Lack of uniformly exponential stabilization for isometric 𝐶₀-semigroups under compact perturbation of the generators in Banach spaces

Guo, Faming; Guo, Ke; Zhang, Chaolun

doi:10.1090/S0002-9939-07-08698-4

Lack of uniformly exponential stabilization for isometric $C_0$-semigroups under compact perturbation of the generators in Banach spaces
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by Faming Guo, Ke Guo and Chaolun Zhang PDF

Proc. Amer. Math. Soc. 135 (2007), 1881-1887 Request permission

Addendum: Proc. Amer. Math. Soc. 137 (2009), 2809-2812.

Abstract:

This paper is concerned with non-uniformly exponential stabilization for infinite-dimensional linear systems under compact feedback in Banach spaces. We prove that a compact perturbation of the generator of an isometric $C_0$-semigroup cannot generate a uniformly exponentially stable $C_0$-semigroup in a Banach space. Finally, examples are provided to illustrate our result.

References

Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
Hector O. Fattorini, The Cauchy problem, Encyclopedia of Mathematics and its Applications, vol. 18, Addison-Wesley Publishing Co., Reading, Mass., 1983. With a foreword by Felix E. Browder. MR 692768
J. S. Gibson, A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim. 18 (1980), no. 3, 311–316. MR 569020, DOI 10.1137/0318022
Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
Fa Lun Huang, Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Differential Equations 104 (1993), no. 2, 307–324. MR 1231471, DOI 10.1006/jdeq.1993.1074
W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl. (4) 152 (1988), 281–330. MR 980985, DOI 10.1007/BF01766154
Kangsheng Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim. 35 (1997), no. 5, 1574–1590. MR 1466917, DOI 10.1137/S0363012995284928
Kangsheng Liu, Zhuangyi Liu, and Bopeng Rao, Exponential stability of an abstract nondissipative linear system, SIAM J. Control Optim. 40 (2001), no. 1, 149–165. MR 1855310, DOI 10.1137/S0363012999364930
Yue Hu Luo, Shao Peng Weng, and De Xing Feng, Stability of linear systems in Banach spaces, Kexue Tongbao (Chinese) 43 (1998), no. 15, 1676–1677 (Chinese). MR 1671371
Luc Miller, Controllability cost of conservative systems: resolvent condition and transmutation, J. Funct. Anal. 218 (2005), no. 2, 425–444. MR 2108119, DOI 10.1016/j.jfa.2004.02.001
K. Ramdani, T. Takahashi, G. Tenenbaum, and M. Tucsnak, A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator, J. Funct. Anal. 226 (2005), no. 1, 193–229. MR 2158180, DOI 10.1016/j.jfa.2005.02.009
Bopeng Rao, Uniform stabilization of a hybrid system of elasticity, SIAM J. Control Optim. 33 (1995), no. 2, 440–454. MR 1318659, DOI 10.1137/S0363012992239879
David L. Russell, Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods, J. Differential Equations 19 (1975), no. 2, 344–370. MR 425291, DOI 10.1016/0022-0396(75)90009-1
D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim. 31 (1995), no. 3, 257–295. MR 1316260, DOI 10.1007/BF01215993
R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc. 105 (1989), no. 2, 375–383. MR 953013, DOI 10.1090/S0002-9939-1989-0953013-0
Ting-yu Zhu, Yan Hu, and Fa-lun Huang, Non-exponential stability of infinite-dimensional linear systems under a generating compact perturbation, Sichuan Daxue Xuebao 40 (2003), no. 2, 217–220 (English, with English and Chinese summaries). MR 1976102