an:03843614
Zbl 0531.93044
Lagnese, John
Boundary stabilization of linear elastodynamic systems
EN
SIAM J. Control Optimization 21, 968-984 (1983).
00147776
1983
j
93D15 35L20 93C20 93C05 74B99 74H99 35B37
elastodynamic systems; boundary control; uniform stabilization
This paper considers vibrations of a finite elastic medium whose motion can be modeled by the linear, autonomous, elastodynamic system \(\rho u_{i,tt}-\sigma_{ij,j}+qu_ i=0, i=1,2,3\), where \(\rho\) (x) is the local density of the medium, \(\sigma_{ij}(x)\) the stress tensor, q(x)u a restoring force (q(x)\(\geq 0)\) and u(x,t) the local displacement. A portion \(\Gamma_ 0\) of the boundary of the medium is clamped and a traction force of the form \(\sigma_{ij}n_ j=-bu_{i,t}\) is exerted on the remainder \(\Gamma_ 1\) of the boundary. It is trivial that if b(x)\(\geq 0\), the energy within the medium is non-increasing in time. The main result of this paper is that the energy decays exponentially if \(b(x)\geq b_ 0>0\) and if \(\Gamma_ 0\) and \(\Gamma_ 1\) satisfy certain geometric conditions. For related results see \textit{G. Chen} [J. Math. Pures Appl. IX. Ser. 58, 249-274 (1979; Zbl 0414.35044) and SIAM J. Control Optimization 19, 106-113 (1981; Zbl 0461.93036)] and the author [J. Differ. Equations 50, 163-182 (1983)].
Zbl 0414.35044; Zbl 0461.93036