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

Abstract

In this paper we present a continuation method which transforms spatially\ndistributed ODE systems into continuous PDE. We show that this continuation can\nbe performed both for linear and nonlinear systems, including multidimensional,\nspace- and time-varying systems. When applied to a large-scale network, the\ncontinuation provides a PDE describing evolution of continuous state\napproximation that respects the spatial structure of the original ODE. Our\nmethod is illustrated by multiple examples including transport equations,\nKuramoto equations and heat diffusion equations. As a main example, we perform\nthe continuation of a Newtonian system of interacting particles and obtain the\nEuler equations for compressible fluids, thereby providing an original\nalternative solution to Hilbert's 6th problem. Finally, we leverage our\nderivation of the Euler equations to control multiagent systems, designing a\nnonlinear control algorithm for robot formation based on its continuous\napproximation.\n