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 non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around $(\overline{\rho},0,\overline{\theta})$, with $\overline{\rho}>0,\overline{\theta}>0$ is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove that the the one-dimensional compressible Navier−Stokes system for non-barotropic fluid linearized around a constant steady state $(\overline{\rho},\overline{v},\overline{\theta})$, with $\overline{\rho}>0,\overline{v}>0,\overline{\theta}>0$ is not null controllable by localized interior control or by boundary control for small time $T.$ Next we consider two-dimensional compressible Navier−Stokes system for barotropic fluid linearized around a constant steady state $(\overline{\rho},0)$. We prove that this system is also not null controllable by localized interior control.

Keywords: Linearized compressible Navier−Stokes System, Null controllability, localized interior control, boundary control, Gaussian Beam

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@article{COCV_2015__21_4_1002_0, author = {Maity, Debayan}, title = {Some controllability results for linearized compressible navier\ensuremath{-}stokes system}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1002--1028}, publisher = {EDP-Sciences}, volume = {21}, number = {4}, year = {2015}, doi = {10.1051/cocv/2014056}, mrnumber = {3395753}, zbl = {1328.35154}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2014056/} }

TY - JOUR AU - Maity, Debayan TI - Some controllability results for linearized compressible navier−stokes system JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 1002 EP - 1028 VL - 21 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014056/ DO - 10.1051/cocv/2014056 LA - en ID - COCV_2015__21_4_1002_0 ER -

%0 Journal Article %A Maity, Debayan %T Some controllability results for linearized compressible navier−stokes system %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 1002-1028 %V 21 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014056/ %R 10.1051/cocv/2014056 %G en %F COCV_2015__21_4_1002_0

Maity, Debayan. Some controllability results for linearized compressible navier−stokes system. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 1002-1028. doi : 10.1051/cocv/2014056. http://www.numdam.org/articles/10.1051/cocv/2014056/

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