  The boundary stabilization problem of the Boussinesq KdV-KdV type system is
investigated in this paper. An appropriate boundary feedback law consisting of
a linear combination of a damping mechanism and a delay term is designed. Then,
first, considering time-varying delay feedback together with a smallness
restriction on the length of the spatial domain and the initial data, we show
that the problem under consideration is well-posed. The proof combines Kato's
approach and the fixed-point argument. Last but not least, we prove that the
energy of the linearized KdV-KdV system decays exponentially by employing the
Lyapunov method.
