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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)].

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 |

### Keywords:

plane elastodynamics; existence; uniqueness; regularity of solutions; boundary data; feedback law; nonlinear contraction semigroups### Citations:

Zbl 0237.35001
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\textit{J. E. Lagnese}, Nonlinear Anal., Theory Methods Appl. 16, No. 1, 35--54 (1991; Zbl 0726.73014)

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### References:

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[10] | Wang, H.K.; Chen, G., Asymptotic behavior of solutions of the one-dimensional wave equation with a nonlinear boundary stabilizer, SIAM J. control optim., 27, 758-775, (1989) · Zbl 0682.93042 |

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