ n A ữợ A0 = E ❱➼ ❞ö ✶✳✾✳ ❈❤♦ A = 0 1  0   0   ⇒ A2 = 0 0 0   0 ✈➔ A3 = 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 −3 ❱➼ ❞ö ✶✳✶✵✳ ❚➼♥❤ ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ A = −1 3  −1 det (A) = (−2).1(−1) + 2.3.2 + (−3).(−1).0 − 2.1.(−3) − 0.3.(−2) − (−1).(−1).2 = −2 + 12 + + + − = 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❝❤ ✤↔♦ A1 ữợ t tr ♥❣❤à❝❤ ✤↔♦✿ ✶✳ ❚➼♥❤ ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ A✿ ✲ ◆➳✉ det(A) = t❤➻ A ❦❤æ♥❣ ❝â ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ A−1✳ ✲ ◆➳✉ det(A) = t❤➻ A ❝â ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ A−1✳ ✷✳ ▲➟♣ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à AT ❝õ❛ A✳ ✸✳ ▲➟♣ ♠❛ tr➟♥ ♣❤ư ❤đ♣ AT : A∗ = (ATij )nn✳ A∗ ✹✳ ❚➼♥❤ ♠❛ tr➟♥ A−1 = det(A) ❱➼ ❞ö ✶✳✶✶✳ ❈❤♦ A = ✶✳ det(A) = ✷✳ AT = ✸✳ A∗ = −2 −2 ✳ ❚➼♥❤ A−1 =8 −2 −4 −6 ✹✳ A−1 = 81 2 −3 = −0, 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 ♥❤➙♥ ❝õ❛ f ỵ ker f ❈❤♦ ❤➺ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥ ✈➔ ❝❤♦ x1(0) = 0✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ t1 > ✈➔ w Rn tỗ t ởt ữủ u = Cpv − 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 = ∈ Rn∞ = Rn ◆➯♥ ❤➺ ✭✷✳✸✮ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✳ ✸❜ ⇐⇒ ✸❝ ❚❤❡♦ ✶ ✈➔ ✷✳ ✸❜ ⇐⇒ ✸❞ ❚❤❡♦ ✶ ✈➔ ✷✳ ✸❞ ⇐⇒ ✸❡ rank[ λE − A α βE B ] = rank −A B = rank[ αE − βA B ] ❱➼ ❞ö ✷✳✹✳ ❳➨t ❤➺ s❛✉ x˙ = 1 = x2 + x1 + −1 0 u u ❚❛ ❝â rank[ B1 JB1 ] = rank 1 ✈➔rank[ B2 =2 BN2 ] = rank −1 0 = < ❱➟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❤❛♥❣✳ ❍➺ ♠ỉ t↔ t✉②➳♥ t➼♥❤ ❜➜t ❜✐➳♥ ❧➔ ❤➺ ♠æ t↔ ❝â t t ữủ ợ tf > x1 (0) R, w R, tỗ t ởt ✤➛✉ ✈➔♦ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ u ∈ Cpv−1 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 ❱➼ ❞ö ✷✳✺✳ ✶✳ ❳➨t ❤➺ ð ✈➼ ❞ư ✭✷✳✹✮ ✤÷đ❝ ❝❤♦ ❜ð✐ x˙1 = 1 = x2 + x1 + −1 0 u u ❍➺ ❝♦♥ ❝❤➟♠ tr➯♥ ❧➔ ❈✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥❤÷ ✤➣ ❝â✱ ♥➯♥ ❤➺ tr➯♥ ❧➔ ❘✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✷✳ ❈❤♦ ◆ ❧ơ② ❧✐♥❤ ✈➔ ①➨t ❤➺ N x˙ = x + Bu ❍➺ tr➯♥ ❝❤➾ ❜❛♦ ỗ õ ổ ữủ q t ợ ũ ♠❛ 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 ❧✐➯♥ tư❝✱ ✤â ❧➔ f (φ) → tr♦♥❣ Rn ✈ỵ✐ ♠å✐ ❞➣② (φi )i∈N ✈ỵ✐ φ1 → tr♦♥❣ Dn ✳ ❱➼ ❞ö ✷✳✻✳ P❤➙♥ ❜è ❉✐r❛❝ ❞❡❧t❛ δα ∈ C n ữủ () = () ợ ♠å✐ φ ∈ Dn , α ∈ R ✳ +∞ (x) = x= ú ỵ ❑❤✐ ❝❤♦ δ ∈ Dn ✈➔ tˆ > ✤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 ˆ ˆ H(t)φ(t)dt =: H(φ) =− R H(t) = ợt < ữợ ✤ì♥ ✈à✳ ❚❛ t➻♠ ✤÷đ❝ q✉❛♥ ❤➺ δ0 = H.˙ ❚❛ ✈ỵ✐t ≥ ❝ơ♥❣ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ sü ❞à❝❤ ❝❤✉②➸♥ ❝õ❛ H ❜➡♥❣ Hα(t) := H(t − α) ✈➔ δα = H˙ α ❍❛✐ ♣❤➙♥ ❜è f1, f2 ∈ C n ❧➔ ♥❤÷ ♥❤❛✉ ♥➳✉ f1(φ) = f2(φ) ✈ỵ✐ ♠å✐ φ ∈ Dn✳ ❚❤❡♦ ✤â✱ x : I Rn , I R ữủ ỷ ỵ ♥❤÷ ♠ët ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ R ❜ð✐ x(t) = ✈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❤ ♥➳✉ ♥â 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) ởt ổ t õ trữợ ♣❤➨♣ ♥❤➙♥ ✈ỵ✐ ❤➔♠ 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✐ ≤ q ≤ ∞ ❧➔ ♠ët sè ♥❣✉②➯♥ ✤õ ❧ỵ♥ −q − 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 [τj−1; τ 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 ♠➝✉ ♠↕❝❤ ✤✐➺♥ ❝õ❛ ♠ët ♠↕❝❤ ✈✐ ♣❤➙♥ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè s❛✉✿ ❍➻♥❤ ✷✳✷✿ ▼↕❝❤ ✤✐➺♥ ❝õ❛ ♠ët ♠↕❝❤ ✈✐ ♣❤➙♥✳ x1 − x4 = u(t), C(x˙ − x˙ ) + (x3 − x2 ) = 0, R x3 = A(x4 − x2 ), x4 = 0, ✷✹ ✈ỵ✐ ✤➛✉ ✈➔♦ ✤✐➺♥ →♣ u(t) = ✈ỵ✐t < 0, ✈ỵ✐t ≥ ❱ỵ✐ x4 = 0, x1 = u(t), x˙ = u˙ ✈➔ x3 = −Ax2 t❛ ✤÷đ❝ x˙ = − (A + 1)x2 + u ˙ CR ❱➻ sü t❤❛② ✤ê✐ tr♦♥❣ ✤✐➺♥ →♣ ✤➛✉ ✈➔♦✱ ✉ ❦❤æ♥❣ ❦❤↔ ✈✐✳ ❈❤♦ u = H ✈➔ A → ∞✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣ x1 − x4 = H, C(x˙ − x˙ ) + (x3 − x2 ) = 0, R x2 = 0, x4 = 0, ✈ỵ✐ ♥❣❤✐➺♠ x1 = H, x2 = 0, x3 = −RC H˙ = −RCδ0✈➔ x4 = 0✳ ❍➺ tr➯♥ ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ số ợ số = = 1✮ ✈➔ iordf = −1✳ Ð ♠é✐ ❣✐→ trà ♥❤➜t q✉→♥ ❜❛♥ ✤➛✉✱ ✈➼ ❞ö x(−1) = t❛ ❝â ♠ët ♥❣❤✐➺♠ ✤ì♥ x ✈ỵ✐ iordx = ✣è✐ ✈ỵ✐ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ♣❤÷ì♥❣ 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 N , J 0 In∞ ❱➟②✱ t❛ ❝â x˙ = Jx1 + f1 , N x˙ = 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 f f z˙ = g1 = X −1 f1 , ✷✺ n (T) t❛ ❝â f ∈ C n (T) ✈➔ s✉② r❛ ♥❣❤➽❛ ❧➔ z ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ g1✳ ❱➻ f ∈ Cimp imp n g1 ∈ Cimp (T) ✈ỵ✐ iordg1 = iordf1 ❜➡♥❣ ❜ê ✤➲ ✭✷✳✹✮✳ ❙û ❞ö♥❣ ♣❤➙♥ r➣ ữủ g1 = g + g1,imp ợ f gˆ1 = gˆ1,j qi ✈➔ (i) g1,imp = ci,j j jZ i=0 jZ ợ q ữợ r g1,imp,j = (i) qj i=0 ci,j δτj ♥➳✉ qj < 0✱ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ g1 ❝â ❞↕♥❣ qi t t (i) gˆ1,j (s)ds + z =c+ ci,j δτj t0 j∈Z i=0 t0 j∈Z t qi t =c+ gˆ1,j (s)ds + t0 j∈Z (i) cij δτj c0j δτj + t0 j∈Z i=1 qi −1 t =c+ gˆ1,j (s)ds + t0 j∈Z (i) c0j Hτj + j∈Z ci,j δτj j∈Z i=0 ✈ỵ✐ c ∈ Rn ❱➟②✱ ♠é✐ ♥❣✉②➯♥ ❤➔♠ z ❝õ❛ g1 ❝â t❤ù tö ①✉♥❣ ❧ü❝ q✲✶ ✈➔ ✈➻ ✈➟② ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✻❛✮✳ ❱ỵ✐ ❣✐→ trà ❜❛♥ ✤➛✉ ❝è ✤à♥❤ z(t0 ) = c + c0j Hτ (t0 ) ✈ỵ✐t0 ∈ R \ T f j j∈Z ✈➔ z(τ1− ) = c + c0j lim− Hτj (t) = c + j∈Z z(τj− ) = c + t→τj c0j , j∈Z,τj ≥τi c0j lim+ Hτj (t) = c + j∈Z t→τj c0j j∈Z,τj ≥τi n (T) t❤ä❛ ♠➣♥ ♠ët ♠ët tr tỗ t ởt ỡ tr Cimp ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x1 (τi− ) = x1,0 , x1 (t0 ) = x1,0 , x1 (τi+ ) = x1,0 ✈ỵ✐ τi ∈ T✳ ❱ỵ✐ ♣❤➛♥ ✤↕✐ sè ✷✳✻❜✱ t❛ sû ❞ö♥❣ n∞ f2 = fˆ2 + f2,imp ∈ Cimp (T) ◆❣❤✐➺♠ ✤ì♥ ❝õ❛ ✭✷✳✻❜✮ ✤÷đ❝ ❝❤♦ ❜ð✐ v−1 (i) (i) n∞ N i fˆ2 + f2,imp ∈ Cimp (T) x2 = − i=0 ✈ỵ✐ iord(x2) ≤ q + v ữỡ tỹ ữ trữợ t t q tr ỵ r x2(t0) = x2,0 ✈ỵ✐ t0 ∈ R \ T ✷✻ ỵ (E, A) q ợ v = ind(E, A)✳ ❈❤♦ xj0 ∈ C([τj−1, τj ], Rn) ✈➔ ❝❤♦ f = fˆ + fimp, fˆ = ˆ i∈Z fi ✱ t↕✐ fˆj = E x˙ j0 − Axj0✳ ❑❤✐ ✤â✱ ❝â ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ✤➙②✿ ✶✳ Pữỡ tr số ợxj1 = xj0 ex = Ax + f ✭✷✳✼✮ n ❝â ♥❣❤✐➺♠ ✤ì♥ x ∈ Cimp ✈ỵ✐ iordx ≤ iordf + v − ✷✳ ❈❤♦x = xˆ + ximp, xˆ = ˆi ❧➔ ♥❣❤✐➺♠ ✤ì♥ ❝õ❛ ✭✷✳✼✮✳ ❑❤✐ ✤â x¯ = x − xˆj−1 ❧➔ ♠ët ♥❣❤✐➺♠ ✤ì♥ i∈Z x ❝õ❛ E x˜˙ = A˜ x + f˜ + Exj,0 δτj , x˜j−1 = 0, t↕✐ xj,0 = xˆj−1(τj ) ✈➔ f¯ = f − fˆj ❇ê ✤➲ ✷✳✻✳ ✣➦t fˆj = E x˙ j0 − Axj0 ❧ü❝ xj0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ E x˙ = Ax + fˆj ✈ỵ✐t ∈ [τj , j+1]ợ ồj Z t ỗ t ❝õ❛ fˆj ❜à ❜✐➳♥ ✤ê✐ t❤❡♦ ❝→❝❤ ♠ët ✤✐➲✉ ❦✐➺♥ ữủ trữợ x(j ) = xj,0 (j ) ữủ ❜✐➳♥ ✤ê✐ ❝❤♦ ♣❤ị ❤đ♣ ✈ỵ✐ ✭✷✳✽✮✳ ❱➻ x˙ = xˆ˙ +x˙ imp + (ˆ xi (τi )) − xˆi−1 (τi )δτ t❛ ❝â i E xˆ˙ + x˙ i mp + (ˆ xi (τj ) − xˆi−1 (τj )δτi ) = A (ˆ x + ximp ) + fˆ + fimp fˆ + fimp + Ax˙ j0 − Axj0 = A (ˆ x + ximp ) + j∈Z,j=i ✣➦t x˜ = x − xˆj−1 ✈➔ f˜ = f − fˆj ✱ ❝â ♥❣❤➽❛ ❧➔ E x˜˙ = A˜ x + f˜ + Exj,0 δτj , x˜j−1 = 0, t↕✐ xj,0 = xˆj−1(τj ) ❳➨t ✭✷✳✾✮ ✈ỵ✐ ❤➔♠ ✤➛✉ ✈➔♦ u ∈ Cp∞(R, Rn) ❱➟②✱ u ∈ Cimp (T) ✈ỵ✐ iord(u) = q ≤ −1 ✈➔ t❛ ❝â t❤➸ t tr ởt ố s rở ú ỵ ✷✳✺✳ ◆➳✉ t❛ ①➨t ♠➞✉ ♣❤➙♥ ❜è ❧✐➯♥ tö❝✱ t❤➻ u ∈ Cpv−1 ❧➔ ✤õ✳ ❍➺ t÷ì♥❣ ù♥❣ tr♦♥❣ ✭❲❈❋✮ ❝â ❞↕♥❣ Ax˙ = Ax + Bu n x˙ = Jx1 + B1 u N x˙ = x2 + B2 u ✷✼ ✭✷✳✶✵❛✮ ✭✷✳✶✵❜✮ ❱➻ iord(u) = q ≤ −1✱ t❛ ❝â iord(x1 = q − ≤ −1) ♥➯♥ x1 ∈ C 0(R, Rn)✱ ♥❣❤➽❛ ❧➔ ❦❤æ♥❣ tỗ t số tr ỗ x1 ỗ số ❝♦♥ ♥❤❛♥❤✮ ❝â ❞↕♥❣ v−1 N i B2 u(i) x2 = i=0 ✈ỵ✐ iord(x2) ≤ q + v − v ứ ỵ t õ x˜2 ♥❣❤✐➺♠ ✤ì♥ ❝õ❛ N x˜˙ = x˜2 + B2 u˜ + N x2j,0 δτj , = x2 − xˆ2,j−1 ❧➔ ♠ët x˜2,j−1 = 0, t↕✐ x2j,0 = xˆ2,j−1(τj ) ✈➔ u˜ = u − uˆj ❱➟②✱ v−1 x˜2 = − v−1 v−1 i i N B2 u˜ − i=0 N i+1 (i) x2j,0 δτj v−1 i =− N i x2j,0 δτj N B2 u˜ − i=0 i=0 (i−1) i i=1 ỵ t ợ q ỗ t u Cp s x(j+ ) x(j ) = w ữợ tr x t↕✐ τj ✮ ✈ỵ✐ τj ∈ T τj x ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ w := ∈ ⊕ Im[ B2 N B2 N v−1 B2 ] ❈❤ù♥❣ ♠✐♥❤✳ ” ⇒ ” δτ j x = δτj x2 ✈➻ δτ x = ✈ỵ✐ u ∈ Cp∞.τj ∈ T ❱➟②✱ j v−1 δτj x2 = x2 (τj+ ) − x2 (τj− ) N i B2 (u(i) (τj+ ) − u(i) (τj− )) =− i=0 ❱➟②✱ δτ x2 ∈ im[B2 N B2 N v−1B2] ” ⇐ ” ❈❤å♥ w0 , , wv−1 s❛♦ ❝❤♦ w = − j u(t) = v−1 i i=0 N B2 wi  w0 + (t − τj )w1 + (t − τj )2 w2 +  (t − τj )υ−1 wυ−1 (υ − 1)! ✈➔ ∆τ x2 = − υ− τ −1 j ( N j B2 wj = w N B2 ∆υj u i) = − j i=0 i=0 ✷✽ ✱ ✈➟② t❛ ❝❤å♥ ✳ ♥➳✉ t ≥ τj , ♥➳✉ t < τj ✭✷✳✶✶✮ ❚❛ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕ Iτ j n : Rn∞ → Cimp ✈ỵ✐ Iτ (W ) := j I2,τj (w) τ −1 (i−1) I2,τj (w) := − δτ j ✈ỵ✐ ợj T N j (w) i=0 ú ỵ r Ix=0(X2(0)) ❜✐➸✉ ❞✐➵♥ ♥❣❤✐➺♠ tr♦♥❣ x(t) t↕✐ t❤í✐ ✤✐➸♠ ❜❛♥ ✤➛✉ t0 = ♥❣✉②➯♥ ♥❤➙♥ ❜ð✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ x2(0) I (w) ỗ tt tr x(t) t j j ỵ ❳➨t ✭✷✳✾✮ ✈ỵ✐ ❝➦♣ ❝❤➼♥❤ q✉② (E, A)✳ ❱ỵ✐ ♠å✐ w Rn tỗ t u Cp s❛♦ ❝❤♦ tr↕♥❣ t❤→✐ ♥❣❤✐➺♠ ❝õ❛ x t↕✐ τj ❦➼ ❤✐➺✉ ❜ð✐ ximpj ✤÷đ❝ ❝❤♦ ❜ð✐ ximpj = Iτ (w) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ j B2 N B2 · · · N (v−1) B2 I2,τj (w) ∈ ♥❣❤➽❛ ❧➔ I2,τ (w)(φ) ∈ j B2 N B2 · · · N (υ−1) B2 ✈ỵ✐ ♠å✐ φ ∈ Dn✳ ✱ ❱➻ ximpj = t õ ximpj = trữợ ỵ ự x2,imp v1 v1 j xˆ2 = − (i) N B2 u˜ ( x2,imp j ✳ ❚÷ì♥❣ tü ✳ N j x2j,0 δτj i − 1) − i=0 ˜ ≤v−2 ❱➔ iord(x) ❱➟② ✈➔ ximpj = i=0 υ−1 (i−1) N i x2j,0 δτj x˜2,imp j = − i=0 ▼➦t ❦❤→❝ t❛ ❝â υ−1 (i−1) I2,τj (w) = δτ j N iw ✳❱➟② x˜2,imp,j = I2,τj (w) i=0 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✳ sỷ r tỗ t ởt t ủ u s❛♦ ❝❤♦ −N x2j,0 = N w✱ υ−1 t❛ ✤➦t x2,j,0 = xˆ2,j−1(τj ) ✈➔ xˆ2,j−1(τj ) = − niB2u˜(i) j−1 (τj ) r➡♥❣ −N x2j,0 = N w i=0 w ∈ Im B2 N B2 · · · N (υ−1) B2 ✷✾ ✳ + Ker(N ) ❚❛ ❝â ♣❤➙♥ t→❝❤ w = w˜ + wˆ ✈ỵ✐ w˜ = Im ✈➔ w˜ = v−1 N j B2 w˜j B2 N B2 · · · N (v−1) B2 ✱ wˆ ∈ Ker(N ) ✈ỵ✐ ♠é✐ w˜j ∈ Rn ✳ ❱➟② ∞ j=0 υ−1 (i−1) I2,τj (w) = δτj N i (w¯ + w) ˆ i=1 υ−1 υ−1 (i−1) δτj N j = N i B2 w˜j i=0 i=1 υ−1 υ−1 (i−1) = δτ j N i+j B2 w˜j i=1 i=0 ❱ỵ✐ I2,τ (w) ∈ Im N B2 N 2B2 · · · N (υ−1)B2 ✳ ▼➦t ❦❤→❝ ♥➳✉ I2,τ (w) ∈ Im N B2 N 2B2 · · · N (υ−1)B2 t❤➻ j j N i w ∈ Im N B2 N B2 · · · N (υ−1) B2 B2 N B2 · · · N (υ−1) B2 = N Im ❱ỵ✐ i = 1, · · · , υ − 1✈➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦t ❜✐➺t N w ∈ N Im ⇔ w ∈ Im B2 N B2 · · · N (υ−1) B2 ✳ B2 N B2 · · · N (υ−1) B2 + Ker(N ) ❚❛ t✐♠ ♠ët ❣✐→ trà u t❤➼❝❤ ❤đ♣ ✤÷đ❝ sû ❞ư♥❣ tr ỵ q ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ①✉♥❣ ❧ü❝ ✭■✲ ✤✐➲✉ ❦❤✐➸♥✮ ♥➳✉ ✈ỵ✐ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x(0), τj ∈ T w Rn tỗ t ởt ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ u ∈ Cp∞ s❛♦ ❝❤♦ ximp,j = Ir (w) j ỵ ♠➺♥❤ ✤➲ s❛✉ ✤➙② t÷ì♥❣ ✤÷ì♥❣ ✶✳ ❍➺ ✷✳✾ ❧➔ ■✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✷✳ ❍➺ ❝♦♥ ♥❤❛♥❤ ✭✷✳✶✵❜✮ ❧➔ ■✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✸✳ Ker(N ) + Im B2 N B2 · · · N (υ−1)B = Rn ✹✳ im(N ) = im N B2 N 2B2 · · · N (υ−1)B2 ✺✳ im(N ) + m(B2) + Ker(N ) = Rn ✱ ✻✳ r❛♥❦ E 0 = n + rank(E)✳ ∞ ∞ A E B ✸✵ ✭✶✮ tứ ximp = ỵ ✷✳✼✳ ✭✸✮ ⇐⇒ ✭✹✮✳ ❱➻✿ ❈❤ù♥❣ ♠✐♥❤✳ Im N B2 N B2 · · · N (υ−1) B2 ximp,2 ✳ N B2 N B2 · · · N (υ−1) B2 = N Im = N (Im N B2 N B2 · · · N (υ−1) B2 + Ker(N )) ✭✹✮ =⇒ ✭✺✮ ❚ø✿ Im(N ) + Im(B2 ) + Ker(N ) = Im = Im N B2 N B2 · · · N (υ−1) B2 B2 N B2 · · · N (υ−1) B2 + Im(B2 ) + Ker(N ) + Ker(N ) = Rn∞ ✭✺✮ =⇒ ✭✸✮ ◆â ❝è ✤à♥❤ r➡♥❣ Im(N n+1) + Im(N j B2) = Im(N j ) ✈ỵ✐ i = 1, · · · , υ − 1✳ ✣✐➲✉ ♥➔② ỡ ỵ Rn = Ker(N ) + Im(B2 ) + Im(N B2 ) + · · · Im(N n−1 B2 ) + Im(N υ ) = Ker(N ) + Im B2 N B2 · · · N (υ−1) B2 ✭✹✮ ⇐⇒ ✭✻✮ ❳➨t ♣❤➛♥ r➣ ❬✺❪ ❝õ❛ ♠❛ tr➟♥ ổ (N, B2) tỗ t tr ổ s V s❛♦ ❝❤♦ (N, B2 ) (V −1 N V, V −1 B2 ) = N11 N12 N 14 ❱ỵ✐ (N11N12) ❧➔ ❤♦➔♥ t♦➔♥ ✤✐➲✉ ❦❤✐➸♥✱♥❣❤➽❛ ❧➔ Im Rn˜ ✳ ◆➯♥ , B21 n ˜1 n ˜2 υ−1 B21 N11 B21 · · · N11 B21 Im(N ) = Im = Im = Im N11 N12 N 22 N B2 N B2 · · · N (υ−1) B2 N11 ❤♦➦❝ N22 = 0✳ ❍ì♥ ♥ú❛✱ ❝è ✤à♥❤ r➡♥❣ ⇐⇒ n ˜2 = ✸✶ = E rank 0 W ET = rank A E B 0 W AT W ET W B  Inf 0    = rank   J  N 0 Inf In∞ N 0 = 2nf + rank    0   B1   N B2 In∞ N B2  N11 N12 0      N22 0   = 2nf + rank    I N N B 11 12 21 n ˜   ❚❛ ❜✐➳t r➡♥❣ r❛♥❦ rank ˜1 N11 N21 = n E 0 A E B E 0 A E B rank = 2nf + n ˜1 + n ˜ + rank N11 −N12 N22 N11 N12 N11 N12 N22 N22 = rank N22 N22 = n + rank(E) = n + nf + rank N22 ✱ ✈➟② t❛ ❝â = n + nf + rank ❱➟② rank In˜ N11 N12 N22 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ N11 N12 N22 N22 ❱➻ N22 ❧➔ ♠❛ tr➟♥ ❧ô② ❧✐♥❤✱ ❞♦ ✤â ❝è ✤à♥❤ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ n˜2 = ❤♦➦❝ N22 = 0✳ ❱➼ ❞ư ✷✳✽✳ ❳➨t ❤❛✐ ❤➺ ♠ỉ t↔ ð ✈➼ ❞ư ✷✳✹ ✤÷đ❝ ❝❤♦ ❜ð✐ x˙ = 1 = x2 + x1 + −1 0 u ❚❛ ❝â t❤➸ ①❡♠ ✤â ❧➔ ❤➺ ❘✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✭✈➼ ❞ư ✷✳✺✮ ❝❤ù ❦❤ỉ♥❣ ❈✲ ✤✐➲✉ ❦❤✐➸♥ ✸✷ ✤÷đ❝✳ ❚❛ ❝â Im(N ) + Ker(N ) + Im(B2 ) = lm 0 + Ker 0 {0} + R + Im 0 0 + Im 1 = R2 ❍➺ tr➯♥ ❧➔ ♠ët ❤➺ ■✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❚❛ ❝â  rank E 0 A E B  I2 0 0    = rank    0 0 I2    I2 0   = = n + rank(E) e2 e2 ỵ ✷✳✽✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ t÷ì♥❣ ✤÷ì♥❣ ✶✳ ❍➺ ✷✳✾ ❧➔ ■✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✷✳ rank EAS∞B = n t↕✐ S∞ ❧➔ ♠ët ♠❛ tr➟♥ ✈ỵ✐ Im(S∞) = Ker(N ) ỗ t F Rm,n s (E, A+BF ) ❧➔ s✉② rë♥❣ ✈➔ υ = ind(E, A+BF ) ≤ 1✱ ✹✳ rank N K∞B2 = n∞ t↕✐ Im(K∞) = Ker(N )✱ ✺✳ rank N 0 In∞ N B2 = n∞ + rank(N ) ✣à♥❤ ♥❣❤➽❛ ✷✳✼✳ ❍➺ ✷✳✾ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♠↕♥❤ ✭❙✲✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✮ ♥➳✉ ♥â ❘✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ ■ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❈â ♥❣❤➽❛ ❧➔ ♥➳✉ rank λE − AB = n ✈ỵ✐ ♠å✐ ❤ú✉ ❤↕♥ λ ∈ C ✈➔ rank EAS∞ B = n t↕✐ Im(S∞ ) = Ker(E)✳ ◗✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ ❦❤→✐ ♥✐➺♠ t➼♥❤ ✤✐➲✉ ữủ õ t ữủ ợ t ữ s❛✉ → R − ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ C − ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ → S − ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ → R − ✈➔ I − ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ → I − ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✸✸ ❑➳t ❧✉➟♥ ❙❛✉ ♠ët t❤í✐ ❣✐❛♥ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ✧❚➼♥❤ ✤✐➲✉ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧✳✤➣ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ♠ët sè ✈➜♥ ✤➲ s❛✉ ✤➙②✿ ✶✳ ❍➺ t❤è♥❣ ❧↕✐ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ t❤✉ë❝ ❧➽♥❤ ✈ü❝ ✤↕✐ sè t✉②➳♥ t➼♥❤✿ ❛✳ ▼❛ tr➟♥ ✈➔ ❝→❝ ❧♦↕✐ ♠❛ tr➟♥✳ ❜✳ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ ♠❛ tr➟♥✳ ❝✳ ✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈➔ ❝→❝ ♣❤➨♣ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤à♥❤ t❤ù❝✳ ✷✳ ợ t tự ỡ ỵ tt ữủ ố ợ ởt ổ t ữủ ữợ ởt ữỡ tr ❝➜♣ ♠ët✳ ✸✳ ❚➻♠ ❤✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❝→❝❤ tt ỳ t q ỡ ỵ tt ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✳ ✹✳ ▼➦❝ ❞ị ✤➣ ❝è ❣➢♥❣✱ s♦♥❣ ❞♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ♥➯♥ tr♦♥❣ ✤➲ t➔✐ ❦❤ä✐ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ rt ữủ õ õ qỵ qỵ t ổ t t ỡ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ✣➻♥❤ ❚r➼✳ ❚♦→♥ ❝❛♦ ❝➜♣ t➟♣ ✶✳ ✣↕✐ sè t✉②➳♥ t➼♥❤ ✈➔ ❤➻♥❤ ❤å❝ ❣✐↔✐ t➼❝❤✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝✳ ✷✵✶✵✳ ❬✷❪ ▲❡♥❛ ❙❝❤♦❧③ ❈♦♥tr♦❧ ❚❤❡♦r② ♦❢ ❙②st❡♠s ▲❡❝t✉r❡ ◆♦t❡s ❚❯ ❇❡r❧✐♥✱ ✷✵✶✹✳ ✸✺