Optimal control of switched networks for nonlinear hyperbolic conservation laws

本计画的目的是研究非线性双曲守恒律网路在模态切换下的最佳控制问题。这种开关被认为是在边界节点和交界处的网络，以及在源项和通量，导致混合系统的偏微分方程。这种类型的网络例如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，守恒律控制问题的分析和数值求解是困难的。然而，最近申请人和其他人在守恒律的最优控制方面取得了令人鼓舞的进展。切换可能导致解中的额外不连续性，这在熵解的上下文中是非常自然的。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、目标函数相对于控制（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，应考虑守恒律切换网络的最优控制问题的适当数值离散化。该项目将从标量守恒律网络开始，然后进行到一维空间中的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.