%ATriggiani, Roberto%AWan, Xiang%BJournal Name: Evolution Equations and Control Theory; Journal Volume: 11; Journal Issue: 6
%D2022%I
%JJournal Name: Evolution Equations and Control Theory; Journal Volume: 11; Journal Issue: 6
%K
%MOSTI ID: 10417429
%PMedium: X
%TFrom low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae
%XWe 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 \begin{document}$ g $\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}, which, moreover, in the canonical case \begin{document}$ \gamma = 0 $\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$ \gamma = 0 $\end{document} or \begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}, since \begin{document}$ \gamma \neq 0 $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$ g $\end{document} "smoother" than \begin{document}$ L^2(\Sigma) $\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}, and [44] for control less regular in space than \begin{document}$ L^2(\Gamma) $\end{document}. 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 [42], [24,Section 9.8.2].

%0Journal Article