The 13 reference contexts in paper A. Golubev E., A. Krishchenko P., А. Голубев Е., А. Крищенко П. (2016) “Решение терминальной задачи управления для аффинной системы при помощи многочленов // Polynomials-Based Terminal Control of Affine Systems” / spz:neicon:technomag:y:2015:i:2:p:101-114

  1. Start
    1880
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    Õ‡ÔÓÏÌËÏ, ̃ÚÓ ÔÓ‰ ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃ÂÈ ÔÓ‰‡ÁÛÏ‚‡ ̨Ú Ì‡ıÓʉÂÌË ÛÔ‡‚ÎÂÌˡ, ÔÂ‚Ӊˇ ̆Â„Ó ‰Ë̇ÏË ̃ÂÒÍÛ ̨ ÒËÒÚÂÏÛ Á‡ ÌÂÍÓÚÓ ̊È ÓÚÂÁÓÍ ‚ÂÏÂÌË ËÁ Á‡‰‡ÌÌÓ„Ó Ì‡ ̃‡Î ̧ÌÓ„Ó ÒÓÒÚÓˇÌˡ ‚ Á‡‰‡ÌÌÓ ÍÓÌ ̃ÌÓ ÒÓÒÚÓˇÌËÂ
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    [1]
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    . Œ‰ËÌ ËÁ ÒÔÓÒÓ·Ó‚  ̄ÂÌˡ ÚÂÏË̇Π̧Ì ̊ı Á‡‰‡ ̃ ‰Îˇ ÒËÒÚÂÏ ‚ˉ‡ (1) ÓÒÌÓ‚‡Ì ̇ ËÒÔÓÎ ̧ÁÓ‚‡ÌËË ÏÌÓ„Ó ̃ÎÂÌÓ‚ ÒÚÂÔÂÌË2n−1[1,2]. – ̄ÂÌË ‡ÁÎË ̃Ì ̊ı ÚÂÏË̇Π̧Ì ̊ı Á‡‰‡ ̃ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ ÏÌÓ„Ó ̃ÎÂÌÓ‚ ‡ÒÒχÚË‚‡ÎÓÒ ̧, ̇ÔËÏÂ, ‚ ‡·ÓÚ‡ı [3,4,5,6,7,8,9,10,11,12,13,14].
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  2. Start
    2002
    Prefix
    Õ‡ÔÓÏÌËÏ, ̃ÚÓ ÔÓ‰ ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃ÂÈ ÔÓ‰‡ÁÛÏ‚‡ ̨Ú Ì‡ıÓʉÂÌË ÛÔ‡‚ÎÂÌˡ, ÔÂ‚Ӊˇ ̆Â„Ó ‰Ë̇ÏË ̃ÂÒÍÛ ̨ ÒËÒÚÂÏÛ Á‡ ÌÂÍÓÚÓ ̊È ÓÚÂÁÓÍ ‚ÂÏÂÌË ËÁ Á‡‰‡ÌÌÓ„Ó Ì‡ ̃‡Î ̧ÌÓ„Ó ÒÓÒÚÓˇÌˡ ‚ Á‡‰‡ÌÌÓ ÍÓÌ ̃ÌÓ ÒÓÒÚÓˇÌË [1]. Œ‰ËÌ ËÁ ÒÔÓÒÓ·Ó‚  ̄ÂÌˡ ÚÂÏË̇Π̧Ì ̊ı Á‡‰‡ ̃ ‰Îˇ ÒËÒÚÂÏ ‚ˉ‡ (1) ÓÒÌÓ‚‡Ì ̇ ËÒÔÓÎ ̧ÁÓ‚‡ÌËË ÏÌÓ„Ó ̃ÎÂÌÓ‚ ÒÚÂÔÂÌË2n−1
    Exact
    [1,2]
    Suffix
    . – ̄ÂÌË ‡ÁÎË ̃Ì ̊ı ÚÂÏË̇Π̧Ì ̊ı Á‡‰‡ ̃ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ ÏÌÓ„Ó ̃ÎÂÌÓ‚ ‡ÒÒχÚË‚‡ÎÓÒ ̧, ̇ÔËÏÂ, ‚ ‡·ÓÚ‡ı [3,4,5,6,7,8,9,10,11,12,13,14]. ¬ ̇ÒÚÓˇ ̆ÂÈ ÒÚ‡Ú ̧ ËÒÒΉÛÂÚÒˇ ÚÂÏË̇Π̧̇ˇ Á‡‰‡ ̃‡, ‰Îˇ ÍÓÚÓÓÈ ÍÓÌ ̃ÌÓ ÒÓÒÚÓˇÌË ÒËÒÚÂÏ ̊ (1) ÒÓ‚Ô‡‰‡ÂÚ Ò Ì‡ ̃‡ÎÓÏ ÍÓÓ‰Ë̇Úy= 0. » ̆ÂÚÒˇ ÏÌÓÊÂÒÚ‚Ó Ì‡ ̃‡Î ̧Ì ̊ı ÒÓÒÚÓˇÌËÈ Ú‡ÍËı, ̃ÚÓ  ̄ÂÌË ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ÏÓÊÌÓ ÔÓÒÚÓËÚ ̧ ÔË ÔÓÏÓ ̆Ë ÏÌÓ„
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  3. Start
    2119
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    Œ‰ËÌ ËÁ ÒÔÓÒÓ·Ó‚  ̄ÂÌˡ ÚÂÏË̇Π̧Ì ̊ı Á‡‰‡ ̃ ‰Îˇ ÒËÒÚÂÏ ‚ˉ‡ (1) ÓÒÌÓ‚‡Ì ̇ ËÒÔÓÎ ̧ÁÓ‚‡ÌËË ÏÌÓ„Ó ̃ÎÂÌÓ‚ ÒÚÂÔÂÌË2n−1[1,2]. – ̄ÂÌË ‡ÁÎË ̃Ì ̊ı ÚÂÏË̇Π̧Ì ̊ı Á‡‰‡ ̃ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ ÏÌÓ„Ó ̃ÎÂÌÓ‚ ‡ÒÒχÚË‚‡ÎÓÒ ̧, ̇ÔËÏÂ, ‚ ‡·ÓÚ‡ı
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    [3,4,5,6,7,8,9,10,11,12,13,14]
    Suffix
    . ¬ ̇ÒÚÓˇ ̆ÂÈ ÒÚ‡Ú ̧ ËÒÒΉÛÂÚÒˇ ÚÂÏË̇Π̧̇ˇ Á‡‰‡ ̃‡, ‰Îˇ ÍÓÚÓÓÈ ÍÓÌ ̃ÌÓ ÒÓÒÚÓˇÌË ÒËÒÚÂÏ ̊ (1) ÒÓ‚Ô‡‰‡ÂÚ Ò Ì‡ ̃‡ÎÓÏ ÍÓÓ‰Ë̇Úy= 0. » ̆ÂÚÒˇ ÏÌÓÊÂÒÚ‚Ó Ì‡ ̃‡Î ̧Ì ̊ı ÒÓÒÚÓˇÌËÈ Ú‡ÍËı, ̃ÚÓ  ̄ÂÌË ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ÏÓÊÌÓ ÔÓÒÚÓËÚ ̧ ÔË ÔÓÏÓ ̆Ë ÏÌÓ„Ó ̃ÎÂ̇ ÒÚÂÔÂÌË2n−2.
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  4. Start
    2648
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    ÍÓÚÓÓÈ ÍÓÌ ̃ÌÓ ÒÓÒÚÓˇÌË ÒËÒÚÂÏ ̊ (1) ÒÓ‚Ô‡‰‡ÂÚ Ò Ì‡ ̃‡ÎÓÏ ÍÓÓ‰Ë̇Úy= 0. » ̆ÂÚÒˇ ÏÌÓÊÂÒÚ‚Ó Ì‡ ̃‡Î ̧Ì ̊ı ÒÓÒÚÓˇÌËÈ Ú‡ÍËı, ̃ÚÓ  ̄ÂÌË ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ÏÓÊÌÓ ÔÓÒÚÓËÚ ̧ ÔË ÔÓÏÓ ̆Ë ÏÌÓ„Ó ̃ÎÂ̇ ÒÚÂÔÂÌË2n−2. ŒÚÏÂÚËÏ, ̃ÚÓ  ̄ÂÌË ‡ÒÒχÚË‚‡ÂÏÓÈ ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ÛÔ‡‚ÎÂÌˡ ÏÓÊÂÚ · ̊Ú ̧ ËÒÔÓÎ ̧ÁÓ‚‡ÌÓ ‰Îˇ  ̄ÂÌˡ Á‡‰‡ ̃Ë ÒÚ‡·ËÎËÁ‡ˆËË ÌÛÎÂ‚Ó„Ó ÔÓÎÓÊÂÌˡ ‡‚ÌÓ‚ÂÒˡ Á‡ ÍÓÌ ̃ÌÓ ‚ÂÏˇ
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    [15,16,17,18]
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    . ¬ ‡Á‰. 1 ‰Îˇ ÒËÒÚÂÏ ‚ÚÓÓ„Ó ÔÓˇ‰Í‡ ‰Ó͇Á ̊‚‡ ̨ÚÒˇ ÌÂÓ·ıÓ‰ËÏ ̊Â Ë ‰ÓÒÚ‡ÚÓ ̃Ì ̊ ÛÒÎӂˡ ÒÛ ̆ÂÒÚ‚Ó‚‡Ìˡ ÏÌÓ„Ó ̃ÎÂ̇ ‚ÚÓÓÈ ÒÚÂÔÂÌË, ÍÓÚÓ ̊È ÓÔ‰ÂΡÂÚ  ̄ÂÌË ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë. œË‚Ó‰ËÚÒˇ  ̄ÂÌË ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ÛÔ‡‚ÎÂÌˡ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ ÏÌÓ„Ó ̃ÎÂÌÓ‚ ‚ÚÓÓÈ Ë ÚÂÚ ̧ÂÈ ÒÚÂÔÂÌË.
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  5. Start
    3938
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    “Ó„‰‡ ËÒÍÓχˇ ÔÓ„‡ÏÏ̇ˇ Ú‡ÂÍÚÓˡy(t),u(t)ÒËÒÚÂÏ ̊ (2) ‰ÓÎÊ̇ Û‰Ó‚ÎÂÚ‚ÓˇÚ ̧ „‡ÌË ̃Ì ̊Ï ÛÒÎÓ‚ËˇÏ y(0) =y0, ̇y(0) = ̇y0Ëy(T) = 0, ̇y(T) = 0, „‰ÂT >0ÌÂÍÓÚÓÓ ÍÓÌ ̃ÌÓ Á̇ ̃ÂÌË ÌÂÁ‡‚ËÒËÏÓÈ ÔÂÂÏÂÌÌÓÈt. Õ‡ÔÓÏÌËÏ, ̃ÚÓ Ù‡ÁÓ‚ ̊Ï „‡ÙËÍÓÏ
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    ÙÛÌ͈ËËφ(t)∈Cn[0, T]‚ Ù‡ÁÓ‚ÓÏ ÔÓÒÚ‡ÌÒÚ‚Â ÒËÒÚÂÏ ̊ (1) ̇Á ̊‚‡ ̨Ú ÍË‚Û ̨, Á‡‰‡ÌÌÛ ̨ Ô‡‡ÏÂÚË ̃ÂÒÍË ÔË ÔÓÏÓ ̆Ë Û‡‚ÌÂÌËÈy(i)= =φ(i)(t),i=0, n−1,t∈[0, T]. ՇȉÂÏ ÏÌÓ„Ó ̃ÎÂÌp(t), Ù‡ÁÓ‚ ̊È „‡ÙËÍp(t) = (p(t), ̇p(t)),t∈[0, T], ÍÓÚÓÓ„Ó ÒÓ‰ËÌˇÂÚ ÚÓ ̃ÍËy= (y0, ̇y0)Ëy= (0,0)‚ Ù‡ÁÓ‚ÓÏ ÔÓÒÚ‡ÌÒÚ‚Â ÒËÒÚÂÏ ̊ (2).
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  6. Start
    4300
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    ՇȉÂÏ ÏÌÓ„Ó ̃ÎÂÌp(t), Ù‡ÁÓ‚ ̊È „‡ÙËÍp(t) = (p(t), ̇p(t)),t∈[0, T], ÍÓÚÓÓ„Ó ÒÓ‰ËÌˇÂÚ ÚÓ ̃ÍËy= (y0, ̇y0)Ëy= (0,0)‚ Ù‡ÁÓ‚ÓÏ ÔÓÒÚ‡ÌÒÚ‚Â ÒËÒÚÂÏ ̊ (2). –‡ÒÒÏÓÚËÏ Ò̇ ̃‡Î‡ ÒÎÛ ̃‡È, ÍÓ„‰‡TÁ‡‰‡ÌÓ. —ӄ·ÒÌÓ
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    [1,2]
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    ÒÛ ̆ÂÒÚ‚ÛÂÚ Ú‡ÍÓÈ Â‰ËÌÒÚ‚ÂÌÌ ̊È ÏÌÓ„Ó ̃ÎÂÌ ÒÚÂÔÂÌË3, ËÏ ̨ ̆ËÈ ‚ˉ p(t) =y0+ ̇y0t+c1t2+c2t3,(3) ̃ÚÓ ‚ ̊ÔÓÎÌÂÌ ̊ ÛÒÎӂˡp(T) = 0, ̇p(T) = 0. ŒÚÏÂÚËÏ, ̃ÚÓ ÔË Î ̨· ̊ı Á̇ ̃ÂÌˡı ÔÓÒÚÓˇÌÌ ̊ı c1Ëc2ÒÔ‡‚‰ÎË‚ ̊ ‡‚ÂÌÒÚ‚‡p(0) =y0Ë ̇p(0) = ̇y0.
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  7. Start
    6366
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    ‚Ó (5). œË Ì‡È‰ÂÌÌ ̊ı Á̇ ̃ÂÌˡıc1ËTÏÌÓ„Ó ̃ÎÂÌ (6) ÔËÏÂÚ ‚ˉ p(t) =y0+ ̇y0t+ y ̇20 4y0 t2.(9) ‘‡ÁÓ‚ ̊È „‡ÙËÍp(t) = (p(t), ̇p(t)),t∈[0, T], ÏÌÓ„Ó ̃ÎÂ̇ (9) ÒÓ‰ËÌˇÂÚ ÚÓ ̃ÍËy= (y0, ̇y0) Ëy= (0,0)̇ Ù‡ÁÓ‚ÓÈ ÔÎÓÒÍÓÒÚË, ÔË ̃ÂÏp(T) = 0, ̇p(T) = 0. “ÂÓÂχ ‰Ó͇Á‡Ì‡. œÓ„‡ÏÏÌÓ ÛÔ‡‚ÎÂÌËÂ, ˇ‚Ρ ̨ ̆ÂÂÒˇ  ̄ÂÌËÂÏ ‡ÒÒχÚË‚‡ÂÏÓÈ ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ‰Îˇ ÒËÒÚÂÏ ̊ (2), Á‡ÔË ̄ÂÚÒˇ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ
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    [1,2]
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    : u(t) = 1 g(p(t)) ( p ̈(t) +f(p(t)) ) ,(10) „‰Âp(t)| ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÏÌÓ„Ó ̃ÎÂÌ, ËÏ ̨ ̆ËÈ ‚ˉ (3), (4) ËÎË (9). œËÏÂ.–‡ÒÒÏÓÚËÏ ÚÂÏË̇Π̧ÌÛ ̨ Á‡‰‡ ̃Û ÛÔ‡‚ÎÂÌˡ ‰Îˇ χÚÂχÚË ̃ÂÒÍÓ„Ó Ï‡ˇÚÌË͇, Û‡‚ÌÂÌˡ ‰‚ËÊÂÌˡ ÍÓÚÓÓ„Ó ËÏ ̨Ú ‚ˉ    x ̇1=x2, x ̇2=csinx1−dx2+u, (11) „‰Âx= (x1, x2) Ú ∈R2| ‚ÂÍÚÓ ÒÓÒÚÓˇÌˡ ÒËÒÚÂÏ ̊;u| ÛÔ‡‚Ρ ̨ ̆ËÈ ÏÓÏÂÌÚ, ÍÓÌÒÚ‡ÌÚ ̊ cËdÔÓÎÓÊËÚÂÎ ̧Ì ̊.
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  8. Start
    9449
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    ≈ÒÎË Á̇ ̃ÂÌËÂT >0ÙËÍÒËÓ‚‡ÌÓ, ÚÓ ÒÛ ̆ÂÒÚ‚ÛÂÚ Â‰ËÌÒÚ‚ÂÌÌ ̊È ÏÌÓ„Ó ̃ÎÂÌ ÔˇÚÓÈ ÒÚÂÔÂÌË p(t) =y0+ ̇y0t+ y ̈0 2 t2+c1t3+c2t4+с3t5,(13) Ù‡ÁÓ‚ ̊È „‡ÙËÍp(t) = (p(t), ̇p(t), ̈p(t)),t∈[0, T], ÍÓÚÓÓ„Ó ÒÓ‰ËÌˇÂÚ ÚÓ ̃ÍËy= (y0, ̇y0, ̈y0) Ëy= (0,0,0)‚ Ù‡ÁÓ‚ÓÏ ÔÓÒÚ‡ÌÒÚ‚Â ÒËÒÚÂÏ ̊ (12), ÔË ̃ÂÏp(T) = 0, ̇p(T) = 0, ̈p(T) = 0
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    [1,2]
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    . œÓÒÚÓˇÌÌ ̊Âc1,c2Ëc3Ë ̆ÛÚÒˇ ËÁ ÒÎÂ‰Û ̨ ̆ÂÈ ÒËÒÚÂÏ ̊ ÎËÌÂÈÌ ̊ı ‡Î„·‡Ë ̃ÂÒÍËı Û‡‚ÌÂÌËÈ:      T3T4T5 3T24T35T4 6T12T220T3           c1 c2 c3      =      −y0− ̇y0T− y ̈0 2 T2 − ̇y0− ̈y0T − ̈y0      ,(14)  ̄ÂÌË ÍÓÚÓÓÈ ÒÛ ̆ÂÒÚ‚ÛÂÚ Ë Â‰ËÌÒÚ‚ÂÌÌÓ, Ú‡Í Í‡Í Ï‡Úˈ‡ ̋ÚÓÈ ÒËÒÚÂÏ ̊ Ì‚ ̊ÓʉÂ̇.
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  9. Start
    11765
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    ‚ ̊ÔÓÎÌÂÌˡ ÛÒÎӂˡy0 ̇y0<0. ƒ‡ÎÂÂ, ‚ ÒÎÛ ̃‡Â ÔÓÎÓÊËÚÂÎ ̧ÌÓÒÚËTÙ‡ÁÓ‚ ̊È „‡ÙËÍp(t) = (p(t), ̇p(t), ̈p(t)),t∈[0, T], ÏÌÓ„Ó ̃ÎÂ̇ (15), (17), (18) ÒÓ‰ËÌˇÂÚ ÚÓ ̃ÍËy= (y0, ̇y0, ̈y0)Ëy= (0,0,0)‚ Ù‡ÁÓ‚ÓÏ ÔÓÒÚ‡ÌÒÚ‚Â ÒËÒÚÂÏ ̊ (12), ÔË ̃ÂÏp(T) = 0, ̇p(T) = 0, ̈p(T) = 0. “ÂÓÂχ ‰Ó͇Á‡Ì‡. œÓ„‡ÏÏÌÓ ÛÔ‡‚ÎÂÌËÂ, ˇ‚Ρ ̨ ̆ÂÂÒˇ  ̄ÂÌËÂÏ ‡ÒÒχÚË‚‡ÂÏÓÈ ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ‰Îˇ ÒËÒÚÂÏ ̊ (12), Òӄ·ÒÌÓ
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    [1,2]
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    Á‡ÔË ̄ÂÚÒˇ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ: u(t) = 1 g(p(t)) ( p(3)(t) +f(p(t)) ) , „‰Âp(t)| ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÏÌÓ„Ó ̃ÎÂÌ, ËÏ ̨ ̆ËÈ ‚ˉ (13), (14) ËÎË (15), (17), (18). 3. —ËÒÚÂÏ ̊ ÔÓˇ‰Í‡n>3 –‡ÒÒÏÓÚËÏ ÒËÒÚÂÏÛ (1) ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó ÔÓˇ‰Í‡n. œ‰ÔÓÎÓÊËÏ, ̃ÚÓ Á̇ ̃ÂÌËÂT >0 Á‡‰‡ÌÓ.
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  10. Start
    12045
    Prefix
    ÏÏÌÓ ÛÔ‡‚ÎÂÌËÂ, ˇ‚Ρ ̨ ̆ÂÂÒˇ  ̄ÂÌËÂÏ ‡ÒÒχÚË‚‡ÂÏÓÈ ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ‰Îˇ ÒËÒÚÂÏ ̊ (12), Òӄ·ÒÌÓ [1,2] Á‡ÔË ̄ÂÚÒˇ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ: u(t) = 1 g(p(t)) ( p(3)(t) +f(p(t)) ) , „‰Âp(t)| ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÏÌÓ„Ó ̃ÎÂÌ, ËÏ ̨ ̆ËÈ ‚ˉ (13), (14) ËÎË (15), (17), (18). 3. —ËÒÚÂÏ ̊ ÔÓˇ‰Í‡n>3 –‡ÒÒÏÓÚËÏ ÒËÒÚÂÏÛ (1) ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó ÔÓˇ‰Í‡n. œ‰ÔÓÎÓÊËÏ, ̃ÚÓ Á̇ ̃ÂÌËÂT >0 Á‡‰‡ÌÓ. “Ó„‰‡ Òӄ·ÒÌÓ
    Exact
    [1,2]
    Suffix
    ÒÛ ̆ÂÒÚ‚ÛÂÚ Â‰ËÌÒÚ‚ÂÌÌ ̊È ÏÌÓ„Ó ̃ÎÂÌ ÒÚÂÔÂÌË2n−1, ËÏ ̨ ̆ËÈ ‚ˉ p(t) = n−1∑ k=0 y (k) 0 k! tk+ ∑n k=1 cktn−1+k,(19) Ù‡ÁÓ‚ ̊È „‡ÙËÍp(t) = ( p(t), ̇p(t), . . . , p(n−1)(t) ) ,t∈[0, T], ÍÓÚÓÓ„Ó ÒÓ‰ËÌˇÂÚ ÚÓ ̃ÍË y= (y0, ̇y0, . . . , y (n−1) 0)Ëy= (0, . . . ,0)‚ Ù‡ÁÓ‚ÓÏ ÔÓÒÚ‡ÌÒÚ‚Â ÒËÒÚÂÏ ̊ (1), ÔË ̃ÂÏ p(T) = 0, ̇p(T) = 0, . . . ,p(n−1)(T) = 0. œÓÒÚÓˇÌÌ ̊Âc1,c2, . . . ,cṅıÓ‰ˇÚÒˇ ËÁ ÒËÒÚÂÏ ̊ ÎËÌÂ
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  11. Start
    14478
    Prefix
    – ̄Ë‚ ̋ÚÛ ÒËÒÚÂÏÛ Ë ÔÓ‰ÒÚ‡‚Ë‚ ̇ȉÂÌÌ ̊ Á̇ ̃ÂÌˡc1,c2,. . .,cn−1‚ ÔÓÒΉÌ Û‡‚ÌÂÌË ÒËÒÚÂÏ ̊ (22), ÔÓÎÛ ̃ËÏ ÓÚÌÓÒËÚÂÎ ̧ÌÓ ÌÂËÁ‚ÂÒÚÌÓ„ÓTÒÎÂ‰Û ̨ ̆ Û‡‚ÌÂÌËÂ: y (n−1) 0T n−1+n! 1!(n−2)! y (n−2) 0T n−2+(n+ 1)! 2!(n−3)! y (n−3) 0T n−3+. . . . . .+ (2n−3)! (n−2)!1! y ̇0T+ (2n−2)! (n−1)! y0= 0.(23) —ӄ·ÒÌÓ
    Exact
    [19]
    Suffix
    ̃ËÒÎÓ ÔÓÎÓÊËÚÂÎ ̧Ì ̊ı ÍÓÌÂÈ ÏÌÓ„Ó ̃ÎÂ̇, ÒÚÓˇ ̆Â„Ó ‚ ΂ÓÈ ̃‡ÒÚË ‡‚ÂÌÒÚ‚‡ (23), ‡‚ÌÓ ̃ËÒÎÛ ÔÂÂÏÂÌ Á̇ÍÓ‚ ‚ ÒËÒÚÂÏ ÍÓ ̋ÙÙˈËÂÌÚÓ‚y0, ̇y0. . . ,y (n−1) 0ËÎË ÏÂÌ ̧ ̄ ̋ÚÓ„Ó ̃ËÒ· ̇ ̃ÂÚÌÓ ̃ËÒÎÓ.
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  12. Start
    15567
    Prefix
    ÛÒÎÓ‚ËÂÏ ÒÛ ̆ÂÒÚ‚Ó‚‡Ìˡ ÏÌÓ„Ó ̃ÎÂ̇ ÒÚÂÔÂÌË2n−2, Ù‡ÁÓ‚ ̊È „‡ÙËÍ ÍÓÚÓÓ„Ó ÒÓ‰ËÌˇÂÚ ÚÓ ̃ÍËy= (y0, ̇y0, . . . , y (n−1) 0)Ë y= (0, . . . ,0)‚ Ù‡ÁÓ‚ÓÏ ÔÓÒÚ‡ÌÒÚ‚Â ÒËÒÚÂÏ ̊ (1), ˇ‚ΡÂÚÒˇ ̇ÎË ̃Ë ÔÓÎÓÊËÚÂÎ ̧ÌÓ„Ó ÍÓÌˇ Û ÏÌÓ„Ó ̃ÎÂ̇, ÒÚÓˇ ̆Â„Ó ‚ ΂ÓÈ ̃‡ÒÚË ‡‚ÂÌÒÚ‚‡ (23). Õ‡ÍÓ̈, ÔÓ„‡ÏÏÌÓ ÛÔ‡‚ÎÂÌËÂ, ˇ‚Ρ ̨ ̆ÂÂÒˇ  ̄ÂÌËÂÏ ‡ÒÒχÚË‚‡ÂÏÓÈ ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ‰Îˇ ÒËÒÚÂÏ ̊ (1), ÒÎÂ‰Û ̨ ̆ÂÂ:
    Exact
    [1,2]
    Suffix
    : u(t) = 1 g(p(t)) ( p(n)(t) +f(p(t)) ) ,(24) „‰Âp(t)| ÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËÈ ÏÌÓ„Ó ̃ÎÂÌ (19), (20) ËÎË (21), (22). «‡ÍÎ ̨ ̃ÂÌË ¬ ̇ÒÚÓˇ ̆ÂÈ ‡·ÓÚ ËÒÒΉӂ‡Ì‡ ÚÂÏË̇Π̧̇ˇ Á‡‰‡ ̃‡ ÛÔ‡‚ÎÂÌˡ ‰Îˇ ‡ÙÙËÌÌ ̊ı ÒËÒÚÂÏ, ËÏ ̨ ̆Ëı ‚ˉ (1).
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  13. Start
    16208
    Prefix
    – ̄ÂÌË ÚÂÏË̇Π̧ÌÓÈ Á‡‰‡ ̃Ë ÔÓÒÚÓÂÌÓ Ì‡ ÓÒÌÓ‚Â ÏÌÓ„Ó ̃ÎÂ̇ ÒÚÂÔÂÌË 2n−2. œ˂‰ÂÌÓ Ú‡ÍÊ ËÁ‚ÂÒÚÌÓ  ̄ÂÌË ÔË ÔÓÏÓ ̆Ë ÏÌÓ„Ó ̃ÎÂ̇ ÒÚÂÔÂÌË2n−1. ŒÚÏÂÚËÏ, ̃ÚÓ ÌÂÓ·ıÓ‰ËÏ ̊Â Ë ‰ÓÒÚ‡ÚÓ ̃Ì ̊ ÛÒÎӂˡ ÒÛ ̆ÂÒÚ‚Ó‚‡Ìˡ Á‡ÏÂÌ ̊ ÔÂÂÏÂÌÌ ̊ı, ÔÂÓ·‡ÁÛ ̨ ̆ÂÈ ‡ÙÙËÌÌÛ ̨ ‰Ë̇ÏË ̃ÂÒÍÛ ̨ ÒËÒÚÂÏÛ Í ‚Ë‰Û (1), ÏÓÊÌÓ Ì‡ÈÚË, ̇ÔËÏÂ, ‚ ÏÓÌÓ„‡ÙËË
    Exact
    [1]
    Suffix
    . ƒ‡Î ̧ÌÂÈ ̄Ë ËÒÒΉӂ‡Ìˡ ÏÓ„ÛÚ · ̊Ú ̧ Ò‚ˇÁ‡Ì ̊ Ò  ̄ÂÌËÂÏ Ì‡ ÓÒÌÓ‚Â ÏÌÓ„Ó ̃ÎÂÌÓ‚ ÒÚÂÔÂÌË2n−2ÚÂÏË̇Π̧Ì ̊ı Á‡‰‡ ̃ ÛÔ‡‚ÎÂÌˡ ÔË ÔÓËÁ‚ÓÎ ̧ÌÓÏ ÍÓÌ ̃ÌÓÏ ÒÓÒÚÓˇÌËË ÒËÒÚÂÏ ̊ (1), Ì ÒÓ‚Ô‡‰‡ ̨ ̆ËÏ Ò Ì‡ ̃‡ÎÓÏ ÍÓÓ‰Ë̇Ú.
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