The Yakubovich Frequency Theorem, in its periodic version and in its general
nonautonomous extension, establishes conditions which are equivalent to
the global solvability of a minimization problem of infinite horizon type,
given by the integral in the positive half-line of a quadratic functional
subject to a control system. It also provides the unique minimizing pair
\lq\lq solution, control\rq\rq~and
the value of the minimum. In this paper we establish less restrictive conditions
under which the problem is partially solvable, characterize the set of
initial data for which the minimum exists, and obtain its value as well a
minimizing pair. The occurrence of exponential dichotomy and the
null character of the rotation number for a nonautonomous
linear Hamiltonian system defined
from the minimization problem are fundamental in the analysis.
