Motivated by problems in control theory concerning decay rates for the damped wave equation \begin{equation} w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0, \end{equation} we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $\mu_\alpha$-relatively dense (where $d\mu_\alpha(x) \approx x^{2\alpha+1}\, dx$) for $\alpha > -1/2$, and ${\rm supp} \mathcal{F}_\alpha(f) \subset [R,R+1]$, then we show \begin{equation} |f|_{L^2_\alpha(\mathbb{R}^+)} \lesssim |f|_{L^2_\alpha(E)}, \end{equation} for all $f\in L^2_\alpha(\mathbb{R}^+)$, where the constants in $\lesssim$ do not depend on $R > 0$. Previous results on PLS theorems for the Fourier-Bessel transform by Ghobber and Jaming (2012) provide bounds that depend on $R$. In contrast, our techniques yield bounds that are independent of $R$, offering a new perspective on such results. This result is applied to derive decay rates of radial solutions of the damped wave equation. This is joint work with Ben Jaye.