  We consider a damped plate equation on an open bounded subset of R^d, or a
smooth manifold, with boundary, along with general boundary operators
fulfilling the Lopatinskii-Sapiro condition. The damping term acts on a region
without imposing a geometrical condition. We derive a resolvent estimate for
the generator of the damped plate semigroup that yields a logarithmic decay of
the energy of the solution to the plate equation. The resolvent estimate is a
consequence of a Carleman inequality obtained for the bi-Laplace operator
involving a spectral parameter under the considered boundary conditions. The
derivation goes first though microlocal estimates, then local estimates, and
finally a global estimate.
