Local stabilization of the compressible Navier–Stokes system, around null velocity, in one dimension

In this paper we study the exponential stabilization of the one dimensional compressible Navier–Stokes system, in a bounded interval (0,π)(0,π), locally around a constant steady state (ρ¯,0), ρ¯>0, by a localized distributed control acting only in the velocity equation. We determine a linear feedback law able to stabilize a nonlinear transformed system. Coming back to the original nonlinear system, we obtain a nonlinear feedback law able to stabilize locally this nonlinear system. To the best of our knowledge, the results of the paper are the first ones providing feedback control laws stabilizing compressible fluid flows.