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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

Debanjana Mitra 1, , Mythily Ramaswamy 1, and  Jean-Pierre Raymond 2,

1. 

T.I.F.R Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Oce, 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.
Keywords: localized interior control., stabilizability, Compressible Navier-Stokes equations.
Mathematics Subject Classification: Primary: 93C20, 93D15; Secondary: 34H05, 93B52, 35B35, 76N2.
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 and 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), 1776-1795. doi: 10.1134/S001226611112007X.

[2]

A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Second edition, Systems and Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.

[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), 2959-2987. doi: 10.1137/110846683.

[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.

[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.

[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), 3813-3849. doi: 10.1016/j.jde.2014.07.010.

[7]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[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), 189-238. doi: 10.1007/s00205-012-0534-3.

[9]

S. Kesavan and J.-P. Raymond, On a degenerate Riccati equation, Control Cybernet., 38 (2009), 1393-1410.

[10]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329. doi: 10.1007/s002050050078.

[11]

D. Mitra, M. Ramaswamy and J.-P. Raymond, Local stabilization of compressible Navier-Stokes equations in one dimension around non-zero velocity, submitted.

[12]

J. Zabczyk, Mathematical Control Theory. An Introduction, Reprint of the 1995 edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

[13]

K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall, Englewood Cliffs, New Jersey, 1995.

show all references

References:
[1]

E. V. Amosova, Exact local controllability for the equations of viscous gas dynamics, Differential Equations, 47 (2011), 1776-1795. doi: 10.1134/S001226611112007X.

[2]

A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Second edition, Systems and Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.

[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), 2959-2987. doi: 10.1137/110846683.

[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.

[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.

[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), 3813-3849. doi: 10.1016/j.jde.2014.07.010.

[7]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[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), 189-238. doi: 10.1007/s00205-012-0534-3.

[9]

S. Kesavan and J.-P. Raymond, On a degenerate Riccati equation, Control Cybernet., 38 (2009), 1393-1410.

[10]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329. doi: 10.1007/s002050050078.

[11]

D. Mitra, M. Ramaswamy and J.-P. Raymond, Local stabilization of compressible Navier-Stokes equations in one dimension around non-zero velocity, submitted.

[12]

J. Zabczyk, Mathematical Control Theory. An Introduction, Reprint of the 1995 edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

[13]

K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall, Englewood Cliffs, New Jersey, 1995.

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