  In this article, we develop a new method to prove both global propagation of
analyticity and unique continuation in finite time for solutions of semilinear
wave-type equations with analytic nonlinearity. It combines control theory
techniques and Galerkin approximation, inspired by Hale-Raugel, to prove that
analyticity in time can be propagated for the nonlinear equation from a zone
where linear observability holds towards the full space.For semilinear wave
equations with Dirichlet boundary condition on a bounded domain, this implies
that analyticity can be propagated to the entire domain from a subset $\omega$
that satisfies the geometric control condition. It also implies the unique
continuation when the solution is assumed to be zero on $\omega$. When the
nonlinearity is assumed to be subcritical and defocusing, we also obtain
observability estimates in the optimal time of the geometric control
condition.For semilinear plate equations, similar propagation of analyticity is
achieved by assuming the controllability of the linear Schr\''odinger equation.
