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

@article{Chowdhury2012ControllabilityAS,
title={Controllability and Stabilizability of the Linearized Compressible Navier-Stokes System in One Dimension},
author={Shirshendu Chowdhury and Mythily Ramaswamy and J. P. Raymond},
journal={SIAM J. Control. Optim.},
year={2012},
volume={50},
pages={2959-2987}
}
• Published 2 October 2012
• Mathematics
• SIAM J. Control. Optim.
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…
30 Citations
Some Controllability Results for Linearized Compressible Navier-Stokes System
In this article, we study the null controllability of linearized compressible Navier-Stokes system in one and two dimension. We first study the one-dimensional compressible Navier-Stokes system for
Boundary null-controllability of 1d linearized compressible Navier-Stokes system by one control force
• Mathematics
• 2022
. In this article, we study the boundary null-controllability properties of the one-dimensional linearized (around ( Q 0 ,V 0 ) with constants Q 0 > 0 ,V 0 > 0) compressible Navier–Stokes equations
Null controllability of the linearized compressible Navier–Stokes equations using moment method
• Mathematics
• 2015
In this paper, we consider one-dimensional compressible Navier–Stokes equations linearized around a constant steady state (Q0, V0), Q0 >  0, V0 >  0 with periodic boundary condition. We explore the
Local exact controllability for Navier-Stokes-Korteweg model in dimension $d\in\{2,3\}$
In this paper, we investigate the local null controllability for a compressible Navier-Stokes-Korteweg with quantum pressure in dimension d ∈ {2, 3}, when the control acts on the whole boundary of a
LARGEST SPACE FOR THE STABILIZABILITY OF THE LINEARIZED COMPRESSIBLE NAVIER-STOKES SYSTEM IN ONE DIMENSION
• Mathematics
• 2015
In this paper we determine the largest space in which the linearized compressible Navier-Stokes system in one dimension, with periodic boundary conditions, is stabilizable with any prescribed
Local boundary controllability to trajectories for the 1d compressible Navier Stokes equations
• Mathematics
• 2018
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
Local exact controllability for the 1-d compressible Navier-Stokes equations
Let us emphasize that the boundary conditions do not appear in the equation (1.1), as frequently happens when controlling hyperbolic equations like the equation of the density. They will be used as
Local exact controllability for the 2 and 3-d compressible Navier-Stokes equations
• Mathematics
• 2015
The goal of this article is to present a local exact controllability result for the 2 and 3-dimensional compressible Navier-Stokes equations on a constant target trajectory when the controls act on

## References

SHOWING 1-10 OF 24 REFERENCES
On the Controllability of the Linearized Benjamin--Bona--Mahony Equation
• S. Micu
• Mathematics
SIAM J. Control. Optim.
• 2001
This work studies the boundary controllability properties of the linearized Benjamin--Bona--Mahony equation and proves a finite controllable result and estimates the norms of the controls needed in this case.
Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers
• Mathematics
• 2010
We study the boundary stabilization of the two-dimensional Navier-Stokes equations about an unstable stationary solution by controls of finite dimension in feedback form. The main novelty is
Local Exact Controllability for the One-Dimensional Compressible Navier–Stokes Equation
• Mathematics
• 2012
In this paper we deal with the isentropic (compressible) Navier-Stokes equation in one space dimension and we adress the problem of the boundary controllability for this system. We prove that we can
Tangential boundary stabilization of Navier-Stokes equations
• Mathematics
• 2006
Introduction Main results Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): $d=3$ Boundary feedback uniform stabilization of the linearized system (3.1.4) via an optimal control problem
Stabilization for the 3D Navier-Stokes system by feedback boundary control
We study the problem of stabilization a solution to 3D Navier-Stokes system given in a bounded domain $\Omega$. This stabilization is carried out with help of feedback control defined on a part
Local exact controllability of the Navier – Stokes system ✩
In this paper we deal with the local exact controllability of the Navier–Stokes system distributed controls s upported in small sets. In a first step, we present a new Carleman ineq for the
Optimal control of unsteady compressible viscous flows
• Engineering, Mathematics
• 2002
The control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can
Some Controllability Results for Linear Viscoelastic Fluids
• Mathematics
SIAM J. Control. Optim.
• 2012
The large time approximate-finite dimensional controllability of the system, with distributed or boundary controls supported by arbitrary small sets, is established and the large time exact controllable of fluids of the same kind with controls support by suitable large sets is proved.
Boundary stabilizability of the linearized viscous Saint-Venant system
• Mathematics
• 2011
We consider a shallow water flow in a channel modeled by the Saint-Venant equations with a viscous term. We are interested in the stabilization of the flow at a steady state. We establish that the