Null controllability of degenerate/singular parabolic equations

The purpose of this paper is to provide a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {u_t} - {\left( {a(x){u_x}} \right)_x} - \frac{\lambda }{{{x^\beta }}}u = 0 $\end{document}, (t, x) ∈ (0, T) × (0, 1), where the diffusion coefficient a(∙) is degenerate at x = 0. Also the boundary conditions are considered to be Dirichlet or Neumann type related to the degeneracy rate of a(∙). Under some conditions on the function a(∙) and parameters β, λ, we prove global Carleman estimates. The proof is based on an improved Hardy-type inequality.