Finite Codimensional Controllability for Evolution Equations
  Motivated by infinite-dimensional optimal control problems with endpoint state constraints, in this Note, we introduce the notion of finite codimensional exact controllability for evolution equations. It is shown that this new controllability is equivalent to the finite codimensionality condition in the literatures to guarantee Pontryagin's maximum principle. As examples, LQ problems with fixed endpoint state constraints for a wave and a heat equation are analyzed, respectively. 
Xu Liu, Qi Lu, Xu Zhang
math.OC