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

Pierre Lissy (1, 2) , Ionel Roventa (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 McTAO - Mathematics for Control, Transport and Applications
3 University of Craiova

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.

Mots clés: control approximation; biorthogonal families; moment problem; fractional Laplacian; hyperbolic equations

Domaines: Optimisation et contrôle [math.OC]

Pierre Lissy, Ionel Roventa. Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method. Mathematical Models and Methods in Applied Sciences, 2020, 30 (3), pp.439-475. ⟨hal-01865920⟩