A SINGULAR CONTROLLABILITY PROBLEM WITH VANISHING VISCOSITY

作者：Bugariu Ioan Florin^*; Micu Sorin

来源：ESAIM: Control, Optimisation and Calculus of Variations , 2014, 20(1): 116-140.

DOI：10.1051/cocv/2013057

• Wave equation
• null-controllability
• vanishing viscosity
• moment problem
• biorthogonals

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 a of the Dirichlet Laplace operator. Through the parameter a 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 epsilon 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 epsilon, under the assumption that alpha is an element of [0, 1)\{1/2}. It follows that, under this assumption, our starting question has a positive answer.

• 出版日期2014-1