  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.
