Symplectic discretizations for optimal control problems in mechanics

机械问题的最优控制在我们受技术影响的日常生活中以及在许多科学问题中无处不在。由于最优控制问题的解析解通常不可用，应用程序依赖于稳健和准确的数值模拟，并且工程师可以直接使用。对于一般的离散化方法，得到的状态方程和伴随方程的近似是不同的。到目前为止的结果表明，某些辛方法可以对这两个方程产生相同的近似，并导致离散化和优化步骤之间的转换，例如，在直接方法和间接方法之间产生了联系。这个项目的一个目标是将这些结果推广到应用于机械系统最优控制问题的完整类辛积分器。这个项目的结果将使我们更深入地理解辛性在机械系统最优控制问题中的作用。此外，这种新的方法将提供一个方便的框架来推导最优控制问题的辛离散化，类似于力学中正动力学问题的变分积分器的使用。这使得精确的近似状态和控制方案--通常通过间接方法获得，但在推导过程中需要复杂的技巧--也可以通过直接方法获得，因此更容易用于工程应用。

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.