Exact Boundary Controllability of a Beam and Mass System

Abstract

In recent years much attention has been focused on the boundary controllability of beams. In the mathematical treatment of such problems, it is often assumed that the beam is free of loads along its length. This ideal situation is not always attainable. A beam might, for example, have small objects attached to it at various points, perhaps as part of the control mechanism. This paper investigates the exact boundary controllability of such a system. More specifically, we consider an Euler-Bernoulli Beam which is clamped at its left end and either pinned (i.e. supported) or free and attached to a point mass at its right end. We also allow the possibly of the beam being pinned at various points along its length, and of having a finite number of point masses mounted at various points along its length. We show that the system may be controlled (i.e. steered to its equilibrium state) in an arbitrarily short time interval by applying a torque to the right end of the beam.