Verified Solution to Optimal Control Problems of Elastic Rod Motion Based on the Ritz Method

To model vibrations in flexible structures, a variational formulation of PDE control problems is considered in the frame of the method of integro-differential relations. This approach allows to estimate a posteriori the quality of finite-dimensional approximations and, as a result, either to refine or coarsen them if necessary. Such estimates also make it possible to correct the input signals. The related control law is regularized via a quadratic cost functional including the discrepancy of the constitutive equations. Procedures for solving optimization problems in dynamics of linear elasticity have been developed based on the Ritz method and FEM. The verification of optimized control for elastic rod motion involves the local and integral error estimates proposed. A FEM solver for mechanical systems with varying distributed parameters and linear boundary conditions of different kinds is presented.