论文标题
最佳控制中的离散伴随隐式同行方法
Discrete Adjoint Implicit Peer Methods in Optimal Control
论文作者
论文摘要
众所周知，在控制普通微分方程的第一个消化的方法中，伴随方法只能在其他订单条件下收敛。对于同行两步方法，我们得出了这种伴随顺序条件，并特别注意边界步骤。对于$ s $ stage方法，如果伴随方法满足订单$ s \！ - \！1 $的条件，我们证明了状态变量的订单$ s $。我们在第一作者等人的早期论文中遇到的边界上删除了一些瓶颈。 [J。计算。应用。 Math。，262：73-86，2014]，详细讨论订单对（3,2）的三阶段方法的构建，包括一些矩阵背景，用于联合前进和伴随顺序条件。突出显示了具有相等差异的节点的影响。事实证明，最有吸引力的方法与BDF有关。构建了三种三阶段方法，其中显示了数值测试中的预期顺序。
It is well known that in the first-discretize-then-optimize approach in the control of ordinary differential equations the adjoint method may converge under additional order conditions only. For Peer two-step methods we derive such adjoint order conditions and pay special attention to the boundary steps. For $s$-stage methods, we prove convergence of order $s$ for the state variables if the adjoint method satisfies the conditions for order $s\!-\!1$, at least. We remove some bottlenecks at the boundaries encountered in an earlier paper of the first author et al. [J. Comput. Appl. Math., 262:73-86, 2014] and discuss the construction of 3-stage methods for the order pair (3,2) in detail including some matrix background for the combined forward and adjoint order conditions. The impact of nodes having equal differences is highlighted. It turns out that the most attractive methods are related to BDF. Three 3-stage methods are constructed which show the expected orders in numerical tests.
