SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 1341-1369, January 2021.
In this paper we analyze the optimal control of initial boundary value problems for entropy solutions of scalar hyperbolic balance laws with pointwise state constraints. Here, we suppose that the initial and the boundary data switch between different $C^1$-functions at certain switching points, where the $C^1$-functions and the switching points are considered as the control. For a class of cost functionals, we prove first order necessary optimality conditions for the corresponding optimal control problem with state constraints. Furthermore, we use a Moreau--Yosida type regularization to approximate the optimal control problem with state constraints. We derive optimality conditions for the regularized problems and finally prove convergence to the solution of the optimal control problem with state constraints.

中文翻译：

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具有状态约束的双曲平衡定律的最优边界控制

SIAM控制与优化杂志，第59卷，第2期，第1341-1369页，2021年1月。
在本文中，我们分析了具有点状态约束的标量双曲平衡定律的熵解的初始边值问题的最优控制。在此，我们假设初始数据和边界数据在某些切换点在不同的$ C ^ 1 $函数之间切换，其中$ C ^ 1 $函数和切换点被视为控制。对于一类成本函数，我们证明了带有状态约束的相应最优控制问题的一阶必要最优性条件。此外，我们使用Moreau-Yosida类型正则化来近似具有状态约束的最优控制问题。我们推导了正则化问题的最优条件，并最终证明了状态约束下最优控制问题解的收敛性。