Some controllability results for linearized compressible navier−stokes system

Maity, Debayan¹

¹ Centre for Applicable Mathematics, TIFR, Post Bag No. 6503, GKVK Post Office, 560065 Bangalore, India.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1002-1028

Résumé

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.

Détail

DOI : 10.1051/cocv/2014056

Classification : 35Q30, 93C20, 93B05
Keywords: Linearized compressible Navier−Stokes System, Null controllability, localized interior control, boundary control, Gaussian Beam