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 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.
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

show all references

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

[1]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[2]

Yuming Qin, Lan Huang, Shuxian Deng, Zhiyong Ma, Xiaoke Su, Xinguang Yang. Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 163-192. doi: 10.3934/dcdss.2009.2.163

[3]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495

[4]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[5]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

[6]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

[7]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609

[8]

Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719

[9]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

[10]

Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021

[11]

Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110

[12]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

[13]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[14]

Igor Kukavica. Interior gradient bounds for the 2D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 873-882. doi: 10.3934/dcds.2001.7.873

[15]

Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409

[16]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004

[17]

Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure & Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373

[18]

Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201

[19]

Tao Wang, Huijiang Zhao, Qingyang Zou. One-dimensional compressible Navier-Stokes equations with large density oscillation. Kinetic & Related Models, 2013, 6 (3) : 649-670. doi: 10.3934/krm.2013.6.649

[20]

Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

[Back to Top]