Exact boundary controllability of a singular/degenerate wave equation via singular Sturm-Liouville theory
We prove exact boundary controllability for a class of one-dimensional singular/degenerate wave equations of Sturm--Liouville type, \[ u_{tt}-(x^αu_x)_x-βx^{α-1}u_x-μx^{α-2}u=0, \qquad x\in(0,1), \] with a Dirichlet condition at the regular endpoint and a boundary control acting at the singular endpoint \(x=0\). The analysis is carried out in the fractional energy space % \[ X=\mathcal H^{ν+1/2}\times \mathcal H^{ν-1/2}, \] associated with the corresponding singular Sturm--Liouville operator. Using the spectral decomposition induced by Bessel functions, we establish admissibility of the boundary observation operator and derive precise lower estimates for the observation coefficients. Exact observability is obtained through Ingham-type inequalities together with an abstract observability result for unitary groups. The proof treats simultaneously the subcritical, critical logarithmic, and limit-point regimes by means of singular Sturm--Liouville theory and boundary traces identified through the Lagrange bracket of the singular Sturm--Liouville expression.