An example of the effect of time delays in boundary feedback stabilization of wave equations

This note is concerned with the effect of time delays in boundary feed-back stabilization schemes for wave equations. The question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays. It will be shown by example that for "almost" arbitrary delays such destablization can indeed occur in certain otherwise stable boundary feedback schemes for both undamped and damped wave equations. Although the examples involve only one spatial dimension, it is to be expected that a similar phenomenon occurs in higher dimensions. This suggests that certain boundary stabilization schemes which have been proposed for various classes of hyperbolic systems (see e.g. Chen[1][2], Lagnese[4][5] and Quinn-Russell[6]) may not be robust to the small delays which might well occur in computing feedback controls.