Local boundary controllability to trajectories for the 1d compressible Navier Stokes equations

Institut de Mathématiques de Toulouse, UMR5219; Université de Toulouse, CNRS, UPS IMT, 31062 Toulouse Cedex 9, France.
ervedoza@math.univ-toulouse.fr

Corresponding author: marc.savel@math.univ-toulouse.fr

Received: 21 September 2015
Revised: 26 September 2016

Abstract

In this article, we show a local exact boundary controllability result for the 1d isentropic compressible Navier Stokes equations around a smooth target trajectory. Our controllability result requires a geometric condition on the flow of the target trajectory, which comes naturally when dealing with the linearized equations. The proof of our result is based on a fixed point argument in weighted spaces and follows the strategy already developed in [S. Ervedoza, O. Glass, S. Guerrero, J.-P. Puel, Arch. Ration. Mech. Anal. 206 (2012) 189–238] in the case of a non-zero constant velocity field. The main novelty of this article is in the construction of the controlled density in the case of possible oscillations of the characteristics of the target flow on the boundary.