A geometric-optics proof of a theorem on boundary control given a convex function

Abstract

In the area of boundary control of hyperbolic equations, the tools of geometric optics have sometimes proven to be very powerful. In geometric optics, authors including Littman [8] and Bardos, Lebeau and Rauch [1] have established under various circumstances that, if every bicharacteristic curve of the hyperbolic equation must cross a point on the boundary where the controls can be applied, then the equation can be controlled--- and the time required is just the maximum time needed for a bicharacteristic to reach that part of the boundary. Now that these results are in place, they allow for theorems on boundary control which do not require new integral inequalities for particular situations. Rather, assumptions are made on the geometry of the domain of the equation. For instance, Gulliver and Littman [3] show that every bicharacteristic will cross the boundary, and hence control will be attained, so long as chords between points of the boundary are unique and the boundary is locally convex. They go on to give several examples of regions where this holds. The present paper uses geometric optics to prove one of the main theorems in the important paper "Inverse/Observability Estimates for Second-Order Hyperbolic Equations with Variable Coefficients" by Lasiecka, Triggiani and Yao [5]. In that paper, the authors use Carleman estimates to show that the equation is controlled if there is a positive function v on which is strictly convex with respect to the metric defined by the coefficients of the equation, if that convex function has non-positive outward normal derivative on the uncontrolled part of the boundary. The time needed for control is a function of the maximum value of v on and a lower bound on its convexity. Here we will show that control in the same time is established by a simpler geometric optics argument---in fact it comes down to a short calculus computation on the value of v along a bicharacteristic of the equation.