We study the infinite-horizon 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. It 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 Poincaré-Bendixson theory; and also the works of 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.

我们研究希尔伯特空间中线性控制系统的无限范围二次调节器问题，其中成本函数在某种意义上是无界的。我们的动机来自于延迟方程，其反馈部分包含离散延迟，或者换句话说，由\(\delta \)泛函给出的测量，其在\(L_{2}\)中是无界的。在可以处理此类（以及许多其他，包括抛物线边界控制问题）方程的抽象上下文中，我们获得了频率定理的一个版本。它保证了唯一最优过程的存在，并表明最优成本由二次李雅普诺夫泛函给出。在我们的相邻工作中，表明此类泛函可用于构造惯性流形，并允许以统一的方式处理和扩展该领域的许多工作。在这里，我们集中讨论延迟方程的应用，特别提到 RA Smith 在收敛定理和庞加莱-本迪克森理论的发展方面的工作；还有 Yu.A. 的作品 Ryabov、RD Driver 和 C. Chicone 关于小延迟方程的惯性流形及其最近由 S. Chen 和 J. Shen 给出的中性型方程的推广。