Bài toán đuổi bắt trong trò chơi tuyến tính với hạn chế tích phân trên thang thời gian

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - LÊ VĂN QUÝ BÀI TOÁN ĐUỔI BẮT TRONG TRỊ CHƠI TUYẾN TÍNH VỚI HẠN CHẾ TÍCH PHÂN TRÊN THANG THỜI GIAN LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2017 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - LÊ VĂN Q BÀI TỐN ĐUỔI BẮT TRONG TRỊ CHƠI TUYẾN TÍNH VỚI HẠN CHẾ TÍCH PHÂN TRÊN THANG THỜI GIAN LUẬN VĂN THẠC SĨ TOÁN HỌC Chuyên ngành: Toán ứng dụng Mã số : 60 46 01 12 NGƯỜI HƯỚNG DẪN KHOA HỌC: PGS.TS Tạ Duy Phượng THÁI NGUYÊN - 2017 ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ❑❤→✐ ♥✐➺♠ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✶ ✹ ✶✳✶ ❚❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❚ỉ ♣ỉ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✹ P❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺ ✹ ✶✳✹✳✶ ✣↕♦ ❤➔♠ ❍✐❧❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✷ ❚➼♥❤ ❝❤➜t ❝õ❛ ✤↕♦ ❤➔♠ ❍✐❧❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✺✳✶ ❍➔♠ t✐➲♥ ❦❤↔ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✺✳✷ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ q tr t tớ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✼ ❍➔♠ ♠ô tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷ ❚rá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✷✻ ✷✳✶ ❍➺ ✤ë♥❣ ❧ü❝ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✳✶ P❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤ë♥❣ ❧ü❝ t✉②➳♥ t➼♥❤ ❜➟❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✳✷ ❈ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤ë♥❣ ❧ü❝ t✉②➳♥ t➼♥❤ ❜➟❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✳✸ ❍➺ ✤ë♥❣ ❧ü❝ t✉②➳♥ t➼♥❤ ❝â ❤❛✐ t❤❛♠ sè ✤✐➲✉ ❦✐➸♥ ✳ ✳ ✳ ✸✶ ✐ ✐✐ ✷✳✷ ❚rá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸ ❚rá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈ỵ✐ t❤ỉ♥❣ t✐♥ ❝❤➟♠ ✈➔ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ tr➼❝❤ ❞➝♥ ✹✹ ✹✺ ▼ð ✤➛✉ ◆❤➡♠ t❤è♥❣ ♥❤➜t ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❤➺ ✤ë♥❣ ❧ü❝ ❧✐➯♥ tư❝ ✭❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✮ ✈➔ ❤➺ ✤ë♥❣ ❧ü❝ rí✐ r↕❝ ✭❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✮✱ ❙t❡❢❛♥ ❍✐❧❣❡r ♥➠♠ ✶✾✽✽✱ tr♦♥❣ ❧✉➟♥ →♥ ❚✐➳♥ s➽ ❝õ❛ ♠➻♥❤✱ ✤➣ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✭t✐♠❡ s❝❛❧❡✮✳ ❚ø ✤â ✤➳♥ ♥❛② ✤➣ ❝â ♠ët sè q✉②➸♥ s→❝❤✱ ❤➔♥❣ ❝❤ö❝ ❧✉➟♥ →♥ t✐➳♥ s➽ ✈➔ ❤➔♥❣ ♥❣➔♥ ❜➔✐ ❜→♦ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❣✐↔✐ t➼❝❤ ✭♣❤➨♣ t♦→♥ ✈✐ ♣❤➙♥✱ t➼❝❤ ♣❤➙♥✮ ✈➔ ❤➺ ✤ë♥❣ ❧ü❝ tr t tớ tớ õ ỵ tr✐➳t ❤å❝ s➙✉ s➢❝✿ ❚❤❛♥❣ t❤í✐ ❣✐❛♥ ❝❤♦ ♣❤➨♣ ♥❣❤✐➯♥ ❝ù✉ ❤❛✐ ♠➦t ❜↔♥ ❝❤➜t ❝õ❛ t❤ü❝ t➳✱ ✤â ❧➔ t➼♥❤ ❧✐➯♥ tư❝ ✈➔ t➼♥❤ rí✐ r↕❝✳ ❚r♦♥❣ t♦→♥ ❤å❝✱ t❤❛♥❣ t❤í✐ ❣✐❛♥ ❝❤♦ ♣❤➨♣ ♥❣❤✐➯♥ ❝ù✉ t❤è♥❣ ♥❤➜t ♥❤✐➲✉ ổ ữợ ũ ởt ❝ỉ♥❣ ❝ư✳ ●✐↔✐ t➼❝❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❤➺ ✤ë♥❣ ❧ü❝ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ ♥❤â♠ t tr ữợ q t ✣➣ ❝â ♠ët sè ❜➔✐ ✈✐➳t ✈➲ ù♥❣ ❞ö♥❣ ❝õ❛ t❤❛♥❣ t❤í✐ ❣✐❛♥ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦✐♥❤ t➳ ✈➽ ♠ỉ✱ ❤➺ s✐♥❤ t❤→✐✱ ❜➔✐ t♦→♥ tè✐ ÷✉✳ ❇➔✐ t♦→♥ ✤✉ê✐ ❜➢t ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝→❝ ❜➔✐ t♦→♥ ❝ì ❜↔♥ ỵ tt trỏ ỡ r t t t ữớ ợ ❧✉ỉ♥ ❝è ❣➢♥❣ ❝❤↕② ❝➔♥❣ ♥❤❛♥❤✱ ❝➔♥❣ ①❛ ♥❣÷í✐ ✤✉ê✐ ❝➔♥❣ tèt✳ ❈á♥ ♥❣÷í✐ ✤✉ê✐ t❤➻ ❝è ❣➢♥❣ ✧♣❤→t r❛ ✧ ♥❤ú♥❣ ✤✐➲✉ ❦✐➸♥ ✤➸ t✐➳♥ ✤➳♥ ♥❣÷í✐ ❝❤↕② ❝➔♥❣ ❣➛♥ ❝➔♥❣ tèt✳ ◆❤÷♥❣ ✤➸ trá ❝❤ì✐ ❦➳t t❤ó❝ t❤➻ t❛ ♣❤↔✐ ✤➦t ❣✐↔ t❤✐➳t ❧➔ ♥❣÷í✐ ✤✉ê✐ ♣❤↔✐ ❝â ❧đ✐ t❤➳ ❤ì♥ ♥❣÷í✐ ❝❤↕② ♥❤÷ ❧➔ ❤↕♥ ❝❤➳ ✈➲ ♥➠♥❣ ❧÷đ♥❣✱ ♥❣÷í✐ ✤✉ê✐ ❧✉ỉ♥ ❜✐➳t ✤÷đ❝ t❤ỉ♥❣ t✐♥ ✈➲ ❜✐➳♥ ✤✐➲✉ ❦✐➸♥ ❝õ❛ ♥❣÷í✐ ❝❤↕②✳✳✳❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ❦➳t t❤ó❝ trá ❝❤ì✐ ♥❤÷ ✈➟②✳ ✶ ✷ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ✤✉ê✐ ❜➢t tr♦♥❣ trá ❝❤ì✐ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ✣÷❛ r❛ ✤✐➲✉ ❦✐➺♥ ✤➸ ❜➔✐ t♦→♥ ❦➳t t❤ó❝ ✈ỵ✐ ❝→❝ ❜✐➳♥ ✤✐➲✉ ❦❤✐➸♥ t❤ä❛ ♠➣♥ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ ✭❤↕♥ ❝❤➳ ♥➠♥❣ ❧÷đ♥❣✮✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ỗ ữỡ ữỡ tr ♥✐➺♠ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ❉ü❛ t❤❡♦ ❬✺❪✱ ❬✻❪✱ ❬✽❪ ✈➔ ♠ët sè t➔✐ ❧✐➺✉ ❦❤→❝✱ ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❝→❝ ✈➜♥ ✤➲ ✈➲ ❣✐↔✐ t➼❝❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✤÷đ❝ tr➻♥❤ ❜➔② ♥❣➢♥ ❣å♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ trá ❝❤ì✐ ✤ê✐ ❜➢t t✉②➳♥ t➼♥❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✤ë♥❣ ❧ü❝ ✈➔ trá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥✱ ❜➔✐ t♦→♥ trá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ ✈➔ tổ t tr t tớ ỵ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❧➔ ❝→❝ ❦➳t q✉↔ ❝❤✉♥❣ ❝õ❛ ❜❛ t→❝ ❣✐↔ ❱✐ ❉✐➺✉ ▼✐♥❤✱ ▲➯ ❚❤à ❚❤ó② ◆❣➔ ✈➔ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❬✸❪✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ tỵ✐ P●❙✳ ❚❙✳ ❚↕ ❉✉② Pữủ ữớ t tớ ữợ t t➻♥❤ ❝❤➾ ❜↔♦✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ✈➔ ❣✐ó♣ ✤ï tr♦♥❣ tr❛♥❣ ❜à ❦✐➳♥ t❤ù❝✱ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ tê♥❣ ❤ñ♣ t➔✐ ❧✐➺✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ P❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ P❤á♥❣ ✣➔♦ t↕♦✱ ❑❤♦❛ ❚♦→♥✲❚✐♥ ✈➔ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❳✐♥ ✤÷đ❝ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❇❛♥ ❝❤✉②➯♥ ♠ỉ♥ ❝ị♥❣ ỗ tr rữớ tr tổ ữ ❨➯♥✱ t➾♥❤ ❍÷♥❣ ❨➯♥✱ ♥ì✐ tỉ✐ ❝ỉ♥❣ t→❝✱ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ♥❤✐➺♠ ✈ư ❤å❝ t➟♣✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❚❤↕❝ s➽ ❱✐ ❉✐➺✉ ▼✐♥❤✱ ❣✐↔♥❣ ✈✐➯♥ ♠ỉ♥ ❚♦→♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ◆ỉ♥❣ ▲➙♠✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❝ị♥❣ ❝ë♥❣ t→❝ ✈➔ ❣✐ó♣ ✤ï tæ✐ ✈➲ ❝❤✉②➯♥ ♠æ♥ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ị♥❣ t→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➦❝ ❜✐➺t ✤➳♥ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥✱ ✸ ❣✐❛ ✤➻♥❤✱ ỗ ỳ ữớ t ❦✐➺♥ t❤✉➟♥ ❧đ✐✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✵ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✶✼ ❍å❝ ✈✐➯♥ ▲➯ ỵ ữỡ t tớ ữỡ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ❉ü❛ t❤❡♦ ❬✺❪✱ ❬✻❪✱ ❬✽❪ ✈➔ ♠ët sè t➔✐ ❧✐➺✉ ❦❤→❝✱ ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❝→❝ ✈➜♥ ✤➲ ✈➲ ❣✐↔✐ t➼❝❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✤÷đ❝ tr➻♥❤ ❜➔②✳ ✶✳✶ ❚❤❛♥❣ t❤í✐ ❣✐❛♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶ ❚❤❛♥❣ t❤í✐ ❣✐❛♥ ✭t✐♠❡ s❝❛❧❡✮ ❧➔ t➟♣ ❝♦♥ ✤â♥❣ tũ ỵ rộ tr t số tỹ R tớ tữớ ữủ ỵ T ✶✳✶ ✶✮ ❈→❝ t➟♣ R, Z, N, [0; 1] ∪ [2; 3] ❧➔ ❝→❝ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➻ ❝❤ó♥❣ ❧➔ ♥❤ú♥❣ t➟♣ ✤â♥❣ tr♦♥❣ R✳ ✷✮ ❈→❝ t➟♣ Q, R\Q; [0, 1) ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➻ ❝❤ó♥❣ ❦❤æ♥❣ ♣❤↔✐ ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ R ❚➟♣ ❝→❝ sè ❤ú✉ t➾ Q✱ t➟♣ ❝→❝ sè ✈æ t➾ R\Q ❦❤æ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➻ ❝❤ó♥❣ t✉② ♥➡♠ tr♦♥❣ R ♥❤÷♥❣ ❦❤ỉ♥❣ ✤â♥❣ tr♦♥❣ R✳ ❚❤➟t ✈➟②✱ tr➯♥ Q ①➨t ❞➣② sè {xn }✿ ✶❀ ✶✱✹❀ ✶✱✹✶❀ ✶✱✹✶✹❀ ✳ ✳ ✳ ❚❛ t❤➜② xn ∈Q✱ √ ♥❤÷♥❣ lim xn = ∈ / Q ♥➯♥ Q ❦❤æ♥❣ ♣❤↔✐ ❧➔ t➟♣ ❝♦♥ ✤â♥❣ tr➯♥ R ❱➻ n→∞ ✈➟② Q ❦❤æ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ❚r➯♥ R\Q ①➨t ❞➣② sè {xn } : √ √ √ √ 3 3; ; ; ; ; n ✹ ✺ ❚❛ t❤➜② xn ∈ R\Q ♥❤÷♥❣ lim xn = ∈ / R\Q ♥➯♥ R\Q ❦❤æ♥❣ ♣❤↔✐ ❧➔ t➟♣ x→∞ ❝♦♥ ✤â♥❣ tr♦♥❣ R ❙✉② r❛ R\Q ❦❤æ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ❚➟♣ ❬✵❀✶✮ ❧➔ ❦❤♦↔♥❣ ♠ð tr♦♥❣ R ♥➯♥ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ✸✮ ▼➦t ♣❤➥♥❣ ♣❤ù❝ C ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➻ C ❦❤ỉ♥❣ ♥➡♠ tr♦♥❣ R, ♠➦❝ ❞ị ♥â ❧➔ t➟♣ õ ổ ổ tr t tớ rữợ t t❛ ♥❤➢❝ ❧↕✐ ♠ët ✈➔✐ ❦✐➳♥ t❤ù❝ ❝õ❛ tæ♣æ✳ ●✐↔ sû (X, τ ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ M ⊂ X ❧➔ ♠ët t➟♣ ❝♦♥ ♥➔♦ ✤â✳ ❚æ♣æ ❝↔♠ s✐♥❤ τM tr➯♥ M tø τ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✳ ❚➟♣ ♠ð tr♦♥❣ τM ❧➔ t➜t ❝↔ ❝→❝ t➟♣ ❝â ❞↕♥❣ σM = M ∩ U tr♦♥❣ ✤â σ ∈ τ ✳ ❑❤✐ ➜② τM = {UM : UM = M ∩ U, U ∈ τ } ❧➔ ♠ët tæ♣æ tr➯♥ M ✳ ❚❤➟t ✈➟② t❛ ❝â ✶✮ ❱➻ ∅ ✈➔ X ✤➲✉ t❤✉ë❝ τ ♥➯♥ ❞➵ t❤➜② ∅ = ∅ ∩ M, M = M ∩ M s✉② r❛ ∅ ✈➔ M ✤➲✉ t❤✉ë❝ τM ✷✮ ●✐↔ sû V1 , V2 ∈ τM ❧➔ ❤❛✐ t➟♣ ❤ñ♣ t tự tỗ t U1 , U2 τ s❛♦ ❝❤♦ V1 = M ∩ U1 ✈➔ V2 = M ∩ U2 ✳ ❚❛ ❝â V1 ∩ V2 = (M ∩ U1 ) ∩ (M ∩ U2 ) = M ∩ (U1 ∩ U2 )✳ ❱➻ U1 ∩ U2 ∈ τ ♥➯♥ s✉② r❛ V1 ∩ V2 ∈ τM ✭t❤❡♦ ✤à♥❤ ♥❣❤➽❛ t➟♣ τM ✮✳ ✸✮ ●✐↔ sû {Vα }α∈I ❧➔ ♠ët ❤å ❜➜t ❦➻ ❝→❝ t➟♣ t❤✉ë❝ τM ✳ ❑❤✐ ✤â t❛ ❝â Uα ✈ỵ✐ Uα ∈ τ ∀α ∈ I ✳ ❱➻ (M ∩ Uα ) = M ∩ α∈I α∈I V α ∈ τM ✳ α∈I Vα = α∈I Uα ∈ τ ♥➯♥ s✉② r❛ α∈I ❚ø ✶✮✱ ✷✮✱ ✸✮ s✉② r❛ τM ❧➔ ♠ët tæ♣æ ✈➔ ❣å✐ ❧➔ tæ♣æ ❝↔♠ s✐♥❤ tø τ tr➯♥ M ✳ ❈➦♣ (M, τM ) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ tæ♣æ ❝↔♠ s✐♥❤ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ (X, τ ) ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t❛ ❧✉æ♥ ❣✐↔ t❤✉②➳t r➡♥❣ t❤❛♥❣ t❤í✐ ❣✐❛♥ T ✤÷đ❝ tr❛♥❣ ❜à ♠ët tỉ♣ỉ ❝↔♠ s✐♥❤ tø tỉ♣ỉ t❤ỉ♥❣ t❤÷í♥❣ ❝õ❛ t➟♣ sè t❤ü❝ ✭tỉ♣ỉ t❤ỉ♥❣ t❤÷í♥❣ tr➯♥ t➟♣ sè t❤ü❝ R ❧➔ tỉ♣ỉ t↕♦ ❜ð✐ ❝→❝ ❦❤♦↔♥❣ ♠ð ❝ị♥❣ ✈ỵ✐ ✻ ❣✐❛♦ ❤ú✉ ❤↕♥ ✈➔ ❤đ♣ ❜➜t ❦➻ ❝õ❛ ❝❤ó♥❣✮✱ ♥❣❤➽❛ ❧➔ ❝→❝ t➟♣ ♠ð ❝õ❛ T ❧➔ ❣✐❛♦ ❝õ❛ ❝→❝ t➟♣ ♠ð tr♦♥❣ R ✈ỵ✐ T ❈→❝ ❦❤→✐ ♥✐➺♠ ❧➙♥ ❝➟♥✱ ❣✐ỵ✐ tửữủ ợ tư❝✳✳✳ tr♦♥❣ tỉ♣ỉ ❝↔♠ s✐♥❤✳ ✶✳✸ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ❝ì ❜↔♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✷ ❈❤♦ T ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ❚♦→♥ tû ♥❤↔② t✐➳♥ ✭❢♦r✇❛r❞ ❥✉♠♣✮ ❧➔ t♦→♥ tû σ:T→T ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ σ(t) := inf{s ∈ T : s > t} ❚♦→♥ tû ♥❤↔② ❧ò✐ ✭❜❛❝❦✇❛r❞ ❥✉♠♣✮ ❧➔ t♦→♥ tû ρ:T→T ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ρ(t) := sup{s ∈ T : s < t} ữợ inf = sup T, sup = inf T ❙✉② r❛ σ(M ) = M ♥➳✉ M ❧➔ ♣❤➛♥ tû ❧ỵ♥ ♥❤➜t ✭♥➳✉ ❝â✮ ❝õ❛ T; ρ(m) = m ♥➳✉ m ❧➔ ♣❤➛♥ tû ♥❤ä ♥❤➜t ✭♥➳✉ ❝â✮ ❝õ❛ T ❱➼ ❞ư ✶✳✷ ✶✮ ❱ỵ✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = Z ✭t❤❛♥❣ t❤í✐ ❣✐❛♥ rí✐ r↕❝✮ t❤➻ σ(t) = t + ✈➔ ρ(t) = t − ✈ỵ✐ ♠å✐ t ∈ T ❳❡♠ ❍➻♥❤ ✶✳✶✭❜✮✳ ✷✮ ❱ỵ✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = R ✭t❤❛♥❣ t❤í✐ ❣✐❛♥ ❧✐➯♥ tư❝✮ t❤➻ σ(t) = ρ(t) = t ✈ỵ✐ ♠å✐ t ∈ T ❳❡♠ ❍➻♥❤ ✶✳✶✭❛✮✳ ❍➻♥❤ ✶✳✶ ✸✷ ❚❛ ❝â z = z σ − µ(t)z ∆ ❚❤❛② ✈➔♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥✱ t❛ ❝â z ∆ (t) = A[z σ (t) − µ(t)z ∆ ] − Bu(t) + Cv(t) ⇔ (I + Aµ(t))z ∆ (t) = Az σ (t) − Bu(t) + Cv(t) A AB AC ⇔ z ∆ (t) = z σ (t) − u(t) + u(t) I + Aµ(t) I + Aµ(t) I + Aµ(t) = −( A)z σ (t) − AB I+Aµ(t) u(t) + AC I+Aà(t) u(t) ỵ ( ( A)) = A ✱ t❛ ❝â ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✷✳✶✳✶✸✮ ❧➔ t z(t) = eA (t, t0 )z0 − eA (t, τ ) t0 t eA (t, τ ) + t0 Bu(τ ) ∆τ I + Aµ(τ ) Cv(τ ) ∆τ I + Aµ(τ ) (∗∗) ❍ì♥ ♥ú❛✱ t❤❡♦ ✣à♥❤ ỵ t õ eA (t, τ ) eA (t, τ ) eA (t, τ ) = = = eA (t, σ(τ )) I + Aµ(τ ) eA (σ(τ ), t0 ).eA (t0 , τ ) eA (σ(τ ), τ ) ❚❤❛② ✈➔♦ ✭✯✯✮✱ t❛ ❝â ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✳✶✸✮ ❝â ❞↕♥❣ t z(t) = eA (t, t0 )z0 − t eA (t, σ(τ ))Bu(τ )∆τ + t0 eA (t, ( ))Cv( ) t0 ỵ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✷ ❚rá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ●✐↔ sû ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❤❛✐ ✤è✐ t÷đ♥❣ x(t) ∈ Rn ✭✤÷đ❝ ❣å✐ ❧➔ ♥❣÷í✐ ✤✉ê✐✮ ✈➔ y(t) ∈ Rn ✭✤÷đ❝ ❣å✐ ữớ ợ t tr t tớ T ✤÷đ❝ ♠ỉ t↔ t÷ì♥❣ ù♥❣ ❜ð✐ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ x∆ (t) = f (x(t), u(t), t) (2.2.1) y ∆ (t) = f (y(t), v(t), t) (2.2.2) ✈➔ ✸✸ tr♦♥❣ ✤â u(t) ∈ Rp ✱ v(t) ∈ Rq ❧➔ ❝→❝ ❤➔♠ ✤♦ ✤÷đ❝ ✈➔ r❞✲❧✐➯♥ tư❝ tr➯♥ T✳ ✣➦t T = s✉♣{t, t ∈ T} ❈→❝ ✤✐➲✉ ❦❤✐➸♥ u(t), v(t) t❤ä❛ ♠➣♥ ❝→❝ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ ✭❤↕♥ ❝❤➳ ♥➠♥❣ ❧÷đ♥❣✮ ❞↕♥❣ T T 2 v(s) ∆s ≤ σ u(s) ∆s ≤ ρ , t0 (2.2.3) t0 tr õ , số trữợ ❈→❝ ❤➔♠ ✤♦ ✤÷đ❝ u(t) ✈➔ v(t) t❤ä❛ ♠➣♥ ✭✷✳✷✳✸✮ ✤÷đ❝ ❣å✐ t÷ì♥❣ ù♥❣ ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ữớ ữớ ợ x(0) = x0 trữợ u(t) ữủ t ✤ë♥❣ ❧ü❝ ✭✷✳✷✳✶✮ s➩ ①→❝ ✤à♥❤ ♠ët q✉ÿ ✤↕♦ x(t) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✷✳✶✮✱ ①✉➜t ♣❤→t tø ✤✐➸♠ ❜❛♥ ✤➛✉ x0 tữỡ ự ợ u(t), t T ữỡ tỹ ợ y(0) = y0 trữợ v(t) ✤÷đ❝ ❝❤å♥✱ t❤❛② ✈➔♦ ✭✷✳✳✷✳✷✮✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✳✷✮ s➩ ①→❝ ✤à♥❤ ♠ët q✉ÿ ✤↕♦ y(t), t ∈ T ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✷✳✷✮✱ ①✉➜t ♣❤→t tø ✤✐➸♠ ❜❛♥ ✤➛✉ y0 t÷ì♥❣ ù♥❣ ✈ỵ✐ ✤✐➲✉ ❦❤✐➸♥ v(t) ❚❛ ♥â✐ trá ❝❤ì✐ ❦➳t t❤ó❝ s❛✉ t❤í✐ ❣✐❛♥ K ♥➳✉ ✈ỵ✐ ♠é✐ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ v(t) ❝õ❛ ♥❣÷í✐ ❝❤↕②✱ ♥❣÷í✐ ✤✉ê✐ ❝â t❤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ u(t) ❝õ❛ ♠➻♥❤ s❛♦ ❝❤♦ ♥❣❤✐➺♠ t÷ì♥❣ ù♥❣ ❝õ❛ ✭✷✳✷✳✶✮ ✈➔ ✭✷✳✷✳✷✮ t❤ä❛ ♠➣♥ x(K) = y(K) (2.2.4) ✣÷❛ ✈➔♦ ❜✐➳♥ ♠ỵ✐ z(t) = F (z(t), u(t), v(t)) = x(t) y(t) f (x(t), u(t), t) g(y(t), v(t), t) ❑❤✐ ữỡ tr õ t t ữợ z ∆ (t) = F (z(t), u(t), v(t), t) (2.2.5) ✸✹ ❍➺ t❤ù❝ ✭✷✳✷✳✹✮ s➩ ①→❝ ✤à♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ R2n ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥✱ tù❝ ❧➔ t➟♣ ❤đ♣ ♥❤ú♥❣ ✤✐➸♠ z(K) t❤ä❛ ♠➣♥ ✭✷✳✷✳✹✮ ❝â ❞↕♥❣ M := {z = x y , x ∈ Rn , y ∈ Rn , x = y} ◆❤÷ ✈➟②✱ ❝â t❤➸ ♣❤→t ❜✐➸✉ ❧↕✐ ❜➔✐ t♦→♥ trá ❝❤ì✐ ♠ët ❝→❝❤ tê♥❣ q✉→t ữ s trữợ ổ M RN ✈ỵ✐ ❞✐♠ M < N − ❚❛ ♥â✐ r➡♥❣ q✉→ tr➻♥❤ ✤✉ê✐ ❜➢t ♠ỉ t↔ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✳✺✮ ✭✈ỵ✐ z ∈ RN ✱ ①✉➜t ♣❤→t tø ✤✐➸♠ z0 ∈ / M s➩ ❦➳t t❤ó❝ s❛✉ t❤í✐ ❣✐❛♥ K ợ ộ ữủ v(t) ❝õ❛ ♥❣÷í✐ ❝❤↕②✱ ♥❣÷í✐ ✤✉ê✐ ❝â t❤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ u(t) s❛♦ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✷✳✷✳✺✮ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ z(K) ∈ M ❚❛ õ q ữợ tổ t ữ s ộ t❤í✐ ✤✐➸♠ t, ✤➸ ①➙② ❞ü♥❣ ❞➣② ✤✐➲✉ ❦❤✐➸♥ u(t) ♥❣÷í✐ ✤✉ê✐ ✤÷đ❝ ❜✐➳t tr↕♥❣ t❤→✐ ❝õ❛ ❤➺ ✭♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✳✺✮✮✱ ❝→❝ ❤↕♥ ❝❤➳ ✭✷✳✷✳✸✮ ✈➔ ✤➦❝ ❜✐➺t✱ ✤✐➲✉ ❦❤✐➸♥ v(t) ❝õ❛ ♥❣÷í✐ ❝❤↕②✳ ❳➨t ❜➔✐ t♦→♥ trá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ❞↕♥❣ z ∆ (t) = Az(t) − Bu(t) + Cv(t), t ≥ t0 (2.2.6) ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✳✻✮ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ❝ỉ♥❣ t❤ù❝ s❛✉ t t z(t) = eA (t, t0 )z0 − eA (t, σ(τ ))Cv(τ )∆τ eA (t, σ(τ ))Bu(τ )∆τ + t0 t0 ❑➼ ❤✐➺✉ L ❧➔ ♣❤➛♥ ❜ò trü❝ ❣✐❛♦ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ M tr♦♥❣ Rn , tù❝ ❧➔ Rn = M ⊕ L✱ ❤❛② M + L = Rn ✈➔ M ∩ L = {0}✳ ❑❤✐ ➜② ✈ỵ✐ ộ z Rn tỗ t t z1 M ✈➔ z2 ∈ L s❛♦ ❝❤♦ z = z1 + z2 ❑➼ ❤✐➺✉ π ❧➔ ♣❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ tø Rn ①✉è♥❣ L✳ ❑❤✐ ➜② ✤✐➲✉ ❦✐➺♥ ❦➳t tú z(K) M tữỡ ữỡ ợ z(K) = ỵ sỷ tt s ữủ tọ tt ỗ t t tỷ t t➼♥❤ ❧✐➯♥ tö❝ F (τ ) : Rq → Rp t❤ä❛ ♠➣♥ πeA (t, τ )BF (τ ) = πeA (t, τ )C ✈ỵ✐ ♠å✐ τ ≥ t0 , τ ∈ T, ✸✺ tr♦♥❣ ✤â π ❧➔ ♣❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ tø Rn ❧➯♥ L ●✐↔ t❤✐➳t ✷ ●✐↔ sû K ❧➔ sè ♥❤ä ♥❤➜t tr♦♥❣ sè ❝→❝ sè t❤ü❝ t ≥ t0 s❛♦ ❝❤♦ χ(K) ≤ ρ✱ tr♦♥❣ ✤â t χ (t) = F (τ )v(τ ) ∆τ sup t t0 v(τ ) ∆τ ≤σ t0 ●✐↔ t❤✐➳t ✸ πeA (K, t0 )z0 ∈ G(K)✱ tr♦♥❣ ✤â K G(K) := K w(τ ) ∆τ ≤ (ρ − χ(K))2 πeA (K, σ(τ ))Bw(τ )∆τ : t0 t0 ❑❤✐ ➜② trá ❝❤ì✐ ❦➳t t❤ó❝ s❛✉ t❤í✐ ❣✐❛♥ K ✳ ❈❤ù♥❣ ♠✐♥❤ ❚ø ●✐↔ t❤✐➳t ✸ s r tỗ t ởt số w(s) s K πeA (K, σ(τ ))Bw(τ )∆τ, πeA (K, t0 )z0 = (2.2.7) t0 tr♦♥❣ ✤â K t0 w(τ ) ∆τ ≤ (ρ − χ(K))2 ●✐↔ sû v(τ ) ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❜➜t ❦➻ ❝õ❛ ♥❣÷í✐ ❝❤↕②✱ tù❝ ❧➔ K v(τ ) ∆τ ≤ σ t0 ❳➙② ❞ü♥❣ ✤✐➲✉ ❦❤✐➸♥ ❝õ❛ ♥❣÷í✐ ✤✉ê✐ ♥❤÷ s❛✉ u(τ ) = F (τ )v(τ ) + w(τ ), ≤ τ ≤ K ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❇✉♥❤✐❛❝♦♣①❦✐ t❛ ❝â K K u(τ ) ∆τ F (τ )v(τ ) + w(τ ) ∆τ = t0 t0 K ≤ K F (τ )v(τ ) t0 ∆τ w(τ ) ∆τ ≤ χ(K) + (ρ − χ(K)) = ρ + t0 ✸✻ ❈❤ù♥❣ tä u(τ ) ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❚❤❡♦ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤ë♥❣ ❧ü❝ t❛ ❝â K πz(K) = π eA (K, t0 )z0 − eA (K, σ(τ ))Bu(τ )∆τ t0 K eA (K, σ(τ ))Cv(τ )∆τ + t0 K = πeA (K, t0 )z0 − πeA (K, σ(τ ))Bu(τ )∆τ t0 K + πeA (K, σ(τ ))Cv(τ )∆τ t0 K = πeA (K, t0 )z0 − πeA (K, σ(τ ))B(F (τ )v(τ ) + w(τ ))∆τ t0 K πeA (K, τ )Cv(τ )∆τ + t0 K = πeA (K, t0 )z0 − πeA (K, σ(τ ))BF (τ )v(τ )∆τ t0 K − K πeA (K, σ(τ ))Bw(τ )∆τ + t0 πeA (K, σ(τ ))Cv(τ )∆τ t0 ❚ø ●✐↔ t❤✐➳t ✶ ✈➔ ✭✷✳✷✳✼✮ t❛ s✉② r❛ K πz(K) = πeA (K, t0 )z0 − πeA (K, σ(τ ))Bw(τ )∆τ = t0 ◆❤÷ ✈➟②✱ t❛ ✤➣ ①➙② ❞ü♥❣ ✤÷đ❝ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ u(t) t❤❡♦ t❤ỉ♥❣ t✐♥ v(t) s❛♦ ❝❤♦ trá ❝❤ì✐ ❦➳t t❤ó❝ s❛✉ t❤í✐ ❣✐❛♥ K ỵ ữủ ự ①➨t ✷✳✷ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ T = R ❝→❝ ❤➺ ✤ë♥❣ ❧ü❝ ✭✷✳✷✳✶✮✱ ✭✷✳✷✳✷✮ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ ✭✷✳✷✳✸✮ trð t❤➔♥❤ x(t) ˙ = A(t)x − B(t)u + C(t)v, t≥0 ✈ỵ✐ ❝→❝ ❤↕♥ ❝❤➳ +∞ +∞ 2 v(s) ds ≤ σ u(s) ds ≤ ρ , 0 õ ỵ s trữớ ủ t ỵ ❦❤✐ T = R ❍➺ q✉↔ ✶ ✭◆✐❦♦❧s❦✐✐✱ ①❡♠ ❬✶✶❪✮ ●✐↔ sû ❝→❝ ❣✐↔ t❤✐➳t s❛✉ ✤➙② ✤÷đ❝ t❤ä❛ ♠➣♥ tt ỗ t t tỷ t t ❧✐➯♥ tö❝ F (s) : Rq → Rp t❤ä❛ πesA BF (s) = πesA C ✈ỵ✐ ♠å✐ s ≥ 0, tr♦♥❣ ✤â π ❧➔ ♣❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ tø Rn ❧➯♥ L ●✐↔ t❤✐➳t ✷ ❈❤♦ ❚ ❧➔ sè ♥❤ä ♥❤➜t tr♦♥❣ sè ❝→❝ sè t❤ü❝ t ≥ s❛♦ ❝❤♦ χ(T ) ≤ ρ ❚r♦♥❣ ✤â t χ (t) = F (s)v(s) ds sup t 0 v(s) ds≤σ ●✐↔ t❤✐➳t ✸ πeT Az0 ∈ G(T )✱ tr♦♥❣ ✤â T T A(T −s) G(T ) := πe w(s) ds ≤ (ρ − χ(T ))2 Bw(s)ds : 0 ❑❤✐ ➜② trá ❝❤ì✐ ❦➳t t❤ó❝ s❛✉ t❤í✐ ❣✐❛♥ T ✳ ◆❤➟♥ ①➨t ✷✳✸ tr÷í♥❣ ❤đ♣ T = Z ❝→❝ ❤➺ ✤ë♥❣ ❧ü❝ ✭✷✳✷✳✷✮ ✈ỵ✐ ❤↕♥ ❝❤➳ ♥➠♥❣ ❧÷đ♥❣ ✭✷✳✷✳✸✮ trð t❤➔♥❤ z(k + 1) = A(k)z(k) − B(k)u(k) + C(k)v(k), k = 0, 1, ợ ữủ u(i) i=0 2 ≤ρ , v(i) ≤ σ2 (2.2.8) i=0 ❍➺ q✉↔ s❛✉ ✤➙② ❧➔ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ỵ T = Z q ✭◆✳ ❨✉✳ ❙❛t✐♠♦✈✱ ①❡♠❬✶✵❪✮ ●✐↔ sû K ❧➔ sè ♥❤ä ♥❤➜t tr♦♥❣ sè ❝→❝ sè tü ♥❤✐➯♥ k1 s❛♦ ❝❤♦ ❝→❝ ❣✐↔ t❤✐➳t s❛✉ ✤➙② ✤÷đ❝ t❤ä❛ ♠➣♥ ●✐↔ t❤✐➳t ỗ t tr F (k) p × q t❤ä❛ ♠➣♥ πAK−1−k BF (k) = πAK−1−k C, ∀k = 0, 1, , K − 1, ✸✽ tr♦♥❣ ✤â π ❧➔ ♣❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ tø Rn ❧➯♥ L ●✐↔ t❤✐➳t ✷ χ(T ) ≤ ρ✱ tr♦♥❣ ✤â K−1 χ (t) = F (k)v(k) sup K−1 v(i) ≤σ k=0 i=0 ●✐↔ t❤✐➳t ✸ πeT Az0 ∈ G(K)✱ tr♦♥❣ ✤â K−1 K K−1−i πA G(K) = p Bw(k) : w(k) ∈ R , k=0 w(k) ≤ (ρ − χ(T ))2 k=0 ❑❤✐ trỏ ỡ t tú s K ữợ rỏ ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈ỵ✐ t❤ỉ♥❣ t✐♥ ❝❤➟♠ ✈➔ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ❳➨t ❜➔✐ t♦→♥ trá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ❞↕♥❣ z ∆ (t) = Az(t) − Bu(t) + Cv(t), t ≥ t0 ; t, t0 ∈ T, z(t0 ) = z0 (2.3.1) Ð ✤➙② z ∈ Rn , ❝→❝ ❤➔♠ u(.), u : T → Rp ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❝õ❛ ♥❣÷í✐ ✤✉ê✐✱ ✈➔ v(.), v : T → Rq ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❝õ❛ ♥❣÷í✐ ❝❤↕②✳ ❈→❝ ♠❛ tr➟♥ A, B ✈➔ C õ số tữỡ ự n ì n, n × p ✈➔ n × q ❈→❝ ✤✐➲✉ ❦❤✐➸♥ t❤ä❛ ♠➣♥ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ s❛✉ T T u(s) ∆s ≤ ρ2 ; t0 v(s) ∆s ≤ σ (2.3.2) t0 ❈→❝ ❤➔♠ ❦❤↔ t➼❝❤ u(.) ✈➔ v(.) t❤ä❛ ♠➣♥ ✭✷✳✸✳✷✮ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ✤✐➲✉ ❦❤✐➸♥ ữủ r trữợ t t ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧➼ ✈➲ ✤✐➲✉ ❦✐➺♥ ✤õ ❦➳t t❤ó❝ trá ❝❤ì✐ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❣✐↔ t❤✐➳t t❤ỉ♥❣ t✐♥ ❧➔✿ ◆❣÷í✐ ✤✉ê✐ ①➙② ❞ü♥❣ t❤ỉ♥❣ t✐♥ u (t) = u (v (t)) tự t ộ ữợ t ①➙② ❞ü♥❣ ❝❤✐➳♥ ❧÷đ❝ ❝õ❛ ♠➻♥❤✱ ♥❣÷í✐ ✤✉ê✐ ✤÷đ❝ ❜✐➳t t❤ỉ♥❣ t✐♥ ✈➲ ✤✐➲✉ ❦❤✐➸♥ v(t) ❝õ❛ ♥❣÷í✐ ✸✾ ❝❤↕②✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ t❤ü❝ t➳ t❤ỉ♥❣ t✐♥ ♠➔ ♥❣÷í✐ ✤✉ê✐ t❤÷í♥❣ ❜à ❝❤➟♠ ❜ð✐ ♠ët ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ♥➔♦ ✤â✳ ❱➻ ✈➟②✱ tr♦♥❣ ♠ö❝ ♥➔② t❛ s➩ ♣❤→t ❜✐➸✉ ✤✐➲✉ ❦✐➺♥ ❦➳t t❤ó❝ trá ❝❤ì✐ ✈ỵ✐ t❤ỉ♥❣ t✐♥ ❝❤➟♠✳ ✣➸ ①➙② ❞ü♥❣ t❤ỉ♥❣ t✐♥ ❝❤➟♠✱ t❛ ✤÷❛ ✈➔♦ ❣✐↔ t❤✐➳t s sỷ tỗ t ởt số T, ν ≥ ✈➔ ♠ët ❤➔♠ r : Tν → T ❦❤æ♥❣ ❣✐↔♠✱ ❞❡❧t❛✲❦❤↔ ✈✐ tr➯♥ T t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ r(t) ≤ t ✈ỵ✐ ♠å✐ t ∈ Tν , tr♦♥❣ ✤â Tν := T ∩ [ν, +∞) ✣➸ ①➙② ❞ü♥❣ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ u(.), ♥❣÷í✐ ✤✉ê✐ ✤÷đ❝ ❜✐➳t t❤ỉ♥❣ t✐♥ ✈➲ ❤➺ ✭✷✳✸✳✶✮✱ t➟♣ ❦➳t t❤ó❝ trá ❝❤ì✐ M, ✈➔ ✤➦❝ ❜✐➺t✱ ♥❣÷í✐ ✤✉ê✐ ✤÷đ❝ ❜✐➳t t❤ỉ♥❣ t✐♥ ✈➲ ✤✐➲✉ ❦❤✐➸♥ ❝õ❛ ♥❣÷í✐ ❝❤↕② t↕✐ t❤í✐ ✤✐➸♠ r(t), tù❝ ❧➔ u(t) = u (v(r(t))) ●✐↔ sû u∗ (t) ❧➔ ♠ët ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ♥➔♦ ✤â ❝õ❛ ♥❣÷í✐ ✤✉ê✐✱ tù❝ ❧➔ T u (s) ∆s ≤ ρ , ❑➼ ❤✐➺✉ ρ := ρ − ∗ 2 t0 T u∗ (s) ∆s t0 π ❧➔ ♣❤➨♣ ❝❤✐➳✉ trü❝ ❣✐❛♦ tø R ①✉è♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ M, M = M1 + M2 ⊆ RN , tr♦♥❣ ✤â M1 ⊂ RN ⊆ RN , M2 ⊆ RN ✱ RN ❧➔ ❝→❝ ❦❤æ♥❣ n ❣✐❛♥ ❝♦♥ ❝õ❛ RN ✈➔ RN = RN ⊕ RN ❑❤✐ ✤â✱ ✤✐➲✉ ❦✐➺♥ ❦➳t t❤ó❝ trá ❝❤ì✐ s➩ ❧➔ z(K) M tữỡ ữỡ ợ z(K) M2 ✣➸ ❦✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ ❦➳t t❤ó❝ trá ❝❤ì✐ t❛ ✤÷❛ ✈➔♦ ❦❤→✐ ♥✐➺♠ ❤✐➺✉ ❤➻♥❤ ❤å❝ P♦♥t✐❛❣✐♥ ♥❤÷ s❛✉ A∗B := {z ∈ Rn , z + B ⊆ A} ỵ sỷ K T sè ♥❤ä ♥❤➜t s❛♦ ❝❤♦ ❝→❝ ❣✐↔ t❤✐➳t s❛✉ ✤➙② ữủ tọ tt ỗ t ởt tr➟♥ ❤➔♠ r❞✲❧✐➯♥ tö❝ F (t) : Rq → Rp s❛♦ ❝❤♦ πeA (K, σ(t))BF (t) = πeA (K, σ (r(t))) C ✈ỵ✐ ♠å✐ t ∈ T, t ≥ υ (2.3.3) ●✐↔ t❤✐➳t ✷ K χ2 (K) = sup K v(s) ∆s≤σ υ F (t)v(r(t))r∆ (t) ∆t ≤ ρ2 (2.3.4) ✹✵ ●✐↔ t❤✐➳t ✸ υ πeA (K, σ(s))Bu∗ (s)∆s ∈ G(K) + (M2 ∗H(K)) , (2.3.5) πeA (K, 0)z0 − tr♦♥❣ ✤â K K G(K) = w(s) ∆τ ≤ (ρ − χ(K))2 , πeA (K, σ(s))Bw(s)∆s : ν ν r(ν) πeA (K, σ(s))Cv(s)∆s H(K) = K K πeA (K, σ(s))Cv(s)∆s : + r(K) v(s) ≤ θ2 ❑❤✐ ➜② trá ❝❤ì✐ ✤✉ê✐ ❜➢t ✭✷✳✸✳✶✮✲✭✷✳✸✳✷✮ ❦➳t t❤ó❝ s❛✉ t❤í✐ ❣✐❛♥ K ❈❤ù♥❣ tỗ t ởt tỡ m M2H(K) ✈➔ ♠ët ❤➔♠ K ❦❤↔ t➼❝❤ w(s) tr➯♥ [υ, K] ∩ T ✈ỵ✐ w(s) ∆s ≤ (ρ − χ(K))2 s❛♦ ❝❤♦ υ υ K πeA (K, σ(s))Bu∗ (s)∆s = πeA (K, 0)z0 − πeA (K, σ(s))Bw(s)∆s + m υ ❱➻ m ∈ M2 ∗H(K) ♥➯♥ m + H(K) ⊆ M2 ●✐↔ sû v(.) ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❜➜t ❦➻ ❝õ❛ ♥❣÷í✐ ✤✉ê✐✱ ❦❤✐ ➜② K r(ν) πeA (K, σ(s))Cv(s)∆s ∈ H(K) πeA (K, σ(s))Cv(s)∆s + r(K) õ tỗ t ởt tỡ m2 M2 s❛♦ ❝❤♦ r(ν) m+ K πeA (K, σ(s))Cv(s)∆s + πeA (K, σ(s))Cv(s)∆s = m2 r(K) ❈❤ù♥❣ tä υ πeA (K, 0)z0 − πeA (K, σ(s))Bu∗ (s)∆s = r(ν) K πeA (K, σ(s))Cv(s)∆s − πeA (K, σ(s))Bw(s)∆s + m2 υ − K πeA (K, σ(s))Cv(s)∆s r(K) (2.3.6) ợ v(.) ữủ ❜➜t ❦➻ ❝õ❛ ♥❣÷í✐ ❝❤↕②✱ t❛ ①➙② ❞ü♥❣ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ♥❣÷í✐ ✤✉ê✐ ♥❤÷ s❛✉  u∗ (t) ♥➳✉ t ∈ [0; υ]T ; u(t) := F (t)v(r(t))r∆ (t) + w(t) ♥➳✉ t ∈ [υ; K] T ❑❤✐ ➜② t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ▼✐♥❦♦✈s❦✐ t❛ ❝â K K F (t)v(r(t))r∆ (t) + w(t) ∆s u(s) ∆s = υ υ K K F (t)v(r(t))r∆ (t) ∆s + ≤ w(t) ∆s υ υ ≤ (ρ − χ(K)) + χ(K) = ρ ❱➟② K υ u(s) ∆s = u(s) ∆s u(s) ∆s + 0 υ K υ u∗ (s) ∆s + = K F (t)v(r(t))r∆ (t) + w(t) ∆s υ υ u∗ (s) ∆s + ρ2 = ρ2 − ρ2 + ρ2 = ρ2 ≤ ❈❤ù♥❣ tä u(.) ❧➔ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✤ë♥❣ ❧ü❝ K πz(K) = πeA (K, 0)z0 − πeA (K, σ(τ ))Bu(τ )∆τ K + πeA (K, σ(τ ))Cv(τ )∆τ ✹✷ υ πeA (t, σ(τ ))Bu∗ (τ )∆τ = πeA (K, 0)z0 − K πeA (K, σ(τ ))B[F (τ )v(r(τ ))r∆ (τ ) + w(τ )]∆τ − ν K πeA (K, σ(τ ))Cv(τ )∆τ + (2.3.7) ❙û ❞ö♥❣ ●✐↔ t❤✐➳t ✶ πeA (K, σ(t))BF (t) = πeA (K, σ (r(t))) C ❚❛ ❝â K πeA (K, σ(τ ))B[F (τ )v(r(τ ))r∆ (τ ) + w(τ )]∆τ ν K K ∆ = πeA (K, σ(τ ))BF (τ )v(r(τ ))r (τ )∆τ + πeA (K, σ(τ ))Bw(τ )∆τ ν ν K πeA (K, σ(r(τ )))Cv(r(τ ))r∆ (τ )∆τ = ν K + πeA (K, σ(τ ))Bw(τ )∆τ (2.3.8) ν ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ ϑ := r(τ ) t❛ ❝â ∆ϑ = r∆ (τ )∆τ t❛ ✤✐ ✤➳♥ r(K) K πeA (K, σ(r(τ )))Cv (r (τ )) r∆ (τ )∆τ = υ πeA (K, σ(ϑ))Cv (ϑ) ∆ϑ r(υ) ❚❤❛② ✈➔♦ ✭✷✳✸✳✽✮ t❛ ✤÷đ❝ K K πeA (K, σ(r(τ )))Cv (r (τ )) r∆ (τ )∆τ + = υ υ r(K) = K πeA (K, σ(ϑ))Cv (ϑ) ∆ϑ + r(υ) πeA (K, σ(τ ))Bw(τ )∆τ πeA (K, σ(τ ))Bw(τ )∆τ υ ✹✸ ❚❤❛② ✈➔♦ ✭✷✳✸✳✼✮ t❛ ✤÷đ❝ K πz(K) = πeA (K, 0)z0 − πeA (K, σ(τ ))Bu(τ )∆τ K πeA (K, σ(τ ))Cv(τ )∆τ + ν πeA (K, σ(τ ))Bu∗ (τ )∆τ = πeA (K, 0)z0 − r(K) − πeA (K, σ(ϑ))Cv(ϑ)∆ϑ r(ν) K − r(ν) πeA (K, σ(τ ))Bw(τ )∆τ + ν πeA (K, σ(τ ))Cv(τ )∆τ r(K) + K πeA (K, σ(τ ))Cv(τ )∆τ + r(ν) πeA (K, σ(τ ))Cv(τ )∆τ r(K) ν πeA (K, σ(τ ))Bu∗ (τ )∆τ = πeA (K, 0)z0 − K − πeA (K, σ(τ ))Bw(τ )∆τ ν K r(ν) πeA (K, σ(τ ))Cv(τ )∆τ πeA (K, σ(τ ))Cv(τ )∆τ + + r(K) ❙û ❞ö♥❣ ✭✷✳✸✳✻✮ t❛ ✤✐ ✤➳♥ ν πeA (K, σ(τ ))Bu∗ (τ )∆τ πz(K) = πeA (K, 0)z0 − K − πeA (K, σ(τ ))Bw(τ )∆τ ν r(ν) + K πeA (K, σ(τ ))Cv(τ )∆τ + πeA (K, σ(τ ))Cv(τ )∆τ r(K) = m2 ❱➟② πz(K) = m2 ∈ M2 ❤❛② trá ❝❤ì✐ ❦➳t tú s tớ K ỵ ữủ ự t r(t) t t ỵ ỵ t ✤➣ tr➻♥❤ ❜➔② ♠ët sè ✤à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❣✐↔✐ t➼❝❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✭❈❤÷ì♥❣ ✶✮✱ ❇➔✐ t♦→♥ trá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥✱ ❜➔✐ t♦→♥ trá ❝❤ì✐ ✤✉ê✐ ❜➢t ✈ỵ✐ t❤ỉ♥❣ t✐♥ ❝❤➟♠ ✈➔ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t→❝ ❣✐↔ ✤➣ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❦➳t t❤ó❝ trá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈ỵ✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥✳ ❈❤♦ ♣❤➨♣ ❤ñ♣ ♥❤➜t ♠ët sè ❦➳t q✉↔ ✤➣ ❜✐➳t tr♦♥❣ trá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ♠ỉ t↔ ❜ð✐ ữỡ tr ữỡ tr s ợ r➔♥❣ ❜✉ë❝ t➼❝❤ ♣❤➙♥ ✳ ✹✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ P❤❛♥ ❍✉② ❑❤↔✐ ✭✶✾✾✼✮✱ ▼ët sè ❜➔✐ t ỵ tt trỏ ỡ s ♣❤➙♥✱ ●✐→♦ tr➻♥❤ ❈❛♦ ❤å❝✱ ❱✐➺♥ ❚♦→♥ ❤å❝✳ ❬✷❪ ❱✐ ❉✐➺✉ ▼✐♥❤ ✭✷✵✶✼✮✱ ❱➲ trá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥✱ tr❛♥❣ ✶✲✻✳ ❬✸❪ ❱✐ ❉✐➺✉ ▼✐♥❤✱ ú ỵ trỏ ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈ỵ✐ t❤æ♥❣ t✐♥ ❝❤➟♠✱ ❇→♦ ❝→♦ t↕✐ ❤ë✐ ♥❣❤à t♦→♥ ❤å❝ ♠✐➲♥ ❚r✉♥❣✲❚➙② ◆❣✉②➯♥✱ t❤→♥❣ ✶✷✲✷✵✶✼✭✤➣ ❣û✐ t♦➔♥ ✈➠♥✮✳ ❚✐➳♥❣ ❆♥❤ ❬✹❪ ❘❛✈✐ ❆❣❛r✇❛❧✱ ▼❛rt✐♥ ❇♦❤♥❡r✱ ❉♦♥❛❧ ♦✬❘❡❣❛♥ ❆❧❧❛♥ P❡t❡rs♦♥ ✭✷✵✵✷✮✱ ✏❉②♥❛♠✐❝ ❡q✉❛t✐♦♥s ♦♥ t✐♠❡ s❝❛❧❡s✿ ❛ s✉r✈❡②✑✱ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛✲ t✐♦♥❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✶✹✶ ✭✶✲✷✮✱ ♣♣✳ ✶✲✷✻✳ ❬✺❪ ▼✳ ❇♦❤♥❡r ❛♥❞ ❆✳ P❡t❡rs♦♥ ✭✷✵✵✶✮✱ ❉②♥❛♠✐❝ ❊q✉❛t✐♦♥s ♦♥ ❚✐♠❡ ❙❝❛❧❡s✿ ❆♥ ■♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❇✐r❦❤☎❛✉s❡r✱ ❇♦st♦♥✳ ❬✻❪ ▼✳ ❇♦❤♥❡r ❛♥❞ ❆✳ P❡t❡rs♦♥ ✭✷✵✵✸✮✱ ❆❞✈❛♥❝❡s ✐♥ ❉②♥❛♠✐❝ ❊q✉❛t✐♦♥s ♦♥ ❚✐♠❡ ❙❝❛❧❡s✱ ❇✐r❦❤☎❛✉s❡r✱ ❇♦st♦♥✳ ❬✼❪ ❍✐❧❣❡r ✭✶✾✽✽✮✱ ❊✐♥ ▼❛ß❦❡tt❡♥❦❛❧❦☎ ✉❧ ♠✐t ❛♥✇❡♥❞✉♥❣ ❛✉❢ ❩❡♥tr✉♠s♠❛♥♥✐♥❣✲❢❛❧t✐❦❡✐t❡♥✱ P❤✳❉✳ ❚❤❡s✐s✱ ❯♥✐✈❡rs✐t☎❛t ❲☎ ✉r③❜✉r❣✳ ✹✺ ✹✻ ❬✽❪ ❏✳ ❏✳ ❉❛❈✉♥❤❛ ✭✷✵✵✹✮✱ ▲②❛♣✉♥♦✈ ❙t❛❜✐❧✐t② ❛♥❞ ❋❧♦q✉❡t ❚❤❡♦r② ❢♦r ◆♦❛✉t♦♥♦♠♦✉s ▲✐♥❡❛r ❉②♥❛♠✐❝ ❙②st❡♠s ♦♥ ❚✐♠❡ ❙❝❛❧❡s✱ P❤✳ ❉✳ ❚❤❡✲ s✐s✱ ❇❛②❧♦r ❯♥✐✈❡rs✐t②✳ ❬✾❪ ❇✳ ❏✳ ❏❛❝s♦♥ ✭✷✵✵✼✮✱ ❆ ●❡♥❡r❛❧ ▲✐♥❡❛r ❙②st❡♠s ❚❤❡♦r② ♦♥ ❚✐♠❡ ❙❝❛❧❡s✿ ❚r❛♥s❢♦r♠s✱ ❙t❛❜✐❧✐t②✱ ❛♥❞ ❈♦♥tr♦❧✱ P❤✳ ❉✳ ❚❤❡s✐s✱ ❇❛②❧♦r ❯♥✐✈❡rs✐t②✳ ❬✶✵❪ P❤❛♥ ❍✉② ❑❤❛✐ ✭✶✾✽✸✮✱ ❖♥ t❤❡ P✉rs✉✐t Pr♦❝❡ss ✐♥ ❉✐❢❢❡r❡♥t✐❛❧ ●❛♠❡s✱ ✽ ❆❝t❛ ▼❛t❤❡♠❛t✐❝❛ ❱✐❡t♥❛♠✐❝❛✱ ✭✶✮✱ ♣♣✳ ✹✶✲✺✼✳ ❬✶✶❪ P❤❛♥ ❍✉② ❑❤❛✐ ✭✶✾✽✺✮✱ ❖♥ ❛♥ ❊❢❢❡❝t✐✈❡ ▼❡t❤♦❞ ♦❢ P✉rs✉✐t ✐♥ ▲✐♥❡❛r ❉✐s❝r❡t❡ ●❛♠❡s ✇✐t❤ ❉✐❢❢❡r❡♥t ❚②♣❡s ♦❢ ❈♦♥str❛✐♥ts ♦♥ ❈♦♥tr♦❧s✱ ❆❝t❛ ▼❛t❤❡♠❛t✐❝❛ ❱✐❡t♥❛♠✐❝❛✱ ✶✵✭✷✮✱ ♣♣✳ ✷✽✷✲✷✾✺✳ ... TRƯỜNG ĐẠI HỌC KHOA HỌC  - LÊ VĂN QUÝ BÀI TOÁN ĐUỔI BẮT TRONG TRỊ CHƠI TUYẾN TÍNH VỚI HẠN CHẾ TÍCH PHÂN TRÊN THANG THỜI GIAN LUẬN VĂN THẠC SĨ TỐN HỌC Chun ngành: Tốn ứng dụng... ✣✐➸♠ t ∈ T ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ trò ♠➟t ♣❤↔✐ ✭r✐❣❤t✲❞❡♥❝❡✮ ♥➳✉ σ(t) = t ✣✐➸♠ t ∈ T ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ trị ♠➟t tr→✐ ✭❧❡❢t✲❞❡♥❝❡✮ ♥➳✉ ρ(t) = t ✣✐➸♠ t ∈ T ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ trò ♠➟t ✭❞❡♥❝❡✮ ♥➳✉ ρ(t) = t... sè f : T → R ✤÷đ❝ ❣å✐ ❧➔ ❝❤➼♥❤ q✉② rt ợ õ tỗ t ỳ ❤↕♥✮ t↕✐ ♠å✐ ✤✐➸♠ trò ♠➟t ♣❤↔✐ tr♦♥❣ T ✈➔ ợ tr õ tỗ t ỳ t ♠å✐ ✤✐➸♠ trò ♠➟t tr→✐ ❝õ❛ T✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✶ ❍➔♠ f : T → R ✤÷đ❝ ❣å✐ ❧➔ r❞✲❧✐➯♥