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

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