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 Gårding 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.

在本文中，受研究具有端点状态约束的无限维系统的最优控制问题的启发，我们引入了有限维（精确/近似）可控性的概念。提出了关于有限维可控性的一些等效准则。特别是，将有限维的精确可控制性简化为得出伴随系统的Gårding型不等式，这对于许多演化方程都是新的。可以针对某些具体问题（因此将其应用于相应的最佳控制问题）验证这种不等式，例如具有随时间和空间变化的势的波动方程。而且，在一些温和的假设下，