The optimization of the motion of an elastic rod by the method of integro-differential relations

The possibility of constructing a solution for given cost functionals and the optimization of the motion of elastic systems with distributed parameters are investigated. The regular integro-differential approach to a broad class of boundary value problems is developed, and a cost functional for the solution obtained is proposed. For the two-dimensional motions of a uniform straight elastic rod, the case of polynomial control is considered. An algorithm for forming the optimal control, which steers the system to the state of minimum total energy at a final time instant, is developed. The parameters of the problem are adjusted so that the period of the lower order mode is comparable with the interval in which the motions are investigated. The results obtained by using the method of separation of variables and the method of integro-differential relations are analyzed and compared.[PUBLICATION ABSTRACT]