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

International audienceWe consider a finite-differences semi-discrete scheme for the approximation of boundary controls for the one-dimensional wave equation. The high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh-size) controllability property of the semi-discrete model in the natural setting. We prove that, by filtering the high frequencies of the initial data in an optimal range, we restore the uniform controllability property. Moreover, we obtain a relation between the range of filtration and the minimal time of control needed to ensure the uniform controllability

Boundary controllability for finite-differences semi-discretizations of a clamped beam equation

This article deals with the boundary observability properties of a space finite-differences semi-discretization of the clamped beam equation. We make a detailed spectral analysis of the system and, by combining numerical estimates with asymptotic expansions, we localize all the eigenvalues of the corresponding discrete operator depending on the mesh size $h$. Then, an Ingham's type inequality and a discrete multiplier method allow us to deduce that the uniform (with respect to $h$) observability property holds if and only if the eigenfrequencies are filtered out in the range $O(1/hˆ4)$

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

We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. The proof is mainly based on a (non-classic) moment method

Time optimal boundary controls for the heat equation

The fact that the time optimal controls for parabolic equations have the bang-bang property has been recently proved for controls distributed inside the considered domain. The aim of this article consists in showing that the boundary controls for the heat equation have the same property, at least in rectangular domains. The main result is proved by combining results and methods from traditionally distinct fields: the Lebeau-Robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems, on one side, and a Remez-type inequality for Müntz spaces and a generalization of Tur{á}n's inequality, on the other side

On the convexity of nonlinear elastic energies in the right Cauchy-Green tensor

We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies which are convex with respect to the right Cauchy-Green tensor , where denotes the gradient of deformation. Examples of such energies exhibiting a blow up for are given

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

We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. The proof is mainly based on a (non-classic) moment method