论文标题
部分积分方程表示向GPDE输入输出系统的扩展
Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems
论文作者
论文摘要
已经表明，部分积分方程（PIE）的存在部分微分方程（PDE）简化了分析，仿真和最佳控制的许多数值方面。但是，以前没有将PIE表示扩展到许多复杂的高阶PDE，例如在投机或基于数据的模型中可能遇到的。在本文中，我们提出了一类此类PDE模型的PIE表示，包括高阶导数，边界值输入和与普通微分方程的耦合。实现此扩展的主要技术贡献是对库奇重复集成规则的概括。通过开源软件Pietools中的PDE建模界面简化了复杂的PDE模型转换为PIE的过程。几个数值测试和插图用于证明结果。
It has been shown that the existence of a Partial Integral Equation (PIE) representation of a Partial Differential Equation (PDE) simplifies many numerical aspects of analysis, simulation, and optimal control. However, the PIE representation has not previously been extended to many of the complex, higher-order PDEs such as may be encountered in speculative or data-based models. In this paper, we propose PIE representations for a large class of such PDE models, including higher-order derivatives, boundary-valued inputs, and coupling with Ordinary Differential Equations. The main technical contribution which enables this extension is a generalization of Cauchy's rule for repeated integration. The process of conversion of a complex PDE model to a PIE is simplified through a PDE modeling interface in the open-source software PIETOOLS. Several numerical tests and illustrations are used to demonstrate the results.
