Symplectic discretizations for optimal control problems in mechanics

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The optimal control of mechanical problems is omnipresent in our technically affected daily living as well as in many scientific questions. As analytical solutions of optimal control problems are in general not available, applications rely on numerical simulations that are robust and accurate, and directly utilizable by engineers. For general discretization methods, the resulting approximation to the state and adjoint equations are different. Results to date suggest that certain symplectic methods can produce the same approximation for both equations and lead to commutation between the discretization and the optimization step, e.g.~yielding a link between direct and indirect methods. One objective of this project to generalize these results to the complete class of symplectic integrators applied to optimal control problems for mechanical systems. The results of this project will lead to a deeper understanding of the role of symplecticity in optimal control problems for mechanical systems. Furthermore, this new approach will provide a convenient framework to derive symplectic discretizations for optimal control problems, similar to the use of variational integrators for forward dynamics problems in mechanics. This makes accurate schemes to approximate state and control -- which are usually obtained via indirect methods but require sophisticated skills in the derivation -- accessible also via direct methods and thus more easily available for engineering applications.