Approximation of the controls for the linear beam equation

Spectral analysis of discrete elliptic operators and applications in control theory

“…As for parabolic problems, their proof makes use of explicit computations on the eigenelements (µ h k , ψ h k ) 1≤|k|≤N of the operator L h . Using the discrete spectral estimates obtained in Section 3 of the present paper, it is very likely that one can adapt the ideas of [18] to obtain similar results for more general second order elliptic operator A h and, more importantly, for non uniform grids.…”

“…1. In [18] the authors consider the problem of null-controllability at the boundary for the semi-discretized in space linear beam equation with hinged boundary conditions and constant diffusion coefficient. They discretize the operator ∂ xxxx with finite differences in 1D on a uniform mesh.…”

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.…”

Uniform indirect boundary controllability of semi-discrete <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations

“…But this is not the case. In fact, as witnessed by various authors, uniform controllability of discrete systems fails to hold also for the semi-discrete models obtained by finite differences or by the classical finite elements methods applied to the single wave equations [10,13,14] as well as to the beam equations [6,22]. This phenomenon is due to the fact that the semi-discrete dynamics lead to high-frequency spurious solutions which propagate with arbitrary small velocity and make the discrete controls diverge when the mesh-size h goes to zero.…”

“…In [21], the author considered a space-discrete scheme with an added numerical vanishing viscous term and proved that this extra numerical damping filters out the high numerical frequencies and ensures the convergence of the sequence of discrete controls to a control of the continuous single wave equation; but again, the result is proved for a large time of controllability T . The hints proposed in [20,21] have been developed later in [6,22] dealing with the uniform controllability of the beam equations. More recently, the problem with finite difference approximations of the wave equation was considered again in [19], where a generalization of the results in [20] was given.…”