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Λ 2.1 2004.2.20 1 Λ 1

2 Ay = u 2 2 A 2 u " # a 11 a 12 A = ; u = a 21 a 22 " # u 1 u 2 y Ay = u (1) A (1) y = A 1 u y A 2 x i i i =1; 2 Ax 1 = 1 x 1 ; Ax 2 = 2 x 2 (2) x 1 x 2 =0 (3) (3) (2) x 1 x 2 x 1 x 1 =1; x 2 x 2 =1 (4) 2 2 (1) x 1 x 2 2 u x 1 x 2 1 u = U 1 x 1 + U 2 x 2 (5) U 1 U 2 u x 1 x 2 U 1 U 2 x 1 x 2 U 1 U 2 (5) U 1 = x 1 u; U 2 = x 2 u (6) x 1 u = x 1 (U 1 x 1 + U 2 x 2 ) = x 1 (U 1 x 1 )+x 1 (U 2 x 2 ) 2

u U 2 x 2 x 2 U 1 x 1 x 1 1: x 1 x 2 u U 1 U 2 (3) (4) x 1 u = U 1 (x 1 x 1 )+U 2 (x 1 x 2 ) = U 1 (6) U 1 U 2 y y x 1 x 2 Y 1 Y 2 y y = Y 1 x 1 + Y 2 x 2 (7) (1) y Y 1 Y 2 (5) (7) (1) A (Y 1 x 1 + Y 2 x 2 )=U 1 x 1 + U 2 x 2 (8) Y 1 Y 2 x 1 x 2 (8) A (Y 1 x 1 + Y 2 x 2 ) = Y 1 (Ax 1 )+Y 2 (Ax 2 ) = Y 1 ( 1 x 1 )+Y 2 ( 2 x 2 ) = ( 1 Y 1 ) x 1 +( 2 Y 2 ) x 2 (9) (8) (9) ( 1 Y 1 ) x 1 +( 2 Y 2 ) x 2 = U 1 x 1 + U 2 x 2 (10) 3

(1) (10) x 1 x 2 (10) 1 Y 1 = U 1 ; 2 Y 2 = U 2 (11) y y Y 1 = U 1 1 ; Y 2 = U 2 2 (11) y = U 1 1 x 1 + U 2 2 x 2 4

3 A 2 A 2 A (2) A 2 x 1 x 2 A 2 x 1 = A (Ax 1 ) = A ( 1 x 1 ) = 1 (Ax 1 ) = 2 1 x 1 A 2 x 1 2 1 x 2 A A 2 A 2 A 2 y + Ay+ y = u (12) y A 2 y (7) A 2 (Y 1 x 1 + Y 2 x 2 )=( 2 1 Y 1) x 1 +( 2 2 Y 2) x 2 (13) (13) (12) (5) (7) ( 2 1 Y 1 + 1 Y 1 + Y 1 ) x 1 +( 2 2 Y 2 + 2 Y 2 + Y 2 ) x 2 = U 1 x 1 + U 2 x 2 (14) x 1 x 2 (14) Y 1 Y 2 2 1 Y 1 + 1 Y 1 + Y 1 = U 1 2 2 Y 2 + 2 Y 2 + Y 2 = U 2 Y 1 = U 1 2 1 + 1 +1 ; Y 2 = U 2 2 2 + 2 +1 (7) (12) y y = U 1 2 1 + 1 +1 x 1 + U 2 2 2 + 2 +1 x 2 A 2 2 n n (12) a 2 A 2 y + a 1 Ay+ a 0 y = b 0 u 5

y y = = b 0 U 1 b 0 U 2 b 0 U n a 2 2 1 + a x 1 + 1 1 + a 0 a 2 2 2 + a x 2 + :::+ 1 2 + a 0 a 2 2 + a x n n 1 n + a 0 nx i=1 b 0 U i a 2 2 + a x i i 1 i + a 0 6

4 dy(t) = u(t) (15) y(t) t u(t) y(t) (15) d y(t) =u(t) (16) u(t) y(t) t u(t) t 1 t 2 ::: u(t 1 ) u(t 2 ) ::: 2 u(t) (16) d A y Ay y(t) (16) (1) d u(t) y(t) 3 d d d e j!t d ej!t = j! e j!t (17) e j!t d j! ej!t (17)!! e j!t u 6 - t 2: 7

y 6 - t d dy 6 - t u 6 - t 3:! e j!t U(j!) u(t) e j!t u(t) U(j!) u(t) = 1 1 U(j!) ej!t d! (18)! (18) (18) U(j!) (6) f (t) g(t) hf; gi hf; gi = 0 f (t) g(t) (19) f (t) f (t) (19) n (19) f (t) t hf; fi 0 n 2 8

(19) he j! 1t ;e j! 2t i =ffi(! 2! 1 ) (20) ffi( )! 1 6=! 2 ffi(! 2! 1 )=0 (20) e j! 1t e j! 2t U(j!) U(j!) = he j!t ;ui (21) = = e j!t u(t) (22) 0 u(t) e j!t (23) 0 (18) (23) (16) u(t) (16) y(t) (18) y(t) Y (j!) y(t) = 1 y(t) (18) (24) (16) d 1 1 1 Z 1 1 1 1 Y (j!) ej!t d! (24) = 1 1 Y (j!) ej!t d! d (Y (j!) ej!t ) d! = 1 1 Y (j!)(j! ej!t ) d! = 1 1 j! Y (j!) ej!t d! = 1 e j!t (25) (26) Y (j!) 1 U(j!) ej!t d! 1 U(j!) ej!t d! 1 U(j!) ej!t d! 1 U(j!) ej!t d! (25)! j! Y (j!)=u(j!) (26) Y (j!)= U(j!) (24) (27) Y (j!) y(t) j! (27) 9

5 d 2 2 n n 2 d 2 2 ej!t = d (j! ej!t ) = (j!) d ej!t = (j!) 2 e j!t e j!t (j!) 2 n n =2 n >2 d 2 y(t) 2 + dy(t) (28) d 2 + y(t) =u(t) (28) y(t) + d y(t) +y(t) =u(t) (29) 2 u(t) y(t) (24) d 2 y(t) = d 1 2 2 2 = 1 = 1 Z 1 1 d 2 1 Y (j!) ej!t d! 2 (Y (j!) ej!t ) d! 1 (j!)2 Y (j!) e j!t d! (29) (18) (24) 1 1 ((j!)2 + j! +1)Y (j!) e j!t d! = 1 1 U(j!) ej!t d! (30) e j!t! (30) Y (j!) ((j!) 2 + j! +1)Y (j!)=u(j!) (31) Y (j!)= U(j!) (j!) 2 + j! +1 (32) 10

y(t) (29) d 2 a 2 y(t) +a d 2 1 y(t) +a d 0 y(t) =b 1 u(t) +b 0 u(t) (33) Y (j!) Y (j!)= b 1 (j!)+b 0 a 2 (j!) 2 + a 1 (j!)+a 0 U(j!) (34) d A ψ! ψ! e j!t ψ! j! ψ! ψ! 11

6 (23) u(t) t!1 t!1 u(t) lim u(t) =1 t!1 c>0 ^u(t) =u(t) e ct (35) lim ^u(t) =0 (36) t!1 u(t) =e t c =2 ^u(t) =e t e 2t = e t (36) (35) ^u(t) ^u(t) ^U (j!) ^u(t) = 1 1 ^U (j!) e j!t d! (37) u(t) =^u(t) e ct u(t) u(t) = ect 1 ^U (j!) e j!t d! (38) e ct u(t) = 1 = 1 1 ^U (j!) e ct e j!t d! 1 ^U (j!) e (c+j!) t d! (39) 12

e ct u(t) ^u(t) ^U (j!)! U(s) e ct 4: ^u(t) u(t) u(t) ^u(t) c>0 c>0 ^u(t) u(t) ^U(j!) (37) (18) (23) ^U(j!)= (35) ^U (j!) = = 0 ^u(t) e j!t (40) u(t) e ct e j!t 0 u(t) e (c+j!) t (41) 0 (39) (41) c + j! ^u(t) ^U(j!)= ^U(Im(c + j!)) j! c + j! ^U U c + j! s (39) u(t) = 1 j = 1 j 1 ^U (j!) e (c+j!) t jd! Z c+j1 c j1 U(s) est ds (42) (41) U(s) = u(t) e st (43) 0 13

(42) (43) 4 (39) (41) (42) (43) s c + j! (43) s (43) s c (38) c s = c j1 s = c + j! c c (42) (42) c 4 14

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