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Abstract
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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