Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using a finite-difference method

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

“…The best result in the case α = 1 (i.e. the wave equation) has recently been obtained by the authors in [26] for boundary controls (the same result would hold for internal controls). It is notably proved an optimal result with respect to the range of filtration of the initial condition, together with precise (but non-optimal) estimates on the minimal time of uniform controllability.…”

“…It is notably proved an optimal result with respect to the range of filtration of the initial condition, together with precise (but non-optimal) estimates on the minimal time of uniform controllability. Even if the spirit of the present article is quite similar to [26], we would like to emphasize that the present study is more intricate and is far from being a direct consequence of the computations made in [26]. This is due to the following facts:…”

“…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.…”

“…• The use of the Paley-Wiener Theorem requires to compensate the explosion of the product R m by an adequate multiplier. Because of the bad behaviour of the product R m emphasized above, the construction given in [26] is not sufficient to ensure the uniform boundedness of the discrete control with respect to h. Here, we use a new construction of the multiplier in the context of semi-discrete control problems. This construction is made of two parts.…”

“…The first part is similar to the multiplier is similar to the one given in [26], and enables to compensate R m (x) for high |x|. The second part is similar to the construction used in [29] in the context of continuous fractional control problems, and is used to compensate the bad behaviour of the product R m (x) for low |x| (this is the main difference with [26]). However, the use we make of this multiplier is quite different from the continuous case.…”