Optimal control of switched networks for nonlinear hyperbolic conservation laws

本项目的目的是对非线性双曲型守恒律网络在模式切换下的最优控制问题进行分析研究和数值逼近。这种切换被考虑在网络的边界节点和结点以及源项和通量中，从而导致了偏微分方程组的混合系统。例如，这种类型的网络出现在交通流模型以及水和天然气网络模型中。由于守恒律的熵解可能产生激波，因此分析和数值求解守恒律的控制问题是困难的。尽管如此，申请者和其他人最近在保护法律的最佳控制方面取得了令人鼓舞的进展。切换可能会导致解决方案中的额外不连续，这在熵解决方案的上下文中是很自然的。一方面，应仔细分析这些最优控制问题，特别是最优控制的存在性、目标函数关于控制(切换时间、边界控制等)的可微性、相应的灵敏度和伴随方程、最优性条件。另一方面，应考虑对守恒律切换网络的最优控制问题进行适当的数值离散。该项目将从标量守恒定律网络开始，然后进入一个空间维度的2×2系统。

The aim of this project is the analytical study and numerical approximation of optimal control problems for networks of nonlinear hyperbolic conservation laws under modal switching. Such switching is considered at the boundary nodes and junctions of the network as well as in the source terms and fluxes, leading to a hybrid system of PDEs. Networks of this type arise for example in traffic flow models as well as in water and gas network models. Since entropy solutions of conservation laws may develop shocks, the analysis and numerical solution of control problems for conservation laws is difficult. Nevertheless, encouraging progress has been achieved recently by the applicants and others for the optimal control of conservation laws. Switching may lead to additional discontinuities in the solution, which is quite natural in the context of entropy solutions. On one hand a careful analysis of these optimal control problems shall be conducted, in particular existence of optimal controls, differentiability properties of the objective function with respect to controls (switching times, boundary controls, etc.), corresponding sensitivity and adjoint equations, optimality conditions. On the other hand the appropriate numerical discretization of optimal control problems for switched networks of conservation laws shall be considered. The project will start with networks of scalar conservation laws and then proceed to 2 × 2-systems in one space dimension.