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This assumption is valid because, for most systems, longitudinal wave speeds are at least 100 times faster than transverse wave speeds. Substitution of uu = 0 into Eq. (2. 73) gives XE (0, 1). 75) gives 2(Ux + 2wx 1 2) Vz = F(t), xE(O,l). 84) where F(t) = F0 + F 1 (t). For large vz, the measured output is y(t) = u(1, t) 111 = -- 2 0 w;dx. 4 Matrix Operator Representation Linear distributed parameter systems ean be east in matrix operator forms that allow study of their eommon underlying strueture and better understanding of their eharacteristie responses.

1 for positive p and m. 96) where integration by parts has been used twiee. Aeeording to the definition of 1t, w1 (0) = w2(0) = 0, so the lower evaluations of the first two terms disappear in 24 Distributed Parameter Models Eq. 96). Upper evaluation of the first term cancels the third term so only the upper evaluation of the second term and the integral remain. Thus, and Ao is symmetric. 97) 0. w 2(x, t) for all x E (0, L). Thus, we obtain the result given in Eq. 88) and Ao is positive definite.

44) The Sturm-Liouville problern Eq. 44) admits a unique solution in w E H 2 ([0, 1]) if we choose ).. , right hand side of Eqs. 277 [9]). w- wE H 1 (0, 1), so z = [w, v, cp, 1/J, iJ, eJT E "D(A). 0 Lemma 19 The operator A in Eq. 39} is densely defined in Z defined in Eq. 38}. }. We first choose a, b, c E 'R such that 2k Pkap (cp- a)2 < €2 /5, 2 ~: (:;[;- b)2 < €2 /5, (Po- 1)2 (7}- c)2 < €2 /5. 45) implying that "D(A) is dense in Z. 0 The following theorem follows directly from semigroup theory [99].