具有随机PDE约束和不确定控制的最优控制
Optimal control with stochastic PDE constraints and uncertain controls
AI总结：
提出一种基于随机函数的拉格朗日框架，将控制分解为确定性分量和零均值随机分量，利用随机配置有限元方法求解，并揭示确定性最优控制或包含矩的成本泛函会导致随机配置点耦合，从而削弱配置法相对于随机Galerkin法的优势。
摘要：
研究了受不确定偏微分方程约束且具有不确定控制的问题的最优控制。定义问题的拉格朗日量以随机函数的形式提出，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。探索了随机维度在定义成本泛函时提供的额外自由度，展示了控制系统响应统计方面的潜力。使用单次随机有限元方法寻找控制问题的近似解。结果表明，当考虑确定性最优控制或成本泛函中包含矩时，将随机配置有限元应用于所提出的问题会导致随机配置点之间的耦合，从而消除了配置法相对于随机Galerkin法的主要优势。通过多个数值示例展示了所提出方法的应用。所提出的框架足够通用，还可以考虑一类逆问题，并给出了此类逆问题的数值示例。
英文摘要：
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 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 obviating 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.
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