Existence and cost of boundary controls for a degenerate/singular parabolic equation

Abstract

<p style='text-indent:20px;'>In this paper, we consider the following degenerate/singular parabolic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} u_t -(x^\alpha u_{x})_x - \frac{\mu}{x^{2-\alpha}} u = 0, \qquad x\in (0,1), \ t \in (0,T), \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0\leq \alpha <1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \mu\leq (1-\alpha)^2/4 $\end{document}</tex-math></inline-formula> are two real parameters. We prove the boundary null controllability by means of a <inline-formula><tex-math id="M3">\begin{document}$ H^1(0,T) $\end{document}</tex-math></inline-formula> control acting either at <inline-formula><tex-math id="M4">\begin{document}$ x = 1 $\end{document}</tex-math></inline-formula> or at the point of degeneracy and singularity <inline-formula><tex-math id="M5">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>. Besides we give sharp estimates of the cost of controllability in both cases in terms of the parameters <inline-formula><tex-math id="M6">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences.</p>