The problem of optimally controlling'vibrations of a beam with a variable cross-section, subject to various boundary conditions, is solved by formulating a maximum principle. The problem is posed as a multiobjective control problem in which functional~ of displacement, velocity, and force make up the performance measure. Compromise solutions are sought within the framework of Pareto optimality, with a view toward minimizing dynamic response of the beam within a specified time and with a minimum expenditure of force. Explicit solutions are given for an example problem, the behavior of which is numerically studied. In particular, the effectiveness of the proposed control law and the effect of various problem parameters on vibration damping are evaluated. Relations between various objectives are studied by means of optimal tradeoff curves that provide a valuable tool for choosing the most suitable combination of objectives.
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