Polynomial decay and control of a 1− d hyperbolic–parabolic coupled system

Abstract

In this paper we consider a linearized model for fluid–structure interaction in one space dimension. The domain where the system evolves consists in two parts in which the wave and heat equations evolve, respectively, with transmission conditions at the interface. First of all we develop a careful spectral asymptotic analysis on high frequencies for the underlying semigroup. It is shown that the semigroup governed by the system can be split into a parabolic and a hyperbolic projection. The dissipative mechanism of the system in the domain where the heat equation holds produces a slow decay of the hyperbolic component of solutions. According to this analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Next, we discuss the problem of null-controllability of the system when the control acts on the boundary of the domain where the heat equation holds. The key observability inequality of the dual system with observation on the heat component is derived though a new Ingham-type inequality, which in turn, thanks to our spectral analysis, is a consequence of a known observability inequality of the same system but with observation on the wave component.