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 $\left\{w,w_t,w_{tt}\right\}$ in the elastic case, and of $\left\{w,w_t,w_{tt},\theta \right\}$ 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.

Let $A$ be a closed linear operator on a Banach space $E$. We study periodic solutions of the differential equation $d^{N}u\left(t\right)/d{t}^N=Au\left(t\right)$ with an arbitrary integer $N\ge 1$.