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Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary. (English) Zbl 0726.73014

In plane elastodynamics the question often arises of prescribing the surface tractions \((g_ 1,g_ 2)\) on a part \(\Gamma_ 1\) of the boundary of a body in order to ensure that the total energy, which is the sum of the strain energy U(t) and the kinetic energy K(t), tends to zero uniformly, provided that E(0)\(\leq M\). If \(u,v\) are the displacement components, an application of Green’s formula yields \[ \dot E=- \int_{\Gamma_ 1}(g_ 1,g_ 2)\cdot (\dot u,\dot v)d\Gamma, \] and this implies that E(t) decays if \((g_ 1,g_ 2)\) are selected so that \((g_ 1,g_ 2)\cdot (\dot u,\dot v)\geq 0\). However, in order to establish uniform asymptotic stability, it is necessary to modify the control functions \((g_ 1,g_ 2)\) and consider the identity \[ \dot E=- \int_{\Gamma_ 1}(g_ 1,g_ 2)\cdot (\dot u,\dot v)d\Gamma -\alpha \int_{\Gamma_ 1}(v_{\tau}\dot u-u_{\tau}\dot v)d\Gamma, \] where \(\alpha\) is a positive constant \((0<\alpha <\mu)\) and \(u_{\tau}\), \(v_{\tau}\) are derivatives in the direction of the tangent to \(\Gamma\).
A second question is to prove existence, uniqueness, and regularity of solutions when the boundary data are chosen according to the feedback law. The proof can be effected by applying the theory of nonlinear contraction semigroups in Hilbert spaces. Then there is a unique pair of functions (\(u,v)\) which are “strong” solutions of the system, in a sense made precise by H. Brezis [J. Math. Pur. Appl., IX. Ser. 51, 1-168 (1972; Zbl 0237.35001)].
Reviewer: P.Villaggio (Pisa)

MSC:

74B20 Nonlinear elasticity
93D15 Stabilization of systems by feedback
35B40 Asymptotic behavior of solutions to PDEs
70J25 Stability for problems in linear vibration theory

Citations:

Zbl 0237.35001
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References:

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