Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems

• 成果类型: Article
• 署名作者: Shivakumar, Sachin; Das, Amritam; Weiland, Siep; Peet, Matthew
• 署名单位: Iowa State University; Eindhoven University of Technology; Arizona State University; Arizona State University-Tempe
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3505954
• 发表日期: 2025
• 页码: 3240-3255
• 关键词: mathematical models; computational modeling; Analytical models; Integral equations; stability analysis; Numerical models; Context modeling; Thermal stability; Space heating; Partial differential equations; LMIs; optimization; partial differential equations (PDEs)

摘要

Partial integral equation (PIE) representation of a partial differential equation (PDE) allows using computationally tractable algorithms for analysis, simulation, and optimal control. However, the PIE representation has not previously been extended to many of the complex, higher order PDEs that may be encountered in speculative or data-based models. In this article, 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 that 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. Numerical tests and illustrations are used to demonstrate the controller synthesis and simulation of PDEs.