  We study the quadratic regulator problem for linear control systems in
Hilbert spaces, where the cost functional is in some sense unbounded. Our
motivation comes from delay equations with the feedback part containing
discrete delays or, in other words, measurements given by $\delta$-functionals,
which are unbounded in $L_{2}$. Working in an abstract context in which such
(and many others, including parabolic boundary control problems) equations can
be treated, we obtain a version of the Frequency Theorem (following the works
of V.A. Yakubovich and A.L. Likhtarnikov), which guarantees the existence of a
unique optimal process and shows that the optimal cost is given by a quadratic
Lyapunov-like functional. In our adjacent works it is shown that such
functionals can be used to construct inertial manifolds and allow to treat and
extend many works in the field in a unified manner. Here we concentrate on
applications to delay equations and especially mention the works of R.A. Smith
on developments of convergence theorems and the Poincar\'{e}-Bendixson theory;
Yu. A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equations
with small delays and their recent generalization for equations of neutral type
given by S. Chen and J. Shen.
