ODE约束最优控制问题的隐式A-稳定对等点三元组
Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
摘要：
针对非线性常微分方程约束最优控制问题，构造了一种新的四阶段两步隐式Peer三元组，并进行了收敛性分析.我们结合了一些标准的Peer方法的内部网格点的超收敛的性质，精心设计的开始和结束的方法，以实现四阶的状态变量和三阶的伴随变量在一个第一次离散，然后优化的方法与A-稳定。三元组的概念强调这三个不同的Peer方法必须满足额外的匹配条件。构建了四个具有实际意义的对等三元组。作为一种基准方法，将仅具有$A(73.35^o)$ -稳定性的向后微分公式BDF 4推广到一种特殊的Peer三元组，给出了高阶BDF型等距节点伴随相容方法.在类的同行三元组，我们发现了一个对角隐式$A(84^o)$稳定的方法与节点对称的$[0,1]$一个共同的中心，表现同样出色。三个建立良好的最优控制问题的数值试验证实了理论研究结果也关于A-稳定性。
英文摘要：
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.
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