论文标题
使用凸优化的延迟和稳定性分析的PDE系统的表示 - 扩展版本
Representation of PDE Systems with Delay and Stability Analysis using Convex Optimization -- Extended Version
论文作者
论文摘要
部分积分方程（PIE）已用于在一个或两个空间维度中表示具有延迟和部分微分方程（PDE）系统的系统。在本文中，我们表明可以将这些结果组合起来，以获得任何适当延迟的合适1D PDE模型的派代表。特别是，我们将这些延迟的PDE系统表示为1D和2D PDE的耦合系统，获得了两个子系统的PIE表示。借助这些PIE子系统的反馈互连，然后以延迟获得1D PDE的2D PIE表示。接下来，基于PIE表示，我们将稳定性分析的问题提出为正算子的凸优化，可以使用Pietools软件套件来解决。我们将结果应用于状态和边界条件延迟的PDE示例。
Partial Integral Equations (PIEs) have been used to represent both systems with delay and systems of Partial Differential Equations (PDEs) in one or two spatial dimensions. In this paper, we show that these results can be combined to obtain a PIE representation of any suitably well-posed 1D PDE model with constant delay. In particular, we represent these delayed PDE systems as coupled systems of 1D and 2D PDEs, obtaining a PIE representation of both subsystems. Taking the feedback interconnection of these PIE subsystems, we then obtain a 2D PIE representation of the 1D PDE with delay. Next, based on the PIE representation, we formulate the problem of stability analysis as convex optimization of positive operators which can be solved using the PIETOOLS software suite. We apply the result to PDE examples with delay in the state and boundary conditions.
