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

Abstract

In this paper, we consider the following degenerate/singular parabolic equation $$u_t-(x^{\alpha }{u}_x{)}_x-\frac{\mu }{x^{2-\alpha }}u=0,x\in (0,1),t\in (0,T),$$ where $0\le \alpha <1$ and $\mu \le (1-\alpha {)}^2/4$ 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 $\alpha$ and $\mu$. The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences.