A Lyapunov analysis of the continuous-time adaptive Bellman–Ford algorithm

Abstract The shortest path problem, one of the most classical graph problems, has been addressed in many different ways suitable for various settings in the fields of computer science and artificial intelligence. In this paper, we revisit a distributed control solution, namely the continuous-time adaptive Bellman–Ford algorithm, to the shortest path problem. While previous work only concerned its global asymptotic stability, we not only prove its global asymptotic stability by formulating a Lyapunov function, but characterize the initial conditions under which the algorithm will converge exponentially, and show that the algorithm is globally ultimately bounded under persistent bounded perturbations based on the proposed Lyapunov function.