2013, 2(1): 1-33. doi: 10.3934/eect.2013.2.1

Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems

1. 

Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9, France, France

Received  May 2012 Revised  October 2012 Published  January 2013

In this paper, we consider two damped wave problems for which the damping terms are allowed to change their sign. Using a careful spectral analysis, we find critical values of the damping coefficients for which the problem becomes exponentially or polynomially stable up to these critical values.
Citation: Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1
References:
[1]

A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping,, J. Differential Equations, 161 (2000), 337. doi: 10.1006/jdeq.2000.3714.

[2]

G. Chen, S. A. Fulling, F. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping,, SIAM J. Appl. Math., 51 (1991), 266. doi: 10.1137/0151015.

[3]

S. Cox and E. Zuazua, The rate at which energy decays in a damped string,, Partial Differential Equations, 19 (1994), 213. doi: 10.1080/03605309408821015.

[4]

P. Freitas, On some eigenvalue problems related to the wave equation with indefinite damping,, J. Differential Equations, 127 (1996), 213. doi: 10.1006/jdeq.1996.0072.

[5]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping,, J. Differential Equations, 132 (1996), 338. doi: 10.1006/jdeq.1996.0183.

[6]

I. Gohberg and M. Krein, "Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Spaces,", 18 of Translations of Mathematical Monographs, 18 (1969).

[7]

B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass,, SIAM J. Control Optim., 39 (2001), 1736. doi: 10.1137/S0363012999354880.

[8]

B.-Z. Guo, J.-M. Wang and S.-P. Yung, On the $C_0$-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam,, Systems Control Lett., 54 (2005), 557. doi: 10.1016/j.sysconle.2004.10.006.

[9]

K. Liu, Z. Liu and B. Rao, Exponential stability of an abstract non-dissipative linear system,, SIAM J. Control Optim., 40 (2001), 149. doi: 10.1137/S0363012999364930.

[10]

J. E. Muoz Rivera and R. Racke, Exponential stability for wave equations with non-dissipative damping,, Nonlinear Anal., 68 (2008), 2531. doi: 10.1016/j.na.2007.02.022.

[11]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", 44 of Applied Math. Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1.

[12]

A. A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in the boundary conditions,, Trudy Sem. Petrovsk., 9 (1983), 190.

show all references

References:
[1]

A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping,, J. Differential Equations, 161 (2000), 337. doi: 10.1006/jdeq.2000.3714.

[2]

G. Chen, S. A. Fulling, F. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping,, SIAM J. Appl. Math., 51 (1991), 266. doi: 10.1137/0151015.

[3]

S. Cox and E. Zuazua, The rate at which energy decays in a damped string,, Partial Differential Equations, 19 (1994), 213. doi: 10.1080/03605309408821015.

[4]

P. Freitas, On some eigenvalue problems related to the wave equation with indefinite damping,, J. Differential Equations, 127 (1996), 213. doi: 10.1006/jdeq.1996.0072.

[5]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping,, J. Differential Equations, 132 (1996), 338. doi: 10.1006/jdeq.1996.0183.

[6]

I. Gohberg and M. Krein, "Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Spaces,", 18 of Translations of Mathematical Monographs, 18 (1969).

[7]

B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass,, SIAM J. Control Optim., 39 (2001), 1736. doi: 10.1137/S0363012999354880.

[8]

B.-Z. Guo, J.-M. Wang and S.-P. Yung, On the $C_0$-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam,, Systems Control Lett., 54 (2005), 557. doi: 10.1016/j.sysconle.2004.10.006.

[9]

K. Liu, Z. Liu and B. Rao, Exponential stability of an abstract non-dissipative linear system,, SIAM J. Control Optim., 40 (2001), 149. doi: 10.1137/S0363012999364930.

[10]

J. E. Muoz Rivera and R. Racke, Exponential stability for wave equations with non-dissipative damping,, Nonlinear Anal., 68 (2008), 2531. doi: 10.1016/j.na.2007.02.022.

[11]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", 44 of Applied Math. Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1.

[12]

A. A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in the boundary conditions,, Trudy Sem. Petrovsk., 9 (1983), 190.

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