论文标题
ode限制的最佳控制问题的隐式A稳定对等三重
Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
论文作者
论文摘要
本文涉及对两步性质的新型隐式同伴三胞质的构建和收敛分析，具有四个阶段，用于非线性ode受限的最佳控制问题。我们结合了某些标准对等方法的内部网格点的超授权的属性，以及精心设计的启动和结束方法，以实现状态变量的订单四，并在第一迪比特化的伴随变量和A稳定性以及A稳定性的方法中为伴随变量订购第三种。概念三重态强调，这三种不同的同行方法必须满足其他匹配条件。构建了四个这样的实际兴趣同行三胞胎。同样作为基准方法，众所周知的向后分化公式BDF4（仅$ a（73.35^o）$ - 稳定，都扩展到一个特殊的同伴三重态，以提供具有等距节点的高阶和BDF类型的相邻方法。在同行三胞胎的类中，我们发现了一个对角线隐式$ a（84^o）$ - 稳定的方法，nodes symmetric in $ [0,1] $ in $ [0,1] $ to common中心表现良好。具有三个良好确定的最佳控制问题的数值测试证实了有关A稳定性的理论发现。
This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasizes that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. Also as a benchmark method, the well-known backward differentiation formula BDF4, which is only $A(73.35^o)$-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit $A(84^o)$-stable method with nodes symmetric in $[0,1]$ to a common center that performs equally well. Numerical tests with three well established optimal control problems confirm the theoretical findings also concerning A-stability.
