STABILIZATION OF THE WAVE-EQUATION BY A D YNAMICAL CONTROLLER

We prove the stabilization of the problem u(tt) = Delta u - au(t) + bw when w (., t) = T(t)w(0) - integral(0)(t)(t-tau)bu(t)(., tau) dr and {T(t)} is the heat semi-group. We also establish that, for a or b cons tants, the spectral abscissa of the operator associated to the coupled system is the best rate decay. On the other hand, in tile one dimensi onal case, we compare the rate of decay with or without coupling (b no t equal 0 or b = 0).