STABILITY OF SOLUTIONS FOR A EULER-BERNOULLI BEAM EQUATION WITH MEMORY AND BOUNDARY CONTROL

Abstract

Over the past years, the flexible beam structure is widely used in modern engineering because of its advantages (e.g. light weight, low energy consumption, etc.), and its control problem becomes one of the hot research topics. A large number of systems can be modeled as mechanical flexible systems such as telephone wires, conveyor belts, crane cables, helicopter blades, robotic arms, mooring lines, marine risers, and so on. However, unwanted vibrations due to the flexibility property and the time-varying disturbances restrict the utility of these flexible systems in different engineering applications. If the flexible beam system cannot be well controlled, the vibration will not only affect the accuracy and efficiency of the system, but also accelerate the equipment fatigue damage, seriously shorten the service life of the materials, and bring production safety risk and economic loss. Therefore, it is very important to effectively control flexible beam systems. Flexible beam systems and their vibration suppression have received great attention in the literatures. Boundary control has several merits for vibration suppression of the flexible beam systems. In this work we study the decay rates for the solutions to the mixed problem for Euler–Bernoulli beam equation with memory term.