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Discussed in this short paper is a controllability question pertaining to a class of sensitivity problems. More specifically, the possibility is considered of transferring a time-invariant linear system from each initial state to any terminal state in a finite time and in such a way that the terminal condition is insensitive to small parameter variations. Partial controllability, suitably defined, leads to a necessary and sufficient condition for the existence of the solution to such a problem. Terminal output insensitivity is also discussed and a similar result is proved when the output transformation is not affected by some parameter variations.