Optimal distributed control of a continuous beam with damping

Optimal control of a damped two-span beam is studied with the objective of minimizing its deflection and velocity in a given period of time with the minimum possible expenditure of force. The beam undergoes transient vibrations and is subject to given displacement and velocity initial conditions. The control is exercised by means of a transverse distributed force. The multiple objectives of the problem lead to a vector performance criterion which is reduced to a scalar one by using the concept of Pareto optimality. Necessary and sufficient conditions of optimality are expressed in the form of a maximum principle which leads to an explicit expression for the control force. The behaviour of the controlled beam is numerically studied which indicates that optimally controlled distributed forces are effective in damping out the dynamic response. Relations between various objectives are studied by means of optimal trade-off curves showing the best performance of the controlled beam.