In  this talk I will present some results on  the Cauchy problem for the gravity water-wave equations, in a domain with flat bottom and in arbitrary space dimension. I will show that if the data are of size $\eps$ in a space of analytic functions which have a holomorphic extension in a strip of size $\sigma$, then the solution exists up to a time of size $C/\eps$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $\sigma - C'\eps t$. This question comes from motivations from control theory which force us to consider analytic solutions. I will actually start the talk with these motivations.

This is joint work with T. Alazard and C. Zuily.