A singular controllability problem with vanishing viscosity

TL;DR

This paper investigates whether controls for a viscous approximation of the 1D wave equation converge to controls of the original conservative system as viscosity vanishes, using spectral analysis and moment problems.

Contribution

It introduces a novel analysis of control convergence in a fractional viscous wave equation with vanishing viscosity, employing moment problems and biorthogonal sequences.

Findings

Controls remain uniformly bounded as viscosity tends to zero.

Spectral methods effectively evaluate control magnitudes for each eigenmode.

The results affirm convergence of controls in the vanishing viscosity limit.

Abstract

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? Our viscous term contains the fractional power of the Dirichlet Laplace operator and it is multiplied by a small parameter devoted to tend to zero. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to the vanishing parameter.