Journal of Scientific Computing

, Volume 78, Issue 3, pp 1571–1600 | Cite as

Stochastic Galerkin Method for Optimal Control Problem Governed by Random Elliptic PDE with State Constraints

  • Wanfang ShenEmail author
  • Liang Ge
  • Wenbin Liu


In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The optimal control minimizes the expectation of a cost functional with mean-state constraints. We first represent the stochastic elliptic PDE in terms of the generalized polynomial chaos expansion and obtain the parameterized optimal control problems. By applying the Slater condition in the subdifferential calculus, we obtain the necessary and sufficient optimality conditions for the state-constrained stochastic optimal control problem for the first time in the literature. We then establish a stochastic Galerkin scheme to approximate the optimality system in the spatial space and the probability space. Then the a priori error estimates are derived for the state, the co-state and the control variables. A projection algorithm is proposed and analyzed. Numerical examples are presented to illustrate our theoretical results.


Stochastic optimal control Stochastic Galerkin method Optimal control problem with state constraints 


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Authors and Affiliations

  1. 1.School of Mathematic and Quantitative EconomicsShandong University of Finance and EconomicsJinanPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of JinanJinanPeople’s Republic of China
  3. 3.KBSUniversity of KentCanterburyUK

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