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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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