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

Denis Nikitin (1) , Carlos Canudas de Wit (1) , Paolo Frasca (1)

1 DANCE - Dynamics and Control of Networks

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.

Mots clés: Control of Large-Scale Networks; Multiagent Systems; PDE

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Systèmes et contrôle [cs.SY]

Denis Nikitin, Carlos Canudas de Wit, Paolo Frasca. A Continuation Method for Large-Scale Modeling and Control: from ODEs to PDE, a Round Trip. 2021. ⟨hal-03140368v2⟩