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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 *

Fan's minimax inequality is extended to the context of metric spaces with global nonpositive curvature. As a consequence, a much more general result on the existence of a Nash equilibrium is obtained.

We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.