In this paper we discuss the use of implicit Runge–Kutta schemes for the time discretization of optimal control problems with evolution equations. The specialty of the considered discretizations is that the discretizations schemes for the state and adjoint state are chosen such that discretization and optimization commute. It is well known that for Runge–Kutta schemes with this property additional order conditions are necessary. We give sufficient conditions for which class of schemes these additional order condition are automatically fulfilled. The focus is especially on implicit Runge–Kutta schemes of Gauss, Radau IA, Radau IIA, Lobatto IIIA, Lobatto IIIB and Lobatto IIIC collocation type up to order six. Furthermore, we also use a SDIRK (singly diagonally implicit Runge–Kutta) method to demonstrate, that for general implicit Runge–Kutta methods the additional order conditions are not automatically fulfilled. Numerical examples illustrate the predicted convergence rates.

中文翻译：

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含进化方程的最优控制问题的隐式 Runge-Kutta 方案

在本文中，我们讨论了使用隐式 Runge-Kutta 方案对具有演化方程的最优控制问题进行时间离散化。所考虑的离散化的特点是选择状态和伴随状态的离散化方案，以便离散化和优化交换。众所周知，对于具有此属性的 Runge–Kutta 方案，附加的阶条件是必要的。我们给出了自动满足这些附加订单条件的方案类别的充分条件。重点特别是高斯、Radau IA、Radau IIA、Lobatto IIIA、Lobatto IIIB 和 Lobatto IIIC 搭配类型的隐式 Runge-Kutta 方案，最高可达六阶。此外，我们还使用 SDIRK（单对角隐式龙格库塔）方法来证明，对于一般的隐式 Runge–Kutta 方法，附加的顺序条件不会自动满足。数值示例说明了预测的收敛速度。