Particle supported control of a fluid–particle system

Abstract

MSC:In this paper we study, from a control theoretic view point, a 1D model of fluid-particle interaction. More precisely, we consider a point mass moving in a pipe filled with a fluid. The fluid is modelled by the viscous Burgers equation whereas the point mass obeys Newton's second law. The control variable is a force acting on the mass point. The main result of the paper asserts that for any initial data there exist a time T > 0 and a control such that, at the end of the control process, the particle reaches a point arbitrarily close to a given target, whereas the velocities of the fluid and of the point mass are driven exactly to zero. Therefore, within this simplified model, we can control simultaneously the fluid and the particle, by using inputs acting on the moving point only. Moreover, the main result holds without any smallness assumptions on the initial data. Alternatively, we can see our results as yielding controllability of the viscous Burgers equation by a moving internal boundary.© 2015 Elsevier Masson SAS. All rights reserved. r é s u m é Dans cet article on étudie la contrôlabilité d'un modèle fluide-particule uni-dimensionnel. Le fluide est modélisé par l'équation de Burgers visqueuse et le mouvement du point matériel est gouverné par la deuxième loi de Newton. Le résultat principal de cet article affirme que pour toute donnée initiale il existe un temps T > 0 et un contrôle tels que, au temps final T , la particule atteint un point arbitrairement proche de la cible donnée et les vitesses du fluide et du point matériel sont nulles. Autrement dit, dans le cadre de ce modèle simplifié, on peut contrôler simultanément le fluide et la particule en agissant seulement sur la particule. De plus, le résultat principal a lieu sans supposer la petitesse des données initiales. Les résultats peuvent également être interpretés comme donnant la contrôlabilité de *

Analysis of a system modelling the motion of a piston in a viscous gas

“…We intended in this way to transpose to this physically motivated model our previous results obtained for a toy model, in which the compressible Navier-Stokes equations are replaced by the viscous Burgers equation. We refer to Vázquez and Zuazua [21,22] for the description and the analysis of the toy model and Liu et al [13] and Cîndea et al [6] for the associated control problems. We quickly realized that the major difficulty to be solved in order to accomplish the proposed goal consists in proving the global in time existence and uniqueness of solutions, in appropriate function spaces.…”

“…The above result can be proved modulo an obvious adaptation of the proof of the corresponding result for α 0 = ζ 0 = 1, u −1 = u 1 = 0 which has been given in Proposition 3.3 from [6], so that we omit the details.…”

Controllability of One-Dimensional Viscous Free Boundary Flows

“…This issue was mended in [26], where the authors introduce a systematic methodology for tackling the null-controllability of parabolic systems in spite of source terms, without requiring Carleman inequalities (they thus use spectral techniques). We also refer to the work of Cindea, Micu, Roventa and Tucsnak [7], where the authors consider a control actuating only on the moving particle: m (t) = [v z ](t, (t)) + u(t). They prove global null-controllability (in large time) for the fluid and particle velocities, and approximate controllability for the particle's position.…”

Local Stabilization of a Fluid--Structure System around a Stationary State with a Structure Given by a Finite Number of Parameters

“…The reader can refer for instance to [28] for the stabilization by a POD approach of a fluid around an airfoil. It is also possible to build a stabilizing feedback control that uses only the state of the structure, see [7] for a 1D and [29] for a 2D fluid-solid interaction problems.…”