From 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

Abstract

<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>