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 −ωt for ω > ω 0 , where ω 0 is an accumulation point of the real eigenvalues of the linearized operator.

Interior local null controllability of one‐dimensional compressible flow near a constant steady state

“…In P H 2 per .I 2 / H 2 per .I 2 /, the well-posedness of (3.20), without control, follows by differentiating the equations twice with respect to x and then using the well-posedness of the new system (similar to (3.20)) in P L 2 .I 2 / L 2 .I 2 / (see Remark 2.5 in [5]). Hence, for any small ı > 0, we obtain a control g 4 Finally, we take a smooth function, which interpolates between .% 4 f , 4 f / at time t D 2 C =2 and .0, 0/ at time t D 2 C , that is,…”

“…The differences arising from (3.17) are also easy to estimate. We thus end up with a bound of the form 4 . , 2 C =2/k H 2 per .I2 / C k 4 .…”

“…To obtain the regularity we used previously, we need to ensure that the singular corner eigenfunctions for the triharmonic Dirichlet problem have at least H 4 regularity. This reduces to showing that the transcendental equation cos. / D 17 48 C 44 2 16 3 C 2 4 (5. 10) has no roots in the strip 2 Ä Re Ä 3, other than the trivial roots at 2 and 3.…”

“…We refer to [2][3][4][5][6] for earlier results on a linearized version of our system. There have been a few results on the null controllability of the nonlinear system.…”

Feedback stabilization of a simplified 1d fluid–particle system

“…This comes from the fact that the corresponding 1-D fluid-solid interaction system satisfied by (v, ρ) is coupling a parabolic equation for the velocity and a hyperbolic equation for the density and, even for the linearized equations, there are few tools developed to obtain the stabilization of such systems. About the controllability and the stabilizability of such a type of system, but without solid interaction, we shall mention [13,8].…”