A singular controllability problem with vanishing viscosity

Facultatea de Stiinte Exacte, Universitatea din Craiova, 200585, Romania
florinbugariu86@yahoo.com; sdmicu@yahoo.com

Received: 6 June 2012
Revised: 27 February 2013

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? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. 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 ε, under the assumption that α∈[0,1)\{½}. It follows that, under this assumption, our starting question has a positive answer.