A smoothing property of a hyperbolic system and boundary controllability
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.
existence of solutions with finite energy
scientific article; zbMATH DE number 1419807