In this paper we investigate the exact controllability of n × n first
order quasilinear hyperbolic systems by m < n internal controls that
are localized in space in some part of the domain. We distinguish two
situations. The first one is when the equations of the system have the
same speed. In this case, we can use the method of characteristics and
obtain a simple and complete characterization for linear systems. Thanks
to a linear test this also provides some sufficient conditions for the
local exact controllability around the trajectories of semilinear
systems. However, when the speed of the equations are not anymore the
same, we see that we encounter the problem of loss of derivatives if we
try to control quasilinear systems with a reduced number of controls. To
solve this problem, as in a prior article by J.-M. Coron and P. Lissy
on a Navier-Stokes control system, we first use the notion of algebraic
solvability due M. Gromov. However, in contrast with this prior article
where a standard fixed point argument could be used to treat the
nonlinearities, we use here a fixed point theorem of Nash-Moser type due
to M. Gromov in order to handle the problem of loss of derivatives.