论文标题
中性延迟方程的频率定理和惯性歧管
Frequency theorem and inertial manifolds for neutral delay equations
论文作者
论文摘要
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，在某种意义上，成本功能是无限的。我们的动机来自带有反馈部分的延迟方程，其中包含离散延迟，换句话说是$δ$ functionals给出的测量值，这些测量值在$ l_ {2} $中不受限制。在抽象的环境中（包括抛物线边界控制问题）可以处理的抽象环境中，我们获得了频率定理的版本（遵循V.A. Yakubovich和A.L. Likhtarnikov的作品），保证存在独特的最佳过程，并显示出最佳成本由Quadratic Lyapunov lyapunov lyapunove lyapunove limitiathional sigiality sightional sight officiational officiational sike。在我们相邻的作品中，这表明这些功能可用于构建惯性流形，并允许以统一的方式对待和扩展现场的许多作品。在这里，我们专注于延迟方程式的应用，尤其是提及R.A.的作品。史密斯关于融合定理的发展和庞加莱·奔驰理论； YU。 A. Ryabov，R.D。Driver和C. Chicone在惯性歧管上的方程式及其延迟的方程式以及他们最近对S. Chen和J. Shen给出的中性类型方程的概括。
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.
