  We study the one-dimensional asymmetric simple exclusion process on the
lattice $\{1, \dots,N\}$ with creation/annihilation at the boundaries. The
boundary rates are time dependent and change on a slow time scale $N^{-a}$ with
$a>0$. We prove that at the time scale $N^{1+a}$ the system evolves
quasi-statically with a macroscopic density profile given by the entropy
solution of the stationary Burgers equation with boundary densities changing in
time, determined by the corresponding microscopic boundary rates. We consider
two different types of boundary rates: the "Liggett boundaries" that correspond
to the projection of the infinite dynamics, and the reversible boundaries, that
correspond to the contact with particle reservoirs in equilibrium. The proof is
based on the control of the Lax boundary entropy--entropy flux pairs and a
coupling argument.
