Some controllability results for linearized compressible navier−stokes system
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 1002-1028.

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 (ρ ¯,0,θ ¯), with ρ ¯>0,θ ¯>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 (ρ ¯,v ¯,θ ¯), with ρ ¯>0,v ¯>0,θ ¯>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 (ρ ¯,0). We prove that this system is also not null controllable by localized interior control.

DOI: 10.1051/cocv/2014056
Classification: 35Q30, 93C20, 93B05
Keywords: Linearized compressible Navier−Stokes System, Null controllability, localized interior control, boundary control, Gaussian Beam
Maity, Debayan 1

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

E.V. Amosova, Exact local controllability for equations of viscous gas dynamics. Differ. Equ. 47 (2011) 1776–1795. | DOI | MR | Zbl

F. Chaves-Silva, L. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control. J. Math. Pures Appl. 101 (2014) 198–222. | DOI | MR | Zbl

S. Chowdhury, Approximate Controllability for Linearized Compressible Navier−Stokes System. J. Math. Anal. Appl. 422 (2015) 1034–1057. | DOI | MR | Zbl

S. Chowdhury, D. Mitra, M. Ramaswamy and M. Renardy, Null controllability of the Linearized Compressible Navier−Stokes System in One Dimension. J. Differ. Equ. 257 (2014) 3813–3849. | DOI | MR | Zbl

S. Chowdhury, M. Ramaswamy and J.P. Raymond, Controllability and stabilizability of the linearized compressible Navier−Stokes System in one dimension. SIAM J. Control Optim. 50 (2012) 2959–2987. | DOI | MR | Zbl

J.M. Coron, Control and Nonlinearity. AMS, Math. Surv. Monogr. 136 (2007). | MR | Zbl

S. Ervedoza, O. Glass, S. Guerrero and J.-P. Puel, Local exact controllability for the one-dimensional compressible Navier−Stokes equation. Arch. Ration. Mech. Anal. 206 (2012) 189–238. | DOI | MR | Zbl

E. Feireisl, Dynamics of Viscous Compressible Fluids. Oxford Lect. Series Math. Appl. 26 (2014). | Zbl

F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach. Asymptot. Anal. 32 (2002) 1-26. | MR | Zbl

P. Martin, L. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control. SIAM J. Control Optim. 51 (2013) 660–684. | DOI | MR | Zbl

S. Micu, On the controllability of the linearized Benjamin−Bona−Mahony equation. SIAM J. Control Optim. 39 (2001) 1677–1696. | DOI | MR | Zbl

S. Micu and E. Zuazua, An Introduction to the Controllability of Partial Differential Equations. Available at http://www.uam.es/personal˙pdi/ciencias/ezuazua/informweb/argel.pdf. | Zbl

J. Ralston, Gaussian beams and the propagation of singularities. Studies in partial differential equations, 206248, MAA Stud. Math. 23. Math. Assoc. America, Washington, DC (1982) 204–248. | MR | Zbl

M. Renardy, A note on a class of observability problems for PDEs. Systems Control Lett. 58 (2009) 183–187. | DOI | MR | Zbl

L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping. Int. J. Tomogr. Stat. 5 (2007) 79–84. | MR

J. Zabczyk, Mathematical control theory. An introduction. Modern Birkhäuser Classics. Reprint of the 1995 edition. Birkhäuser Boston, Inc., Boston, MA (2008). | MR | Zbl

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