Optimal Trajectory Control of Geometrically Exact Strings with Space-Time Finite Elements

In this contribution, we present a variational space-time formulation which generates an optimal feed-forward controller for geometrically exact strings. More concretely, the optimization problem is solved with an indirect approach, and the space-time finite element method translates the problem to a set of algebraic equations. Thereby, only the positional field and the corresponding adjoint variable field are approximated by continuous shape functions, which makes the discretization of a velocity field unnecessary. In addition, the variational formulation can be solved using commercial or open source finite element packages. The entire approach can also be interpreted as a multiple-shooting method for solving the optimality conditions based on the semi-discrete problem. The performance of our approach is demonstrated by a numerical test.