Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping. (English) Zbl 0699.93070
Control and estimation of distributed parameter systems, 4th Int. Conf., Vorau/Austria 1988, ISNM 91, 211-236 (1989).
[For the entire collection see Zbl 0682.00026.]
Estimates on the rate of decay of the viscoelastic energy E(t) are obtained for solutions of a dynamic Kirchhoff plate equation which is subject to both viscoelastic damping and mechanical dissipation induced by the action of feedback controls in bending and twisting moments and shear force applied along a part of the edge of the plate, the remaining portion being clamped. If it is assumed that the monotonically decreasing relaxation kernel D(s) exhibits an algebraic rate of decay of the form \(s^ m[D(s)-D_{\infty}]\in L^ 1(0,\infty)\) (where \(D_{\infty}=\lim_{s\to \infty}D(s))\) and that the geometry of the plate is suitably restricted, it is proved that the energy satisfies \(t^{m+1}E(t)\in L^ 1(0,\infty)\).
Estimates on the rate of decay of the viscoelastic energy E(t) are obtained for solutions of a dynamic Kirchhoff plate equation which is subject to both viscoelastic damping and mechanical dissipation induced by the action of feedback controls in bending and twisting moments and shear force applied along a part of the edge of the plate, the remaining portion being clamped. If it is assumed that the monotonically decreasing relaxation kernel D(s) exhibits an algebraic rate of decay of the form \(s^ m[D(s)-D_{\infty}]\in L^ 1(0,\infty)\) (where \(D_{\infty}=\lim_{s\to \infty}D(s))\) and that the geometry of the plate is suitably restricted, it is proved that the energy satisfies \(t^{m+1}E(t)\in L^ 1(0,\infty)\).
Reviewer: J.E.Lagnese
MSC:
93D15 | Stabilization of systems by feedback |
93C20 | Control/observation systems governed by partial differential equations |