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

2020/01/30 by Biccari, Umberto, Hernández-Santamaría, Víctor, Vancostenoble, Judith
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2001.11403

Abstract

In this paper, we consider the following degenerate/singular parabolic equation u_t -(x^αu_x)_x - \fracμx^2-α u =0, x∈ (0,1), t ∈ (0,T), where 0≤ α<1 and μ≤ (1-α)²/4 are two real parameters. We prove the boundary null controllability by means of a H¹(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.

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