## 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

Zbl 0237.35001
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### References:

 [1] Brezis, H., Operateurs maximaux monotones, (1973), North-Holland Amsterdam · Zbl 0252.47055 [2] Browder, F., Problèmes non-linéaires, (1966), Université de Montréal Montréal · Zbl 0153.17302 [3] Conrad, F.; Leblond, J.; Marmorat, J.P., Energy decay estimates for a beam with nonlinear boundary feedback, () · Zbl 0825.93317 [4] Conrad, F.; Leblond, J.; Marmorat, J.P., Stabilization of second order evolution equations by unbounded nonlinear feedback, Proc. 5th IFAC symp. on control of distributed parameter systems, (1989), Perpignan, France [5] Duvaut, G.; Lions, J.-L., LES inéquations en Mécanique et en physique, (1972), Dunod Paris · Zbl 0298.73001 [6] Grisvard, P., Elliptic problems in nonsmooth domains, (1985), Pitman London · Zbl 0695.35060 [7] Lagnese, J.E., Boundary stabilization of thin plates, (1989), SIAM Publications Philadelphia, PA · Zbl 0682.73009 [8] Lasiecka, L., Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. diff. eqns, 79, 340-381, (1989) · Zbl 0694.35102 [9] Lasiecka, L., Asymptotic behavior of solutions to the plate equations with nonlinear dissipation occurring through the shear forces and bending moments, Appl. math. optim., 21, 167-190, (1990) · Zbl 0686.73017 [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 [11] Zuazua, E., Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. control optim., 28, 466-477, (1990) · Zbl 0695.93090
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