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

In this paper, we consider the following degenerate/singular parabolic equation\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*} $\end{document}where \begin{document}$ 0\leq \alpha <1 $\end{document} and \begin{document}$ \mu\leq (1-\alpha)^2/4 $\end{document} are two real parameters. We prove the boundary null controllability by means of a \begin{document}$ H^1(0,T) $\end{document} control acting either at \begin{document}$ x = 1 $\end{document} or at the point of degeneracy and singularity \begin{document}$ x = 0 $\end{document} . Besides we give sharp estimates of the cost of controllability in both cases in terms of the parameters \begin{document}$ \alpha $\end{document} and \begin{document}$ \mu $\end{document} . The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences.