Abstract Orthogonal collocation on finite elements (OCFE) has been used universally to approximate ODEs to date. For PDEs, this contribution presents a novel discretization scheme applying the methodology of OCFE to discretize both space and time domain simultaneously, named as space–time orthogonal collocation on finite elements (ST-OCFE). Due to the existence of boundary conditions, the selection of discrete points and the constitution of discretized algebraic equations are different in space and time domain. Furthermore, for solving optimal control problems constrained by PDEs, a discretize-then-optimize algorithm based on ST-OCFE and quasi-sequential approach is proposed. The formulation of discretized optimization problems and the procedure of sensitivity computation are deduced. In the algorithm, diverse types of PDEs, state constraints, and general control parameterization are considered. The proposed method has the advantages of generality, higher numerical accuracy, and easy handling of state constraints, as demonstrated by three examples.

中文翻译：

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一种基于有限元时空正交配置的PDE约束优化拟序贯算法

摘要 迄今为止，有限元正交搭配 (OCFE) 已被普遍用于近似 ODE。对于偏微分方程，这一贡献提出了一种新的离散化方案，应用 OCFE 的方法同时离散化空间和时域，称为有限元上的时空正交搭配 (ST-OCFE)。由于边界条件的存在，离散点的选取和离散代数方程的构成在空间和时间域上是不同的。此外，针对偏微分方程约束的最优控制问题，提出了一种基于ST-OCFE和拟序贯方法的离散-然后-优化算法。推导了离散优化问题的公式化和灵敏度计算过程。在算法中，不同类型的偏微分方程、状态约束、并考虑一般控制参数化。所提出的方法具有通用性、较高的数值精度和易于处理状态约束的优点，如三个例子所示。