A Continuation Method for Large-Scale Modeling and Control: From ODEs to PDE, a Round Trip

• 成果类型: Article
• 署名作者: Nikitin, Denis; Canudas-de-Wit, Carlos; Frasca, Paolo
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA)
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3122387
• 发表日期: 2022
• 页码: 5118-5133
• 关键词: mathematical models; Transforms; robots; TOPOLOGY; statistics; sociology; Writing; Control of large-scale networks; multiagent systems; partial differential equation (PDE)

摘要

In this article, we present a continuation method, which transforms spatially distributed ordinary differential equation (ODE) systems into a continuous partial differential equation (PDE). We show that this continuation can be performed for both linear and nonlinear systems, including multidimensional, space-varying, and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing the evolution of a continuous-state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples, including transport equations, Kuramoto equations, and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original solution to Hilbert's sixth problem. Finally, we leverage our derivation of Euler equations to solve a control problem multiagent systems, by designing a nonlinear control algorithm for robot formation based on its continuous approximation.