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We study the quadratic regulator problem for linear control systems inHilbert spaces, where the cost functional is in some sense unbounded. Ourmotivation comes from delay equations with the feedback part containingdiscrete 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 canbe treated, we obtain a version of the Frequency Theorem (following the worksof V.A. Yakubovich and A.L. Likhtarnikov), which guarantees the existence of aunique optimal process and shows that the optimal cost is given by a quadraticLyapunov-like functional. In our adjacent works it is shown that suchfunctionals can be used to construct inertial manifolds and allow to treat andextend many works in the field in a unified manner. Here we concentrate onapplications to delay equations and especially mention the works of R.A. Smithon developments of convergence theorems and the Poincar\'{e}-Bendixson theory;Yu. A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equationswith small delays and their recent generalization for equations of neutral typegiven by S. Chen and J. Shen.

Frequency theorem and inertial manifolds for neutral delay equations