<p style='text-indent:20px;'>We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>. Optimal interior and boundary regularity results were given in [<xref ref-type="bibr" rid="b1">1</xref>], after [<xref ref-type="bibr" rid="b41">41</xref>], when <inline-formula><tex-math id="M2">\begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}</tex-math></inline-formula>, which, moreover, in the canonical case <inline-formula><tex-math id="M3">\begin{document}$ \gamma = 0 $\end{document}</tex-math></inline-formula>, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [<xref ref-type="bibr" rid="b19">19</xref>], [<xref ref-type="bibr" rid="b17">17</xref>], [<xref ref-type="bibr" rid="b24">24</xref>,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether <inline-formula><tex-math id="M4">\begin{document}$ \gamma = 0 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M5">\begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}</tex-math></inline-formula>, since <inline-formula><tex-math id="M6">\begin{document}$ \gamma \neq 0 $\end{document}</tex-math></inline-formula> is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with <inline-formula><tex-math id="M7">\begin{document}$ g $\end{document}</tex-math></inline-formula> "smoother" than <inline-formula><tex-math id="M8">\begin{document}$ L^2(\Sigma) $\end{document}</tex-math></inline-formula>, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [<xref ref-type="bibr" rid="b17">17</xref>]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [<xref ref-type="bibr" rid="b22">22</xref>], [<xref ref-type="bibr" rid="b23">23</xref>], [<xref ref-type="bibr" rid="b37">37</xref>] for control smoother than <inline-formula><tex-math id="M9">\begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}</tex-math></inline-formula>, and [<xref ref-type="bibr" rid="b44">44</xref>] for control less regular in space than <inline-formula><tex-math id="M10">\begin{document}$ L^2(\Gamma) $\end{document}</tex-math></inline-formula>. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [<xref ref-type="bibr" rid="b42">42</xref>], [<xref ref-type="bibr" rid="b24">24</xref>,Section 9.8.2].</p>

An optimal, complete, continuous theory of the Luenberger dynamic compensator (or state estimator or state observer) is obtained for the recently studied class of heat-structure interaction partial differential equation (PDE) models, with structure subject to high Kelvin-Voigt damping, and feedback control exercised either at the interface between the two media or else at the external boundary of the physical domain in three different settings. It is a first, full investigation that opens the door to numerous and far reaching subsequent work. They will include physically relevant fluid-structure models, with wave- or plate-structures, possibly without Kelvin-Voigt damping, as explicitly noted in the text, all the way to achieving the ultimate discrete numerical theory, so critical in applications. While the general setting is functional analytic, delicate PDE-energy estimates dictate how to define the interface/boundary feedback control in each of the three cases.

We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L‐T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.