Approximation of the controls for the beam equation with vanishing viscosity (Q2814440)

The authors present a finite difference semi-discrete scheme for the 1D beam equation written as \(u^{\prime \prime }(t,x)+u_{xxxx}(t,x)=0\) in \( (0,T)\times (0,1)\) with the hinged boundary conditions \( u(t,0)=u(t,1)=u_{xx}(t,0)=0\) and \(u_{xx}(t,1)=v(t)\). The initial conditions \( u(0,x)=u^{0}(x)\) and \(u^{\prime }(0,x)=u^{1}(x)\) are considered. The authors recall the notion of null-controllability in finite time for this problem and they express the classical duality condition for this null-controllability property. They also recall a condition on the initial data in terms of their Fourier series under which the problem is null-controllable in finite time. The main purpose of the paper is to study a null-controllability property for a finite difference scheme associated to the above problem. The main result of the paper proves that this null-controllability property is ensured together with the convergence of the scheme, if a vanishing numerical viscosity is added. The proof essentially relies on biorthogonality properties for special classes of exponential functions. The paper ends with the presentation of numerical results obtained through this numerical scheme, for which the authors precise the errors.