Approximation of the controls for the beam equation with vanishing viscosity

Abstract

We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.

Boundary Controllability for Finite-Differences Semidiscretizations of a Clamped Beam Equation

“…More precisely, from the spectral analysis of the problem and the study of the corresponding observability inequality, two possibilities to ensure the convergence of the scheme have been proposed in [10]: by filtering out the spurious high frequencies or by adding an extra boundary control. An alternative method to ensure the uniform controllability result consists in adding a vanishing numerical viscosity and has been analyzed in [1].…”

Controllability of the space semi-discrete approximation for the beam equation

“…We are able to obtain a uniformly bounded family of controls for the perturbed problem (8) with the property that any weak limit of it is a control for the continuous problem. More precisely, the following result holds (see Bugariu et al (2013)).…”

Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method

“…Many possibilities have been proposed to overcome this difficulty: a Tychonov regularization of the HUM cost functional (see [21,49]), a change of the numerical scheme (mixed finite elements [9], vanishing viscosity [6,32] and other type of finite difference schemes [37]), the introduction of non-uniform meshes ( [18,19]), an approximation of discrete controls [12] (which does not lead exactly the discrete solution to zero, but converges to an exact control of the continuous problem), and finally an appropriate filtering technique, introduced in [31] and notably used in [13,34,26] in the context of wave or beam equation, which consists in relaxing the control requirement by controlling only the low-frequency part of the solution. This later approach will be considered in this paper.…”