3 research outputs found
A singular controllability problem with vanishing viscosity
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
Uniform controllability for the beam equation with vanishing structural damping
summary:This paper is devoted to studying the effects of a vanishing structural damping on the controllability properties of the one dimensional linear beam equation. The vanishing term depends on a small parameter . We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls as goes to zero. It is shown that for any time sufficiently large but independent of and for each initial data in a suitable space there exists a uniformly bounded family of controls in acting on the extremity . Any weak limit of this family is a control for the beam equation. This analysis is based on Fourier expansion and explicit construction and evaluation of biorthogonal sequences. This method allows us to measure the magnitude of the control needed for each eigenfrequency and to show their uniform boundedness when the structural damping tends to zero