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

Umberto Biccari, Víctor Hernández-Santamaría, Judith Vancostenoble. Existence and cost of boundary controls for a degenerate/singular parabolic equation (2022). Mathematical Control and Related Fields, Vol. 12, No.2, pp. 495-530, doi: 10.3934/mcrf.2021032

Abstract. In this paper, we consider the following degenerate/singular parabolic equation $u_t -(x^\alpha u_{x})_x – \frac{\mu}{x^{2-\alpha}} u=0, (x,t)\in (0,1)\times(0,T)$ , where $0\leq\alpha\lt1$ and $u\leq(1-\alpha)^2/4$ 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.

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