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Boundary stabilization of linear elastodynamic systems. (English) Zbl 0531.93044
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 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)].

MSC:
93D15 Stabilization of systems by feedback
35L20 Initial-boundary value problems for second-order hyperbolic equations
93C20 Control/observation systems governed by partial differential equations
93C05 Linear systems in control theory
74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
35B37 PDE in connection with control problems (MSC2000)
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