  Let $u(t)=-Fx(t)$ be the optimal control of the open-loop system
$x'(t)=Ax(t)+Bu(t)$ in a linear quadratic optimization problem. By using
different complex variable arguments, we give several lower and upper estimates
of the exponential decay rate of the closed-loop system $x'(t)=(A-BF)x(t)$.
Main attention is given to the case of a skew-Hermitian matrix $A$.
  Given an operator $A$, for a class of cases, we find a matrix $B$ that
provides an almost optimal decay rate.
  We show how our results can be applied to the problem of optimizing the decay
rate for a large finite collection of control systems $(A, B_j)$, $j=1, \dots,
N$, and illustrate this on an example of a concrete mechanical system. At the
end of the article, we pose several questions concerning the decay rates in the
context of linear quadratic optimization and in a more general context of the
pole placement problem.
