We suggest a global perspective on dynamic network flow problems that takes advantage of the similarities to port-Hamiltonian dynamics. Dynamic minimum cost flow problems are formulated as open-loop optimal control problems for general port-Hamiltonian systems with possibly state-dependent system matrices. We prove well-posedness of these systems and characterize optimal controls by the first-order optimality system, which is the starting point for the derivation of an adjoint-based gradient descent algorithm. Our theoretical analysis is complemented by a proof of concept, where we apply the proposed algorithm to static minimum cost flow problems and dynamic minimum cost flow problems on a simple directed acyclic graph. We present numerical results to validate the approach.

翻译：我们提出了一种基于动态网络流问题与port-Hamiltonian动力学相似性的全局视角。将动态最小费用流问题表述为一般port-Hamiltonian系统（可能具有状态依赖系统矩阵）的开环最优控制问题。我们证明了此类系统的适定性，并通过一阶最优性系统刻画最优控制，这为推导基于伴随的梯度下降算法奠定了基础。通过概念验证实验补充理论分析：将所提算法应用于简单有向无环图上的静态与动态最小费用流问题。数值实验结果验证了该方法的有效性。