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

To model vibrations in flexible structures, a generalized variational formulation of PDE control problems is considered in the frame of the method of integro-differential relations. This approach allows us 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 related control signals. The procedures for solving optimization problems in dynamics of linear elasticity have been proposed based on the Ritz method and FEM. An original FEM solver for mechanical systems withvarying distributed parameters is described. The resulting control law is regularized via a quadratic cost functional including the discrepancy of the constitutive equations. The verification of optimized control for elastic rod motion involves local and integral error estimates proposed.

POD basis updates for nonlinear PDE control

In the present paper a semilinear boundary control problem is considered. For its numerical solution proper orthogonal decomposition (POD) is applied. POD is based on a Galerkin type discretization with basis elements created from the evolution problem itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. Therefore, different POD basis update strategies which avoid this problem of unmodelled dynamics are compared numerically.