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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 |
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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