**Original manuscript:** 2020/03/09

We investigate a few problems in the theory of linear time-invariant systems that arise when the open-loop system possesses imaginary eigenvalues. The problems are: (1) the infinite horizon minimum energy optimal control problem; (2) the matter of defining the controllability Gramian for systems that are not stable; (3) the regularity as a distribution of the Fourier transform of the state impulse response. Apart from separately investigating these problems, we also indicate how they are interconnected.

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Last Updated: Fri Jul 10 08:03:49 2020