A smoothing property of a hyperbolic system and boundary controllability
type: Item
instance of: scholarly article
title: A smoothing property of a hyperbolic system and boundary controllability
zbMATH Open document ID: 0974.74042
DOI: 10.1016/S0377-0427(99)00286-1
author: S. Taylor
published in: Journal of Computational and Applied Mathematics
publication date: 2001-12-12T00:00:00Z
review text: A general method of controllability of a strictly hyperbolic system is used for finite Timoshenko beams when physical parameters vary along the beam. Two cases are studied: 1) the beam is clamped at one end and free at the other end; 2) the beam models small motions of a hinged arm, which is hinged at one end and free at the other end. The governing equations of the Timoshenko beam are \(\rho\ddot w+(K(\psi-x'))'=0\) and \(I_\rho\dot\psi-(EI\psi')'+K(\psi-w')=0\), where \(w\) is the transverse displacement of the beam, \(\psi\) is the rotation angle, \(E\) is Young modulus, \(\rho\) is mass density, \(I\), is momentum of inertia, \(I_\varrho\) is polar momentum of inertia, and finally \(K\) is the shear modulus. The corresponding initial and boundary conditions are added. The control functions are the force \(f\) and the torque \(\tau\) in both the cases analysed. For each case, the author proves uniqueness theorems and existence of solutions with finite energy. In addition, he also considers a semi-infinite beam under similar conditions; for both these cases the author formulates an auxiliary optimal control problem for which he proves existence and uniqueness of solution and investigates a smoothing procedure. Using these smoothing, it is possible to obtain a controllability theorem for each case.
reviewed by: Ştefan Zanfir
Mathematics Subject Classification ID: 74M05
Mathematics Subject Classification ID: 93C20
zbMATH DE Number: 1419807
zbMATH Keywords: finite Timoshenko beam
zbMATH Keywords: boundary control
zbMATH Keywords: semi-infinite Timoshenko beam
zbMATH Keywords: controllability
zbMATH Keywords: uniqueness
zbMATH Keywords: existence of solutions with finite energy
zbMATH Keywords: optimal control problem
zbMATH Keywords: smoothing procedure
MaRDI profile type: Publication
cites work: Nonlinear feedback control of Timoshenko beam
cites work: Q4204078
cites work: Q4877087
cites work: Partial differential equations. 4th ed
cites work: Boundary Control of the Timoshenko Beam
cites work: Q4851730
cites work: Control of Planar Networks of Timoshenko Beams
cites work: Q3994379
cites work: Exact Controllability, Stabilization and Perturbations for Distributed Systems
cites work: Exact boundary controllability of a hybrid system of elasticity
cites work: Smoothing evolution equations and boundary control theory
cites work: Q4283316
cites work: Boundary control of a Timoshenko beam attached to a rigid body: planar motion
cites work: Dynamic boundary control of the Timoshenko beam
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Optimal control of a Timoshenko beam by distributed forces
description: scientific article; zbMATH DE number 1419807