  The aim of this paper is to design a feedback operator for stabilizing in
infinite time horizon a system modeling the interactions between a viscous
incompressible fluid and the deformation of a soap bubble. The latter is
represented by an interface separating a bounded domain of $\mathbb{R}^2$ into
two connected parts filled with viscous incompressible fluids. The interface is
a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the
incompressible Stokes equations in time-dependent domains. The mean curvature
of the surface defines a surface tension force which induces a jump of the
normal trace of the Cauchy stress tensor. The response of the fluids is a
velocity trace on the interface, governing the time evolution of the latter,
via the equality of velocities. The data are assumed to be sufficiently small,
in particular the initial perturbation, that is the initial shape of the soap
bubble is close enough to a circle. The control function is a surface tension
type force on the interface. We design it as the sum of two feedback operators:
one is explicit, the second one is finite-dimensional. They enable us to define
a control operator that stabilizes locally the soap bubble to a circle with an
arbitrary exponential decay rate, up to translations, and up to non-contact
with the outer boundary.
