Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls
Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls
复制标题
稀疏控制状态约束椭圆最优控制问题的二阶稳定性分析
DOI：
10.1137/130917314
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
F. Tröltzsch
中科院分区：
文献类型：
--
作者：
E. Casas;F. Tröltzsch
讨论了一类半线性椭圆型偏微分方程具控制和状态约束的最优控制问题。本文的主要创新之处在于将控制的$L^1$范数作为目标泛函的一部分，最终导致最优控制函数的稀疏性。分析了二阶充分最优性条件。应用它们证明了控制的正则化参数为零的L^2的最优解的收敛性。估计了相应的收敛速度。
An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the $L^1$-norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. Second-order sufficient optimality conditions are analyzed. They are applied to show the convergence of optimal solutions for vanishing $L^2$-regularization parameter for the control. The associated convergence rate is estimated.