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Optimal control with stochastic PDE constraints and uncertain controls

机译：具有随机PDE约束和不确定控制的最优控制

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摘要

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby forgoing the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented.
机译：解决了具有不确定性和不确定性的偏微分方程约束的问题的最优控制。定义问题的拉格朗日方程是根据随机函数来假定的，控制函数可能分解为未知的确定性分量和已知的零均值随机分量。探索了由随机维度在定义成本函数中提供的额外自由度，从而证明了控制系统响应的统计方面的范围。单次随机有限元方法用于找到控制问题的近似解。结果表明，当考虑确定性最优控制或在成本函数中包含矩时，将随机配置有限元方法应用于所提问题会导致随机配置点之间的耦合，从而放弃了配置方法的主要优势。考虑的问题的随机Galerkin方法。通过大量的数值示例说明了所提出方法的应用。提出的框架足够笼统，也可以考虑一类反问题，并且还提供了这种类型的数值示例。

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