This paper studies theexact controllability of a finite dimensional system obtained bydiscretizing in space and time the linear 1-D wave system with aboundary control at one extreme. It is known that usual schemesobtained with finite difference or finite element methods are notuniformly controllable with respect to the discretizationparameters h and Δt. We introduce an implicit finitedifference scheme which differs from the usual centered one byadditional terms of order h 2 and Δt 2. Using a discreteversion of Ingham's inequality for nonharmonic Fourier series andspectral properties of the scheme, we show that the associatedcontrol can be chosen uniformly bounded in L2(0,T) and in sucha way that it converges to the HUM control of the continuous wave,i.e. the minimal L 2-norm control. The results are illustratedwith several numerical experiments.

The control of the surface of water in a long canal bymeans of a wavemaker is investigated. The fluid motion is governed by the Korteweg-de Vries equation in Lagrangian coordinates.The null controllability of the elevation of the fluid surface is obtained thanks to a Carleman estimate and some weighted inequalities. The global uncontrollability is also established.