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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ◆●❯❨➍◆ ❱❿◆ ❇➐◆❍ ❇⑨■ ❚❖⑩◆ ✣■➋❯ ❑❍■➎◆ ✣×Đ❈ ❍➏ ✣❐◆● ▲Ü❈ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ◆●❯❨➍◆ ❱❿◆ ❇➐◆❍ ❇⑨■ ❚❖⑩◆ ✣■➋❯ ❑❍■➎◆ ✣×Đ❈ ❍➏ ✣❐◆● ▲Ü❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ữớ ữợ ❦❤♦❛ ❤å❝ ●❙✳❚❙❑❍ ❱Ô ◆●➴❈ P❍⑩❚ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ữủ ró ỗ ố t ♥➠♠ ✷✵✶✺ ◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❱➠♥ ❇➻♥❤ ✐✐ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët tổ ổ ữủ sỹ ữợ ✈➔ ❣✐ó♣ ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ ●❙✳ ❚❙❑❍ ❱ơ ◆❣å❝ P❤→t ✭❱✐➺♥ ❚♦→♥ ❤å❝ ❱✐➺t ◆❛♠✮✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ t❤➛② ✤➣ ❞➔♥❤ ❝❤♦ tæ✐✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❜❛♥ ❧➣♥❤ ✤↕♦ ♣❤á♥❣ s❛✉ ✣↕✐ ❤å❝✱ qỵ t ổ ợ ✷✵✶✺✮ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ t t tr t ỳ tự qỵ ❜→✉ ❝ơ♥❣ ♥❤÷ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ t tợ ỳ ữớ ổ ✤ë♥❣ ✈✐➯♥✱ ❤é trñ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❚❤→✐ ♥❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✺ ◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❱➠♥ ❇➻♥❤ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✐ ▲í✐ ❝↔♠ ì♥ ✐✐ ▼ö❝ ❧ö❝ ✐✐✐ ▼ð ✤➛✉ ✶ ▼ët sè ❦➼ ❤✐➺✉ ✈✐➳t t➢t ✸ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✹ ✶✳✶ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤✐➲✉ ❦❤✐➸♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❇➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❈→❝ ❜ê ✤➲ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷ ❈→❝ t✐➯✉ ❝❤✉➞♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❤➺ ✤ë♥❣ ❧ü❝ ✶✵ ✷✳✶ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➲✉ ❦❤✐➸♥ t✉②➳♥ t➼♥❤ rí✐ r↕❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➲✉ ❦❤✐➸♥ ❝â ❤↕♥ ❝❤➳ tr➯♥ ✤✐➲✉ ❦❤✐➸♥ ✳ ✳ ✷✺ ✐✈ ✷✳✸✳✶ ❍➺ ✤✐➲✉ ❦❤✐➸♥ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸✳✷ ❍➺ ✤✐➲✉ ❦❤✐➸♥ rí✐ r↕❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ❑➳t ❧✉➟♥ ❝❤✉♥❣ ✸✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✻ ✶ ▼ð ✤➛✉ ❇➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❤➺ ✤ë♥❣ ❧ü❝ ❧➔ ❜➔✐ t õ ự q trồ tr ỵ tt ❦❤✐➸♥ t♦→♥ ❤å❝ ✤÷đ❝ ♣❤→t tr✐➸♥ tø ♥❤✐➲✉ t❤➟♣ ❦✛ ❣➛♥ ✤➙②✳ ❘➜t ♥❤✐➲✉ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tr♦♥❣ ❦❤♦❛ ❤å❝✱ ❝ỉ♥❣ ♥❣❤➺✱ ❦➽ t❤✉➟t ✈➔ ❦✐♥❤ t➳ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤ë♥❣ ❧ü❝ ✈➔ ❝➛♥ ✤➳♥ ❝→❝ ❝ỉ♥❣ ❝ư t♦→♥ ❤å❝ ✤➸ t➻♠ ❧í✐ ❣✐↔✐✳ ❚➼♥❤ ữủ ự ợ ữủ s ữợ t õ ❤➺ t❤è♥❣ ✤÷đ❝ ✤✐➲✉ ❦❤✐➸♥ ✈➲ ❝→❝ ✈à tr➼ ♠♦♥❣ ♠✉è♥✳ ◆â✐ ♠ët ❝→❝❤ ❝ư t❤➸ ❤ì♥✿ ❈❤♦ ♠ët ❤➺ t❤è♥❣ ♠ỉ t↔ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➲✉ ❦❤✐➸♥ ✈➼ ❞ư ❞↕♥❣ ✭✵✱✶✮ ❝→❝ ✈à tr➼ ♠♦♥❣ ♠✉è♥ ❝➛♥ ✤✐➲✉ ❦❤✐➸♥ ❝õ❛ ❤➺ t❤è♥❣✱ ♥❤÷ tr↕♥❣ t❤→✐ x0 , x1 ✤÷đ❝ trữợ t ữủ u(t) s ữợ t ❤➺ t❤è♥❣ ✭✵✱✶✮ ✤÷đ❝ ✤✐➲✉ ❦❤✐➸♥ tø tr↕♥❣ t❤→✐ x0 s tr t x1 tr ởt tớ tũ ỵ ❝è ✤à♥❤✮ ♥➔♦ ✤â✱ tù❝ ❧➔✱ q✉ÿ ✤↕♦ ❝õ❛ ❤➺ t❤è♥❣ ✭✵✱✶✮ ①✉➜t ♣❤→t tø tr↕♥❣ t❤→✐ x0 t↕✐ t❤í✐ ✤✐➸♠ t0 s➩ ❝❤✉②➸♥ ✤➳♥ tr↕♥❣ t❤→✐ x1 t↕✐ t❤í✐ ✤✐➸♠ t1 ✳ ❉ü❛ ✈➔♦ ♠ö❝ ✤➼❝❤ ✤✐➲✉ ❦❤✐➸♥ ❝õ❛ ❤➺ t❤è♥❣✱ ♥❣÷í✐ t❛ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❦❤→✐ ♥✐➺♠ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ♥❤÷✿ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➲ ✵✱ ✤↕t ✤÷đ❝ tø ♠ët ✈à tr➼ ❝❤♦ trữợ ữủ t ữủ ỹ ữủ ọ ữợ ỵ tữ ❦➳t ✷ q✉↔ q✉❛♥ trå♥❣ ❝õ❛ ❘✳ ❑❛❧♠❛♥ tø ♥❤ú♥❣ ♥➠♠ ✻✵✱ tr♦♥❣ ✤â ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✤✐➲✉ ❦❤✐➸♥ ✤↕✐ sè ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ ✤ì♥ ❣✐↔♥✳ ❚ø ✤â ✤➳♥ ♥❛② ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❤➺ ✤ë♥❣ ❧ü❝ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t tr tr t ởt ữợ q trồ ỵ tt ữủ ỹ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜↔♥ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➲✉ ❦❤✐➸♥✱ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ ♠ët sè ❜ê ✤➲ ❜ê trđ✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ❝→❝ t✐➯✉ ❝❤✉➞♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❤➺ ✤ë♥❣ ❧ü❝✳ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ●❙✳❚❙❑❍ ❱ơ ◆❣å❝ P❤→t ♥❣÷í✐ t❤➛② ✤➣ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦ ❝❤♦ tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥ ✈➔ ❝→❝ t❤➛② ❝æ tr♦♥❣ tr÷í♥❣ ✣↕✐ ❍å❝ ❙÷ P❤↕♠✲ ✣❍❚◆ ❝ơ♥❣ ♥❤÷ ❝→❝ t❤➛② ❝ỉ ✤➣ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ❝❛♦ ❤å❝ ❦❤â❛ ✷✵✶✸✲✷✵✶✺✳ ▼➦❝ ❞ị ✤➣ ❝è ❣➢♥❣ r➜t ♥❤✐➲✉ ♥❤÷♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚æ✐ rt õ ữủ ỳ ỵ õ õ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥✳ ✸ ▼ët sè ❦➼ ❤✐➺✉ ✈✐➳t t➢t R ❚➟♣ ❝→❝ sè t❤ü❝✳ C ❚➟♣ ❝→❝ sè ♣❤ù❝✳ RN ❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n ❝❤✐➲✉✳ X, Y, U ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỉ ❤↕♥ ❝❤✐➲✉ ✈ỵ✐ ❝❤✉➞♥ ||x|| Xk = X × X × · · · × X✳ X∗ ❑❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ ❝õ❛ X ✳ < x∗ , x > ●✐→ trà ❝õ❛ ♣❤✐➳♠ ❤➔♠ x∗ ∈ X ∗ t↕✐ x✳ l2 ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② (x1 , x2 , · · · ) ✈ỵ✐ ❝❤✉➞♥ ∞ s=0 |xs | < +∞ Lp ([0, t]), Rm ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ x(.) : [0, T ] → R ✈ỵ✐ ❝❤✉➞♥ ||x|| = m t p x(s) ds A∗ ❚♦→♥ tû ❧✐➯♥ ❤ñ♣ A I t tỷ ỗ t detA t❤ù❝ ❝õ❛ ♠❛ tr➟♥ A✳ rank A ❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ A✳ A , AT ❈❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ A✳ L(X, Y ) ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝õ❛ t➜t ❝↔ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ →♥❤ ①↕ X ✈➔♦ Y ✳ M∗ ◆â♥ ❝ü❝ ❞÷ì♥❣ ❝õ❛ M t↕✐ 0✳ M0 ❚➟♣ ❝ü❝ ❝õ❛ M t↕✐ ✵✳ p ✹ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤✐➲✉ ❦❤✐➸♥ ♥❤➡♠ ♠ư❝ ✤➼❝❤ sû ❞ư♥❣ ❝❤♦ ❝❤÷ì♥❣ s❛✉✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ữỡ ỗ ữỡ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❤➺ ✤✐➲✉ ❦❤✐➸♥✱ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ ♠ët sè ❜ê ✤➲ ❜ê trđ✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ❧➜② ❝❤õ ②➳✉ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪ ✈➔ ❬✷❪✳ ✶✳✶ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤✐➲✉ ❦❤✐➸♥ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➲✉ ❦❤✐➸♥ ❝â ❞↕♥❣ x˙ = f (t, x(t), u(t)), t ≥ 0, ✭✶✳✶✮ tr♦♥❣ ✤â x(t) ∈ Rn ❧➔ ✈➨❝tì tr↕♥❣ t❤→✐✱ u(t) ∈ Rm ❧➔ ✈➨❝tì ✤✐➲✉ ❦❤✐➸♥ ✈➔ ❤➔♠ f t❤ä❛ ♠➣♥✿ f (t, x, u) : R+ × Rn × Rm −→ Rn ●✐↔ sû u(t) ❧➔ ❝→❝ ❤➔♠ ✤✐➲✉ ❦❤✐➸♥ ❧✐➯♥ tö❝ tr➯♥ [0, +∞) ✈➔ ❤➔♠ f (t, x, u) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❧✐➯♥ tö❝ t❤❡♦ t ✈➔ ▲✐♣s❝❤✐t③ t❤❡♦ x, u tù❝ ❧➔ ∃l1 , l2 > ✷✷ ❚❛ ❝â k0 −1 F (k0 , i + 1)B(i)B (i)F (k0 , i + 1)D−1 (k0 )x1 x(k0 ) = i=0 = D(k0 )D−1 (k0 )x1 = x1 ❚ø ✤â s✉② r❛ r➡♥❣ ✈ỵ✐ ✤✐➲✉ ❦❤✐➸♥ ①→❝ ✤à♥❤ tr➯♥ ❤➺ s➩ ✤÷đ❝ ❝❤✉②➸♥ tø tr↕♥❣ t❤→✐ s❛♥❣ tr↕♥❣ t❤→✐ ❜➜t ❦➻ x1 ∈ Rn ♥➔♦ ✤â✱ ♥â✐ ❝→❝❤ ❦❤→❝✱ ❤➺ ❧➔ ●❘✳ ◆❤➟♥ ①➨t✳ ữ ợ tử số tr♦♥❣ ♠❛ tr➟♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✭✷✳✶✶✮ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ sè ❝❤✐➲✉ ❦❤æ♥❣ ❣✐❛♥ n✳ ❚ø ❝→❝❤ ✤à♥❤ ♥❣❤➽❛ t➟♣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➲ ✵✱ t❛ ❞➵ t❤➜② r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❤↕♥❣ ✭✷✳✶✶✮ ❝ô♥❣ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ❤➺ ❧➔ ●◆❈✱ ①♦♥❣ ♥â ❦❤æ♥❣ ♣❤↔✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥✳ ❱➼ ❞ö s❛✉ s➩ ❝❤ù♥❣ tä ✤✐➲✉ ✤â✿ ❱➼ ❞ö ✷✳✷✳✸✳ ❳➨t ❤➺ x1 (k + 1) x2 (k + 1) x (k + 1) ❚❛ ❝â = x1 (k) − x2 (k) + 2x3 (k) + u1 (k) + 2u2 (k), = −x1 (k) + x2 (k) − 2x3 (k) − u1 (k) − 2u2 (k), = 2x1 (k) − 2x2 (k) + 4x3 (k) + 2u1 (k) + 4u2 (k), −1 1 −1 A= , B = −1 −2 −2 12 rank[B, AB, A2 B] = rank −1 −2 −6 −12 12 24 2 −2 , 36 72 −36 −72 = < 3, 72 144 ✷✸ ❑❤✐ ✤â ❤➺ ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❤↕♥❣ ✭✷✳✶✶✮ ①♦♥❣ ❞➵ ❦✐➸♠ tr❛ ✤÷đ❝ ❤➺ ❧➔ ●◆❈✳ ❚❤➟t ✈➟② ∀x0 = {x1 (0), x2 (0), x3 (0)} ❜➜t ❦➻✱ t❛ t➻♠ ✤÷đ❝ x1 (1) x2 (1) x (1) ♥➳✉ t❛ ❝❤å♥ = x1 (0) − x2 (0) + 2x3 (0) + u1 (0) + 2u2 (0), = −x1 (0) + x2 (0) − 2x3 (0) − u1 (0) − 2u2 (0), = 2x1 (0) − 2x2 (0) + 4x3 (0) + 2u1 (0) + 4u2 (0), u1 (0) = x2 (0) − x1 (0), u (0) = −x3 (0), t❤➻ t❛ s✉② r❛ ✤÷đ❝ x1 = {0, 0, 0} ❤❛② ❤➺ ❝❤✉②➸♥ ✤÷đ❝ tø tr↕♥❣ t❤→✐ ❜➜t ❦➻ ✈➲ s❛✉ t❤í✐ ❣✐❛♥ t õ ỵ rớ r ❧➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➲ ✵ ❤♦➔♥ t♦➔♥ ❦❤✐ ✈➔ tỗ t số k0 s ImF (k0 , 0) ⊆ ImC(k0 ) ✭✷✳✶✷✮ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❤➺ ❧➔ ●◆❈✱ ❦❤✐ ✤â C = Rn tr♦♥❣ ✤â Ck = [−F (k, 0)]−1 (Rk ) ❚❤❡♦ ❜ê ✤➲ ✶✳✸✳✷✳✭✤à♥❤ ❧➼ ❇❛✐r❡ ✈➲ ♣❤↕♠ trò✮ s➩ ❝â ♠ët sè k0 > s❛♦ ❝❤♦ Ck0 = Rn , ✤✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔ [−F (k0 , 0)]−1 Rk0 = Rn tø ✤â s✉② r❛ ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✷✮✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ ✭✷✳✶✷✮ t❤ä❛ ♠➣♥ t❤➻ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ✈➲ t➼♥❤ ●◆❈✱ ❤➺ s➩ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ s k0 ữợ ❱➼ ❞ư ✷✳✷✳✺✳ ❳➨t ❤➺ rí✐ r↕❝ x1 (k + 1) x2 (k + 1) x (k + 1) ❚❛ ❝â = x1 (k) + kx3 (k) + (k − 2)u(k), = 2x1 (k) + k x2 (k), = x3 (k) + (2 − k)u(k) 1 k k − 2 , B(k) = , A(k) = k 0 2−k 0 C(3) = [B(2), A(2)B(1), A(2)A(1)B(0)] = 0 −2 −14 ❉♦ ✤â rankC(3) = < 3✱ ✤✐➲✉ ❦✐➺♥ ❤↕♥❣ ✭✷✳✶✶✮ ❦❤æ♥❣ t❤ä❛ ♠➣♥✳ ❳♦♥❣ t❛ ❝â −1 rankF (2, 0) = rank[B(1), A(1)B(0)] = rank −4 = ❱➟② ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✷✮ t❤ä❛ ♠➣♥✳ ❚ø ✤â✱ ❤➺ ✤➣ ❝❤♦ ❧➔ ●◆❈✱ ♠➦❝ ❞ị ♥â ❦❤ỉ♥❣ ❧➔ ●❘✳ ✷✺ ✷✳✸ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➲✉ ❦❤✐➸♥ ❝â ❤↕♥ ❝❤➳ tr➯♥ ✤✐➲✉ ❦❤✐➸♥ ✷✳✸✳✶ ❍➺ ✤✐➲✉ ❦❤✐➸♥ ❧✐➯♥ tö❝ ❳➨t ❤➺ ✤✐➲✉ ❦❤✐➸♥ ❞ø♥❣ ❝â ❤↕♥ ❝❤➳ x(t) ˙ u(t) = Ax(t) + Bu(t), t ≥ 0, m ✭✷✳✶✸✮ n ∈ Ω ⊆ R , x(t) ∈ R , tr õ t ỗ Ω ①→❝ ✤à♥❤ ❝→❝ t➟♣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ữ s ã t ữủ R = t>0 Rt ✱ tr♦♥❣ ✤â t Rt = eA(t−s) Bu(s)ds, u(.) ∈ UΩ x= • ❚➟♣ ✤✐➲✉ ❦❤✐➸♥ ✈➲ ✵✿ C = ∪t>0 Ct tr♦♥❣ ✤â t Ct = x=− e−As Bu(s)ds, u(.) ∈ UΩ ✣à♥❤ ❧➼ s❛✉ ✤➙② ❝❤♦ t❛ t✐➯✉ ❝❤✉➞♥ ✈➲ t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ữủ ợ tt t ỗ t ỵ sỷ t ỗ t ✭✷✳✶✸✮ ❧➔ ✤↕t ✤÷đ❝ ✤à❛ ♣❤÷ì♥❣ i) rank[B, AB, , An1 B] = n ii) ổ tỗ t tỡ r ♥➔♦ ❝õ❛ ♠❛ tr➟♥ A ✱ ù♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ t❤ü❝✱ ♥➡♠ tr♦♥❣ ♠➦t ♥â♥ ✤è✐ ♥❣➝✉ ❞÷ì♥❣ (BΩ)+ ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ●✐↔ sû ❤➺ ❧➔ ▲❘✳ ❑❤✐ ✤â ❤➺ ✭✷✳✶✸✮ ✈ỵ✐ Ω = Rn ❧➔ ●❘✳ ❑❤✐ ✤â ❤➺ ❦❤æ♥❣ ❝â ❤↕♥ ❝❤➳ ✤✐➲✉ ❦❤✐➸♥ ❧➔ s tữỡ ữỡ ợ i)✳ ✣➸ ❝❤➾ r❛ ✤✐➲✉ ❦✐➺♥ ii)✱ t❛ ❣✐↔ sû ự r tỗ t tỡ x0 (B)+ s❛♦ ❝❤♦ x0 = ✈➔ A x0 = λx0 , R s tỗ t↕✐ ❧➙♥ ❝➟♥ V (0) ⊆ R tr♦♥❣ ✤â R ❧➔ t➟♣ ✤↕t ✤÷đ❝ ❝õ❛ ❤➺ ✭✷✳✶✸✮ tø ✤â s✉② r r ợ ộ x Rn s tỗ t ♠ët > s❛♦ ❝❤♦ x ∈ R ✈➔ ❞♦ ✤â s➩ t➻♠ ✤÷đ❝ ♠ët t❤í✐ ❣✐❛♥ T > 0✱ ♠ët ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ u(.) ∈ UΩ s❛♦ ❝❤♦ T x= eA(T −s) Bu(s)ds ❱➻ x0 ∈ (BΩ)+ ✱ t❛ ❝â < x0 , x >= < x, x > = T < x0 , eA(T −s) Bu(s) > ds = T eλ(T −s) < x0 , Bu(s) > ds ≤ 0 ❉♦ ✤â t❛ ♥❤➟♥ ✤÷đ❝ < x0 , x >≤ ✈ỵ✐ x ∈ Rn ✳ ✣✐➲✉ ♥➔② ❝❤➾ ①↔② r❛ ❦❤✐ x0 = 0✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ x0 = ✣✐➲✉ ❦✐➺♥ ✤õ✿ ●✐↔ sû ❝â ❝→❝ ✤✐➲✉ ❦✐➺♥ i), ii) s ự rữợ t✐➯♥ t❛ ❝❤➾ r❛ r➡♥❣ intR = ∅✳ ❚❤➟t ✈➟②✱ tø ✤✐➲✉ ❦✐➺♥ ❤↕♥❣ ❑❛❧♠❛♥ ✐✮ s✉② r❛ ❤➺ t✉②➳♥ t➼♥❤ ❞ø♥❣ ❦❤æ♥❣ ❝â ❤↕♥ ❝❤➳ tr➯♥ ✤✐➲✉ ❦❤✐➸♥✿ x(t) ˙ = Ax + Bu, t ≥ 0, u(t) ∈ Rk = spΩ, x(t) ∈ Rn , ✭✷✳✶✹✮ k ≥ m ≥ n, ❧➔ ●❘✳ ❉♦ ✤â tø ❜ê ✤➲ ✶✳✸✳✶ ✭✤à♥❤ ❧➼ ♣❤↕♠ trò ❇❛✐r❡✮ t❤➻ ❤➺ s➩ ❧➔ ●❘ s❛✉ ♠ët t❤í✐ ❣✐❛♥ T > ♥➔♦ ✤â✳ ❳➨t →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ lT ①→❝ ✤à♥❤ ❜ð✐ T lT u = eA(T −s) Bu(s)ds, u ∈ UΩ T ✷✼ ❱➻ t➟♣ ✤↕t ✤÷đ❝ ❝õ❛ ❤➺ ✭✷✳✶✹✮✱ R2.14 ❧➔ trị♥❣ ✈ỵ✐ Rn ♥➯♥ t❛ ❝â ImlT = Rn , s✉② r❛ →♥❤ ①↕ lT (.) ❧➔ →♥❤ ①↕ tr➔♥✱ t❤❡♦ ✤à♥❤ ❧➼ →♥❤ ①↕ ♠ð t❤➻ ♥â ❝ô♥❣ ❧➔ →♥❤ ①↕ ♠ð✳ ❱➻ t➟♣ Ω ①➨t tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Rk = sp t ỗ õ tr rộ ❦➨♦ t❤❡♦ t➟♣ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ UΩT ❝ơ♥❣ s➩ ❝â ♣❤➛♥ tr♦♥❣ ❦❤→❝ ré♥❣✳ ❚ø ✤â t❛ s✉② r❛ intlT (UΩ T ) = ∅ ❱➻ lT (UΩT ) = R ✲ t➟♣ ✤↕t ✤÷đ❝ ❝õ❛ ❤➺ ✭✷✳✶✸✮ ✈ỵ✐ ❤↕♥ ❝❤➳ u(t)✱ ♥➯♥ t❛ ❝â intR = ∅✱ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣➸ ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ t❛ s➩ ❧➔♠ ♥❤÷ s❛✉✿ ❦➼ ❤✐➺✉ K = conR = {λx : x ∈ R, λ > 0}, ❧➔ ♥â♥ s✐♥❤ ❜ð✐ t➟♣ ✤↕t ✤÷đ❝ R✳ ❱➻ R✱ t ỗ õ K õ ỗ intK = ∅✳ ◆❣♦➔✐ r❛ ❞➵ ❦✐➸♠ tr❛ ✤÷đ❝ r➡♥❣ eAt K ⊆ K, ∀t ∈ [0, T ] ●✐↔ sû r➡♥❣ ∈ / intR t❤❡♦ ✤à♥❤ ❧➼ t→❝❤ t ỗ t s r ữủ K = Rn ✳ ⑩♣ ❞ö♥❣ ❜ê ✤➲ ✶✳✸✳✸✭✤à♥❤ ❧➼ ❑r❡✐♥✲❘✉t♠❛♥✮ ❝❤♦ ❤➺ ❝→❝ ♠❛ tr➟♥ tü ❣✐❛♦ ❤♦→♥ {eAt }, t ∈ [0, T ] s➩ t➻♠ ✤÷đ❝ ♠ët ✈➨❝tì ❝❤✉♥❣ ❝õ❛ {eAt } s❛♦ ❝❤♦ x0 ∈ K + , eA t x0 = λ(t)x0 , λ(t) ∈ R, ∀t ∈ [0, T ] ▲➜② ✤↕♦ ❤➔♠ ❤❛✐ ✈➳ ✈➔ ❝❤♦ t = t❛ ❝â x0 ∈ K + , A x0 = λ (0)x0 , λ (0) ∈ R ▼➦t ❦❤→❝✱ ✈➻ x0 ∈ K + , ∀x ∈ R t❛ ❝â T < x0 , eA(T −s) Bu(s)ds >≤ 0, ∀u(.) ∈ UΩ ✭✷✳✶✺✮ ✷✽ ❚ø ✤➙② t❛ s✉② r❛ < x0 , eA(T −s) Bu >≤ 0, ∀u ∈ Ω, ∀s ∈ [0, T ] ❇➙② ❣✐í ❝❤♦ s = T t❛ ❝â < x0 , Bu >≤ 0, ∀u ∈ Ω, tù❝ ❧➔ x0 (B)+ t ủ ợ t ữủ ✤✐➲✉ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✐✐✮✳ ❱➟② ✤✐➲✉ ❣✐↔ sû ∈ / intR ❧➔ s❛✐✱ tù❝ ❧➔✱ t❛ s➩ ❝â ∈ intR✱ ❤➺ ❧➔ ●❘✳ ◆❤➟♥ ①➨t✳ Rn õ ỗ t t r t t ữủ R ụ õ ỗ ỡ ỳ intR tữỡ ữỡ ợ R = Rn ✱ ❤➺ ❧➔ GR✳ ❱➟② ❝→❝ ✤✐➲✉ ❦✐➺♥ i)✱ii) tr♦♥❣ ✤à♥❤ ❧➼ ✷✳✸✳✶✳ s➩ ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ❤➺ ❧➔ ●❘ ❝❤ù ❦❤æ♥❣ ♣❤↔✐ ❧➔ ▲❘ ♥ú❛✳ ✷✳ ✣è✐ ✈ỵ✐ ✈✐➺❝ ①➨t t➼♥❤ ●❘ t❛ ❝â t❤➸ ①➨t t➼♥❤ ●◆❈ ❜➡♥❣ ✈✐➺❝ t❤❛② ✤ê✐ t♦→♥ tû Lt (.) ❜➡♥❣ t Lt (u) = − e−As Bu(s)ds ❑❤✐ ✤â ❝→❝ ❦ÿ t❤✉➟t ❝❤ù♥❣ ♠✐♥❤ ỵ ú t ữủ t õ ỵ s ỵ ợ tt ỵ ❝➛♥ ✈➔ ✤õ ✤➸ ❤➺ ✭✷✳✶✸✮ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➲ ❦❤ỉ♥❣ ✤à❛ ♣❤÷ì♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤✐➲✉ ❦✐➺♥ i), ii) tọ õ ỗ t ❤➺ s➩ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➲ ❦❤ỉ♥❣ ❤♦➔♥ t♦➔♥✳ ✷✾ ❱➼ ❞ư ✷✳✸✳✸✳ ❳➨t t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ x˙1 = x1 + 3x2 + u, x˙ = −2x + u, u ≥ 2 ❚❛ ❝â ,B = , A= −2 (BΩ) = {(x1 , x2 ) ∈ R2 : x1 = x2 } ❚❛ t❤➜② rank[A/B] = ✈➔ λ(A ) = {1, −2}✳ ❉♦ ✤â (BΩ)+ = {(x1 , x2 ) ∈ R2 : x2 ≥ −x1 , x2 ∈ R} ❈→❝ ✈➨❝tì r✐➯♥❣ ❝õ❛ A ù♥❣ ✈ỵ✐ λ = ❧➔ {(x1 , x2 ) ∈ R2 : x2 = 0, x1 ∈ R}, ✈➔ ù♥❣ ✈ỵ✐ λ = −2 ❧➔ {(x1 , x2 ) ∈ R2 : x1 = 0, x2 ∈ R} ◆❤÷ ✈➟② ❝→❝ ✈➨❝tì r✐➯♥❣ ❝õ❛ A ù♥❣ ợ tr r tữỡ ự tr õ (BΩ)+ ✳ ❍➺ ✤➣ ❝❤♦ ❦❤ỉ♥❣ ❧➔ ▲❘✳ ❱➼ ❞ư ✷✳✸✳✹✳ ❳➨t t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ x˙1 = x1 + u2 , x˙ = x − x − u , 2 ✈ỵ✐ Ω = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ∈ R} ❚❛ ❝â 0 ,B = , A= −1 −1 ✸✵ (BΩ) = {(x1 , x2 ) ∈ R2 : x1 ∈ R, x2 ≤ 0} ❚❛ t❤➜② rank[A/B] = ✈➔ λ(A ) = {−1, 1}✳ ❉♦ ✤â (BΩ)+ = {(x1 , x2 ) ∈ R2 : x2 ≤ 0, x1 = 0} ❈→❝ ✈➨❝tì r✐➯♥❣ ❝õ❛ A ù♥❣ ✈ỵ✐ λ = ❧➔ {(x1 , x2 ) ∈ R2 : x2 = 0, x1 ∈ R}, ✈➔ ù♥❣ ✈ỵ✐ λ = −1 ❧➔ {(x1 , x2 ) ∈ R2 : x1 = −1/2x2 } ◆❤÷ ✈➟② ❦❤ỉ♥❣ ❝â ✈➨❝tì r✐➯♥❣ ♥➔♦ ❝õ❛ A ♥➡♠ tr♦♥❣ (BΩ)+ ✳ ❍➺ ❧➔ ▲❘✳ ❱➻ Ω ❧➔ ♥â♥ ♥➯♥ ❤➺ ❝ơ♥❣ ❧➔ ●❘✳ ✷✳✸✳✷ ❍➺ ✤✐➲✉ ❦❤✐➸♥ rí✐ r↕❝ ❳➨t ❤➺ rí✐ r↕❝ ❝â ❤↕♥ ❝❤➳ ✤✐➲✉ ❦❤✐➸♥ ♥❤÷ s❛✉ x(k + 1) = A(k)x(k) + B(k)u(k), k ∈ Z+ , x(k) ∈ Rn , u(k) ∈ Ω ⊆ Rm , ✭✷✳✶✻✮ tr♦♥❣ ✤â Ω t trữợ tr Rn , ú ỵ r trữớ ủ ợ tr➯♥ t➟♣ ✤✐➲✉ ❦❤✐➸♥ u(k) ∈ Ω t❤➻ t➟♣ ✤↕t ✤÷đ❝ ✈➔ t➟♣ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➲ ✵ s➩ ✤÷đ❝ ①→❝ ✤à♥❤ t÷ì♥❣ ù♥❣ ❜ð✐ k−1 Rk = F (k, i + 1)B(i)u(i), u(i) ∈ Ω , x= i=0 Ck = {x : −F (k, 0)x ∈ Rk )} = {x : −F (k, 0)x ∈ C(k)Ωk } ❚❛ s➩ r ữợ ởt số t t ỡ ❜↔♥ ❝→❝ t➟♣ ✤↕t ✤÷đ❝ ✈➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➲ ✵ ❝õ❛ ❤➺ rí✐ r↕❝✳ ✸✶ ✣à♥❤ ❧➼ s❛✉ ✤➙② ❝❤➾ r❛ t✐➯✉ ❝❤✉➞♥ ✤➸ ❤➺ ✭✷✳✶✻✮ ✈ỵ✐ ❝→❝ ♠❛ tr số t ữủ t ỵ ✷✳✸✳✺✳ ❳➨t ❤➺ tr♦♥❣ ✤â ❣✐↔ sû A(.), B(.) ❧➔ tr số t ỗ ❍➺ ❧➔ ✤↕t ✤÷đ❝ ✤à❛ ♣❤÷ì♥❣ ✭✈➔ ❤♦➔♥ t♦➔♥ ♥➳✉ õ ỗ i) ∃k0 > : rank[B, AB, , Ak0 −1 B] = n ii) ổ tỗ t tỡ r A ù♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ ❦❤ỉ♥❣ ➙♠✱ ♥➡♠ tr♦♥❣ (BΩ)+ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❤➺ ❧➔ ▲❘✱ ❦❤✐ ✤â ❤➺ ❦❤ỉ♥❣ ❝â ❤↕♥ ❝❤➳ tr➯♥ ✤✐➲✉ ❦❤✐➸♥✱ tù❝ ❧➔ ✈ỵ✐ u(k) ∈ Rk = spΩ s➩ ❧➔ ●❘✳ ✣➦t W = spΩ✳ ❚ø ✤â s✉② r❛ ❤➺ ✭✷✳✶✻✮ ✈ỵ✐ u(k) W s s k ữợ ỳ k0 > ♥➔♦ ✤â✳ ❑❤✐ ✤â t❛ ❝â [B, AB, , Ak0 −1 B](W k ) = Rn , ❤❛② ❧➔✱ ✤✐➲✉ ❦✐➺♥ ✐✮ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ✐✐✮✱ t❛ ❣✐↔ sû ♣❤↔♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ∃x0 = 0, A x0 = λx0 , λ ≤ 0, x0 ∈ (BΩ)+ ❱➻ ❤➺ ❧➔ ▲❘✱ ợ tũ ỵ x1 Rn s õ ♠ët > s❛♦ ❝❤♦ x1 ❧➔ ✤↕t ✤÷đ❝ tø ✵ ❜ð✐ ♠ët ✤✐➲✉ ❦❤✐➸♥ ♥➔♦ ✤â✱ tù❝ ❧➔✱ s➩ ❝â ♠ët ❞➣② ✤✐➲✉ ❦❤✐➸♥ (u(0), u(1), , u(k0 − 1)) s❛♦ ❝❤♦ k0 −1 x1 = F (k0 , i + 1)B(i)u(i) i=0 ổ ữợ t❤ù❝ tr➯♥ ✈ỵ✐ x0 ✱ ♥❤➟♥ ①➨t r➡♥❣ ✈➻ x0 ∈ (BΩ)+ , t❛ ❝â k0 −1 < x0 , x1 > = < x0 , F (k0 , i + 1)Bu(i) > i=0 k0 −1 = < F (k0 , i + 1)x0 , Bu(i) > i=0 k0 −1 λk0 −i−1 < x0 , Bu(i) >≥ = i=0 ❚ø ✤â s✉② r❛ x0 = ✈➻ x1 ❧➔ tũ ỵ tr Rn õ t ợ ❣✐↔ t❤✐➳t ❦❤→❝ ❦❤ỉ♥❣ ❝õ❛ x0 ✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ ✐✮✱ ✐✐✮ t❤ä❛ ♠➣♥✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❤➺ ❧➔ ▲❘✳ ❚÷ì♥❣ tü tø ✤✐➲✉ ❦✐➺♥ ✐✐✮ s✉② r❛ intR = ∅✳ ✣➦t K = conR✱ ❦❤✐ ✤â intK = ∅ ✈➔ ❞➵ ❦✐➸♠ tr❛ ✤÷đ❝ r➡♥❣ AK ⊆ K ❱➟②✱ ♥➳✉ ❣✐↔ sû ♣❤↔♥ ❝❤ù♥❣ ❧➔ ❤➺ ❦❤æ♥❣ ♣❤↔✐ ▲❘✱ tù❝ ❧➔✱ ∈ / intR✱ ❦❤✐ ✤â K = Rn ✈➔ ❦❤✐ ✤â ❝â t❤➸ →♣ ❞ö♥❣ ❜ê ✤➲ ✶✳✸✳✸✳✭✤à♥❤ ❧➼ ❑r❡✐♥ ✲ ❘✉t♠❛♥✮ ❝❤♦ ♠❛ tr➟♥ ❤➡♥❣ sè A ✈➔ ♥â♥ K t❛ ❝â ∃x0 ∈ K + : A x0 = λx0 , λ ≥ ✣✐➲✉ ✈ø❛ ♥❤➟♥ ✤÷đ❝ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✐✐✮ ✈➻ BΩ ⊆ K ⇒ K + ⊆ (BΩ)+ ❚÷ì♥❣ tü t❛ ❝â t✐➯✉ ❝❤✉➞♥ t ữủ ỵ ❳➨t ❤➺ tr♦♥❣ ✤â ❣✐↔ sû A(.), B(.) ❧➔ ❝→❝ tr số t ỗ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✤à❛ ♣❤÷ì♥❣ ✈➲ ✵ ✭ ✈➔ t õ ỗ ❦❤✐ ✭✷✳✶✻✮ ✸✸ i) ImF (k0 , 0) ⊆ ImC(k0 ), k0 > ii) ổ tỗ t tỡ r A ự ợ tr r ữỡ tr (BΩ)+ ❚ø ✤à♥❤ ❧➼ tr➯♥ t❛ ♥❤➟♥ t❤➜② r➡♥❣✱ ố ợ rớ r ữủ ●❘ ✈➔ ●◆❈ ❦❤→❝ ♥❤❛✉ ð ✤✐➲✉ ❦✐➺♥ ✐✐✮ ♠➔ ð ✤â ❣✐→ trà r✐➯♥❣ ❧➔ ❦❤ỉ♥❣ ➙♠ ❤♦➦❝ ❞÷ì♥❣✳ ❱➼ ❞ö s❛✉ ❝❤➾ r❛ ✤✐➲✉ ♥➔②✳ ❱➼ ❞ö ✷✳✸✳✼✳ ❳➨t t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✭✷✳✶✻✮ tr♦♥❣ ✤â 0 ,B = , A= Ω = {(u1 , u2 ) ∈ R2 : u2 ≥ 0, u1 ∈ R} ❚❛ ❝â (BΩ)+ = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 = 0}, λ(A) = {4, 2} ❱➨❝tì r✐➯♥❣ ❝õ❛ A ù♥❣ ✈ỵ✐ λ = ❧➔ {(x1 , x2 ) ∈ R2 : x2 = 2x1 } ❱➨❝tì r✐➯♥❣ ❝õ❛ A ù♥❣ ✈ỵ✐ λ = ❧➔ {(x1 , x2 ) ∈ R2 : x2 = 0, x1 ∈ R} ◆❤÷ ✈➟② t❛ t❤➜② t➟♣ ❝→❝ ✈➨❝tì r✐➯♥❣ ❝õ❛ ❝õ❛ A ù♥❣ ✈ỵ✐ λ = ♥➡♠ tr♦♥❣ (BΩ)+ ✳ ❍➺ ✤➣ ❝❤♦ ❦❤ỉ♥❣ ❧➔ ●❘✳ ❱➼ ❞ư ✷✳✸✳✽✳ ❳➨t t➼♥❤ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ rí✐ r↕❝ x1 (k + 1) = x2 (k + 1) = u (k) ≥ i −2x1 (k) + u2 (k) 2x1 (k) + 3x2 (k) − u1 (k) ✸✹ ❚❛ ❝â −2 0 ,B = , λ(A) = {3, −2} A= −1 ❚❛ ❝â (BΩ)+ = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≤ 0} ❱➨❝tì r✐➯♥❣ ù♥❣ ợ tr ữỡ = {(x1 , x2 ) ∈ R2 : 2x2 = 5x1 } ◆❤÷ ✈➟② ❦❤ỉ♥❣ ❝â ✈➨❝tì r✐➯♥❣ ♥➔♦ ❝õ❛ ❣✐→ trà r✐➯♥❣ ❞÷ì♥❣ ♥➡♠ tr♦♥❣ t➟♣ tr➯♥ ♥➡♠ tr♦♥❣ (BΩ)+ ✳ ❱➟② ❤➺ ❧➔ ●❘✱ ✈➔ ✈➻ λ = > ♥➯♥ ❤➺ ❧➔ ●◆❈✳ ✸✺ ❑➳t ❧✉➟♥ ❝❤✉♥❣ ◆❤ú♥❣ ✈➜♥ ✤➲ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ • ◆❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤✐➲✉ ❦❤✐➸♥✱ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ✈➔ ♠ët sè ❜ê ✤➲ ❝ì ❜↔♥✳ • P❤➛♥ trå♥❣ t➙♠ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❧↕✐ ❦✐➳♥ t❤ù❝ ✈➲ ❝→❝ t✐➯✉ ❝❤✉➞♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝ ❝õ❛ ❤➺ ✤ë♥❣ ❧ü❝ ỗ t t ữủ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝✱ rí✐ r↕❝ ✈➔ ❝â ❤↕♥ ❝❤➳✳ ❉♦ ✈➜♥ ✤➲ ✤÷đ❝ ✤➲ ❝➟♣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ t÷ì♥❣ ✤è✐ ♣❤ù❝ t↕♣✱ ❤ì♥ ♥ú❛ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ ❦❤↔ ♥➠♥❣ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ♠➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ♥❤ú♥❣ þ ❦✐➳♥ ✤â♥❣ ❣â♣ q✉þ ❜→✉ ❝õ❛ t❤➛② ❝æ ❣✐→♦ ✈➔ ♥❤ú♥❣ ♥❣÷í✐ q✉❛♥ t➙♠ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✸✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❚❤➳ ❍♦➔♥ ✱ P❤↕♠ P❤✉ ✭✷✵✵✸✮✱ ✈➔ ỵ tt ỡ s ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ ❬✷❪ ❱ơ ◆❣å❝ P❤→t✱ ✭✷✵✵✶✮✱ ◆❤➟♣ ♠ỉ♥ ỵ tt t ố ●✐❛✳ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❑❛❧♠❛♥ ❘✳❊✳ ❈♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ t❤❡♦r② ♦❢ ♦♣t✐♦♥❛❧ ❝♦♥tr♦❧ ❇♦❧ ❙♦❝✳ ▼❛t❤✳ ▼❡①✐❝❛♥❛✱ ✺✭✶✾✻✵✮✱✶✵✷✲✶✶✾✳ ❬✹❪ ❩❛❜❝③②❦ ❏✳ ▼❛t❤❡♠❛t✐❝❛❧ ❈♦♥tr♦❧ ❚❤❡♦r② ❇✐r❦❤❛✉s❡r✱ ✶✾✾✷✳
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