In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.

中文翻译：

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无约束稀疏控制的状态约束半线性椭圆优化问题

在本文中，我们考虑与半线性椭圆方程方程有关的最优控制问题，其中状态受点约束，但对控制没有明确约束。成本函数中包含一个术语，可促进最佳控制的稀疏性。我们证明了最优控制的存在，并推导了一阶和二阶最优条件。此外，我们为最佳控制以及相关的伴随状态和Lagrange乘子建立一些规律性结果。