4 ENGINEERING BACKGROUN D i(t) f e(t) £ n FIGURE 1.1 . Blac k box . c(t) c(t) FIGURE 1.2 . Now we describe the mathematical setu p we use in these chapters. W e stick t o single-input single-outpu t system s (SIS O i n th e literature) , sinc e generalizatio n is straightforward . Le t u s suppos e tha t al l inpu t function s li e i n £ 2 [0,oo] an d that ou r particula r bo x B send s the m t o function s whic h gro w n o faste r tha n e + / c t namely , (1.1C) B: L 2 [0,oo] - L 2 [0,oc #] where L2[0,oo X] 4 {/ : e~ Kt f{t) i s in L 2 }. Note tha t (1.1C ) implicitl y use s causality. Time invariance say s tha t B com - mutes wit h shift s (LIT) Bf a = {Bf) a fo r al l a 0 where f a stand s fo r shifte d / tha t is , f a {t) = f(t a). Fo r the first fe w chapter s stability wil l mea n (LIS) £:L 2 [0,oo]-*L 2 [0,oo] and strictly stable wil l mea n tha t fo r som e e 0 w e hav e B: £ 2 [0, oo, e) » L2[0, oo, e] and B i s a bounde d operator . Furthermore , th e linea r operator s arising fro m physica l system s wil l alway s ma p real-value d function s ont o real - valued functions . In studyin g a syste m i t i s very commo n t o Laplac e o r Fourie r transfor m al l functions an d wor k i n u the frequenc y domain. " Her e the Laplac e transfor m o f a function / i s / ( * ) ^ L / ( * ) = r f(t)e' st dt Jo and th e usua l Fourie r transfor m i s /(iw) , th e Laplac e transfor m / restricte d t o the imaginar y axis . The invers e Laplac e transfor m i s -j pk+ioo / W = 5 - / f(s)ds 27r Jk-ioo
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