Rapid pointwise stabilization of vibrating strings and beams

Alia Barhoumi

doi:10.5269/bspm.v27i2.10210

Abstract

Applying a general construction and using former results on the ob-servability we prove, under rather general assumptions, a rapid pointwise stabilization of vibrating strings and beams.

Highlights

Alia BARHOUMI abstract: Applying a general construction and using former results on the observability we prove, under rather general assumptions, a rapid pointwise stabilization of vibrating strings and beams
Remark 1.4 It follows from some results of Komornik and Loreti that the system (1.5) can not be exactly controllable for some exceptional choices of the parameters A, B, C, D: see [12] and [13] for explicit counter examples concerning an equivalent observability problem
In a complex Hilbert space H, where A is a linear operator defined on some linear subspace D(A) of H, with values in H and B is a linear operator, defined on some linear subspace D(B) of H with values in another Hilbert space G

Summary

Introduction

Alia BARHOUMI abstract: Applying a general construction and using former results on the observability we prove, under rather general assumptions, a rapid pointwise stabilization of vibrating strings and beams. Given an arbitrarily large positive number ω, there exist two linear operators (P, Q) : Hξ → Dξ−1 and a positive constant M such that setting v(t) := (P yt + Qy)(t, ξ) the problem (1.1) is well posed in Hξ and its solutions satisfy the inequality (y, yt) Hξ ≤ M e−ωt (y0, y1) Hξ for all (y0, y1) ∈ Hξ and t ≥ 0.

Results

Conclusion