We consider mixed problems for the Kirchhoff elastic and
thermoelastic systems, subject to boundary control in the clamped
boundary conditions BC (clamped control). If $w$
denotes the
elastic displacement and $\theta$
the temperature, we establish
sharp regularity of $\{w,w_t,w_{tt}\}$
in the elastic case, and
of $\{w,w_t,w_{tt},\theta\}$
in the thermoelastic case. Our
results complement those by Lagnese and Lions (1988), where sharp
(optimal) trace regularity results are obtained for the
corresponding boundary homogeneous cases. The passage from the
boundary homogeneous cases to the corresponding mixed problems
involves a duality argument. However, in the present case of
clamped BC, and only in this case, the duality argument in
question is both delicate and technical. In this respect, the
clamped BC are “exceptional” within the set of canonical
BC (hinged, clamped, free BC). Indeed, it produces new phenomena
which are accounted for by introducing new, untraditional factor
(quotient) spaces. These are critical in describing both interior
regularity and exact controllability of mixed elastic and
thermoelastic Kirchhoff problems with clamped controls.