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arXiv · 2001.11403

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

Umberto Biccari·Víctor Hernández-Santamaría·Judith Vancostenoble
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Abstract

In this paper, we consider the following degenerate/singular parabolic equation $$ u_t -(x^\alpha u_{x})_x - \frac{\mu}{x^{2-\alpha}} u =0, \qquad x\in (0,1), \ t \in (0,T), $$ where $0\leq \alpha <1$ and $\mu\leq (1-\alpha)^2/4$ 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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Umberto Biccari, Víctor Hernández-Santamaría, Judith Vancostenoble. 2020-01-30. Existence and cost of boundary controls for a degenerate/singular parabolic equation. https://arxiv.org/abs/2001.11403

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