In this talk, the finite codimensional exact controllability for evolution equations is introduced. This new controllability notion is shown to be equivalent to finite codimensionality condition for some set.  Therefore, it can be used to verify Pontryagin's maximum principle for some optimal control problems with endpoint state constraints. As examples, LQ problems with fixed endpoint constraints for a wave and heat equations are studied. This is a joint work with Qi Lv and Xu Zhang. 

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