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March  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

Farah Abdallah 1, , Denis Mercier 1, and  Serge Nicaise 1,

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.
Keywords: Indefinite damping, spectral analysis., Riesz basis.
Mathematics Subject Classification: Primary: 35L05, 35B40; Secondary: 93D20, 35P1.
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), 337-357. doi: 10.1006/jdeq.2000.3714.  Google Scholar

[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), 266-301. doi: 10.1137/0151015.  Google Scholar

[3]

S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Partial Differential Equations, 19 (1994), 213-243. doi: 10.1080/03605309408821015.  Google Scholar

[4]

P. Freitas, On some eigenvalue problems related to the wave equation with indefinite damping, J. Differential Equations, 127 (1996), 213-243. doi: 10.1006/jdeq.1996.0072.  Google Scholar

[5]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J. Differential Equations, 132 (1996), 338-352. doi: 10.1006/jdeq.1996.0183.  Google Scholar

[6]

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

[7]

B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 39 (2001), 1736-1747. doi: 10.1137/S0363012999354880.  Google Scholar

[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), 557-574. doi: 10.1016/j.sysconle.2004.10.006.  Google Scholar

[9]

K. Liu, Z. Liu and B. Rao, Exponential stability of an abstract non-dissipative linear system, SIAM J. Control Optim., 40 (2001), 149-165. doi: 10.1137/S0363012999364930.  Google Scholar

[10]

J. E. Muoz Rivera and R. Racke, Exponential stability for wave equations with non-dissipative damping, Nonlinear Anal., 68 (2008), 2531-2551. doi: 10.1016/j.na.2007.02.022.  Google Scholar

[11]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

A. A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in the boundary conditions, Trudy Sem. Petrovsk., 9 (1983), 190-229. Russian. English transl. in J. Soviet Math. 33 (1986), 1311-1342.  Google Scholar

show all references

References:
[1]

A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping, J. Differential Equations, 161 (2000), 337-357. doi: 10.1006/jdeq.2000.3714.  Google Scholar

[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), 266-301. doi: 10.1137/0151015.  Google Scholar

[3]

S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Partial Differential Equations, 19 (1994), 213-243. doi: 10.1080/03605309408821015.  Google Scholar

[4]

P. Freitas, On some eigenvalue problems related to the wave equation with indefinite damping, J. Differential Equations, 127 (1996), 213-243. doi: 10.1006/jdeq.1996.0072.  Google Scholar

[5]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J. Differential Equations, 132 (1996), 338-352. doi: 10.1006/jdeq.1996.0183.  Google Scholar

[6]

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

[7]

B.-Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 39 (2001), 1736-1747. doi: 10.1137/S0363012999354880.  Google Scholar

[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), 557-574. doi: 10.1016/j.sysconle.2004.10.006.  Google Scholar

[9]

K. Liu, Z. Liu and B. Rao, Exponential stability of an abstract non-dissipative linear system, SIAM J. Control Optim., 40 (2001), 149-165. doi: 10.1137/S0363012999364930.  Google Scholar

[10]

J. E. Muoz Rivera and R. Racke, Exponential stability for wave equations with non-dissipative damping, Nonlinear Anal., 68 (2008), 2531-2551. doi: 10.1016/j.na.2007.02.022.  Google Scholar

[11]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

A. A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in the boundary conditions, Trudy Sem. Petrovsk., 9 (1983), 190-229. Russian. English transl. in J. Soviet Math. 33 (1986), 1311-1342.  Google Scholar

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