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

TL;DR

This paper introduces a continuation method that transforms large-scale spatially distributed ODE systems into PDEs, enabling analysis and control of complex systems like fluids and multiagent formations.

Contribution

The paper presents a novel continuation approach applicable to linear and nonlinear systems, including multidimensional, space- and time-varying cases, with applications to fluid dynamics and multiagent control.

Findings

Successfully transforms ODE networks into PDEs respecting spatial structure

Derives Euler equations from particle systems as an alternative to classical methods

Designs a nonlinear control algorithm for robot formation based on PDE approximation

Abstract

In this paper we present a continuation method which transforms spatially distributed ODE systems into continuous PDE. We show that this continuation can be performed both for linear and nonlinear systems, including multidimensional, space- and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing evolution of 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 alternative solution to Hilbert's 6th problem. Finally, we leverage our derivation of the Euler equations to control multiagent systems, designing a nonlinear control algorithm for robot formation based on its continuous approximation.