Pointwise boundary stabilizability of hyperbolic evolution equations: Two-dimensional hybrid elastic structures

AbstractA two-dimensional hybrid elastic structure of a rectangular membrane linked with two rib strings on the boundary sides that have rigid bodies and pointwise controllers attached at the corner points is modelled as an abstract hyperbolic evolution equation. By an analysis of energy decay and ω-limit sets with a dissipative feedback, the following necessary and sufficient condition for stabilizability in the energy space is obtained: vi2 − vj2 ≠ 2n(πl2)2, for any positive integer n and for any i > j, where v0 = 0 and vi (i = 1, 2,…) are the increasing positive roots of the transcendental equation cot(l1μ) = 12(μa − aμ), in which a > 0 is a structural constant, l1 × l2 is the size of the region. If this condition is satisfied, then the stabilization is achieved by a pointwise boundary damping feedback. Otherwise the stabilization can be achieved by combining an additional boundary damping feedback controller locally distributed in the interior of one boundary string