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

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 largescale 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.

Synchronization of Spin-Torque Oscillators via Continuation Method

“…The main idea now is to use a PDE approximation of the original ODE system (1). A general description of our method to derive PDE approximations for nonlinear systems of ODEs is given in [17]. Here we propose a self-contained example application of this method to obtain a Kuramoto PDE.…”

“…It is now possible to perform continuation for the coupled system in the same way it was done for Kuramoto oscillators in the previous section. According to [17], the continuation of ( 20) is performed in several steps:…”

“…We approach this problem by utilizing a recently developed continuation method, which transforms a network of coupled ODEs into a single PDE. The general continuation method was introduced by the authors in [17] and was applied in [18] to derive an urban traffic PDE model and in [19] to stabilize a Stuart-Landau laser chain. To demonstrate applicability of the method to the problem of network synchronization, in Section II we start with a toy example of analysis of the Kuramoto model synchronization on a ring.…”

Multi-directional continuous traffic model for large-scale urban networks

“…In the end, we will be able to describe the whole urban area due to a unified description of the traffic behaviour at any intersection. This unified description will be obtained using the continuation method that has just been introduced in [43].…”

“…The NEWS model is derived from the CTM in Section 5, where we also show how to obtain a PDE from an ODE (ordinary differential equation) using a new continuation method presented in [43]. In Section 6, mathematical properties of the NEWS model are analysed: the conservation law property, hyperbolicity and boundedness of its state (four-dimensional density).…”