TY - JOUR
T1 - Exponential stability of an abstract nondissipative linear system
AU - Liu, Kangsheng
AU - Liu, Zhuangyi
AU - Rao, Bopeng
PY - 2002
Y1 - 2002
N2 - In this paper we consider an abstract linear system with perturbation of the form dy/dt = Ay + εBy on a Hilbert space H, where A is skew-adjoint, B is bounded, and ε is a positive parameter. Motivated by a work of Freitas and Zuazua on the one-dimensional wave equation with indefinite viscous damping [P. Freitas and E. Zuazua, J. Differential Equations, 132 (1996), pp. 338-352], we obtain a sufficient condition for exponential stability of the above system when B is not a dissipative operator. We also obtain a Hautus-type criterion for exact controllability of system (A, G), where G is a bounded linear operator from another Hilbert space to H. Our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping, as well as the exponential stabilization of the elastic systems with noncolocated observation and control.
AB - In this paper we consider an abstract linear system with perturbation of the form dy/dt = Ay + εBy on a Hilbert space H, where A is skew-adjoint, B is bounded, and ε is a positive parameter. Motivated by a work of Freitas and Zuazua on the one-dimensional wave equation with indefinite viscous damping [P. Freitas and E. Zuazua, J. Differential Equations, 132 (1996), pp. 338-352], we obtain a sufficient condition for exponential stability of the above system when B is not a dissipative operator. We also obtain a Hautus-type criterion for exact controllability of system (A, G), where G is a bounded linear operator from another Hilbert space to H. Our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping, as well as the exponential stabilization of the elastic systems with noncolocated observation and control.
KW - Exact controllability
KW - Exponential stability
KW - Hautus-type criterion
KW - Indefinite damping
KW - Linear elastic system
KW - Stabilization
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U2 - 10.1137/S0363012999364930
DO - 10.1137/S0363012999364930
M3 - Article
AN - SCOPUS:0036203101
SN - 0363-0129
VL - 40
SP - 149
EP - 165
JO - SIAM Journal on Control and Optimization
JF - SIAM Journal on Control and Optimization
IS - 1
ER -