TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* NGUYỄN THỊ PHƯƠNG BÁN KÍNH ĐIỀU KHIỂN ĐƯỢC CỦA HỆ ĐIỀU KHIỂN TUYẾN TÍNH KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích HÀ NỘI – 2018 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* NGUYỄN THỊ PHƯƠNG BÁN KÍNH ĐIỀU KHIỂN ĐƯỢC CỦA HỆ ĐIỀU KHIỂN TUYẾN TÍNH KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn khoa học ThS Trần Thị Thu HÀ NỘI – 2018 ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✶ ▲í✐ ❝❛♠ ✤♦❛♥ ✷ ▲í✐ ♥â✐ ✤➛✉ ✸ ▼ët sè ❦➼ ❤✐➺✉ ✻ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✽ ✶✳✶ ✣↕✐ sè t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷ ●✐↔✐ t➼❝❤ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ⑩♥❤ ①↕ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹ ❍➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✺ ▼ët sè t✐➯✉ ❝❤✉➞♥ ①➨t t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ ❝â r➔♥❣ ❜✉ë❝ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ❇→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ ✶✾ ✷✳✶ ✣à♥❤ ♥❣❤➽❛ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❈→❝❤ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ❱➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữợ t tr ♥❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ ❑➳t ❧✉➟♥ ✸✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✶ ✐✐ ▲í✐ ❝↔♠ ì♥ rữợ tr õ ❡♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ tr✉②➲♥ ❝❤♦ ❡♠ ♥✐➲♠ ❝↔♠ ❤ù♥❣ ũ ỳ tr tự qỵ t ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ♥❤✐➺♠ ✈ö ❦❤â❛ ❤å❝✳ ✣➦❝ ❜✐➺t ❤ì♥ ♥ú❛✱ ❡♠ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❝ỉ s r ữớ trỹ t ữợ ❞➝♥✱ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤✱ ❣✐ó♣ ✤ï ❡♠ ✤➸ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❉♦ ❜✉ê✐ ✤➛✉ ❧➔♠ q✉❡♥ ✈ỵ✐ ❝ỉ♥❣ t→❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ♥➯♥ ❜↔♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ♠➔ ❜↔♥ t❤➙♥ ❝❤÷❛ t❤➜② ✤÷đ❝✳ ❱➻ ✈➟②✱ ❡♠ r➜t ữủ ỳ ỵ õ õ t ❝æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ✤➛② ✤õ ✈➔ ❝❤➼♥❤ ①→❝ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ ♥❣➔② ✹ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽✳ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ Pữỡ ữợ sỹ ữợ t➟♥ t➻♥❤ ❝õ❛ ❝æ ❣✐→♦ ❚❤s✳ ❚r➛♥ ❚❤à ❚❤✉ ❦❤â❛ ❧✉➟♥ ❝❤✉②➯♥ ♥❣➔♥❤ t♦→♥ ❣✐↔✐ t➼❝❤ ✈ỵ✐ ✤➲ t➔✐ ✏❇→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤✑ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❜ð✐ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❜↔♥ t❤➙♥✱ ❦❤ỉ♥❣ trò♥❣ ✈ỵ✐ ❜➜t ❝ù ❦❤â❛ ❧✉➟♥ ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ✤➣ ❣❤✐ tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❍➔ ◆ë✐✱ ♥❣➔② ✹ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽✳ ❙✐♥❤ ✈✐➯♥ Pữỡ õ ỵ t ỵ tt õ ỗ ố tứ trữợ sỹ tỹ ✤✐➲✉ ❦❤✐➸♥ ❝ì ❤å❝ ❜➢t ✤➛✉ ✤÷đ❝ ♣❤➙♥ t➼❝❤ ❜➡♥❣ ❝♦♥ ✤÷í♥❣ t♦→♥ ❤å❝✳ ❚➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ỹ ữủ ữ ỳ ỵ tữ ❦➳t q✉↔ q✉❛♥ trå♥❣ ❝õ❛ ❘✳❑❛❧♠❛♥ tø ♥❤ú♥❣ ♥➠♠ ✻✵✱ tr♦♥❣ ✤â ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤↕✐ sè ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❤➺ t✉②➳♥ t➼♥❤ ✤ì♥ ❣✐↔♥✳ ❚ø ✤â ✤➳♥ ♥❛② ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤➣ tr t ởt ữợ ữủ ự ởt ữợ q trồ ỵ tt ỹ t ữủ ự ợ ữủ s ữợ t ✤ë♥❣ ❝õ❛ ♥â ❤➺ t❤è♥❣ ✤÷đ❝ ✤✐➲✉ ❦❤✐➸♥ ✈➲ ❝→❝ ✈à tr➼ ♠♦♥❣ ♠✉è♥✳ ❑❤✐ ✤➣ ❝â ♠ët ❤➺ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ t❛ ♥❤✐➵✉ ❤➺ ✤â ♠ët ❧÷đ♥❣✱ ♠ët t❤æ♥❣ sè ♥➔♦ ✤â✱ ❧✐➺✉ r➡♥❣ ❤➺ ✤â ❝â ❝á♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥ú❛ ❤❛② ❦❤ỉ♥❣❄ ✣â ❧➔ ❜➔✐ t♦→♥ ❞➝♥ ✤➳♥ ❦❤→✐ ♥✐➺♠ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❇→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝❤➼♥❤ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø ♠ët t➟♣ ❝→❝ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✤➳♥ ♠ët t➟♣ ❝→❝ ❤➺ ✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ ❦❤ỉ♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ◆❤➟♥ t❤➜② ✤÷đ❝ sü ♣❤→t tr✐➸♥ ✈➔ ♥❤ú♥❣ ù♥❣ ❞ư♥❣ t♦ ❧ỵ♥ ➜② ❝❤ó♥❣ tỉ✐ ✤➣ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ✏ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ t t ữợ sỹ ữợ ❞➝♥ ❝õ❛ ❝æ ❣✐→♦ ❚❤s✳ ❚r➛♥ ❚❤à ❚❤✉✳ ◆ë✐ ❞✉♥❣ õ ỗ õ ữỡ ữỡ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✿ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ✤↕✐ sè t✉②➳♥ t➼♥❤✱ ❣✐↔✐ t➼❝❤ ❤➔♠✱ →♥❤ ①↕ ✤❛ trà✱ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ ✈➔ ♠ët sè t✐➯✉ ❝❤✉➞♥ ①➨t t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ ❝â r➔♥❣ ❜✉ë❝✳ ❈❤÷ì♥❣ ✷✿ ❇→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤✿ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ t➻♠ ❤✐➸✉ ❝→❝ ❦➳t q✉↔ ✈➲ ❝→❝❤ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ q✉❛ ❝→❝ s→❝❤✱ t↕♣ ❝❤➼ ❜➡♥❣ ❚✐➳♥❣ ❆♥❤✳ ◗✉❛ ✤â ❝❤ó♥❣ tỉ✐ t➼♥❤ t♦→♥ ♠ët sè ✈➼ ❞ư →♣ ❞ư♥❣ ✤➸ ♥❣÷í✐ ✤å❝ ❞➵ ❤➻♥❤ ❞✉♥❣✳ ❈✉è✐ ❝ò♥❣ ❝❤ó♥❣ tỉ✐ ữợ t tr ỵ tt ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❚➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝❤ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤✳ ✸✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ✲ ◆❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝❤ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤✳ ✲ Ù♥❣ ❞ö♥❣ ❝õ❛ ❝→❝❤ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ t❤ỉ♥❣ q✉❛ ❝→❝ ✈➼ ❞ư ❝ư t❤➸✳ ✹✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✲ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ ❍➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤✳ ✲ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ❇→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❚ê♥❣ ❤đ♣ ❦✐➳♥ t❤ù❝ t❤✉ t❤➟♣ ✤÷đ❝ q✉❛ ♥❤ú♥❣ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐ ✈➔ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❣✐↔✐ t➼❝❤✳ ✻✳ ✣â♥❣ ❣â♣ ❝õ❛ ✤➲ t➔✐ ❳➙② ❞ü♥❣ ❦❤â❛ ❧✉➟♥ t❤➔♥❤ ♠ët t➔✐ ❧✐➺✉ tê♥❣ q✉❛♥ tèt ❝❤♦ s✐♥❤ ✈✐➯♥ ✈➲ ✤➲ t➔✐ ✏❇→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤✑✳ ❉♦ ❧➔ ❧➛♥ ✤➛✉ t✐➯♥ t❤ü❝ t➟♣ ♥❣❤✐➯♥ ❝ù✉✱ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ✈➔ ♥➠♥❣ ❧ü❝ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❜➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sõt rt ữủ ỳ õ õ ỵ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❜↕♥ ✤å❝ ✤➸ ✤➲ t➔✐ ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ✈➔ ✤↕t ❦➳t q✉↔ ❝❛♦ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✺ ❇↔♥❣ ❦➼ ❤✐➺✉ K ❚r÷í♥❣ Kn ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì (Kn )∗ ❑❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❝õ❛ Km×n , Mat(m × n, K) ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ ❝➜♣ x A∗ , AT , A−1 A R ❤♦➦❝ C✳ ❈❤✉➞♥ ❝õ❛ tỡ n x tr Kn mìn tr trữớ K Kn ▼❛ tr➟♥ ❧✐➯♥ ❤ñ♣✱ ❝❤✉②➸♥ ✈à✱ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♠❛ tr➟♥ ❈❤✉➞♥ ❝õ❛ ♠❛ tr➟♥ A σ(A) ●✐→ trà ❦➻ ❞à ❝õ❛ ♠❛ tr➟♥ ❆✳ σmin (A) ●✐→ trà ❦➻ ❞à ♥❤ä ♥❤➜t ❝õ❛ ♠❛ tr➟♥ F : Km ⇒ Kn ❚♦→♥ tû ✤❛ trà dom F ▼✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ Im F ▼✐➲♥ ↔♥❤ ❝õ❛ F Ker F t F gr F ỗ t ❝õ❛ F ∗ , F −1 ❚♦→♥ tû ❧✐➯♥ ❤ñ♣✱ ♥❣÷đ❝ ❝õ❛ G◦F ❚♦→♥ tû ✤❛ trà ❤đ♣ t❤➔♥❤ ❝õ❛ (A, B) ▼❛ tr➟♥ ❣❤➨♣ ❜ð✐ ♠❛ tr➟♥ (A|B) ▼❛ tr➟♥ ❝â ❞↕♥❣ ◆❤✐➵✉✳ ✻ A F F F A F G ✈➔ F ✈➔ ♠❛ tr➟♥ (B, AB, , An−1 B) B A ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ ❑✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ ❑❛❧♠❛♥ ✈ỵ✐ ♥❂✷ t❛ ❝â  rank(A|B) = rank(B, AB) = rank  2  =2 ♥➯♥ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❤↕♥❣ ❑❛❧♠❛♥ t❤➻ ❤➺ ✤➣ ❝❤♦ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❤♦➔♥ t♦➔♥✳ ❚ø ❤❛✐ ✤à♥❤ ❧➼ tr➯♥ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✤à♥❤ ❧➼ s❛✉ ✣à♥❤ ❧➼ ✶✳✺✳✸✳ ❈❤♦ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ✭✶✳✶✶✮✱ ❦❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✐✳ ❍➺ ❧➔ GC✳ ✐✐✳ ❍➺ ❧➔ LC✳ ✐✐✐✳ rank(A|B) = n✳ ✐✈✳ rank(A − λI, B) = n, ∀λ ∈ C ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ✤ë❝ ❣✐↔ q✉❛♥ t➙♠ ❝â t❤➸ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✹✱✺❪✳ ❙❛✉ ♥➔② ♥â✐ ✤➳♥ ❤➺ ✭✶✳✶✶✮ ✤✐➲✉ ❦❤✐➸♥ ữủ q ữợ õ ữủ t ✶✽ ❈❤÷ì♥❣ ✷ ❇→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ ❚❛ ❝â ❝➦♣ (A, B) ✈➔ t❛ ♠♦♥❣ ♠✉è♥ ♥➳✉ ♥❤✐➵✉ (A, B) ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ s❛✉ ✤â ♥❤✐➵✉ (A, B) (A, B) (A, B) ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ▼ët ❦➳t q✉↔ ✤➣ ❜✐➳t ✤â ❧➔✱ (A, B) ♠ët ❧÷đ♥❣ ✤õ ❜➨✱ ❤❛② A=A+ B=B+ t❤➻ t❛ t❤➜② (A, B) ✈➝♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❈➙✉ ❤ä✐ ✤➦t r ợ ữ t ✤õ ❜➨✳ ❉ü❛ ✈➔♦ ❜➔✐ t♦→♥ ✤â✱ t❤➯♠ ♥ú❛ ▲❡❡ ▼❛r❦✉s ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ t➟♣ ❝→❝ ❤➺ ❦❤ỉ♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❧➔ t➟♣ ✤â♥❣✱ ♥➯♥ ♥❣÷í✐ t❛ ✤➣ ✤à♥❤ ♥❣❤➽❛ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ♥❤ä ♥❤➜t tø ♠ët t➟♣ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✤➳♥ t➟♣ ❝→❝ ❤➺ ❦❤ỉ♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❈ư t❤➸ ❝❤ó♥❣ tỉ✐ s tr ữợ ♥❣❤➽❛ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ❈❤♦ ❝➦♣ (A, B) (A, B) ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ♥❤✐➵✉ (A, B) = (A, B) + (∆1 , ∆2 ) = (A + ∆1 , B + ∆2 ) ✶✾ ✈ỵ✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ (∆1 , ∆2 ) Knì(n+m) (A, B) Pữỡ ✤â ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ rK (A, B) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ rK (A, B) = inf{ (∆1 , ∆2 ) : (∆1 , ∆2 ) ∈ Kn×(n+m) s❛♦ ❝❤♦ ✈ỵ✐ · ❝õ❛ (A, B) + (∆1 , ∆2 ) ❦❤æ♥❣ ❧➔ ♠ët ❝❤✉➞♥ ❝õ❛ ♠❛ tr➟♥✳ ❙è ❝➦♣ ♠❛ tr➟♥ (A, B) ✭✷✳✶✮ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝} rK (A, B) ❝❤♦ ❜✐➳t t❛ ❝➛♥ ♥❤✐➲✉ ❜❛♦ ♥❤✐➯✉ ✤➸ ❦❤ỉ♥❣ ♣❤→ ✈ï t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ◆❤➟♥ ①➨t ✷✳✶✳✶✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❦➳t q✉↔ ✤➛✉ t✐➯♥ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❧➔ ❝õ❛ ❊✐s✐♥❣✱ ♥❣÷í✐ ❝❤ù♥❣ ♠✐♥❤ ❝ỉ♥❣ t❤ù❝ rC (A, B) = inf σmin ([A − λI, B]) λ∈C ✈ỵ✐ σmin ✭✷✳✷✮ ❧➔ ❣✐→ trà ❦➻ ❞à ♥❤ä ♥❤➜t ❝õ❛ ♠❛ tr➟♥ ✈➔ ❝❤✉➞♥ tr♦♥❣ ✭✷✳✷✮ ❧➔ ❝❤✉➞♥ ♣❤ê ❤♦➦❝ ❝❤✉➞♥ ❋r♦❜❡♥✐✉s✳ ❚ê♥❣ q✉→t ❤ì♥ ①➨t ♥❤✐➵✉ ❝➜✉ tró❝ ❞↕♥❣ (A, B) tr♦♥❣ ✤â (A, B) = (A, B) + D∆E D ∈ Kn×l , E ∈ Kq×(n+m) ❧➔ ❝→❝ ♠❛ tr➟♥ ✤➣ ❝❤♦ ✈➔ ♠❛ tr➟♥ ♥❤✐➵✉✳ ❈→❝ ♠❛ tr➟♥ ❝➜✉ tró❝ ♥❤✐➵✉ ❧➔ D∆E ❑❤✐ D, E D∆E = (∆1 , ∆2 ) D, E ✭✷✳✸✮ ∆ ❧➔ ①→❝ ✤à♥❤ ❝➜✉ tró❝ ❝õ❛ ❧➔ ❝→❝ ♠❛ tr➟♥ ✤ì♥ ✈à ✈➔ ∆ = (∆1 , ∆2 ), tù❝ t❤➻ ♥❤✐➵✉ ❞↕♥❣ ✭✷✳✸✮ trð ✈➲ ♥❤✐➵✉ ✤➣ ❜✐➳t ð ✤à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✷✳ ❈❤♦ ♠ët ữủ (A, B) ã tr Klìq , ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ❝❤à✉ ♥❤✐➵✉ ❞↕♥❣ ✭✷✳✸✮ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ ✷✵ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ D,E rK (A, B) = inf{ ∆ : ∆ ∈ Kl×q s❛♦ ❝❤♦ (A, B) = (A, B) + D∆E ❦❤ỉ♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝} ✭✷✳✹✮ ◆➳✉ ❝➦♣ (A, B) Klìq ợ t t t (A, B) = (A, B) + DE ữủ ợ ♠å✐ D,E rK (A, B) = +∞ ◆❤➟♥ ①➨t ✷✳✶✳✷✳ ❑❤✐ D, E ❧➔ ❝→❝ ♠❛ tr➟♥ ✤ì♥ ✈à ✈➔ ∆ = (∆1 , ∆2 ) t❤➻ ❝æ♥❣ t❤ù❝ ✭✷✳✹✮ trð ✈➲ ❝æ♥❣ t❤ù❝ ✭✷✳✶✮✳ ✷✳✷ ❈→❝❤ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❙❛✉ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ❊✐s✐♥❣✱ ❝â ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ✤÷❛ r❛ ♠ët sè ❝ỉ♥❣ t❤ù❝ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❚r♦♥❣ ♣❤↕♠ õ ú tổ t ữỡ ợ sû ❞ư♥❣ →♥❤ ①↕ ✤❛ trà t✉②➳♥ t➼♥❤ t❤ỉ♥❣ q✉❛ t rữợ t t õ ởt ✈➔✐ ✤à♥❤ ♥❣❤➽❛ s❛✉ ❚❛ ✤à♥❤ ♥❣❤➽❛✱ ∀λ ∈ C, t♦→♥ tû ✤ì♥ trà t✉②➳♥ t➼♥❤ Wλ : Kn+m −→ Kn z −→ Wλ (z) = (A − λI, B)z ✈➔ t♦→♥ tû ✤❛ trà t✉②➳♥ t➼♥❤ EWλ−1 D : Kl ⇒ Kq u −→ (EWλ−1 D)(u) = E(Wλ−1 (Du)) tr♦♥❣ ✤â Wλ−1 : Kn ⇒ Kn+m ❧➔ t♦→♥ tû ✤❛ trà ♥❣÷đ❝ ❝õ❛ ✷✶ Wλ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣à♥❤ ❧➼ ✷✳✷✳✶✳ ◆➳✉ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ K=C t❤➻ sup EWλ−1 D rCD,E (A, B) = ✭✷✳✺✮ λ∈C ❈❤ù♥❣ ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✭✷✳✺✮ t❛ ❝❤ù♥❣ ♠✐♥❤ sup EWλ−1 D rCD,E (A, B) ≥ λ∈C ✈➔ rCD,E (A, B) ≤ sup EWλ−1 D λ∈C ✲ ❚❛ ❝â (A, B) ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥➯♥ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❍❛✉t✉s t❛ ❝â rank(A − λI, B) = n, ❚ø ✤â s✉② r❛ rank Wλ = n C õ t ỵ ✶✳✶✳✷ t❛ ❝â Wλ ❧➔ t♦➔♥ →♥❤✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ✶✳✸✳✸✐✐✐ t❛ ❝â Wr∗ ❧➔ ✤ì♥ →♥❤ ❤❛② W1∗−1 ❧➔ ✤ì♥ trà ∀λ ∈ C ●✐↔ sû r➡♥❣ ❝➦♣ ❦❤✐➸♥ ữủ ợ (A, B) ợ Clìq (A, B) = (A, B) + D∆E ❧➔ ❦❤æ♥❣ ✤✐➲✉ ♥➔♦ ✤â✳ ❚ø ♥❤ú♥❣ ✤✐➲✉ tr➯♥ t❛ ❝â ∃λ0 ∈ C Gλ0 ,∆ = Wλ0 + D∆E : Cn+m → C ❦❤æ♥❣ ❧➔ s❛♦ ❝❤♦ t♦→♥ tû t✉②➳♥ t➼♥❤ t♦➔♥ →♥❤✳ ❱➻ ✈➟②✱ t❛ ❝â ∃y0∗ ∈ (Cn )∗ , y0∗ = s❛♦ ❝❤♦ (Gλ0 ,∆ )∗ (y0∗ ) = (Wλ0 + D∆E)∗ (y0∗ ) = Wλ∗0 (y0 )∗ + (E ∗ ∆∗ E ∗ )(y0∗ ) = 0✳ ❚ø Wλ∗−1 ❧➔ ✤ì♥ trà t❛ ❝â y0∗ = −(Wλ∗−1 E ∗ ∆∗ )(D∗ (y0∗ )) ✈➔ ❞♦ ✤â ✷✷ (D∗ (y0∗ )) = 0✳ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ ❑❤✐ ✤â D∗ (y0∗ ) = D∗ (−(Wλ∗−1 E ∗ ∆∗ )(D∗ (y0∗ )) = −(D∗ Wλ∗−1 E ∗ )(∆∗ (D∗ (y0∗ ))) 0 ❱➻ ✈➟②✱ tø ✭✶✳✸✮ t❛ ❝â E ∗ )(∆∗ Wλ∗−1 E ∗ )(∆∗ (D∗ (y0∗ ))) < D∗ (y0∗ ) = (D∗ Wλ∗−1 0 ≤ D∗ Wλ∗−1 E∗ ❚ø ∆∗ (D∗ (y0∗ )) ≤ D∗ Wλ∗−1 E∗ ⊂ dom E = Cn+m Im Wλ−1 D∗ (y0∗ ) ✭✷✳✻✮ ♥➯♥ t❤❡♦ ✭✶✳✾✮ t❛ ❝â (EWλ−1 )∗ = Wλ−1∗ E ∗ = Wλ∗−1 E ∗ 0 Im D ⊂ dom(EWλ−1 ) = Cn ∆∗ ❍ì♥ ♥ú❛✱ tø Wλ ❧➔ t♦➔♥ →♥❤✱ ✈➔ ✈➻ ✈➟② tø ✭✶✳✾✮ t❛ ❝â t❤➸ ✈✐➳t (EWλ−1 D)∗ = D∗ (EWλ−1 )∗ = D∗ Wλ∗−1 E ∗ 0 ❉♦ ✤â✱ tø ✭✶✳✼✮ t❛ ❝â ∗ = ∆ ≥ 1 = ≥ −1 −1 ∗ D∗ Wλ∗−1 E EW D sup EW D λ0 λ λ∈C ❚ø ✤à♥❤ ♥❣❤➽❛ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ t❛ ❝â rCD,E (A, B) ≥ sup EWλ−1 D λ∈C ✲ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ♥❣÷đ❝ ❧↕✐ rCD,E (A, B) ≤ sup EWλ−1 D λ∈C ✷✸ ✭✷✳✼✮ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❱ỵ✐ ♠å✐ ε>0 s❛♦ ❝❤♦ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ sup EWλ−1 D − 2ε > t❛ ❝â ∃λε ∈ C s❛♦ ❝❤♦ λ∈C D∗ Wλ∗−1 E ∗ = EWλ−1 D ≥ sup EWλ−1 D − ε ε ε λ∈C ❚ø D∗ Wλ∗−1 E∗ ε ✈➟② t❛ ❝â ✤ì♥ trà ♥➯♥ ❝❤✉➞♥ ❝õ❛ ♥â ❧➔ ❝❤✉➞♥ ❝õ❛ t♦→♥ tû ✈➔ ✈➻ E ∗) ∃vε∗ ∈ (Cq )∗ : vε∗ = 1, vε∗ ∈ dom(D∗ Wλ∗−1 ε s❛♦ ❝❤♦ < sup EWλ−1 D − 2ε ≤ EWλ−1 D − ε ≤ (D∗ Wλ−1 E ∗ )(vε∗ ) ε ε λ∈C ✣➦t u∗ε = −Wλ∗−1 (E ∗ (vε∗ )) = 0, ε Wλ∗ε (u∗ε ) = −E ∗ (vε∗ ) ✈➔ t❛ ❝â D∗ (u∗ε ) = −(D∗ Wλ∗−1 E ∗ )(vε∗ ) = ỵ t õ h ∈ Cl h = 1, (D∗ (u∗ε ))h = D∗ (u∗ε ) ❉♦ ✤â✱ t❛ ❝â t❤➸ ①→❝ ✤à♥❤ ∆ε ∈ Cl×q ∆ε = s❛♦ ❝❤♦ ✳ ❜➡♥❣ ❝→❝❤ ✤➦t D∗ (uε ) hvε∗ ❑❤✐ ✤â✱ ❤✐➸♥ ♥❤✐➯♥ t❛ ❝â ∆ε = 1 ≤ = −1 ∗ )(v ∗ ) D∗ (uε ) (D∗ Wλ∗−1 E sup EW D − 2ε ε λ ε λ∈C ❚ø ✭✶✳✶✵✮ t❛ ❝â ❉♦ ✤â ✭✈ỵ✐ (∆∗ε D)(u∗ε ) = ∆∗ε (D∗ (u∗ε )) = D∗ (u∗ε )∆ε = vε∗ ✳ (E ∗ ∆∗ε D∗ )(u∗ε ) = E ∗ (vε∗ ) ✈➔ ✈➻ ✈➟② Wλ∗ε (u∗ε ) + (E ∗ ∆∗ε D∗ )(u∗ε ) = u∗ε = 0✮ ❤❛② t÷ì♥❣ ✤÷ì♥❣ Wλ + D ❚ø ♥❤ú♥❣ ✤✐➲✉ tr➯♥ t❛ ❝â ❝➦♣ ❜à ♥❤✐➵✉ ε E ❦❤æ♥❣ ❧➔ t♦➔♥ →♥❤✳ (A, B) ✈ỵ✐ (A, B) + D∆ε E ❦❤ỉ♥❣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ❉♦ ✤â✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❛ ❝â ✷✹ ❧➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ rCD,E (A, B) ≤ ∆ε ≤ sup EWλ−1 D − 2ε λ∈C ❈❤♦ ε→0 t❛ ♥❤➟♥ ✤÷đ❝ ❜➜t ✤➥♥❣ t❤ù❝ ♥❣÷đ❝ ❧↕✐ rCD,E (A, B) ≤ ∆ε ≤ sup EWλ−1 D ✭✷✳✽✮ λ∈C ❚ø t õ ỵ ữủ ự ♠✐♥❤✳ ◆❤➟♥ ①➨t ✷✳✷✳✶✳ ❈ỉ♥❣ t❤ù❝ ✭✷✳✺✮ t÷ì♥❣ tü ♥❤÷ ❝æ♥❣ t❤ù❝ ✤➣ ❜✐➳t ✤➸ t➼♥❤ ❜→♥ ❦➼♥❤ ê♥ ✤à♥❤ ❝â ❝➜✉ tró❝ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✈➔ ❝❤à✉ ♥❤✐➵✉ ❝â ❝➜✉ tró❝ ❞↕♥❣ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❈❤♦ t♦→♥ tû t✉②➳♥ t➼♥❤ G ∈ Cm×p , A x = Ax, A + D∆E FG (z) = Gz ❣✐↔ ♥❣❤à❝❤ ✤↔♦ ▼♦♦r❡✲P❡♥r♦s❡ tê♥❣ q✉→t ❦❤✐ ♥â FG+ ❝õ❛ tr♦♥❣ ✤â FG ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ FG+ (y) =   z s❛♦ ❝❤♦ Gz = y, z = d(0, FG−1 (y))  φ ♥➳✉ y ∈ Im FG ♥➳✉ y∈ / Im FG ✭✷✳✾✮ ❍➺ q✉↔ ✷✳✷✳✶✳ ữủ ố ợ ❦❤ỉ♥❣ ❝â ❝➜✉ tró❝ (A, B) (A, B) ∈ Cn×(n+m) (A, B) + (∆1 , ∆2 ) ✤÷đ❝ ❝❤♦ ❜ð✐ rC (A, B) = sup Wλ+ ✭✷✳✶✵✮ λ∈C tr♦♥❣ ✤â ❝õ❛ Wλ+ = Wλ∗ (Wλ Wλ∗ )−1 ❦➼ ❤✐➺✉ ❧➔ ❣✐↔ ♥❣❤à❝❤ ✤↔♦ ▼♦♦r❡✲P❡♥r♦s❡ Wλ = (A − λI, B) ✷✺ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ ❈ỉ♥❣ t❤ù❝ ✭✷✳✶✵✮ ❝❤➼♥❤ ❧➔ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❊✐s✐♥❣✳ E ∗ E Ker Wλ ⊂ Ker Wλ ❍➺ q✉↔ ✷✳✷✳✷✳ ●✐↔ sû ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❝➦♣ λ ∈ C ❑❤✐ ✤â✱ (A, B) ✤è✐ ✈ỵ✐ ♥❤✐➵✉ ❝â ❝➜✉ tró❝ ❞↕♥❣ (A, B) + D∆E, ∆ ∈ Cl×q (A, B) ợ tr õ D Cnìl , E Cqì(n+m) ữủ ổ tự rCD,E (A, B) = sup EWλ+ D ✭✷✳✶✶✮ λ∈C tr♦♥❣ ✤â ❝õ❛ Wλ+ = Wλ+ (Wλ Wλ∗ )−1 ❦➼ ❤✐➺✉ ❧➔ ❣✐↔ ♥❣❤à❝❤ ✤↔♦ ▼♦♦r❡✲P❡♥r♦s❡ Wλ = (A − λI, B) ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ E ∗ Ev ∈ Ker Wλ λ∈C ✈➔ ❜➜t ❦➻ v ∈ Ker Wλ = Wλ−1 (0) t❛ ❝â ✈➔ ❞♦ ✤â (EWλ+ Du)∗ Ev = u∗ D∗ (Wλ Wλ∗ )−1∗ Wλ E ∗ Ev = ❤❛② (EWλ+ Du)∗ ∈ (E Ker Wλ )⊥ ▼➦t ❦❤→❝ tø ❉♦ ✤â Wλ Wλ+ Du, ∀u ∈ Cl , t❛ t❤➜② Wλ+ Du ∈ Wλ−1 (Du) Wλ−1 (Du) = Wλ+ Du + Wλ−1 (0) = Wλ+ Du + Ker Wλ ✈➔ ✈➻ ✈➟② (EWλ−1 D)(u) = (EWλ−1 )(Du) = EWλ+ Du + E Ker Wλ ❚ø ✤â s✉② r❛ d(0, (EWλ−1 D)(u)) = EWλ+ Du ✈ỵ✐ ♠å✐ λ C ứ ỵ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ◆❤➟♥ ①➨t ✷✳✷✳✷✳ ✐✳ ❑❤✐ D, E ❧➔ ❝→❝ ♠❛ tr➟♥ ✤ì♥ ✈à ✈➔ ✭✷✳✶✶✮ trð t❤➔♥❤ ❝æ♥❣ t❤ù❝ ✭✷✳✶✵✮✳ ✷✻ ∆ = (∆1 , ∆2 ) t❤➻ ❝æ♥❣ t❤ù❝ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ ✐✐✳ ❈ỉ♥❣ t❤ù❝ ✭✷✳✶✶✮ tê♥❣ q✉→t ❤ì♥ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ❑❛r♦✇ ✈➔ ❑r❡ss♥❡r tr♦♥❣ ❬✶✵❪✳ ✐✐✐✳ ❈ỉ♥❣ t❤ù❝ ✭✷✳✺✮ ❝á♥ ✤÷đ❝ ♠ð rë♥❣ ❝❤♦ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ ❝❤à✉ ♥❤✐➵✉ ✤❛ ❝➜✉ tró❝✳ ❈❤✐ t✐➳t ✤ë❝ ❣✐↔ q✉❛♥ t➙♠ ❝â t❤➸ ✤å❝ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶✶❪✳ ✷✳✸ ❱➼ ❞ö  (A,  B) 1  , B =  ✳ A= 0 ❳➨t ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤  tr♦♥❣ ✤â ❝❤♦ ❜ð✐ x (t) = Ax(t) + Bu(t),  ✲ ❘ã r➔♥❣✱ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❑❛❧♠❛♥✱ ❤➺ tr➯♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➻  rank(B, AB) = rank  0 rank(A|B) =  =2 ✲ ●✐↔ sû ♠❛ tr➟♥ ✤✐➲✉ ❦❤✐➸♥ ✳ (A, B) ❝❤à✉ ♥❤✐➵✉ ❞↕♥❣ B) + D∆E     (A, B)  (A,   δ + δ1 + δ2 1 0 1  ✳   ❤❛②  ✈ỵ✐ D =   , E =  + δ1 δ1 δ2 0 0 1 ❑❤✐ ✤â✱ ♠❛ tr➟♥ E ❦❤æ♥❣ ❝â ❤↕♥❣ ✤➛② ✤õ t❤❡♦ ❝ët ✈➔ ✈➻ ✈➟② ❝æ♥❣ t❤ù❝ ✭✸✳✶✵✮ ❝õ❛ ❑❛r♦✇ ✈➔ ❑r❡ss♥❡r tr♦♥❣ ❬✶✵❪ ❦❤æ♥❣ t❤➸ sû ❞ö♥❣ ✤➸ t➼♥❤ t♦→♥ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➳♥ ữủ ợ vC t õ v E(A − λI, B)−1 D(v) = E(A − λI, B)−1   v ✷✼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣➦t ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣     p   v   (A − λI, B)−1   = q    v r t❤➻       p   p     v  −λ 1     = [A − λI, B]  q  q  =   −λ   v r r ❤❛② ❉♦ ✤â     v −λp + q + r  = ✳ v p − λq + r q − λp + r = p − λq + r = v ✳ ❱➻ ✈➟②✱ ✈ỵ✐ ♠é✐ v∈C ❜➔✐ t♦→♥ t➼♥❤ d(0, E[A − λI]−1 D(v)) ✈➲ t➼♥❤ ❦❤♦↔♥❣ ❝→❝❤ tø ❣è❝ ✤➳♥ ✤÷í♥❣ tr♦♥❣ C2 ♠➔ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ x2 − (λ − 1)x1 = 2v   x1 = v + (λ + 1)q õ õ t ữủ t ữợ ữủ ữ ợ x2 = ( + 1)v + (λ2 − 1)q ❚❛ t❤➜② r➡♥❣ ♥➳✉ ●✐↔ sû λ = −1 λ = −1 ✈➔ ❝❤♦ C t❤➻ ✤÷í♥❣ ♥➔② ✤÷đ❝ ✤÷❛ ✈➲ ✤✐➸♠ ❝❤✉➞♥ ✈❡❝tì ·   v   ∞ t❤➻ t❛ s✉② r❛   x1 2|v| ≤ |x2 |+|λ−1||x1 | ≤ (1+|λ−1|) max{|x1 |, |x2 |} = (1+|λ−1|)   x2 ❚ø ✤â t❛ ❝â ✣➥♥❣ t❤ù❝ ♥➔② ①↔② r❛ ♥➳✉   x 2|v|  1 ≥ ✳ + |λ − 1| x2 2v iϕ x2 = ✈➔ x1 = e x2 + |λ − 1| ✷✽ ✈➔ ϕ ✤÷đ❝ ∞ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝❤å♥ s❛♦ ❝❤♦ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ −(λ − 1)eiϕ = |λ − 1|✳ ❱➻ ✈➟② E[A − λI, B]−1 D = sup d(0, E[A − λI, B]−1 D(v)) |v|=1     ♥➳✉ λ = −1 |λ − 1| + =   1 ♥➳✉ λ = −1 ỵ t ữủ rCD,E (A, B) = ữợ t tr ỵ tt ữủ õ ❜➲♥ ✈ú♥❣ ✭❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✮ ♥â✐ r✐➯♥❣ ✤➳♥ ♥❛② ✈➝♥ ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩✱ ❝â ♥❤✐➲✉ ❜➔✐ t♦→♥ ✤÷đ❝ ✤➦t r❛ ✈➝♥ ❝❤÷❛ ❝â ❧í✐ ❣✐↔✐ tè✐ ÷✉ ♥❤➜t ♥❤÷ ✐✳ ❚➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❝→❝ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ t✉②➳♥ t➼♥❤ ❝❤à✉ ♥❤✐➵✉ tr➯♥ ✤✐➲✉ ❦❤✐➸♥✳ ✐✐✳ ❚➼♥❤ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❝→❝ ❤➺ ❦❤æ♥❣ ❝❤➢❝ ❝❤➢♥ ✭✉♥❝❡rt❛✐♥ s②st❡♠✮✱ ❤➺ ❝❤✉②➸♥ tr↕♥❣ t❤→✐ ✭s✇✐t❝❤❡❞ s②st❡♠✮✳ ❈❤ó♥❣ tỉ✐ s➩ ❝è ❣➢♥❣ ✤å❝✱ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤✐➺♥ ❦❤â❛ ❧✉➟♥ t❤➔♥❤ ♠ët t➔✐ ❧✐➺✉ tèt ❝❤♦ s✐♥❤ ✈✐➯♥ q✉❛♥ t➙♠✳ ✷✾ ❑➳t ❧✉➟♥ ❚r➯♥ ✤➙② ❧➔ t♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ✏❇→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤✑ ✳ ❚r♦♥❣ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔② ❡♠ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❤✐➸✉ ❜✐➳t ❝õ❛ ♠➻♥❤ ♠ët ❝→❝❤ ❤➺ t❤è♥❣✱ rã r➔♥❣ ✈➲ ❜→♥ ❦➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤✱ ❝→❝ ✈➼ ❞ư ữợ t tr ✤➣ ✤➟t ✤÷đ❝ ♠ư❝ ✤➼❝❤ ✈➔ ♥❤✐➺♠ ✈ư ✤➲ r❛✳ ❚✉② ♥❤✐➯♥ ❞♦ ❦✐➳♥ t❤ù❝ ❜↔♥ t❤➙♥ ✈➔ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳✱ ♠ët ♣❤➛♥ ✈➻ ✤➙② ❧➔ ❧➛♥ ✤➛✉ t✐➯♥ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ ♥➯♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ t❤✐➳✉ sât✳ ❊♠ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ♥❤ú♥❣ ✤â♥❣ õ qỵ t ổ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ ✤➛② ✤õ t ỡ rữợ t tú õ ♥➔②✱ ♠ët ❧➛♥ ♥ú❛ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ ❝æ ❣✐→♦ ❚❤✳❙ ❚r➛♥ ❚❤à t t ữợ ú ù ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✸✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ✲ ❚➔✐ ❧✐➺✉ t ỡ s ỵ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ◆❳❇ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✳ ❬✷❪ ◆❣✉②➵♥ ❚❤➳ ❍♦➔♥✱ P❤↕♠ P❤✉ ✭✷✵✵✾✮✱ ❈ì sð ♣❤÷ì♥❣ tr ỵ tt ❞ö❝✳ ❬✸❪ ◆❣✉②➵♥ P❤ö ❍② ✭✷✵✵✻✮✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❤➔ ①✉➜t ❜↔♥ ❑❤♦❛ ❤å❝ ❑ÿ t❤✉➟t✳ ❬✹❪ ❱ô ◆❣å❝ P❤→t ổ ỵ tt t ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✺❪ ◆❣✉②➵♥ ❚❤à ◗✉ý♥❤ ✭✷✵✶✽✮✱ ▼ët sè t✐➯✉ ❝❤✉➞♥ ①➨t t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤✱ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✤↕✐ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✳ ❬✻❪ P❤❛♥ ỗ rữớ số t t ữ ♣❤↕♠ ❍➔ ◆ë✐ ✷✳ ✸✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à P❤÷ì♥❣ ❬❇❪ ✲ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ ❬✼❪ ❘✳ ❈r♦ss ✭✶✾✾✽✮✱ ▼✉❧t✐✈❛❧✉❡❞ ▲✐♥❡❛r ❖♣❡r❛t♦rs✱ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ◆❡✇ ❨♦r❦✳ ❬✽❪ ❘✳ ❊✐s✐♥❣ ✭✶✾✽✹✮✱ ❇❡t✇❡❡♥ ❝♦♥tr♦❧❧❛❜❧❡ ❛♥❞ ✉♥❝♦♥tr♦❧❧❛❜❧❡✱ ❙②s✲ t❡♠s ❈♦♥tr♦❧ ▲❡t t✳✺✱ ✷✻✸✲✷✻✹✳ ❬✾❪ ❉✳ ❍✐♥r✐❝❤s❡♥✱ ❆✳ ❏✳ Pr✐t❝❤❛r❞ ✭✶✾✽✻✮✱ ❙t❛❜✐❧✐t② r❛❞✐✉s ❢♦r str✉❝✲ t✉r❡❞ ♣❡rt✉r❜❛t✐♦♥s ❛♥❞ t❤❡ ❛❧❣❡❜r❛✐❝ ❘✐❝❛t✐ ❡q✉❛t✐♦♥✱ ❙②st❡♠s ❈♦♥tr♦❧ ▲❡tt✳✽✱ ✶✵✺✲✶✶✸✳ ❬✶✵❪ ▼✳ ❑❛r♦✇✱ ❉✳ ❑r❡ss♥❡r ✭✷✵✵✾✮✱ ❖♥ t❤❡ str✉❝t✉r❡❞ ❞✐st❛♥❝❡ t♦ ✉♥❝♦♥tr♦❧❧❛❜✐❧✐t②✱ ❙②st❡♠ ❈♦♥tr♦❧ ▲❡tt✳✺✽✱ ✶✷✽✲✶✸✷✳ ❬✶✶❪ ◆✳ ❑✳ ❙♦♥✱ ❉✳ ❉✳ ❚❤✉❛♥ ✭✷✵✶✵✮✱ ❚❤❡ st✉❝t✉r❡❞ ❞✐st❛♥❝❡ t♦ ✉♥❝♦♥✲ tr♦❧❧❛❜✐❧✐t② ✉♥❞❡r ♠✉❧t✐✲♣❡rt✉r❜❛t✐♦♥s✿ ❆♥ ❛♣♣r♦❛❝❤ ✉s✐♥❣ ♠✉❧t✐ ✲ ✈❛❧✉❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦rs✱ ❙②st❡♠ & ❈♦♥tr♦❧ ▲❡tt❡rs ✺✾✱ ✹✼✻✲✹✽✸✳ ❬✶✷❪ ❏❡r③② ❩❛❜❝③②❦ ✭✶✾✾✷✮✱ ▼❛t❤❡♠❛t✐❝❛❧ ❈♦♥tr♦❧ ❚❤❡♦r② tr t ăsr sts ...TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* NGUYỄN THỊ PHƯƠNG BÁN KÍNH ĐIỀU KHIỂN ĐƯỢC CỦA HỆ ĐIỀU KHIỂN TUYẾN TÍNH KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn