The construction of optimal control for the motion of elastic bodies by using the method of integro-differential relations

The opportunities of modeling and optimization of motion of elastic systems with distributed parameters are investigated. A regular integro-differential approach, which reduces a wide class of linear initialboundary value problems to a conditional minimization of non-negative quadratic functionals is developed, and a cost function of approximate solutions obtained is proposed. For longitudinal motions of a uniform straight elastic rod, the case of polynomial control of the motion of its end is considered. An algorithm of constructing an optimal control that steers the system to the state of a minimal mechanical energy at the final time instant. The parameters of the problem are adjusted so that the time of transition processes would be comparable with the interval, on which the motions are investigated. The analysis and comparison of the results obtained by using the method of integro-differential relations for a one-dimensional model of a thin elastic rod and a proposed approximate three-dimensional model of a prismatic beam.