Controllability and Stabilizability of the Linearized Compressible Navier--Stokes System in One Dimension

Abstract

In this paper we consider the one-dimensional compressible Navier--Stokes system linearized about a constant steady state $(Q_0, 0)$ with $Q_0 > 0$. We study the controllability and stabilizability of this linearized system. We establish that the linearized system is null controllable for regular initial data by an interior control acting everywhere in the velocity equation. We prove that this result is sharp by showing that the null controllability cannot be achieved by a localized interior control or by a boundary control acting only in the velocity equation. On the other hand, we show that the system is approximately controllable. We also show that the system is not stabilizable with a decay rate $e^{-\omega t}$ for $\omega > \omega_0$, where $\omega_0$ is an accumulation point of the real eigenvalues of the linearized operator.