  In this article, we examine $L^2$ well-posedness and stabilization property
of the dispersion-generalized Benjamin-Ono equation with periodic boundary
conditions. The main ingredient of our proof is a development of
dissipation-normalized Bourgain space, which gains smoothing properties
simultaneously from dissipation and dispersion within the equation. We will
establish a bilinear estimate for the derivative nonlinearity using this space
and prove the linear observability inequality leading to small-data
stabilization.
