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.