Bài toán ổn định hóa hệ phương trình vi phân phi tuyến có trễ

I HC THI NGUYấN TRNG I HC S PHM NGUYN MINH TRANG BI TON N NH HểA H PHNG TRèNH VI PHN PHI TUYN Cể TR Chuyờn ngnh: Toỏn gii tớch Mó s: 60.46.01.02 LUN VN THC S TON HC NGI HNG DN KHOA HC: GS TSKH V NGC PHT THI NGUYấN - 2016 ổ tr s t ợ t t õ ữỡ tr t õ tr ữủ t tự tổ ổ trũ ợ ổ tr ổ ố t ữớ t r ỡ ổ tọ ỏ t ỡ tợ ụ Pt ữớ ữợ t t t ữợ tổ ỳ t qỵ tổ õ t t ổ ụ tọ ỏ t ỡ t tợ ỏ t ổ t trữớ sữ ú ù t tổ tr sốt q tr t ự tổ ụ ỷ ỡ t tợ ổ ụ t t ủ tổ tr sốt q tr t t ữớ t r ử ỡ ử ởt số ỵ t tt ỡ s t ữỡ tr t õ Pữỡ t õ trủ t õ ữỡ tr t õ tr ữỡ tr õ tr ữỡ tr t ổtổổ õ tr ữỡ tr t ổ ổtổổ õ tr t t r ỵ tt t ỹ t õ õ trỏ rt q trồ ỹ ự t tố tr t ởt ữợ ự ổ t t tr ỵ tt ữỡ tr ỵ tt tố ự ởt tr ỳ t t q trồ ỵ tt t ỹ ữủ sỷ tr ỹ ỡ t ỵ t tt õ ởt tữủ ởt tố ữủ t tr t õ ọ ỳ trú tố ổ tố t s ợ tr t õ ỹ ự t tố ữủ t tứ ố t t tr t ởt ữợ ự ổ t t tr ỵ tt ữỡ tr ỵ tt tố ự ứ ỳ t s s ợ sỹ t tr ỵ tt ự t t t tố ữớ t t ự t x(t) = f (t, x(t), u(t)), t 0(0.1) t õ t ữủ u(t, x) = h(t, x) s ỹ x(t) = f (t, x(t), h(t, x(t))) = F (t, x(t)) t t tr t r t õ tờ qt tữớ ữủ ổ õ ợ t ữủ q st ữ õ ởt tố t ữủ s tố ự ợ õ tr t tố ữủ t tr t ỡ s t t õ ỵ tt ỹ tr ỳ t q t t ữớ t ự t tr ự t õ tố ữủ tr tr ữỡ ữỡ tr ỡ s t ữỡ tr ữỡ tr ỵ tt t õ q ữỡ tr t ữỡ tr t õ tr ữỡ tr t ổtổổ õ tr ữỡ tr t ổ ổtổổ õ tr ởt số ỵ t tt R+ ủ số tỹ ổ Rn ổ n < x, y > xT y ổ ữợ tỡ x, y x tỡ x Rnìr ổ tr n ì r AT tr A I tr ỗ t (A) tr r max (A) = max{Re : (A)} (A) tr ữủ (A) = max (AT A) L2 ([0, t], Rn ) tr à(A) = max (A + AT ) ổ t tr [0, t] tr tr Rn A0 tr ổ A>0 tr ữỡ C([h, 0], Rn ) ổ tử tr [h, 0] tr tr Rn à(A) xt = sups[h,0] x(t + s) BM + (0, ) ủ tr ổ tr [0, ) ữỡ ỡ s t ữỡ tr ởt số tự ỡ s t ữỡ tr ữỡ t õ trủ ữỡ ữủ tr tứ t ữỡ tr ữỡ tr ổ t ữỡ tr rớ r x(t) = f (t, x(t), u(t)), t 0, x(k + 1) = f (k, x(k), u(k)), k = 0, 1, 2, tr õ x(t)(x(k)) Rn tỡ tr t u(t)(u(k)) Rm , n m, tỡ f (t, x, u) : R+ ì Rn ì Rm Rn ố tữủ tr ổ ỹ ữủ ổ t ữ ỳ ỳ õ t q trồ ự ự õ t ữ sỹ r tố ữ t ởt tố ởt ổ t ữủ ổ t ữỡ tr t t sỹ r u(t) x = f (t, x, u) x(t) ởt tr ỳ t tố t s tố r õ ỳ t t t ố ổ tữớ ởt tố õ tứ tr s tr õ t tỹ ữỡ ữợ t ự ỳ t tố r ữớ t t t ữủ t õ t tố ữ r ú t t t õ t õ t õ t õ ỡ s t t õ ỵ tt ỹ tr ỳ t q t t ữớ t ự t tr ự t õ tố ởt tr ỳ t t q trồ ỵ tt t ỹ ữủ sỷ tr ỹ ỡ t ỵ t õ ởt tữủ ởt tố ữủ t ởt tr t õ ọ ỳ trú tố ổ tố t ỷ ỳ ú t õ 2b(p + )eh y(t h(t)) y(t) ) y(t h(t)) 3 + b2 (p + )2 e2h y(t) , (1 ) (1 < B(t)B T (t)y(t), y(t) > B y(t) , < (A (t) + AT (t))y(t), y(t) > 2à(A ) y(t) , < A1, (t)AT1, (t)y(t), y(t) > (A1, ) y(t) õ V (t, xt ) < [P (t) + AT (t)P (t) + P (t)A (t) P (t)Q(t)P (t) + I]y(t), y(t) > [ 2a(p + ) b2 (p + )2 e2h c(p + )2 B ] y(t) (1 ) P (t) ú t õ V (t, xt ) [ 2a(p + ) b2 (p + )2 e2h c(p + )2 B ] y(t) (1 ) õ (i1 , i2 , i3 ) t õ V (t, yt ) 0, t ỡ ỳ t t y(t, ) N > : y(t, ) N , t tr x(t, ) ú t ữủ x(t, ) N et , t 0, s r sỹ ụ õ số N t tứ t ú t õ V (t, yt ) V (0, y0 ), t r tứ s r y(t) V (t, yt ) V (0, y0 ) ứ V (0, y0 ) (p + + h ) y(t) N t tr x(t) ú t ữủ x(t, ) N et , t ỵ ữủ ự t t ổ ổtổổ õ tr x(t) = A(t)x(t) + A1 (t)x(t h(t)) + B(t)u(t) + f (t, x(t), x(t h(t)), u(t)), 1 ợ (t) C([ , 0], R2 ) tr h(t) = t 2 2sin ( ) e sint a(t) A(t) = , A1 (t) = b(t) e cost cos t + B(t) = sin t + 1 x1 (t)sin[x2 (t h(t))] x2 (t h(t))sin[tx1 (t h(t))] f (t, ) = u2 (t)cos[tx(t)] õ 1 a(t) = (cos4 t + 4cos2t + 4)et 4et , 2 1 b(t) = ( sin4 t cos2t + 1)et 4et t f (t, x(t), x(t h(t)), u(t)) ữủ 1 x(t) + x(t h(t)) + u(t) , 1 1 tr õ a = , b = , c = , h = , = = ú t õ 2 sint a(t) + A (t) = A1, (t) = cost b(t) + f (t, x(t), x(t h(t)), u(t)) à(A ) = 1, B = 3, (A1, ) = , = , = 2, =4+ 11 , = 16 cos (t) + 4cos2t + Q(t) = , sin t cos2t + ữủ t e P (t) = 0, t e t R+ , ú t õ t tr ữủ tt ỵ õ õ ữủ t (1 4e )(cos t + 2) u(t) = x(t) t (1 4e )( sin t + 1) ú ỵ r ổ tở số a, b, c t f (.), t ỵ ú ợ ọ t i1 i3 é ỵ ữợ t t rss ú t s t ữủ t q tốt ỡ ỵ s ú ợ sỹ t tũ ỵ t số a, b, c ổ ọ ợ số ữỡ , , h, i , i = 1, 2, 3, a, b, c ú t t P (t) = P (t) + I, = a + b2 + c2 , 2h (1 ) 1e à(A) = sup à(A(t)), tR+ (A1 ) = sup (A1 (t)), tR+ Q(t) = B(t)B T (t) A1 (t)AT1 (t) I, 2h (1 ) 1e M = p + + h + 2h2 , N = = + + + he2h + M , + B +2à(A) + (A1 ) 2h (1 ) 1e ỵ > tt r tỗ t số ữỡ , 1, 2, ởt tr số P BM + (0, ) tọ ữỡ tr t s P (t) + AT (t)P (t) + P (t)A(t) P (t)Q(t)P (t) + 2( + )P (t) + I = (RDE2) õ ữủ ợ ữủ u(t) = B T (t)[P (t) 2I]x(t) ỡ ỳ x(t, ) tọ x(t, ) N et , t ự sỷ u(t) = K(t)x(t) õ K(t) = 12 B T (t)[P (t) 2I], t ợ õ ú t t rss s V (t, xt ) = V1 + V2 + V3 + V4 , V1 =< P (t)x, x >, V2 = x(t) , t V3 = e2(st) x(s) ds, th(t) t e2(s+ht) x(s) dsdr V4 = h t+rh(t+r) t r x(t) V (t, xt ) xt , t 0, ợ số ữỡ , V (t, xt ) t t t x(t) ú t õ V + V = < P (t)x(t), x(t) > +2 < P (t)x(t), x(t) > +2 < x(t), x(t) > = < (P (t) + AT (t)P (t) + P (t)A(t) P (t)B(t)B T (t)[P (t) 2I])x(t), x(t) > + < P (t)A1 (t)x(t h(t)), x(t) > + < P (t)f (t, x(t)x(t h(t)), u(t)), x(t) > + < (A(t) + AT (t) B(t)B T (t)[P (t) 2I])x(t), x(t) > + < A1 x(t h(t)), x(t) > +2 < f (t, x(t)x(t h(t)), u(t)), x(t) > = < [P + AT P + P A P BB T P ]x(t), x(t) > + < P (t)B(t)B T (t)x(t), x(t) > + < [2 B(t)B T (t) + (A(t) + AT (t))]x(t), x(t) > +2 < P (t)A1 (t)x(t h(t)), x(t) > + < A1 (t)x(t h(t)), x(t) > + < P (t)f (t, x(t)x(t h(t)), u(t)), x(t) > V (t, xt ) = 2V3 (t, xt ) + 2V3 (t, xt ) + x(t) x(t) e2h(t) (1 h(t)) x(t h(t)) e2h (1 ) x(t h(t)) 2 V (t, xt ) = 2V4 (t, xt ) + he2h x(t) 2e 2h (1 ) x(t + s h(t + s)) ds h 2V4 (t, xt ) + he2h x(t) é õ t õ V (t, xt )+2V (t, xt ) = V (t, x) + 2V1 (t, x) + V (t, x) + 2V2 (t, x) + V (t, xt ) + 2V3 (t, xt ) + V (t, xt ) + 2V4 (t, xt ) < [P (t) + AT (t)P (t) + P (t)A(t) P (t)B(t)B T (t)P (t)]x(t), x(t) > + < P (t)B(t)B T (t)x(t), x(t) > + < [2 B(t)B T (t) + (A(t) + AT (t))]x(t), x(t) > + < P (t)A1 (t)x(t h(t)), x(t) > + < A1 (t)x(t h(t)), x(t) > + < P (t)f (t, x(t)x(t h(t)), u(t)), x(t) > + 2[< P (t)x(t), x(t) > + x(t) ] + x(t) e2h (1 ) x(t h(t)) + he2h x(t) 2 < [P (t) + AT (t)P (t) + P (t)A(t) P (t)B(t)B T (t)P (t) + 2P (t)]x(t), x(t) > + < P BB T x(t), x(t) > + < [2 BB T + (A(t) + AT (t))]x(t), x(t) > +2 < P (t)A1 (t)x(t h(t)), x(t) > 2h (1 ) < x(t h(t)), x(t h(t)) > 2h (1 ) 1e < x(t h(t)), x(t h(t)) > 1e + < P (t)f (t, x(t)x(t h(t)), u(t)), x(t) > + (2 + 1e + he2h ) x + < A1 x(t h(t)), x > 2h (1 ) x(t h(t)) ỷ t ữủ < P (t)A1 (t)x(t h(t)), x(t) > 1e 2h (1 ) < x(t h(t))x(t h(t)) > 3 < P (t)A1 (t)AT1 (t)P (t)x(t), x(t) >, 2h (1 ) 1e < A1 (t)x(t h(t)), x(t) > 2h (1 ) < x(t h(t)), x(t h(t)) > 3 < A1 (t)AT1 (t)x(t), x(t) > 2h (1 ) 1e 1e t t õ < P (t)f (t, x(t)x(t h(t)), u(t)), x(t) > xT (t)P (t) f (t, x(t)x(t h(t)), u(t)), x(t) 2a xT (t)P (t) x(t) + 2b xT (t)P (t) x(t h(t)) + 2c xT (t)P (t) u(t) tử sỷ ú t õ < P (t)f (t, x(t)x(t h(t)), u(t)), x(t) > a xT (t)P (t) + x(t) + b2 xT (t)P (t) 2h (1 ) 1e 2h (1 ) x(t h(t)) + c2 xT (t)P (t) + u(t) 3 b2 + c2 ]P2 (t)x(t), x(t) > < [ a + 2h (1 ) 1e 2h (1 ) 1e + x(t) + x(t h(t)) + < [P (t) 2I]B(t)B T (t)[P (t) 2I]x(t), x(t) > 1e + 2 < [P (t) + 2P (t) + I]x(t), x(t) > + x(t) + 2h (1 1e ) x(t h(t)) + < [ P (t)B(t)B T (t)P (t) P (t)B(t)B T (t) + B(t)B T (t)]x(t), x(t) > tứ t õ V (t, xt ) + 2V (t, xt ) < [P (t) + AT (t)P (t) + P (t)A(t) + 2( + )P (t)]x(t), x(t) > < [ 43 P (t)B(t)B T (t)P (t) P (t)A1 (t)AT1 (t)P (t) 2h (1 ) 1e 2 P (t)]x(t), x(t) > + < [3 B(t)B T (t) + (A(t) + AT (t)) + 2h A1 (t)AT1 (t)]x(t), x(t) > (1 ) 1e +(2 + + + he2h + ) x(t) ú ỵ r < B(t)B T (t)x(t), x(t) > B x(t) , < (A(t) + AT (t))x(t), x(t) > 2à(A) x(t) , < A1 (t)AT1 (t)x(t), x(t) > (A1 ) x(t) , t V (t, xt ) + 2V (t, xt ) < [P (t) + AT (t)P (t) + P (t)A(t) P (t)Q(t)P (t) + 2( + )P (t)]x(t), x(t) > +[2 + + + he2h + + B +2à(A) + (A1 )] x(t) 2h (1 ) e < [P (t) + AT (t)P (t) + P (t)A(t) P (t)Q(t)P (t) +2( + )P (t) + I]x(t), x(t) > ứ Pt ú t õ V (t, xt ) + 2V (t, xt ) 0, t V (t, xt ) 2V (t, xt ), t 0, ỷ t số N t tứ t ú t õ V (t, xt ) V (0, x0 )e2t , t t tứ V (t, xt ) t õ x(t) V (t, xt ), t 0, s r x(t, ) V (0, x0 ) t e , t ỡ ỳ t õ V (0, x0 ) (p + + h ) + e2(s+h) x(s) dsdr, h rh(r) ((p + + h ) + 2h2 M 2, x(t, ) N et , t ỵ ữủ ự ú ỵ r số tr t t t tr ự ổ tọ r ọ i1 i3 õ õ ữủ ợ t tũ ỵ t t ổ ổtổổ ợ tr t x = A(t)x(t) + A1 (t)x(t h(t)) + B(t)u(t) + f (t, x(t), x(t h(t)), u(t)), ợ tr C([ , 0], R2 ) tr t h(t) ữ tr 2 2x2 (t)sin[tx1 (t h(t))] 2u1 (t)cos[t x(t)] f (t, x(t), x(t h(t)), u(t)) = 2x1 (t h(t))sin[x(t)x1 (t h(t))] a(t) A(t) = , b(t) sint A1 (t) = 2cost , 2(sin t + 3) B(t) = , 2 (cos t + 4) 11 a(t) = e4t ( sin4 t + ) 5e4t , 2 b(t) = e4t cos4 t 5e4t ìợ ữủ f (t, x(t), x(t h(t)), u(t)) f (t, x(t), x(t h(t)), u(t)) x(t) + x(t h(t)) + u(t) 1 h = , = , a = 2, b = 2, c = 2, 2 à(A) = 2, B = 8, (A1 ) = ợ = = , 16 = 3e , 2 =( 991 3), 64e = 1, õ = 16, = 10, 3sin t + 11 Q(t) = cos t ữủ t 4t e + P (t) = 0, t R e4t õ ữủ ợ ữủ 4t (sin t + 3)( e ) u(t) = x(t) 1 (cos2 t + 4)( e4t ) t tr ữủ s r ữỡ tr ữỡ tr õ tr ữỡ tr t õ q r t t õ ữỡ tr õ tr ữỡ tr t ổtổổ õ tr ữỡ tr t ổ ổtổổ õ tr ợ ự tt t t ụ Pt ổ ỵ tt t ố trt t t rt qts rr r Pt t stt ts rt qts t qts r t ss ts Pt r r t stt tr ssts t r rtr ts tr r ... Qi = P Qi P , i = 1, 2, 3, S j = P Sj P , j = 1, 2, 3, 4, R1 = P R1 P , R2 = P R2 P t rss ữ s Vi (t, xt ), t 0, V (t, xt ) = i=1 õ V1 (t, xt ) =xT (t)P x(t), t t e2(st) xT (s)Q1 x(s)ds +... (t)[P (A + BK) + (A + BK)T P + 2P P + a2 F T F ]x(t) + d2 xT (t h(t))GT Gx(t h(t)) tử Vi (t, xt ), i = 2, 3, 4, t t ữủ V (t, xt ) 2V2 (t, xt ) + xT (t)[Q1 + Q2 ]x(t) e2h1 xT (t