This paper is concerned with first- and second-order optimality conditions as well as the stability for non-smooth semilinear optimal control problems involving the \(L^1\)-norm of the control in the cost functional. In addition to the appearance of the \(L^1\)-norm leading to the non-differentiability of the objective and promoting the sparsity of the optimal controls, the non-smoothness of the nonlinear coefficient in the state equation causes the same property of the control-to-state operator. Exploiting a regularization scheme, we derive C-stationarity conditions for any local optimal control. Under a structural assumption on the associated state, we define the curvature functional for the part not including the \(L^1\)-norm of controls of the objective for which the second-order necessary and sufficient optimality conditions with minimal gap are shown. Furthermore, under a more restrictive structural assumption imposed on the mentioned state, an explicit formula for the curvature is established and thus the explicit second-order optimality conditions are stated. Finally, the Lipschitz stability of local solutions with respect to the sparsity parameter is shown.

本文关注一阶和二阶最优性条件以及涉及成本函数中控制的\(L^1\)范数的非光滑半线性最优控制问题的稳定性。除了\(L^1\)范数的出现导致目标的不可微性并促进最优控制的稀疏性之外，状态方程中非线性系数的非光滑性也导致了同样的性质控制到状态操作符。利用正则化方案，我们推导出任何局部最优控制的C平稳条件。在关联状态的结构假设下，我们定义了不包括目标控制的\(L^1\)范数的部分的曲率泛函，其中显示了具有最小间隙的二阶充分必要最优条件。此外，在对上述状态施加更严格的结构假设的情况下，建立了显式的曲率公式，从而给出了显式的二阶最优性条件。最后，显示了局部解相对于稀疏参数的 Lipschitz 稳定性。