Sharp geometric characterization of uniform stabilization for the damped wave equation

Let $V$ satisfy the assumptions in~ and Assumption~, and let $b\in L^\infty(\mathbf{R}^d)$ be non-negative. Let Equation~ denote the damped wave equation, and let uniform stability mean stability in the sense of Definition~. The Geometric Control Condition and the Turning Point Condition are the conditions referred to as~ and~, respectively.

Uniform stabilization conjecture. Equation~ is uniformly stable if and only if $b$ satisfies the Geometric Control Condition~ and the Turning Point Condition~.

The preceding discussion presents these two geometric conditions as the kinetic- and potential-regime components of the Dynamical Stabilization Condition, and suggests that their conjunction is sharp. The supplied text does not state that this characterization has been proved or disproved.