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The coupled PDE system of a composite (sandwich) beam revisited

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  • In this paper we consider the coupled PDE system which describes a composite (sandwich) beam, as recently proposed in [H.1], [H-S.1]: it couples the transverse displacement $w$ and the effective rotation angle $\xi$ of the beam. We show that by introducing a suitable new variable $\theta$, the original model in the original variables $\{w,\xi\}$ of the sandwich beam is transformed into a canonical thermoelastic system in the new variables $\{w,\theta\}$, modulo lower-order terms. This reduction then allows us to re-obtain recently established results on the sandwich beam--which had been proved by a direct, ad hoc technical analysis [H-L.1]--simply as corollaries of previously established corresponding results [A-L.1], [A-L.2], [L-T.1]--[L-T.5] on thermoelastic systems. These include the following known results [H-L.1] for sandwich beams: (i) well-posedness in the semigroup sense; (ii) analyticity of the semigroup when rotational forces are not accounted for; (iii) structural decomposition of the semigroup when rotational forces are accounted for; and (iv) uniform stability.
    In addition, however, through the aforementioned reduction to thermoelastic problems, we here establish new results for sandwich beams, when rotational forces are accounted for. They include: (i) a backward uniqueness property (Section 4), and (ii) a suitable singular estimate, critical in control theory (Section 5). Finally, we obtain a new backward uniqueness property, this time for a structural acoustic chamber having a composite (sandwich) beam as its flexible wall (Section 6).
    Mathematics Subject Classification: 35, 47F, 49, 93.


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