CONTROL OF PLANAR NETWORKS OF TIMOSHENKO BEAMS

The present study is concerned with the questions of controllability and stabilizability of planar networks of vibrating beams consisting of several Timoshenko beams connected to each other by rigid joints at all interior nodes of the system. Some of the exterior nodes are either clamped or free; controls may be applied at the remaining exterior nodes and/or at interior joints in the form of forces and/or bending moments. For a given configuration, is it at all possible to drive all vibrations to the rest configuration in a given finite time interval by means of controls acting at some or all of the available (nonclamped) nodes of the network and, if so, where should such controls be placed? Alternatively, a control objective is to construct energy absorbing boundary-feedback controls that will guarantee uniform energy decay. It is demonstrated that if such a network does not contain closed loops and if at most one of the exterior nodes is clamped, exact controllability and uniform stabilizability of the network is indeed possible by means of controls placed at the free exterior nodes of the system. On the other hand, examples are presented to demonstrate that when a closed loop is present in the network or if the network has more than one clamped exterior node, it may happen that approximate control of the network to its rest configuration is not possible even if controls are placed at every available node of the system.