  On a compact Riemannian manifold with boundary, we prove a spectral
inequality for the bi-Laplace operator in the case of so-called "clamped"
boundary conditions , that is, homogeneous Dirichlet and Neumann conditions
simultaneously. We also prove a resolvent estimate for the generator of the
damped plate semigroup associated with these boundary conditions. The spectral
inequality allows one to observe finite sums of eigenfunctions for this
fourth-order elliptic operator, from an arbitrary open subset of the manifold.
Moreover, the constant that appears in the inequality grows as exp(C$\mu$ 1/4)
where $\mu$ is the largest eigenvalue associated with the eigenfunctions
appearing in the sum. This type of inequality is known for the Laplace
operator. As an application, we obtain a null-controllability result for a
higher-order parabolic equation. The resolvent estimate provides the spectral
behavior of the plate semigroup generator on the imaginary axis. This type of
estimate is known in the case of the damped wave semigroup. As an application ,
we deduce a stabilization result for the damped plate equation, with a log-type
decay. The proofs of both the spectral inequality and the resolvent estimate
are based on the derivation of different types of Carleman estimates for an
elliptic operator related to the bi-Laplace operator: in the interior and at
some boundaries. One of these estimates exhibits a loss of one full derivative.
Its proof requires the introduction of an appropriate semi-classical calculus
and a delicate microlocal argument.
