In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

中文翻译：

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半线性椭圆状态约束的最优控制问题的拉格朗日乘数法

在本文中，我们将增强拉格朗日方法应用于具有点状态约束的一类半线性椭圆最优控制问题。我们显示了原始变量的子序列到原始问题的局部解的强收敛性，以及伴随状态的弱收敛性和与状态约束相关的乘子的弱*收敛性。此外，我们显示了原始问题的局部解的任意小邻域中存在平稳点。另外，给出了各种数值结果。