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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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