# American Institute of Mathematical Sciences

June  2015, 5(2): 259-290. doi: 10.3934/mcrf.2015.5.259

## Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension

 1 T.I.F.R Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Oce, Bangalore-560065, India, India 2 Institut de Mathématiques de Toulouse, Université Paul Sabatier & CNRS, 31062 Toulouse Cedex, France

Received  May 2014 Revised  February 2015 Published  April 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 exponential decay rate, by an interior control acting only in the velocity equation. As a consequence, it also follows that this largest space for the stabilizability with any prescribed exponential decay rate is also the largest one for the null controllability of the same system.
Citation: Debanjana Mitra, Mythily Ramaswamy, Jean-Pierre Raymond. Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension. Mathematical Control & Related Fields, 2015, 5 (2) : 259-290. doi: 10.3934/mcrf.2015.5.259
##### References:
 [1] E. V. Amosova, Exact local controllability for the equations of viscous gas dynamics,, Differential Equations, 47 (2011), 1776. doi: 10.1134/S001226611112007X. Google Scholar [2] A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, Second edition, (2007). doi: 10.1007/978-0-8176-4581-6. Google Scholar [3] 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. doi: 10.1137/110846683. Google Scholar [4] S. Chowdhury, D. Maity, M. Ramaswamy and J.-P. Raymond, Local stabilization of the compressible Navier-Stokes system, around a null velocity, in one dimension,, to appear in J. Differential Equations., (). doi: 10.1016/j.jde.2015.02.025. Google Scholar [5] S. Chowdhury and D. Mitra, Null controllability for linearized compressible Navier-Stokes equations by moment method,, J. Evol. Equ., (2014). doi: 10.1007/s00028-014-0263-1. Google Scholar [6] S. Chowdhury, D. Mitra, M. Ramaswamy and M. Renardy, Null controllability of the linearized compressible Navier-Stokes system in one dimension,, J. Differential Equations, 257 (2014), 3813. doi: 10.1016/j.jde.2014.07.010. Google Scholar [7] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Texts in Applied Mathematics, (1995). doi: 10.1007/978-1-4612-4224-6. Google Scholar [8] S. Ervedoza, O. Glass, S. Guerrero and J.-P. Puel, Local exact controllability for the one-dimensional compressible Navier-Stokes equation,, Arch. Rational Mech. Anal., 206 (2012), 189. doi: 10.1007/s00205-012-0534-3. Google Scholar [9] S. Kesavan and J.-P. Raymond, On a degenerate Riccati equation,, Control Cybernet., 38 (2009), 1393. Google Scholar [10] G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity,, Arch. Rational Mech. Anal., 141 (1998), 297. doi: 10.1007/s002050050078. Google Scholar [11] D. Mitra, M. Ramaswamy and J.-P. Raymond, Local stabilization of compressible Navier-Stokes equations in one dimension around non-zero velocity,, submitted., (). Google Scholar [12] J. Zabczyk, Mathematical Control Theory. An Introduction,, Reprint of the 1995 edition, (1995). doi: 10.1007/978-0-8176-4733-9. Google Scholar [13] K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control,, Prentice Hall, (1995). Google Scholar

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##### References:
 [1] E. V. Amosova, Exact local controllability for the equations of viscous gas dynamics,, Differential Equations, 47 (2011), 1776. doi: 10.1134/S001226611112007X. Google Scholar [2] A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, Second edition, (2007). doi: 10.1007/978-0-8176-4581-6. Google Scholar [3] 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. doi: 10.1137/110846683. Google Scholar [4] S. Chowdhury, D. Maity, M. Ramaswamy and J.-P. Raymond, Local stabilization of the compressible Navier-Stokes system, around a null velocity, in one dimension,, to appear in J. Differential Equations., (). doi: 10.1016/j.jde.2015.02.025. Google Scholar [5] S. Chowdhury and D. Mitra, Null controllability for linearized compressible Navier-Stokes equations by moment method,, J. Evol. Equ., (2014). doi: 10.1007/s00028-014-0263-1. Google Scholar [6] S. Chowdhury, D. Mitra, M. Ramaswamy and M. Renardy, Null controllability of the linearized compressible Navier-Stokes system in one dimension,, J. Differential Equations, 257 (2014), 3813. doi: 10.1016/j.jde.2014.07.010. Google Scholar [7] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Texts in Applied Mathematics, (1995). doi: 10.1007/978-1-4612-4224-6. Google Scholar [8] S. Ervedoza, O. Glass, S. Guerrero and J.-P. Puel, Local exact controllability for the one-dimensional compressible Navier-Stokes equation,, Arch. Rational Mech. Anal., 206 (2012), 189. doi: 10.1007/s00205-012-0534-3. Google Scholar [9] S. Kesavan and J.-P. Raymond, On a degenerate Riccati equation,, Control Cybernet., 38 (2009), 1393. Google Scholar [10] G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity,, Arch. Rational Mech. Anal., 141 (1998), 297. doi: 10.1007/s002050050078. Google Scholar [11] D. Mitra, M. Ramaswamy and J.-P. Raymond, Local stabilization of compressible Navier-Stokes equations in one dimension around non-zero velocity,, submitted., (). Google Scholar [12] J. Zabczyk, Mathematical Control Theory. An Introduction,, Reprint of the 1995 edition, (1995). doi: 10.1007/978-0-8176-4733-9. Google Scholar [13] K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control,, Prentice Hall, (1995). Google Scholar
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