Discrete optimal control for Birkhoffian systems

In this paper, we investigate a discrete variational optimal control for mechanical systems that admit a Birkhoffian representation. Instead of discretizing the original equations of motion, our research is based on a direct discretization of the Pfaff–Birkhoff–d’Alembert principle. The resulting discrete forced Birkhoffian equations then serve as constraints for the minimization of the objective functional. In this way, the optimal control problem is transformed into a finite-dimensional optimization problem, which can be solved by standard methods. This approach yields discrete dynamics, which is more faithful to the continuous equations of motion and consequently yields more accurate solutions to the optimal control problem which is to be approximated. We illustrate the method numerically by optimizing the control for the damped oscillator.