Null controllability of the linearized compressible Navier Stokes system in one dimension

In this paper we consider the one-dimensional compressible Navier–Stokes equations linearized around a constant steady state (Q0,V0)(Q0,V0), Q0>0Q0>0, V0>0V0>0, with periodic boundary conditions in the interval I2π:=(0,2π)I2π:=(0,2π). We explore the controllability of this linearized system using a control only for the velocity equation. We prove that the linearized system with homogeneous periodic boundary conditions is null controllable in H˙per1(I2π)×L2(I2π) by a localized interior control when time is sufficiently large, where H˙per1(I2π) denotes the Sobolev space of periodic functions with mean value zero. We show null controllability of the system by proving an observability inequality with the help of two types of Ingham inequality.We also consider the analogous problem with Dirichlet boundary conditions rather than periodicity. For this case, we show approximate controllability and null controllability in the case of creeping flow.