  In this paper, motivated by the study of optimal control problems for
infinite dimensional systems with endpoint state constraints, we introduce the
notion of finite codimensional (exact/approximate) controllability. Some
equivalent criteria on the finite codimensional controllability are presented.
In particular, the finite codimensional exact controllability is reduced to
deriving a G{\aa}rding type inequality for the adjoint system, which is new for
many evolution equations. This inequality can be verified for some concrete
problems (and hence applied to the corresponding optimal control problems), say
the wave equations with both time and space dependent potentials. Moreover,
under some mild assumptions, we show that the finite codimensional exact
controllability of this sort of wave equations is equivalent to the classical
geometric control condition.
