This paper concerns the relation between exact controllability and stabilizability with arbitrary decay rate in infinite dimensional spaces. It appears that in several cases the notions are equivalent, but there are a lot of situations when additional conditions are needed, for example in Banach spaces. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces initiated by V. I. Korobov during the 70th of the past century in Kharkov State University.