A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the $ L^1 $-norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints. To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.

中文翻译：

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具有混合控制状态约束的热方程的稀疏最优控制

考虑了热方程的稀疏最优控制问题，给出了控制的点向边界和混合的点向控制​​状态约束。标准的二次跟踪类型的函数应最小化，包括Tikhonov正则项和考虑稀疏性的控件的$ L ^ 1 $-范数。拉格朗日乘子的存在性和规则性特别强调混合控制状态约束。为此，证明了希尔伯特空间中线性规划问题的对偶定理，并将其应用于给定的最优控制问题。