Một số bài toán về tính bền vững của hệ động lực tuyến tính chịu nhiễu

ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN ĐỖ ĐỨC THUẬN MỘT SỐ BÀI TỐN VỀ TÍNH BỀN VỮNG CỦA HỆ ĐỘNG LỰC TUYẾN TÍNH CHỊU NHIỄU Chun ngành: Tốn Giải tích Mã số TĨM TẮT LUẬN ÁN TIẾN SĨ TỐN HỌC HÀ NỘI - 2012 Cơng trình hồn thành tại: Trường Đại học Khoa học Tự nhiên, Đại học Quốc gia Hà nội Người hướng dẫn khoa học: GS TSKH Nguyễn Khoa Sơn Phản biện 1: GS TSKH Phạm Kỳ Anh Phản biện 2: GS TSKH Vũ Ngọc Phát Phản biện 3: PGS TS Nguyễn Sinh Bảy Luận án bảo vệ trước hội đồng chấm luận án Tiến sĩ cấp nhà nước họp …………………………………………………………………………… …………………………………………………………………………… …………… Vào hồi ……… giờ…… ngày……… tháng……… năm……… Có thể tìm hiểu luận án tại: - Thư viện Quốc gia Việt Nam - Trung tâm Thông tin - Thư viện, Đại học Quốc gia Hà nội L˝I CAM OAN Tæi xin cam oan Ơy l cổng trnh nghiản cứu ca riảng tổi C¡c k‚t qu£, sŁ li»u lu“n ¡n l trung thỹc v chữa tng ữổc cổng b bĐt cø cæng tr…nh n o T¡c gi£ lu“n ¡n Ø i ức Thun LIC MèN Du dt trản ữớng to¡n håc, ln t⁄o nhœng thß th¡ch gióp tỉi tü håc häi, t…m tỈi v s¡ng t⁄o, â l nhœng g… tỉi may m›n ÷ỉc ti‚p nh“n tł ng÷íi thƒy ¡ng k‰nh cıa m…nh, GS TSKH Nguy„n Khoa Sìn Thy Sỡn khổng nhng  hữợng dÔn tn tnh m cặn truyãn cho tổi nhiãu kinh nghiằm quỵ bĂu nghiản cứu khoa hồc cụng nhữ cuc sng Tổi xin gòi n Thy lặng bit ỡn sƠu sc nhĐt Tỉi cơng b y tä lỈng bi‚t ìn ‚n GS TS Nguyn Hu Dữ Thy cõ nhng ch dÔn quỵ b¡u chuy¶n mỉn v nghi¶n cøu khoa håc ÷ỉc l m vi»c vỵi Thƒy gióp tỉi mð rºng vŁn ki‚n thøc cıa m…nh v thu ÷ỉc mºt sŁ k‚t qu£ âng gâp v o lu“n ¡n Tæi xin gòi tợi GS TSKH Phm Ký Anh, PGS TS Vơ Ho ng Linh v c¡c Thƒy Cỉ gi¡o Khoa To¡n - Cì - Tin håc tr÷íng ⁄i håc Khoa hồc Tỹ nhiản - HQGHN lặng bit ỡn sƠu sc, nhng ngữới  dy dỉ v ch bÊo tn tnh tổi,  giúp ù rĐt nhiãu tổi n ÷ỉc ÷íng to¡n håc nh÷ b¥y gií Tỉi xin ch¥n th nh c£m ìn c¡c Thƒy Cỉ Hºi çng ph£n bi»n v c¡c Thƒy Cỉ tr¶n Vi»n To¡n hồc, nhng ngữới  ồc v cho nhng nhn xt, gõp ỵ quỵ giĂ lun Ăn ữổc tt hỡn ii Tổi cụng xin gòi lới cÊm ỡn tợi PGS TS Nguy„n Thà B⁄ch Kim, c¡c Thƒy Cæ gi¡o Khoa To¡n - Tin øng dưng tr÷íng ⁄i håc B¡ch Khoa H Nºi, nhœng ng÷íi ln ıng hº nhi»t t…nh, t⁄o i•u ki»n thu“n lỉi v sfin s ng gióp ï tỉi thíi gian n y Lu“n ¡n n y ữổc ho n th nh dữợi sỹ ng viản, chia sÃ, giúp ù lợn lao ca B, Mà, ngữới thƠn v bn b Tổi xin gòi lới cÊm ỡn v d nh mân qu n y cho t§t c£! H Nºi, ng y 24 th¡ng n«m 2011 T¡c giÊ iii Mửc lửc Danh mửc cĂc kỵ hiằu v chœ vi‚t t›t Líi nâi ƒu KI N THÙC CHU N BÀ 1.1 To¡n tß a trà tuy‚n t‰n 1.2 Tnh iãu khin ữổc c 1.3 Tnh iãu khin ÷ỉc c 1.4 Sü Œn ành mơ cıa h» H C´ R NG BU¸C V˛I MI N THAM Să I UKHI NBNHI U 2.1 KhoÊng cĂch gia cĂc nân 2.2 BĂn knh iãu khin ữổc 2.3 V‰ dö H I UKHI NTUY NT NHCHÀUNHI U C U TRểC 3.1 BĂn knh iãu khin ữổc dữợi nhiu cĐu trú 3.2 BĂn knh iãu khin ữổc dữợi a nhi„u c§u t 3.3 V‰ dư iv 3.4 Thu“t to¡n t‰nh to¡n 71 B NK NHTO N NHV C CÙNGDÖNG CÕA N´ 4.1 B¡n k‰nh to n ¡nh 4.2 B¡n k‰nh Œn ành hâa 4.3 B¡n k‰nh Œn ành cıa 4.4 C¡c b¡n k‰nh i•u khi”n K‚t lu“n Danh mưc cỉng tr…nh khoa håc cıa t¡c gi£ li¶n quan lu“n ¡n T ILI UTHAMKH O v ‚n DANHMệCC CKịHI UV CHVI TT T Trữớng C hoc R K Kn m gr F ker F Im F dom F (A) ^(A) min[A] Gi¡ trà k… dà nhä nh§t cıa A A A y ? M P; Q Pb; Qb f Thang thíi gian Delta ⁄o h m cıa f S Mi•n Œn ành mơ •u cıa thang thíi gian T vi M— U Trong thüc ti„n, câ nhi•u vĐn ã ca k thut, cỡ hồc, vt lỵ, sinh håc, kinh t‚ ÷ỉc mỉ t£ bði c¡c h» ºng lüc H» ºng lüc câ th¶m c¡c bi‚n iãu khin th s ữổc gồi l hằ iãu khin Lỵ thuyt iãu khin ữổc phĂt trin t khoÊng 150 nôm trữợc Ơy cĂc iãu khin cỡ hồc cn v câ th” ÷ỉc mỉ t£ mºt c¡ch to¡n håc CĂc tnh chĐt nh tnh ca hằ iãu khin ữổc quan tƠm nhiãu nhĐt l tnh iãu khin ữổc, tnh Œn ành v t‰nh Œn ành hâa ÷ỉc Nâi mºt cĂch ỡn giÊn, hằ ữổc gồi l iãu khin ữổc nu tỗn ti mt iãu khin chuyn hằ t mt trng thĂi ban u cho trữợc sang mt trng th¡i mong muŁn cuŁi cịng H» ÷ỉc gåi l Œn ành ti»m c“n n‚u måi quÿ ⁄o cıa nâ chuy”n dƒn v• tr⁄ng th¡i dłng thíi gian ti‚n vỉ cịng v h» ÷ỉc gåi l Œn ành hâa ữổc nu tỗn ti mt iãu khin ngữổc ( iãu khi”n phö thuºc v o bi‚n tr⁄ng th¡i) ” bi‚n nâ th nh mºt h» Œn ành ti»m c“n Hi»n nay, vĐn ã ang ữổc quan tƠm l tnh chĐt cıa c¡c h» ºng lüc chàu £nh h÷ðng cıa nhi„u Phn lợn cĂc tnh chĐt "tt" ca cĂc hằ ng lüc cơng nh÷ c¡c Łi t÷ỉng to¡n håc nâi chung ãu bÊo to n cĂc tham s cĐu tróc cıa h» ho°c Łi t÷ỉng chàu nhi„u b† V‰ dử: tnh iãu khin ữổc ca mt hằ iãu khin tuyn tnh lỵ thuyt iãu khin; tnh n nh tiằm cn ca nghiằm phữỡng trnh vi phƠn; tnh °t ch¿nh (well-posedness) cıa mºt h» ph÷ìng tr…nh tuy‚n t‰nh, t‰nh hºi tö cıa mºt thu“t to¡n gi£i t‰ch sŁ; t‰nh kh£ nghàch cıa mºt ma tr“n vuæng ⁄i sŁ tuy‚n t‰nh; t‰nh ch‰nh qui metric cıa mºt ¡nh x⁄ gi£i t‰ch Sü b£o to n cĂc tnh chĐt nh tnh n y dữợi Ênh hững ca nhiu ữổc gồi l sỹ bãn vng CĂc nh to¡n håc mong muŁn t…m ÷ỉc mºt ành l÷ỉng nh‹m ¡nh gi¡ kh£ n«ng b£o to n c¡c t‰nh chĐt nh tnh ca hằ thng dữợi Ênh hững ca nhi„u, ÷ỉc gåi l c¡c b¡n k‰nh b£o to n Łi vỵi t‰nh Œn ành ti»m c“n cıa h» tuy‚n tnh, xuĐt phĂt t hai b i bĂo ông trản t⁄p ch‰ Systems & Control Letters [45, 46], c¡c t¡c gi£ D Hinrichsen v A.J Pritchard ¢ ph¡t tri”n mºt hữợng nghiản cứu mợi l hữợng nghiản cứu n nh vœng cıa c¡c h» ºng lüc düa tr¶n bi”u di„n cıa h» khỉng gian tr⁄ng th¡i v sß dưng khĂi niằm bĂn knh n nh Hữợng nghiản cứu n y  thu hút ữổc sỹ quan tƠm ca nhiãu nh to¡n håc v… t‰nh hi»u qu£ cıa nâ công nhữ cĂc ứng dửng kắ thut (xem [7, 13, 25, 26, 47, 49, 50, 52, 53, 55, 68, 74, 76, 84, 97]) Dữợi dng ỡn giÊn nhĐt, bĂn knh n nh cõ cĐu trúc ca mt hằ phữỡng trnh vi ph¥n tuy‚n t‰nh Œn ành ti»m c“n x = Ax ữổc nh nghắa l s lợn nhĐt cho måi h» chàu nhi„u x = (A + D E)x vÔn cặn n nh tiằm cn mt k k < ; ð ¥y l ma tr“n nhi„u, D v E l c¡c ma tr“n c§u tróc nhi„u v k k l mt chu'n ma trn cho trữợc Mt cĂch t÷ìng ÷ìng, b¡n k‰nh Œn ành câ th” ÷ỉc ành ngh¾a bði rK(A; D; E) = inffk k : l q 2K ; A + D E khæng Œn ành ti»m c“ng: Khi K = C ta câ ành ngh¾a cıa b¡n k‰nh Œn ành phøc r C(A; D; E) v K = R ta câ ành ngh¾a cıa b¡n k‰nh Œn ành thüc r R(A; D; E) Tł nh nghắa ta thĐy rng rC(A; D; E) rR(A; D; E): Cỉng thøc b¡n k‰nh Œn ành phøc ÷ỉc D Hinrichsen v A.J Pritchard [46] ữa nôm 1986 DANH MÖC C˘NG TR NH KHOA H¯C CÕA T CGI LI NQUAN NLU N N Nguyen Khoa Son and Do Duc Thuan (2008), 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