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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 \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].
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Abstract representation of the SMGTJ equation under rough boundary controls: Optimal interior regularity
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.
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- Award ID(s):
- 2205508
- PAR ID:
- 10418850
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematical Methods in the Applied Sciences
- ISSN:
- 0170-4214
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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