✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖ ◆●❯❨➍◆ ◆❍❾❚ ❍❖⑨◆● ❇⑩❖ ❈⑩❖ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ◆●⑨◆❍✿ ❙× P❍❸▼ ❚❖⑩◆ ❚❘➐◆❍ ✣❐ ✣⑨❖ ❚❸❖✿ ✣❸■ ❍➴❈ ✣➋ ❚⑨■ ❚➑◆❍ ◗❯❆◆ ❙⑩❚ ✣×Đ❈ ❈Õ❆ ❍➏ ▼➷ ❚❷ ●✐→♦ ữợ t ✶ ♥➠♠ ✷✵✷✵ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝ì sð ✸ ✹ ✽ ✶✳✶ ▼❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❍➺ ♠æ t↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹ ❍➺ ❜➜t ❜✐➳♥ t✉②➳♥ t➼♥❤ t❤❡♦ t❤í✐ ❣✐❛♥ ✲ ▲❚■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔ ✷✳✶ ✷✳✷ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ởt số ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✽ ✶✺ ✶✺ ✶✻ ✷✼ ✷ ▲í✐ ❝↔♠ ì♥ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥✱ tỉ✐ ✤➣ ♥❤➟♥ ✤÷đ❝ sü ✤ë♥❣ ✈✐➯♥✱ ú ù t t ỵ t ổ ✈➔ ❜↕♥ ❜➧✳ ◆❤➙♥ ✤➙② tỉ✐ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t✳ ✣➛✉ t✐➯♥✱ tỉ✐ ①✐♥ ❣û✐ ỡ t ỵ t ổ tr ❑❤♦❛ ❚♦→♥ tr÷í♥❣ ✣↕✐ ❤å❝ s÷ ♣❤↕♠ ✣➔ ◆➤♥❣ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② s✉èt ❜❛ ♥➠♠ r÷ï✐ ✤➸ tỉ✐ ❝â ✤÷đ❝ ♥➲♥ t↔♥❣ tr✐ t❤ù❝ ❝ơ♥❣ ♥❤÷ ❦✐♥❤ ♥❣❤✐➺♠ số qỵ tr tổ s ♥➔②✳ ✣➦❝ ❜✐➺t✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❚❙✳ r ữớ t ỗ ❤ù♥❣✱ ❣✐↔♥❣ ❞↕② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ♥➲♥ t↔♥❣✱ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♠ët ❝→❝❤ tèt t ú ợ t tổ ọ ữủ t❤ù❝ ❧➔♠ ✈✐➺❝ ❦❤♦❛ ❤å❝✱ sü ♥❤✐➺t t➻♥❤✱ t➼♥❤ ❝➞♥ t❤➟♥ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥❤ú♥❣ ❜➔✐ ❤å❝ ❜ê ➼❝❤ tr♦♥❣ ❝✉ë❝ sè♥❣✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ❧✉æ♥ q✉❛♥ t➙♠ ✤ë♥❣ ✈✐➯♥✱ ❦❤➼❝❤ ❧➺ t✐♥❤ t❤➛♥ ❝❤ó♥❣ tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ỷ ỡ t ỵ t ổ tr ỗ õ tớ qỵ t õ ỵ ♥❤ú♥❣ ✤✐➸♠ ❝á♥ t❤✐➳✉ sât ❣✐ó♣ tỉ✐ rót ✤÷đ❝ ❦✐♥❤ ♥❣❤✐➺♠ ❝❤♦ ❦❤â❛ ❧✉➟♥ ❝ơ♥❣ ♥❤÷ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ♥➔②✳ ❘➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❝❤➾ ❜↔♦ t➟♥ t ỵ t ổ t ỡ ỡ ữủ ỵ tt ❦❤✐➸♥ ✈➔ t➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❈✉è✐ t❤➳ ❦✛ ✶✽✱ ❝✉ë❝ ❝→❝❤ ♠↕♥❣ ❝æ♥❣ ♥❣❤✐➺♣ ❧➛♥ t❤ù ♥❤➜t ♥ê r❛ ổ t ữợ ♠→② ♠â❝ ✈➔ ♥➠♥❣ ❧÷đ♥❣ ❝❤♦ ❝ỉ♥❣ ♥❣❤✐➺♣ ❞➺t✱ ❝→❝ ♥❣➔♥❤ ❝ỉ♥❣ ♥❣❤✐➺♣ ♥❤÷ ❝❤➳ t↕♦ ♠→②✱ ❣✐❛♦ t❤ỉ♥❣ ✈➟♥ t tỷ r ữợ ổ t ữợ õ tỹ õ ◆➲♥ ✤↕✐ ❝ỉ♥❣ ♥❣❤✐➺♣ ❤✐➺♥ ✤↕✐ ❝➔♥❣ ♥❣➔② ❝➔♥❣ ✤÷đ❝ ♥➙♥❣ ❝❛♦ ♠ù❝ ✤ë tü ✤ë♥❣ ❤â❛ ✈ỵ✐ ♠ư❝ ✤➼❝❤ ♥➙♥❣ ❝❛♦ ♥➠♥❣ s✉➜t ❧❛♦ ✤ë♥❣✱ ❣✐↔♠ ❝❤✐ ♣❤➼ s↔♥ ①✉➜t✱ ❣✐↔✐ ♣❤â♥❣ ❝♦♥ ♥❣÷í✐ r❛ ❦❤ä✐ ♥❤ú♥❣ ✈à tr➼ ❧➔♠ ✈✐➺❝ ✤ë❝ ❤↕✐✱✳✳✳ ❉ü❛ tr➯♥ ♥❤✉ ❝➛✉ t❤ü❝ t➳ õ ỵ tt r ũ ❞↕♥❣ ❝õ❛ ❤➺ t❤è♥❣ ✤✐➲✉ ❦❤✐➸♥ ❝â tø t❤í✐ ❝ê ✤↕✐✱ t❤➳ ♥❤÷♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝❤➼♥❤ t❤ù❝ ❝õ❛ ❧➽♥❤ ✈ü❝ ♥➔② ❜➢t ✤➛✉ ✈ỵ✐ ♠ët ♣❤➙♥ t➼❝❤ ✤ë♥❣ ❤å❝ ❝õ❛ ❤➺ ✤✐➲✉ tè❝ ❧✐ t➙♠✱ ✤÷đ❝ t❤ü❝ ❤✐➸♥ ❜ð✐ ♥❤➔ t ỵ s r ợ tỹ ✤➲ ❖♥ ●♦✈❡r♥♦rs ✭❤➺ ✤✐➲✉ tè❝✮✳ ❙❛✉ ✤â ❝â r➜t ♥❤✐➲✉ ♥❤➔ ♥❤➔ ❦❤♦❛ ❤å❝ ✤➣ ✤â♥❣ ❣â♣ ✈➔♦ sü t tr ỵ tt tỹ ỳ ♥❣÷í✐ t✐➯♥ ♣❤♦♥❣ ♥❤÷ ❧➔✿ ✰ P✐❡rr❡✲❙✐♠♦♥ ▲❛♣❧❛❝❡ ✭✶✼✹✾✲✶✽✷✼✮ ♣❤→t ♠✐♥❤ r❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❩ tr♦♥❣ ❝æ♥❣ tr➻♥❤ ✈➲ ỵ tt st ổ ữủ sỷ ❞ư♥❣ ✤➸ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ rí✐ r↕❝ tr♦♥❣ tớ ỵ tt P ✤ê✐ ❩ ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ t÷ì♥❣ ✤÷ì♥❣ tr♦♥❣ ♠✐➲♥ ❣✐í✐ ❣✐❛♥ rí✐ r↕❝ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤÷đ❝ ✤➦t t➯♥ t❤❡♦ ❝❤➼♥❤ t➯♥ æ♥❣✳ ✰ ❆❧❡①❛♥❞❡r ▲②❛♣✉♥♦✈ ✭✶✽✺✼✕✶✾✶✽✮ sỹ ỵ tt ❜➲♥ ✈ú♥❣✳ ✰ ❍❛r♦❧❞ ❙✳ ❇❧❛❝❦ ✭✶✽✾✽✕✶✾✽✸✮✱ ♣❤→t ♠✐♥❤ r❛ ỗ ✶✾✷✼✳ ➷♥❣ ✤➣ t❤➔♥❤ ❝æ♥❣ tr♦♥❣ ✈✐➺❝ ♣❤→t tr✐➸♥ ❝→❝ ỗ ỳ ỳ ♥➠♠ ✶✾✸✵✳ ✰ ❍❛rr② ◆②q✉✐st ✭✶✽✽✾✕✶✾✼✻✮✱ ♣❤→t tr✐➸♥ t✐➯✉ ❝❤✉➞♥ qst tố ỗ ỳ ♥➠♠ ✶✾✸✵✳ ✹ ✰ ❘✐❝❤❛r❞ ❊✳ ❇❡❧❧♠❛♥ ✭✶✾✷✵✕✶✾✽✹✮✱ ♣❤→t tr✐➸♥ q✉② ❤♦↕❝❤ ✤ë♥❣ tø ♥❤ú♥❣ ♥➠♠ ✶✾✹✵✳ ✰ ❆♥❞r❡② ❑♦❧♠♦❣♦r♦✈ ỗ t tr rr rrt r ỗ t tr rr t r❛ t❤✉➟t ♥❣ú ✭✤✐➲✉ ❦❤✐➸♥ ❤å❝✮ ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✶✾✹✵✳ ✰ ❏♦❤♥ ❘✳ ❘❛❣❛③③✐♥✐ ✭✶✾✶✷✕✶✾✽✽✮ ❣✐ỵ✐ t❤✐➺✉ ✤✐➲✉ ❦❤✐➸♥ ❦ÿ t❤✉➟t sè ✈➔ ❜✐➳♥ ✤ê✐ ③ ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✶✾✺✵✳ Ptr ợ t ỵ ỹ ỵ ỵ tt tỹ ❧➔ ♠ët ♥❤→♥❤ ❧✐➯♥ ♥❣➔♥❤ ❝õ❛ ❦ÿ t❤✉➟t ✈➔ t♦→♥ ❤å❝✱ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♥❤ ✈✐ ❝õ❛ ❝→❝ ❤➺ t❤è♥❣ ✤ë♥❣ ❧ü❝✳ ✣➛✉ r❛ ♠♦♥❣ ♠✉è♥ ❝õ❛ ♠ët ❤➺ t❤è♥❣ ữủ tr t trữợ ởt ♥❤✐➲✉ ❜✐➳♥ ✤➛✉ r❛ ❝õ❛ ❤➺ t❤è♥❣ ❝➛♥ t✉➙♥ t❤❡♦ ởt tr t trữợ t tớ ởt ✤✐➲✉ ❦❤✐➸♥ ❝→❝ ✤➛✉ ✈➔♦ ❝❤♦ ❤➺ t❤è♥❣ ✤➸ ✤↕t ✤÷đ❝ ❤✐➺✉ q✉↔ ♠♦♥❣ ♠✉è♥ tr➯♥ ✤➛✉ r❛ ❤➺ t❤è♥❣✳ ✣➸ ❤✐➸✉ ✈➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝→❝ ❤➺ t❤è♥❣ ♣❤ù❝ t↕♣✱ ❝➛♥ ♣❤↔✐ t❤✐➳t ❧➟♣ ❝→❝ ♠æ ❤➻♥❤ t♦→♥ ❤å❝ ✤à♥❤ t➼♥❤ ❝õ❛ ♥❤ú♥❣ ❤➺ t❤è♥❣ ♥➔②✳ ❇ð✐ ✈➻ ❝→❝ ❤➺ t❤è♥❣ ❝❤ó♥❣ t❛ ❝➛♥ q✉❛♥ t➙♠ ❧➔ ♥❤ú♥❣ ❤➺ t❤è♥❣ ✤ë♥❣ ✈➲ ❜↔♥ ❝❤➜t✱ ❞♦ ✤â ♥❣÷í✐ t❛ t❤÷í♥❣ ❞ị♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ổ t ú r ỵ tt ❦❤✐➸♥ ❤✐➺♥ ✤↕✐✱ ♥❣÷í✐ t❛ sû ❞ư♥❣ ♠ỉ t↔ ❦❤ỉ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ tr♦♥❣ ♠✐➲♥ t❤í✐ ❣✐❛♥✱ ♠ët ♠ỉ ❤➻♥❤ t ởt tố t ỵ ữ ♠ët ❝ö♠ ✤➛✉ ✈➔♦✱ ✤➛✉ r❛ ✈➔ ❝→❝ ❜✐➳♥ tr↕♥❣ t q ợ ữỡ tr tr t ởt ✣➸ trø✉ t÷đ♥❣ ❤â❛ tø sè ❧÷đ♥❣ ✤➛✉ ✈➔♦✱ ✤➛✉ r❛ ✈➔ tr↕♥❣ t❤→✐ ✤➳♥ ❝→❝ ❜✐➳♥✱ ❝→❝ ❜✐➸✉ t❤ù❝ ♥❤÷ ✈❡❝t♦r✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ số ữủ t ữợ tr ỳ tự tr ❝❤➾ ❝â t❤➸ t❤ü❝ ❤✐➺♥ ❦❤✐ ❤➺ t❤è♥❣ ✤ë♥❣ ❧ü❝ ❧➔ t✉②➳♥ t➼♥❤✮✳ ❇✐➸✉ ❞✐➵♥ ❦❤æ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ ✭❝á♥ ❣å✐ ❧➔ ✧①➜♣ ①➾ ♠✐➲♥ t❤í✐ ❣✐❛♥ ✧✮ ❝✉♥❣ ❝➜♣ ♠ët ❝→❝❤ t❤ù❝ ♥❣➢♥ ❣å♥ ✈➔ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ t ữợ t tố ợ ✈➔♦ ✈➔ ✤➛✉ r❛✳ ❱ỵ✐ ❝→❝ ✤➛✉ ✈➔♦ ✈➔ ✤➛✉ r❛✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❝â ❝→❝❤ ✈✐➳t ❦❤→❝ ❝❤♦ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ♠➣ ❤â❛ t♦➔♥ ❜ë t❤æ♥❣ t✐♥ ✈➲ ♠ët ❤➺ t❤è♥❣✳ ❑❤ỉ♥❣ ❣✐è♥❣ ♥❤÷ ①➜♣ ①➾ ♠✐➲♥ t➛♥ sè✱ ✈✐➺❝ sû ❞ư♥❣ ❜✐➸✉ ❞✐➵♥ ❦❤ỉ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ ❦❤ỉ♥❣ ❜à ❣✐ỵ✐ ❤↕♥ ✈ỵ✐ ❤➺ t❤è♥❣ ❜➡♥❣ ❝→❝ t❤➔♥❤ ♣❤➛♥ t✉②➳♥ t➼♥❤ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ③❡r♦ ❜❛♥ ✤➛✉✳ ✧❑❤æ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐✧ ✤➲ ❝➟♣ ✤➳♥ ❦❤æ♥❣ ❣✐❛♥ ♠➔ ❝→❝ ❤➺ trö❝ ❧➔ ❝→❝ ❜✐➳♥ tr↕♥❣ t❤→✐✳ ❚r↕♥❣ t❤→✐ ❝õ❛ ❤➺ t❤è♥❣ ❝â t❤➸ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ♥❤÷ ♠ët ✈❡❝t♦r tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤â✳ ❚r♦♥❣ ✤â✱ ❦❤↔ ♥➠♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ q✉❛♥ s→t ✤÷đ❝ ❧➔ ♥❤ú♥❣ tr t ởt tố trữợ ❦❤✐ q✉②➳t ✤à♥❤ ❧♦↕✐ ✤✐➲✉ ❦❤✐➸♥ tèt ♥❤➜t ✤÷đ❝ sû ❞ö♥❣✱ ❤♦➦❝ ①❡♠ ①➨t ❧♦↕✐ ♥➔♦ ❝â ❦❤↔ ♥➠♥❣ ❦❤✐➸♥ ❤♦➦❝ ê♥ ✤à♥❤ ✤÷đ❝ ❤➺ t❤è♥❣✳ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❧➔ ❦❤↔ ♥➠♥❣ t→❝ ✤ë♥❣ ✈➔♦ ❤➺ t❤è♥❣ ✤➸ ✤↕t ✤÷đ❝ tr↕♥❣ t❤→✐ ✺ ✤➦❝ ❜✐➺t ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ♠ët t➼♥ ❤✐➺✉ ✤✐➲✉ ❦❤✐➸♥ t❤➼❝❤ ❤ñ♣✳ ◆➳✉ ♠ët tr↕♥❣ t❤→✐ ❧➔ ❦❤ỉ♥❣ t❤➸ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ t❤➻ s➩ ❦❤ỉ♥❣ ❝â t➼♥ ❤✐➺✉ ♥➔♦ ❝â t❤➸ ❝â ❦❤↔ ♥➠♥❣ ✤✐➲✉ ❦❤✐➸♥ tr↕♥❣ t❤→✐ ✤â✳ ◆➳✉ ♠ët tr↕♥❣ t❤→✐ ❧➔ ❦❤æ♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ♥❤÷♥❣ ❝→❝ ✤➦❝ t➼♥❤ ✤ë♥❣ ❤å❝ ❝õ❛ ♥â ❧➔ ê♥ ✤à♥❤✱ t❤➻ tr↕♥❣ t❤→✐ ✤â ✤÷đ❝ ①❡♠ ❧➔ ❝â ❦❤↔ ♥➠♥❣ ê♥ ✤à♥❤ ❤â❛✳ ❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❧➔ ❦❤↔ ♥➠♥❣ ✧q✉❛♥ s→t✧✱ t❤ỉ♥❣ q✉❛ ✈✐➺❝ ✤♦ ❧÷í♥❣ ✤➛✉ r❛✱ ✈➔ tr↕♥❣ t❤→✐ ❝õ❛ ♠ët ❤➺ t❤è♥❣✳ ◆➳✉ ♠ët tr↕♥❣ t❤→✐ ❧➔ ❦❤ỉ♥❣ t❤➸ q✉❛♥ s→t ✤÷đ❝✱ ❜ë ✤✐➲✉ ❦❤✐➸♥ s➩ ❦❤ỉ♥❣ ❜❛♦ ❣✐í ❝â t❤➸ ①→❝ ✤à♥❤ ❤➔♥❤ ✈✐ ♥â ✈➔ ❞♦ ✤â ❦❤æ♥❣ t❤➸ sû ❞ö♥❣ ♥â ✤➸ ê♥ ✤à♥❤ ❤â❛ ❤➺ t❤è♥❣✳ ❚✉② ♥❤✐➯♥✱ t÷ì♥❣ tü ♥❤÷ ✤✐➲✉ ❦✐➺♥ ê♥ ✤à♥❤ ❤â❛ ð tr➯♥✱ ♥➳✉ ♠ët tr↕♥❣ t❤→✐ ❦❤ỉ♥❣ t❤➸ q✉❛♥ s→t ✤÷đ❝✱ ♥â ✈➝♥ ❝â t❤➸ ✤÷đ❝ ♣❤→t ❤✐➺♥✳ ◆❤➻♥ tø ♠ët ✤✐➸♠ q✉❛♥ s→t ❤➻♥❤ ❤å❝✱ ❝→❝ tr↕♥❣ t❤→✐ ❝õ❛ ♠é✐ ❜✐➳♥ ❝õ❛ ❤➺ t❤è♥❣ ✤÷đ❝ ✤✐➲✉ ❦❤✐➸♥✱ ♠é✐ tr↕♥❣ t❤→✐ ❝õ❛ ♠é✐ ❜✐➳♥ ♥➔② ♣❤↔✐ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ q✉❛♥ s→t ✤÷đ❝ ✤➸ ✤↔♠ ❜↔♦ ❤➔♥❤ ✈✐ tèt tr♦♥❣ ❤➺ ✈á♥❣ ❦➼♥✳ ✣â ❧➔✱ ♥➳✉ ♠ët tr♦♥❣ ❝→❝ ❣✐→ trà ❣è❝ ❝õ❛ ❤➺ t❤è♥❣ ✈ø❛ ❦❤ỉ♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❧↕✐ ✈ø❛ ❦❤ỉ♥❣ q✉❛♥ s→t ✤÷đ❝✱ ♣❤➛♥ ✤ë♥❣ ❤å❝ ♥➔② s➩ ✈➝♥ ❝á♥ ❦❤ỉ♥❣ ①❡♠ ①➨t ✤÷đ❝ tr♦♥❣ ❤➺ ✈á♥❣ ❦➼♥✳ ◆➳✉ ♠ët ❣✐→ trà ❣è❝ ❦❤æ♥❣ ê♥ ✤à♥❤✱ ❝→❝ ✤➦❝ t➼♥❤ ✤ë♥❣ ❤å❝ ❝õ❛ ❣✐→ trà ❣è❝ ♥➔② s➩ ✤÷đ❝ ①✉➜t ❤✐➺♥ tr➯♥ ❤➺ ✈á♥❣ ❦➼♥ ♠➔ ❞♦ ✤â s➩ ❦❤æ♥❣ ê♥ ✤à♥❤ ✤÷đ❝✳ ❈→❝ ❝ü❝ ❦❤ỉ♥❣ q✉❛♥ s→t ✤÷đ❝ t❤➻ ổ ữủ tr ỗ t tr ổ tr t ỵ ✤➲ t➔✐ ❚r♦♥❣ t❤í✐ ✤↕✐ ❤✐➺♥ ♥❛②✱ ❞ị ð ❜➜t ❝ù ✈à tr➼ ♥➔♦✱ ❧➔♠ ❜➜t ❝ù ❝æ♥❣ ✈✐➺❝ ❣➻✱ ộ ữớ ú t t ợ ◆â ❧➔ ❦❤➙✉ q✉❛♥ trå♥❣ ❝✉è✐ ❝ò♥❣ q✉②➳t ✤à♥❤ sü t❤➔♥❤ ❤❛② ❜↕✐ tr♦♥❣ ♠å✐ ❤♦↕t ✤ë♥❣ ❝õ❛ ❝❤ó♥❣ t❛✳ q st ữợ q trồ ♣❤➙♥ t➼❝❤ tr↕♥❣ t❤→✐ ❝õ❛ sü ✈✐➺❝✱ sü ✈➟t ✤➸ tø ✤â ✤÷❛ r❛ ❝→❝❤ t❤ù❝ ✧✤✐➲✉ ❦❤✐➸♥✧ ✤ó♥❣ ♥❤➜t✱ ❝â ❤✐➺✉ q✉↔ ♥❤➜t✳ ❚❛ ❦❤æ♥❣ t❤➸ ✧✤✐➲✉ ❦❤✐➸♥✧ tèt ổ q st t tr ỵ tt ❦❤✐➸♥ ✲ ♠ët tr♦♥❣ ♥❤ú♥❣ ✈➜♥ ✤➲ ✤÷đ❝ r➜t ♥❤✐➲✉ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ❤✐➺♥ ♥❛② q✉❛♥ t➙♠ ❜ð✐ t➼♥❤ ù♥❣ ❞ö♥❣ rë♥❣ r➣✐ ✈➔ ♠↕♥❤ ♠➩ ❝õ❛ ♥â tr♦♥❣ ❝→❝ ♥❤â♠ ♥❣➔♥❤ ❝ì ❤å❝✱ ✤✐➺♥ ❤å❝✱ ❝ỉ♥❣ ♥❣❤➺ ❤â❛✱✳✳✳ t❤➻ ✧t➼♥❤ q✉❛♥ s→t ✤÷đ❝✧ ❝➔♥❣ ✤â♥❣ ♠ët ✈❛✐ trá ổ ũ q trồ ỡ ỳ tr ỵ tt t q st ữủ ởt tữợ ✤♦ ✤➸ ❜✐➳t ✤÷đ❝ ❝→❝ tr↕♥❣ t❤→✐ ❜➯♥ tr♦♥❣ ❝õ❛ ♠ët ❤➺ t❤è♥❣ tèt ♥❤÷ t❤➳ ♥➔♦ ❜ð✐ ❝→❝ ❦➳t q✉↔ ✤➛✉ r❛ ❜➯♥ ♥❣♦➔✐ ❝õ❛ ♥â✳ ❚❤æ♥❣ q✉❛ ✤â ❣✐ó♣ ❝❤♦ ✈✐➺❝ ✈➟♥ ❤➔♥❤ ❤➺ t❤è♥❣ tèt ❤ì♥✱ ❤✐➺✉ q✉↔ ❤ì♥✳ ❉ü❛ ✈➔♦ ♥❤ú♥❣ ♣❤➙♥ t➼❝❤ tr➯♥✱ ❝❤ó♥❣ t❛ t❤➜② ✤÷đ❝ r➡♥❣ ✧t➼♥❤ q✉❛♥ s→t ✤÷đ❝✧ ❝â ✈❛✐ trá q trồ t tr ỵ tt ụ ữ tr tớ ợ ố ❤✐➸✉ rã ❤ì♥ ✈➲ ✈❛✐ trá ✈➔ ù♥❣ ❞ư♥❣ ❝õ❛ q st ữủ tr ổ t ữợ sỹ ủ ỵ ữợ ❍↔✐ ❚r✉♥❣✱ tæ✐ ❝❤å♥ ✤➲ t➔✐ ✿ ✧❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧ ❝❤♦ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳ ✸✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❚r♦♥❣ t❤í✐ ✤↕✐ ❤✐➺♥ ♥❛②✱ ✧✤✐➲✉ ❦❤✐➸♥ ✲ q✉❛♥ s→t✧ ❤➛✉ ♥❤÷ ♣❤õ sâ♥❣ ①✉♥❣ q✉❛♥❤ ❝❤ó♥❣ t❛✳ ❚✉② ♥❤✐➯♥✱ ➼t ❛✐ ♥❤➟♥ r❛ ❝ơ♥❣ ♥❤÷ ❤✐➸✉ ❜✐➳t ✈➲ t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ♥â✳ ◆❤➟♥ t❤➜② ✧❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠æ t↔✧ ❧➔ ♠ët ✤➲ t➔✐ ❝â t➼♥❤ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝❛♦ ❞♦ ✤â ♠➔ tæ✐ ❝❤å♥ ✤➲ t➔✐ ✧❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧ ♥❤➡♠ ♠ư❝ ✤➼❝❤ ❣✐ỵ✐ t❤✐➺✉ ✤➳♥ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➔ ♥➲♥ t↔♥❣ ♥❤➜t ✈➲ ✧❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧✱ ♠♦♥❣ ♠✉è♥ ❝â t❤➸ t↕♦ ✤÷đ❝ sü ❤ù♥❣ t❤ó✱ ♠ð rë♥❣ ♣❤↕♠ ✈✐ ù♥❣ ❞ư♥❣ ❝õ❛ õ ụ ữ t r ữợ ợ õ tr♦♥❣ ♠ët sè ❧➽♥❤ ✈ü❝ tr♦♥❣ ①➣ ❤ë✐✳ ✹✳ ◆❤✐➺♠ ✈ư ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❈ị♥❣ ợ sỹ ữợ tổ t❤✉ t❤➟♣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t➔✐ ❧✐➺✉ ❝õ❛ ♥❤ú♥❣ t trữợ q q st ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧✳ ❚ø ❦✐➳♥ t❤ù❝ ❝➠♥ ❜↔♥ ✤â✱ tỉ✐ sû ❞ư♥❣ ❝❤ó♥❣ ❦❤→✐ q✉→t✱ ❝❤➢❝ ❧å❝ ✈➔ ✤÷❛ r❛ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝➠♥ ❜↔♥ ✈➔ ♥➲♥ t↔♥❣ ♥❤➜t ✈➲ ✧❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧✳ ✻✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ sû ❞ư♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❜ê trđ ❝❤õ ②➳✉ ❧➔ ♠❛ tr➟♥ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✳ ✣è✐ t÷đ♥❣ →♣ ❞ư♥❣ ❝❤➼♥❤ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤✳ ✼ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝ì sð ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ♥❤➡♠ ♠ư❝ ✤➼❝❤ ❤➺ t❤è♥❣ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ✤↕✐ sè t✉②➳♥ t➼♥❤ ỵ tt t ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❜➔✐ ✈✐➳t ❧➔ ✧❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø t➔✐ ❧✐➺✉ ❬✷❪✳ ✶✳✶ ▼❛ tr➟♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ m × n ❧➔ ❤❛✐ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❛ ❣å✐ ♠ët ♠❛ tr➟♥ ❝ï ✭❝➜♣✮ m ì n ởt số ỗ m ì n sè t❤ü❝ ✤÷đ❝ ✈✐➳t t❤➔♥❤ m ❤➔♥❣ n ❝ët ❝â ❞↕♥❣ ♥❤÷ s❛✉✿ a11 a12 a1n a21 a22 a2n am1 am2 amn ❚r♦♥❣ ✤â ❝→❝ sè t❤ü❝ aij , j = 1, n ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♣❤➛♥ tû ❝õ❛ ♠❛ tr➟♥✱ ❝❤➾ sè i ❝❤➾ t❤ù tü ❤➔♥❣ ✈➔ ❝❤➾ sè j ❝❤➾ t❤ù tü ❝ët ❝õ❛ ♣❤➛♥ tỷ aij tr tr ỵ tr t❤÷í♥❣ ✤÷đ❝ ✈✐➳t tr♦♥❣ ❞➜✉ ♥❣♦➦❝ ✈✉ỉ♥❣✿ a11 a12 ··· a1n a21 A= ✳✳ ✳ a22 ··· a2n ✳✳ ✳ ✳✳✳ ✳✳ ✳ am1 am2 · · · amn ởt ỵ sỷ ♥❣♦➦❝ ✤ì♥ ❧ỵ♥ t❤❛② ❝❤♦ ❞➜✉ ♥❣♦➦❝ ✈✉ỉ♥❣✿ a11 a12 · · · a1n a21 A= ✳✳ ✳ a22 · · · a2n ✳✳ ✳ ✳ ✳ ✳ ✳✳ ✳ am1 am2 · · · amn ❈→❝ ♣❤➨♣ t♦→♥ ❝õ❛ ♠❛ tr➟♥ P❤➨♣ ❝ë♥❣✳ ❚ê♥❣ A + B ❝õ❛ tr ũ tữợ m ì n 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 , ≤ i ≤ m, ≤ j ≤ n ❑➼ ❤✐➺✉✿ C = A + B ✳ ❱➼ ❞ö✿ + 0 0 = 1+0 3+0 1+5 1+7 0+5 0+0 = P❤➨♣ ♥❤➙♥ ♠ët sè t❤ü❝ ✈ỵ✐ ♠ët ♠❛ tr➟♥✳ ❚➼❝❤ α❆ ❝õ❛ sè α ✈ỵ✐ ♠❛ tr➟♥ ❆ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ❜➡♥❣ ❝→❝❤ ♥❤➙♥ ♠é✐ ♣❤➛♥ tỷ ợ ()i,j = ì Ai,j ❑➼ ❤✐➺✉✿ C = αA = [αAij ]mn ✳ ❱➼ ❞ö✿ −3 −2 = 2.1 2.8 2.4 2.(−2) 2.(−3) 2.5 = 16 −6 −4 10 P❤➨♣ ♥❤➙♥ ♠❛ tr➟♥ ✈ỵ✐ ♠❛ tr➟♥✳ ❈❤♦ A = [aij ]mn ❝ï m × n ✈➔ ♠❛ tr➟♥ B = [bik ]nq ❝ï n × q ✳ ❚➼❝❤ ❝õ❛ ❆ ✈ỵ✐ ❇ ❧➔ ♠ët ♠❛ tr➟♥ C = [cik ]mq ù m ì q ợ n cik = aij bjk = a1j b1k + ai2 b2k + + ain bnk , i = 1, m, k = 1, q j=1 ❑➼ ❤✐➺✉✿ C = A.B ❤❛② C = AB ✳ ❱➼ ❞ö✿ 3 1.1 + 3.1 + 2.3 1.3 + 3.(−1) + 2.2 10 2 7 1 −1 = 2.1 + 4.1 + 7.3 2.3 + 4.(−1) + 7.2 = 27 16 3.1 + 5.1 + 6.3 3.3 + 5.(−1) + 6.2 ✾ 26 16 ❈❤✉②➸♥ ✈à✳ ❈❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ ❆ ❝ï m × n ❧➔ ♠❛ tr➟♥ AT ❝ï m × n t↕♦ r❛ ❜➡♥❣ ❝→❝❤ ❝❤✉②➸♥ ❤➔♥❣ t❤➔♥❤ ❝ët ✈➔ ❝ët t❤➔♥❤ ❤➔♥❣✿ (AT )Ti,j = Aj,i ❑➼ ❤✐➺✉✿ AT ✳ ❱➼ ❞ö✿ A= −6 ❚❛ ❝â✿ AT = 2 −6 ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ❧➔ tê♥❣ ✤↕✐ sè ❝õ❛ n! ✭n ❣✐❛✐ t❤ø❛✮ sè ❤↕♥❣✱ ♠é✐ sè ❤↕♥❣ ❧➔ t➼❝❤ ❝õ❛ n ♣❤➛♥ tû ❧➜② tr➯♥ ❝→❝ ❤➔♥❣ ✈➔ ❝→❝ ❝ët ❦❤→❝ ♥❤❛✉ ❝õ❛ ♠❛ tr➟♥ A✱ ♠é✐ t➼❝❤ ✤÷đ❝ ♥❤➙♥ ✈ỵ✐ ♣❤➛♥ tû ❞➜✉ ❧➔ +1 ❤♦➦❝ −1 t❤❡♦ ♣❤➨♣ t❤➳ t↕♦ ❜ð✐ ❝→❝ ❝❤➾ sè ❤➔♥❣ ✈➔ ❝❤➾ sè ❝ët ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ t➼❝❤✳ ●å✐ Sn ❧➔ ♥❤â♠ ❝→❝ ❤♦→♥ ✈à ❝õ❛ ♥ ♣❤➛♥ tû 1, 2, , n t❛ ❝â✿ n det(A) = sgn(σ) ai,σ(i) ✭❈æ♥❣ t❤ù❝ ▲❡✐❜♥✐③t✮ i=1 σ∈Sn ⑩♣ ❞ư♥❣ ✈ỵ✐ ❝→❝ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ ✶✱✷✱✸ t❛ ❝â✿ ❛✮ det a = a a11 a12 ❜✮ det a21 a22 = a11 a22 − a12 a21 a11 a12 a13 ❝✮ det a21 a22 a23 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 − a13 a22 a31 − a12 a21 a33 − a11 a23 a32 a31 a32 a33 ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ A ❧➔ ❝➜♣ ❝❛♦ ♥❤➜t ❝õ❛ t➜t ❝↔ ❝→❝ ✤à♥❤ t❤ù❝ ❝♦♥ ❦❤→❝ ❝õ❛ ♠❛ tr➟♥ A✳ ❑➼ r(A) rank(A) ữợ tr ổ ❝â ❤↕♥❣ ❜➡♥❣ ✵✳ ◆➳✉ ❆ ❧➔ ♠❛ tr➟♥ ❝ï m × n t❤➻ ≤ r(A) ≤ min(m, n)✳ ❱➼ ❞ö✿ A = 0 2 ⇒ rank(A) = 0 −1 ✶✵ ❍➻♥❤ ✶✳✶✿ ❇✐➸✉ ❞✐➵♥ ❝õ❛ ♠ët ❤➺ ✤✐➲✉ ❦❤✐➸♥ tê♥❣ q✉→t✳ ◆➳✉ Fx˙ ❧➔ t❤æ♥❣ t❤÷í♥❣✱ ♣❤÷ì♥❣ tr➻♥❤ tr↕♥❣ t❤→✐ ✭✶✳✶❛✮ ❝â t❤➸ ✤÷đ❝ ♣❤→t ❜✐➸✉ ❧↕✐ ❜➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✭❖❉❊✮✿ x˙ = (t, x, u) sỷ ỵ ❤➔♠ ➞♥✳ Ð ✤➙②✱ ❝❤ó♥❣ t❛ ❝ơ♥❣ s➩ ❝❤♦ ♣❤➨♣ r➡♥❣ Fx˙ ❧➔ ❦❤✉②➳t ❜➟❝✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ✭✶✳✶❛✮ ỗ ữỡ tr số ♥â✐ ❝→❝❤ ❦❤→❝✱ Fx˙ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕✐ sè ✭❉❆❊✮✳ ❚r♦♥❣ ♣❤÷ì♥❣ ❞✐➺♥ ✤✐➲✉ ❦❤✐➸♥✱ ❤➺ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❤➺ ♠æ t↔✳ ◆❤ú♥❣ ❤➺ ❝â ❞↕♥❣ ✭✶✳✶✮ ①✉➜t ❤✐➺♥ ✈ỵ✐ ❝→❝ ✈➼ ❞ư tr♦♥❣ ❝ì ❤å❝✱ ❦ÿ t❤✉➟t ✤✐➺♥ ✈➔ ❝æ♥❣ ♥❣❤➺ ❤â❛✳ ✶✳✹ ❍➺ ❜➜t ❜✐➳♥ t✉②➳♥ t➼♥❤ t❤❡♦ t❤í✐ ❣✐❛♥ ✲ ▲❚■ ❍➺ ❜➜t ❜✐➳♥ t✉②➳♥ t➼♥❤ t❤❡♦ t❤í✐ ❣✐❛♥ ✭▲❚■✮ ✤÷đ❝ ♠ỉ t↔ ❜➡♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝â ❞↕♥❣ s❛✉ ✤➙②✿ ✭✶✳✷✮ E x˙ = Ax + Bu, x(0) = x0 , y = Cx ✈ỵ✐ E, A ∈ Rn,n , B ∈ Rn,m , C ∈ Rp,n ✳ ❑❤æ♥❣ ❣✐↔♠ tê♥❣ q✉→t✱ ❝❤ó♥❣ t❛ ❣✐↔ sû r➡♥❣ r = rank(E) < n õ tỗ t tr ổ s✉② ❜✐➳♥ T, W ∈ Rn,n s❛♦ ❝❤♦ ❤➺ ❜✐➳♥ tữỡ ữỡ ợ x = Jx1 + B1 u, x1 (0) = x1,0 ✭✶✳✸❛✮ N x˙ = x2 + B2 u, x2 (0) = x2,0 ✭✶✳✸❜✮ y = C x1 + C x2 ✭✶✳✸❝✮ ✈ỵ✐ W ET = Inf 0 N , W AT = ✶✸ J 0 In∞ CT = C1 C2 , W BT = B1 B2 , T −1 x = x1 x2 ✈➔ ✤➦t ν = ind(E, A) ❚❛ ❣å✐ ✭✶✳✸❛✮ ❧➔ ❤➺ ❝♦♥ ❝❤➟♠ ❝õ❛ ❝❤✐➲✉ nf ✈➔ ✭✶✳✸❜✮ ❧➔ ❤➺ ❝♦♥ ♥❤❛♥❤ ❝õ❛ ❝❤✐➲✉ n∞ ✳ ❑❤✐ ✤â✱ t❛ ❜✐➳t r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr↕♥❣ t❤→✐ ❝õ❛ ✭✶✳✸✮ ❧➔✿ t x1 (t) = eJt x1 (0) + eJ(t−s) B1 u(s)ds, (t > 0) ν−1 N i B2 u(i) x2 (t) = − i=0 ❉♦ ✤â✱ ♠ët ữủ ợ ởt ✤✐➸♥✮ t❤ä❛ ♠➣♥ u ∈ Cpν−1 (I, Rm ) ❱ỵ✐ ❜➜t ❦ý t > 0✱ tr↕♥❣ t❤→✐ ✤→♣ ù♥❣ x(t) = T [xT1 xT2 ]T ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❜ð✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x1 (0)✱ ✤✐➲✉ ❦❤✐➸♥ ✤➛✉ ✈➔♦ u(s)✱ ≤ s ≤ t ✈➔ t❤í✐ ❣✐❛♥ t✳ ✣➦❝ ❜✐➺t✱ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x2 (0) ❧➔ ♥❤➜t q✉→♥ ✭tù❝ ❧➔ x2 (0) ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✮ ✈➔ ❝❤➾ x1 (0) ❝â t❤➸ ✤÷đ❝ ❝❤å♥ tị② þ✳ ❚❤❡♦ s❛✉ ✤â✱ t❛ ❦➼ ❤✐➺✉ ❜➡♥❣ ˜ t➟♣ t✐➳♣ ❝➟♥ ❝õ❛ ✭✶✳✸✮ tø ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ❦❤æ♥❣ x1 (0) = ✭✈➔ x2 (0) ♥❤➜t R q✉→♥✮✳ ✶✹ ❈❤÷ì♥❣ ✷ ❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔ ❈❤÷ì♥❣ ♥➔② ❧➔ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ✤➲ t tr ữỡ ợ t ✤à♥❤ ♥❣❤➽❛ ✈➲ ✧❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t ự ỵ q ❝→❝ ✤à♥❤ ♥❣❤➽❛ ♥➯✉ tr➯♥✳ ◆ë✐ ❞✉♥❣ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶❪✳ ✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❍➺ ♠ỉ t↔ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥ s→t ✤÷đ❝ ❤♦➔♥ t♦➔♥ ✭❈ ✲ q✉❛♥ s→t ✤÷đ❝✮ ♥➳✉ tr↕♥❣ t❤→✐ ❜❛♥ ✤➛✉ x(0) ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❜ð✐ u(t) ✈➔ y(t) ✈ỵ✐ ≤ t < ∞ ✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ✈ỵ✐ ♠ët ❤➺ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ tr↕♥❣ t❤→✐ x(t) ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t tø u ✈➔ y ❜➡♥❣ ❝→❝❤ q✉❛♥ s→t tr↕♥❣ t❤→✐ ❜❛♥ ✤➛✉ ✈➔ ①➙② ❞ü♥❣ ✤→♣ ù♥❣ ❤➺ ✈ỵ✐ ❜➜t ❦ý t ≥ ✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ❍➺ ✭✶✳✶✮ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ ♥➳✉ ✤➛✉ r❛ ❦❤ỉ♥❣ y(t) ≡ ✈ỵ✐ u(t) ≡ ❦➨♦ t❤❡♦ ❤➺ ❝❤➾ ❝â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ x(t) ≡ ✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ❍➺ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥ s→t ✤÷đ❝ ❜➯♥ tr♦♥❣ ❜ë t✐➳♣ ❝➟♥ ✭❘ ✲ q✉❛♥ s→t ✤÷đ❝✮ ♥➳✉ tr↕♥❣ t❤→✐ ❜➜t ❦ý tr♦♥❣ ❜ë t✐➳♣ ❝➟♥ ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❜ð✐ y(t) ✈➔ u(t) ✈ỵ✐ t ≥ ✳ ✶✺ ❉♦ ✤â✱ t❛ ❝➛♥ ♠ët ♣❤➨♣ ❝❤✐➳✉ ①➜♣ ①➾ ❧➯♥ ❝→❝ ❜✐➳♥ ♠➔ ❧✐➯♥ ❦➳t ✈ỵ✐ ♣❤➛♥ ✤ë♥❣ ❧ü❝ ❝õ❛ ❤➺ ✳ ❱ỵ✐ ❤➺ tê♥❣ q✉→t ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤✱ ✤✐➲✉ ♥➔② ❧➔ ❦❤â t➻♠ ✤÷đ❝✳ ◆❤➟♥ ①➨t ✷✳✶✳ ✶✳ ❘ ✲ q✉❛♥ s→t ✤÷đ❝ ✤ỉ✐ ❧ó❝ ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥ s→t ✤÷đ❝ ✤ë♥❣ ❧ü❝ ❤ú✉ ❤↕♥✳ ✷✳ ❚r♦♥❣ ❦❤✐ ❈✲ q✉❛♥ s→t ✤÷đ❝ ♣❤↔♥ →♥❤ ❦❤↔ ♥➠♥❣ ①➙② ❞ü♥❣ ❧↕✐ ❝õ❛ t♦➔♥ ❜ë tr↕♥❣ t❤→✐ x(t) tứ r ữủ ũ ợ ❦❤✐➸♥✱ ❘ ✲ q✉❛♥ s→t ✤÷đ❝ t✐➯✉ ❜✐➸✉ ❝❤♦ ❦❤↔ ♥➠♥❣ ①➙② ❞ü♥❣ ❧↕✐ ❞✉② ♥❤➜t tr↕♥❣ t❤→✐ t✐➳♣ ❝➟♥✳ ❉♦ ✤â✱ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ ❦➨♦ t❤❡♦ ❘ ✲ q✉❛♥ s→t ✤÷đ❝✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ❍➺ ♠ỉ t↔ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ①✉♥❣ q✉❛♥ s→t ✤÷đ❝ ✭■ ✲ q✉❛♥ s→t ✤÷đ❝✮ ♥➳✉ ❤➔♥❤ ✈✐ ①✉♥❣ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr↕♥❣ t❤→✐ x(t) ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t tø ❤➔♥❤ ✈✐ ①✉♥❣ ❝õ❛ ✤➛✉ r❛ ✈➔ ❤➔♥❤ ✈✐ ♥❤↔② ❝õ❛ ✤➛✉ ✈➔♦✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ▼ët ❤➺ ♠ỉ t↔ ✭✶✳✷✮ ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥ s→t ✤÷đ❝ ♠↕♥❤ ♠➩ ✭❙ ✲ q✉❛♥ s→t ✤÷đ❝✮ ♥➳✉ ♥â ❧➔ ❘ ✲ q✉❛♥ s→t ✤÷đ❝ ✈➔ ■ ✲ q✉❛♥ s→t ữủ ởt số ỵ ỵ t ♠ët ❤➺ ♠ỉ t↔ t✉②➳♥ t➼♥❤ t❤ỉ♥❣ t❤÷í♥❣ ❝â ❞↕♥❣✿ E x˙ = Ax + Bu, x(0) = x0 y = Cx ✶✳ ✣➦t u(t) ≡ 0✳ ❑❤✐ ✤â✱ y(t) ≡ ✈ỵ✐ ♠å✐ t ≥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ C1 C1 J x˜0 ∈ ker C2 C1 J nf −1 ⊕ ker C2 N C2 N ν−1 , x˜ = T −1 x = x1 x2 ♥❤÷ tr♦♥❣ ✭✶✳✸✮✳ ✷✳ ❍➺ ❝♦♥ ❝❤➟♠ ✭✶✳✸❛✮ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ rank λE − A C ✈ỵ✐ ♠å✐ λ ❤ú✉ ❤↕♥ ∈ C ✸✳ ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✶✻ =n ❛✮ ❍➺ ❝♦♥ ♥❤❛♥❤ ✭✶✳✸❜✮ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝✳ ❜✮ C2 C2 N rank C2 N ν−1 = n∞ ❝✮ ker N C2 = {0} ❞✮ rank N C2 = n∞ ❡✮ rank E C = n ❢✮ ❱ỵ✐ ❤❛✐ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤ ❜➜t ❦ý P1 ✈➔ Q1 t❤ä❛ ♠➣♥ P1 EQ1 = I 0 ✤➦t CP1 = C˜1 C˜2 ❑❤✐ ✤â✱ C˜2 ❝â ❤↕♥❣ tè✐ ✤❛ t❤❡♦ ❝ët✱ rank(C˜2 ) = n − rank(E)✳ ✹✳ ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ❛✮ ❍➺ ✭✶✳✷✮ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝✳ ❜✮ ❍➺ ❝♦♥ ❝❤➟♠ ✈➔ ♥❤❛♥❤ ✭✶✳✸❛✮ ✈➔ ✭✶✳✸❜✮ ✤➲✉ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝✳ ❝✮ rank λE − A ❞✮ rank αE − βA C C = n ✈ỵ✐ ♠å✐ λ ∈ C ✈➔ rank E C = n = n ✈ỵ✐ ♠å✐ (α, β) ∈ C2 \ {(0; 0)} ự ỵ t sỷ t q s ỵ ❝❤➟♠ ✭✶✳✸❛✮ ❧➔ ❈ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ∀λ ∈ C, rank λE − A B = n, λ ❤ú✉ ❤↕♥ ✷✳ ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ✤➙✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ❛✮ ❍➺ ❝♦♥ ❝❤➟♠ ✭✶✳✸❜✮ ❧➔ ❈ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❜✮ rank B2 N B2 N ν−1 B2 = n∞ ❝✮ rank N B2 = n∞ ❞✮ rank E B = n∞ ❡✮ ❱ỵ✐ ♠❛ tr➟♥ ❦❤ỉ♥❣ s✉② ❜✐➳♥ P1 ✈➔ Q1 ❜➜t ❦ý t❤ä❛ ♠➣♥ E = Q1 I 0 P1 ; QB = ˜1 B ˜2 B ❑❤✐ ✤â✱ B˜2 ❝â ❤↕♥❣ tè✐ ✤❛ t❤❡♦ ❤➔♥❣ ✱ rank(B˜2 ) = n − rank(E) ✸✳ ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ❛✮ ❍➺ ✭✶✳✸✮ ❧➔ ❈ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❜✮ ❍➺ ❝♦♥ ❝❤➟♠ ✈➔ ♥❤❛♥❤ ✭✶✳✸❛✮ ✈➔ ✭✶✳✸❜✮ ✤➲✉ ❧➔ ❈ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❝✮ rank B1 JB1 J nf −1 B1 = nf ✈➔ rank B2 N B2 N ν−1 B2 = n∞ ❞✮ rank λE − A B = n ✈ỵ✐ ♠å✐ λ ❤ú✉ ❤↕♥ ∈ C ✈➔ rank B2 N B2 N ν−1 B2 = n∞ ❡✮ rank αE − βA B = n ✈ỵ✐ ♠å✐ (α, β) ∈ C2 \ {(0; 0)} ❈❤ù♥❣ ♠✐♥❤ ✶✳ ❱ỵ✐ u(t) ≡ ♣❤÷ì♥❣ tr➻♥❤ tr↕♥❣ t❤→✐ ❝õ❛ ✭✶✳✸✮ ✤÷đ❝ ❝❤♦ ❜ð✐ x1 (t) = eJt x1 (0) ν−1 (i−1) N i x2 (0)δ0 x2 (0) = − i=1 y(t) = y1 (t) + y2 (t) = C1 x1 (t) + C2 x1 (t) ❉♦ ✤â✿ y(t) ≡ ⇔ C1 x1 (t) + C2 x2 (t) = 0, ∀t ≥ ⇔ C1 x1 (t) = C2 x2 (t) = ✶✽ , ∀t ≥ ✈ỵ✐ ❜✐➸✉ t❤ù❝ ❝✉è✐ ❝ị♥❣ t÷ì♥❣ ✤÷ì♥❣ t❤❡♦ tø ❦❤❛✐ tr✐➸♥ tr♦♥❣ ♣❤➛♥ trì♥ ✈➔ ①✉♥❣✳ ◆➳✉ y1 (t) = C1 x1 (t) = C1 eJt x1 (0) ≡ 0, ∀t ≥ ✱ ❦❤✐ ✤â ❝ô♥❣ ❝â✿ (i) y1 (t) ≡ 0, ∀i = 0, , nf − 1, ∀t ≥ (i) ✣➦❝ ❜✐➺t✱ t❛ ❝â y1 (0) = ✈➔ ❞♦ ✤â✿ C1 C1 J C1 J nf −1 C1 x1 (0) = ⇒ x1 (0) ∈ ker C1 J C1 J nf −1 ❍ì♥ ♥ú❛✱ ν−1 (i−1) C2 N i x2 (0)δ0 y2 (t) = C2 x2 (t) = − ≡ ⇔ C2 N i x2 (0), i = 0, , ν − i=1 C2 C2 N ⇔ x2 (0) ∈ ker C2 N ν−1 ❉♦ ✤â✿ C1 C1 J x˜(0) = ∈ ker x2 (0) x1 (0) C1 J nf −1 C2 ⊕ ker C2 N C2 N ν−1 ✷✳ ❍➺ ❝♦♥ ❝❤➟♠ ❧➔ ♠ët ❤➺ ▲❚■ t✐➯✉ ❝❤✉➞♥✳ ❉♦ ✤â✿ ✭✶✳✸❛✮ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ ⇔ (J, C1 )❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ ⇔ rank λI − J = nf , C1 ❍ì♥ ♥ú❛✱ rank λE − A C = rank λW ET − W AT CT = rank λI − J 0 λN − I C1 = n∞ + rank ✶✾ C2 λI − J C1 ∀λ ∈ C ✸✳ ❚❤❡♦ ♥❤÷ ✤à♥❤ ♥❣❤➽❛ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ❝♦♥ ♥❤❛♥❤ ❝â ♥❣❤➽❛ ❧➔✿ ✤➸ t ≥ ✈ỵ✐ u(t) ≡ ❦➨♦ t❤❡♦ r➡♥❣ x2 (0) = 0✳ ❚❤❡♦ ♥❤÷ ✶ ✤✐➲✉ tữỡ ữỡ ợ C2 C2 N ker C2 N ν−1 = {0} ✈➔ ❞♦ ✤â 3a ⇔ 3b✳ ❍ì♥ ♥ú❛✱ C2 C2 N rank C2 N ν−1 = n∞ ⇔ rank C T N T C T (N T )ν−1 C2 = n∞ 2 ⇔ N T ξ˙2 = ξ2 + C T u ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ ỵ tr ữủ ỵ t õ 3b 3f ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✹✳ ❚ø ❝❤ù♥❣ ♠✐♥❤ ✶ ✈➔ tữỡ tỹ ữ ự ỵ t❛ ❝â ✤♣❝♠✳ ❱➼ ❞ư ✷✳✶✳ ❳➨t ❤➺ ♠ỉ t↔✿ x˙ = 1 x1 + = x2 + −1 u u y = x1 ❱➻✿ rank C1 C1 J = rank 1 = = nf ✈➔ rank C2 C2 N = rank 0 0 = < n∞ t❛ t❤➜② r➡♥❣ ❤➺ ❝♦♥ ❝❤➟♠ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ tr♦♥❣ ❦❤✐ ❤➺ ❝♦♥ t ổ ỵ t ởt ❤➺ t✉②➳♥ t➼♥❤ t❤ỉ♥❣ t❤÷í♥❣ ❝õ❛ ❞↕♥❣ ✭✶✳✷✮✳ ❑❤✐ ✤â ❤➺ ✭✶✳✷✮ ❧➔ ❘ ✲ q✉❛♥ s→t ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➺ ❝♦♥ ❝❤➟♠ ✭✶✳✸❛✮ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝✱ tù❝ ❧➔✿ rank λE − A = n ✈ỵ✐ ♠å✐ λ ❤ú✉ ❤↕♥ ∈ C C ❈❤ù♥❣ ♠✐♥❤ ợ ữỡ tr tr t t ý x(t) = T x1 (t) x2 (t) ❝â ❞↕♥❣✿ t Jt eJ(t−s) B1 u(s)ds x1 (t) = e x1 (0) + ν−1 N i B2 u(i) (t) x2 (t) = − i=1 y(t) = y1 (t) + y2 (t) = C1 x1 (t) + C2 x2 (t) tù❝ ❧➔✱ x2 (t) ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❜ð✐ u(t) ✈➔ y1 (t) = C1 x1 (t) = y(t) − C2 x2 (t) ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❜ð✐ y(t) ✈➔ u(t)✳ ❉♦ ✤â✱ ♠ët ♣❤÷ì♥❣ tr➻♥❤ tr↕♥❣ t❤→✐ x(t) ❝â t❤➸ ✤÷đ❝ ①➙② ❞ü♥❣ ❧↕✐ tø y(t) ✈➔ u(t) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x1 (t) ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❜ð✐ y1 (t) ✈➔ u(t)✱ tù❝ ❧➔ ❤➺ ❝♦♥ ❝❤➟♠ ✭✶✳✸❛✮ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝✳ ❍➺ q✉↔ ✷✳✶✳ ❍➺ ✭✶✳✷✮ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➺ ❧➔ ❘ ✲ q✉❛♥ s→t ✤÷đ❝ ✈➔ rank E C = n ✣➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t ❦➨♣ ❝❤♦ ■ ✲ q✉❛♥ s→t ✤÷đ❝ t❛ sû ❞ư♥❣ ❜ê ✤➲ s❛✉✿ ❇ê ✤➲ ✷✳✶✳ ❳➨t ♠ët ❤➺ t❤ỉ♥❣ t❤÷í♥❣ tr♦♥❣ ❲❈❋ ✭✶✳✸✮✳ ❑❤✐ ✤â✱ ♣❤➛♥ ①✉♥❣ ❝õ❛ ✤➛✉ r❛ t↕✐ τj ∈ T ❧➔ yimp,j ≡ 0, ∀j ∈ Z ✈ỵ✐ u(t) ≡ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✿ C2 C2 N x2j,0 ∈ ker C2 N ν−1 , x2j,0 = xˆ2,j−1 (τj ), τj ∈ T ❈❤ù♥❣ ♠✐♥❤ ❱➻✿ yimp,j = Cximp,j = C2 x2,imp,j t❛ ❝â✿ ν−1 (i) C2 N i x2j,0 δτj , ∀τj ∈ T yimp,j = − i=1 ✷✶ ❉♦ ✤â✿ ✈ỵ✐i = 1, , ν − yimp,j ≡ ⇔ C2 N i x2j,0 = C2 C2 N ⇔ x2j,0 ∈ ker C2 N ν−1 ỵ t õ ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✶✳ ❍➺ ✭✶✳✷✮ ❧➔ ■ ✲ q✉❛♥ s→t ✤÷đ❝✳ ✷✳ ❍➺ ❝♦♥ ♥❤❛♥❤ ✭✶✳✸❜✮ ❧➔ ■ ✲ q✉❛♥ s→t ✤÷đ❝✳ C2 C2 N ✸✳ ker C2 N ν−1 C2 N ✹✳ ker ∩ Im(N ) = {0} = ker(N ) C2 N ν−1 ✺✳ ker(N ) ∩ ker(C2 ) ∩ Im(N ) = {0} ✻✳ ✣➦t✿ ( T NT N11 21 T N22 , T C21 )nn˜˜ 12 ❧➔ ❦❤❛✐ tr✐➸♥ ❑❛❧♠❛♥ ❬✺❪ ❝õ❛ (N T , C2T ) ✱ ✈ỵ✐ (N11 , C21 ) ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝✳ ❑❤✐ ✤â✱ ❤♦➦❝ n ˜ = ❤♦➦❝ N22 = ✈➔ rank(N1 1) = rank E A N11 N21 ✼✳ rank E = n + rank(E) C ự ỵ tr t õ sỷ t q s ỵ t s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✶✳ ❍➺ E x˙ = Ax + Bu ❧➔ ■ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✷✳ ❍➺ ❝♦♥ ♥❤❛♥❤ N x˙ = x2 + B2 u ❧➔ ■ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✸✳ ker(N ) + Im B2 N B2 N ν−1 B = Rn∞ ✹✳ Im(N ) = Im N B2 N B2 N ν−1 B2 ✺✳ Im(N ) + Im(B2 ) + ker(N ) = Rn∞ ✻✳ rank E 0 A E B = n + rank(E) ✷✷ ❈❤ù♥❣ ♠✐♥❤ ✯ ●✐↔ sû r➡♥❣ u ∈ Cp∞ ✳ ❉ü❛ ✈➔♦ ximp = x2,imp ✈➔ yimp = C1 x1,imp + C2 x2,imp = C2 x2,imp = y2,imp ❉♦ ✤â✱ t❛ ❝â ⇔ 2✳ ✯2⇔4 (i−1) ν−1 i i=1 N x2j,0 δτj ❱ỵ✐ u(t) ≡ t❛ ❝â x2,imp = − ≡ ⇔ N x2j,0 = C2 C2 N ❉♦ ✤â tø ❜ê ✤➲ ✭✷✳✶✮ t❛ ✤÷đ❝ ⇔ ker(N ) = ker C2 N ν−1 ✯3⇔4 C2 N ●✐↔ sû r➡♥❣ ⇔ = ker(N ) ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ❜➜t ❦ý C2 N ν−1 C2 w∈ ∩ Im(N ) C2 N tỗ t s ❝❤♦ C2 w = N β ∈ ker ⇒ β ∈ ker C2 N ν−1 C2 N C2 N ν−1 ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû r➡♥❣ C2 C2 N ker C2 N ν−1 ❑❤✐ ✤â✱ ✈ỵ✐ ❜➜t ❦ý w ∈ ker C2 N ∩ Im(N ) = {0} t❛ ❝â✿ C2 N ν−1 C2 N w ∈ ker ⊆ ker(N ) C2 N ν−1 ✷✸ = ker(N ) ❱➻ ker(N ) ⊆ ker C2 N t❤✉ ✤÷đ❝ ✤➥♥❣ t❤ù❝✳ C2 N ν−1 ✯3⇔5 CN2 ●✐↔ sû r➡♥❣ ker(N ) = ker ✳ ❱➻ ker(N ) + Im(N T ) = Rn∞ t❛ ❝â✿ C2 N ν−1 Im(N T ) = Im N T C2T (N T )ν−1 C2T ❑❤✐ ✤â ✹✱ ✺✱ ✻ ✈➔ tữỡ ữỡ t ỵ ữủ ú ỵ r ữủ q s→t ✤÷đ❝ ❧➔ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ❦➨♣✱ tù❝ ❧➔ ♥❤ú♥❣ t q s ú ỵ ỵ ✤è✐ ♥❣➝✉✮✳ ❳➨t ♠ët ❤➺ ♠æ t↔ t✉②➳♥ t➼♥❤ ❝õ❛ ❞↕♥❣ ✭✶✳✷✮✳ ◆❤ú♥❣ ✤✐➲✉ s❛✉ ✤➙② ❧➔ ✤ó♥❣✿ ✶✳ ❍➺ ✭✶✳✷✮ ❧➔ ❈ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➺ ❦➨♣ ✤÷đ❝ ❝❤♦ ❜ð✐✿ ✭✷✳✶✮ E T ξ˙ = AT ξ + C T u, y = B T ξ ❧➔ ❈ ✲ q✉❛♥ s→t ✤÷đ❝✳ ✷✳ ❍➺ ✭✶✳✷✮ ❧➔ ❘ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➺ ❦➨♣ ✭✷✳✶✮ ❧➔ ❘ ✲ q✉❛♥ s→t ✤÷đ❝✳ ✸✳ ❍➺ ✭✶✳✷✮ ❧➔ ■ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➺ ❦➨♣ ✭✷✳✶✮ ❧➔ ■ ✲ q✉❛♥ st ữủ ứ ỳ t q trữợ õ t õ ✤÷đ❝ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ q✉❛♥ s→t ữủ ữủ ổ t tổ q sỡ ỗ s ❘ ✕ q✉❛♥ s→t ✤÷đ❝ ❚ø ❈ ✕q✉❛♥ s→t ✤÷đ❝ ⇒ ❙ ✕ q✉❛♥ s→t ✤÷đ❝ ⇒ ■ ✕ q✉❛♥ s→t ✤÷đ❝ ❇ê ✤➲ ✷✳✷✳ ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✶✳ ❍➺ E x˙ = Ax+ Bu ❧➔ ■ ✲ q✉❛♥ s→t ✤÷đ❝✳ E T ✷✳ rank T∞ A = n ✱ ✈ỵ✐ Im(T∞ ) = ker(E T ) ✳ C ✷✹ ❘ ✕ q✉❛♥ s→t ✤÷đ❝ ■ ✕ q✉❛♥ s→t ✤÷đ❝ E ✸✳ rank K∞ = n∞ ✱ ✈ỵ✐ Im(K∞ ) = ker(N T ) C2 ◆❤➟♥ ①➨t ✷✳✷✳ ợ sỹ tỗ t ởt ỗ F Rm,p s❛♦ ❝❤♦ (E, A + BF C) ❧➔ ✤➲✉ ✤➦♥ ✈➔ ❝õ❛ ❝❤➾ sè ν ≤ t❛ ❝➛♥ ■ ✲ q✉❛♥ s→t ✤÷đ❝ ✈➔ ■ ✲ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✷✺ ❑➌❚ ▲❯❾◆ ❙❛✉ ♠ët t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ✧❚➼♥❤ q✉❛♥ s→t ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✧✱ ữợ sỹ ữợ r tổ ✤➣ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ♠ët sè ✈➜♥ ✤➲ s❛✉ ✤➙②✿ ✶✳ ❍➺ t❤è♥❣ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ t❤✉ë❝ ❧➽♥❤ ✈ü❝ ✤↕✐ sè t✉②➳♥ t➼♥❤✳ ✷✳ ●✐ỵ✐ t❤✐➺✉ tự ỡ ỵ tt q st ữủ ố ợ ởt ổ t ữủ ữợ ữỡ tr ởt ❚➻♠ ❤✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❝→❝❤ ❝❤✐ t✐➳t ♥❤ú♥❣ t q ỡ ỵ tt tt q st ✤÷đ❝ ❝õ❛ ❤➺ ♠ỉ t↔✳ ▼➦❝ ❞ị ✤➣ ❝è ❣➢♥❣ s♦♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ♥➯♥ tr♦♥❣ ✤➲ t➔✐ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ ♠♦♥❣ ♥❤➟♥ ữủ ỳ õ õ qỵ t ổ t ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✷✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▲❡♥❛ ❙❝❤♦❧③✳ ❈♦♥tr♦❧ ❚❤❡♦r② ♦❢ ❉❡s❝r✐♣t♦r ❙②st❡♠s ▲❡❝t✉r❡ ◆♦t❡s✳ ❚❯ ❇❡r❧✐♥ ✭❲❙ ✷✵✶✹✴✶✺✮✳ ❬✷❪ ✣➟✉ ❚❤➳ ❈➜♣✳ ✣↕✐ ❙è ❚✉②➳♥ ❚➼♥❤✳ ◆❤➔ ①✉➜t ❜↔♥ ❣✐→♦ ❞ö❝✱ ✷✵✵✽✳ ✷✼