SIAM Journal on Control and Optimization, Volume 58, Issue 6, Page 3486-3507, January 2020.
In this paper, we investigate optimal control problems subject to a semilinear elliptic partial differential equation. The cost functional contains a term that measures the size of the support of the control, which is the so-called $L^0$-norm. We provide necessary and sufficient optimality conditions of second order. The sufficient second-order condition is obtained by analyzing a partially convexified problem. Interestingly, the structure of the problem yields second-order conditions with different bilinear forms for the necessary and for the sufficient conditions.

中文翻译：

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成本函数中具有$ L ^ 0 $项的最优控制问题的一阶和二阶条件

SIAM控制与优化杂志，第58卷，第6期，第3486-3507页，2020
年1月。在本文中，我们研究受半线性椭圆型偏微分方程约束的最优控制问题。成本函数包含一个用于度量控件支持大小的术语，即所谓的$ L ^ 0 $ -norm。我们提供了二阶最优条件。通过分析部分凸问题获得足够的二阶条件。有趣的是，对于必要条件和充分条件，问题的结构产生具有不同双线性形式的二阶条件。