ổn định mũ cho hệ suy biến có trễ

β− I TV N II BÀI TOÁN VÀ CÁC KI N TH C C B N Ta kí hi u Cn ,h := C ([ −h,0] , n ) không gian Banach ánh x liên t c t o n [ − h,0] vào chu n c a n , xt := x ( t + s ) , ∀s ∈ [ −h,0] Khi ó v i ∀xt ∈ Cn ,h c xác nh b i xt := sup x ( t + s ) = φ C V i ma tr n A s∈[ − h ,0 ] có ph n t thu c t p s th c ,ta có A := λmax ( AT A ) A n a nhóm xác nh d ng ( A ≥ ) n u xT Ax ≥ , A c g i c g i xác nh ng ( A > ) n u xT Ax > Xét h ph ng trình vi phân suy bi n khơng ch c ch n có tr (uncertain singular time-delay system) sau : d Ex ( t ) = ( A + ∆A ) x ( t ) + ( Ad + ∆Ad ) x ( t − h ) x ( t ) = φ ( t ) , t ∈ [ − h,0] (1) ây x ( t ) ∈ n , bi n tr ng thái bi n i u n E , A, Ad ma tr n h ng v i ph n t s th c v i s chi u thích h p < rankE = r < n φ ( t ) ∈ Cn ,h hàm i u ki n ban u ∆A, ∆Ad ma tr n th i gian b t bi n,và ∆A = DFH , ∆Ad = DFH d (2a) T F F≤I (2b) Trong ó D, Dd , H , H d ma tr n h ng có s chi n thích h p F , Fd ma tr n h ng không ch c ch n Tr c h t,chúng ta s a nh ngh a b cho h suy bi n tr th i gian sau Ex ( t ) = Ax ( t ) + Ad x ( t − h ) x ( t ) = φ ( t ) , t ∈ [ −h,0] ( 3) nh ngh a 1: i H (3) c g i regular n u a th!c "c tr ng det ( sE − A ) không #ng nh t b ng không ii H (3) c g i impulse-free n u deg ( det ( sE − A ) ) = rankE B N u h (3) regular impulse-free ,khi ó v i t t c$ hàm i u ki n ban u φ ( t ) ,h (3) t#n t i nh t nghi m [ 0,+∞ ) nh ngh a 2: i Cho β > Nghi m không c a h (3) c g i β − n nh m% n u t#n t i s N > cho v i b t k& nghi m x ( t ) c a h (3) v i hàm i u ki n ban u φ (t ) , u tho$ mãn x ( t ) ≤ Ne − β t φ ( t ) C ii H (3) c gi β − n nh m% n u nghi m không c a h β − n nh m% B 2: Gi$ s r ng a ( ⋅) ∈ n , b ( ⋅) ∈ n N ( ⋅) ∈ n ×n xác nh t p Ω Khi a b a b ó v i b t k& ma tr n X ∈ n ×n , Y ∈ n ×n , Z ∈ X Y ≥ , ta có b t 'ng th!c sau: ∗ Z a −2 Ω a ( s )N ( s ) b ( s ) ds ≤ a a a(s) T Ω b(s) T b X ∗ nb × nb tho$ mãn Y − N (s) Z a(s) b( s) ds B 3: Cho ma tr n Q ≥ ,n u ph n t qi n m ng chéo c a Q b ng khơng ,thì c t hàng i qua qi c%ng b ng không B Cho Ma tr n i x!ng ()c nh d ng Q *+, -)c ma tr n M , N , -., s , chi u /01 ch h p,ta -., b t 'ng th!c sau: T M N + N T M ≤ N T QN + M T Q −1 M III M T S K T QU CHÍNH nh lý Cho β > H suy bi n có ch m (singular time-delay system) (3) regular ,impulse-free β − n nh m% ,n u t#n t i ma tr n i x!ng Q > 0, X ≥ 0, Z > P, Y tho$ mãn PE = E T PT ≥ , ( 4a ) L PAd − Y + hAT ZAd ∗ − Q + hAdT ZAd ( 4b ) ,do V ( yt ) + β V ( yt ) ≤ −λmin ( H ) y T ( t ) y ( t ) ≤ −λmin ( H ) y1T ( t ) y1 ( t ) = −λmin ( H ) y1 ( t ) ≤ ;1ch phân hai v , (29) t , suy : λmin ( P11 ) y1 ( t ) ≤ V ( yt ) ≤ V ( y0 ) e −2 β t ( ) n t,ta ( c V ( yt ) ≤ V ( y0 ) e−2 β t ,nên t , (18) ta ≤ λmax ( P11 ) + hλmax Q + h A Z A + A Z Ad + Ad Z Ad T ( 29 ) T T ) e −2 β t φ C y1 ( t ) ≤ ( λmax ( P11 ) + hλmax ( Q ) + h A Z A + A Z Ad + Ad Z Ad T λmin ( P11 ) T T ) e− β t φ Nh v y ta %,th t v y T , (12) *+, (14) /02, ,ta - T P22 + P22 + Q22 + e β h P22 Ad 22 Q22−1 Ad 22T P22T < C ( 30 ) nh >%, Suy P22 + P22T + Q22 + e β h P22 Q22−1 P22T + e β h P22 Ad 22 Q22−1 Ad 22T P22T − P22 Q22−1 P22T < + Q22 ) Q22−1 ( P22T + Q22T ) + P22 22 (e (P (e A )Q (e ρ (e A ) < βh d 22 −1 22 βh βh Ad 22 ) Q22−1 ( e β h Ad 22 ) − Q22−1 P22T < T Ad 22 ) − Q22−1 < T ( 31) βh d 22 Do ρ ( e β h Ad 22 ) < , *+,Q22 > nên t#n /i s ,D > , *+δ ∈ ( 0,1) cho : ( 32 ) e β kh Ad 22 k ≤ Dδ k , k = 0,1,2, V i b t 9&,t ≥ ,luôn t#n /i s , nguyên d T , (17) ,suy r ng y2 ( t ) = − Ad 21 y1 ( t − h ) − Ad 22 y2 ( t − h ) = ( − Ad 22 ) y2 ( t − kh ) − k k i =1 Suy y2 ( t ) ≤ ( Ad 22 ) φ k C + Ad 21 (−A ) i −1 d 22 ng k cho ( k − 1) h ≤ t ≤ kh Ad 21 y1 ( t − ih ) D N e− β t φ 1−δ = Ad 22e β h Ad 22e β h ⋅ ⋅ ⋅ Ad 22e β h e − β kh φ C k C + Ad 21 k D N e− β t φ 1− δ C *+ Trong , hi = kh , i =1 N= ( λmax ( P11 ) + hλmax ( Q ) + h A Z A + A Z Ad + Ad Z Ad T T λmin ( P11 ) A2,kh ≥ t ,nên; y2 ( t ) ≤ T (A ) d 22 k e β kh e − β t φ K t h p (32) *+, (33),ta -.: y2 ( t ) ≤ C + Ad 21 D N e− βt φ 1−α D D + Ad 21 N e− β t φ 1− δ 1−δ ) C ( 33) C ( 34 ) T , (30) *+, (34) ,suy i u 30$i ch!ng minh.Nh v y ch!ng minh Bây gi , ta B a b , , quan /C ng 7Dng nh =B, cho Q + ε HH T + ε −1M T M < nh Cho β > H suy bi n có ch m khơng ch c ch n(singular uncertain time-delay system) (1) regular ,impulse-free β − n nh m% ,n u t#n t i ma tr n i x!ng Q > 0, X ≥ 0, Z > , P, Y , *+, s , d ng ε ,v i P , =+, không suy bi n, tho$ mãn ( 35a ) EPT = PE T ≥ , PH T εD h PAdT 0 ∗ − hZ ε hD ∗ ∗ ∗ − hI ∗ ∗ ∗ ∗ − hI PAT + PT A PAd Ad PT − Y hPAT ∗ − e −2 β h Q ∗ +Q + hX + Y + Y T