中立型延迟方程的频率定理与惯性流形
Frequency theorem and inertial manifolds for neutral delay equations
AI总结：
本文针对希尔伯特空间中无界代价的二次调节器问题，通过频率定理证明最优过程存在且代价由二次泛函给出，并应用于延迟方程，为构造惯性流形提供统一框架。
摘要：
我们研究了希尔伯特空间中线性控制系统的二次调节器问题，其中代价泛函在某种意义上是无界的。我们的动机来自具有离散延迟的反馈部分的延迟方程，或者换句话说，由δ泛函给出的测量，这些测量在L2中是无界的。在可以处理此类方程（以及许多其他方程，包括抛物型边界控制问题）的抽象背景下，我们获得了频率定理的一个版本（遵循V.A. Yakubovich和A.L. Likhtarnikov的工作），该定理保证了唯一最优过程的存在性，并表明最优代价由二次类李雅普诺夫泛函给出。在我们的相邻工作中，我们证明了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。这里我们专注于延迟方程的应用，并特别提及R.A. Smith关于收敛定理和庞加莱-本迪克森理论发展的著作；Yu. A. Ryabov、R.D. Driver和C. Chicone关于小延迟方程惯性流形的著作，以及S. Chen和J. Shen给出的中立型方程的最新推广。
英文摘要：
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 $δ$-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 Poincaré-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.
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