Probability Distribution Control of Finite-State Markov Chains with Wasserstein Costs and Application to Operation of Car-Sharing Services

This study investigates an optimal control problem of discrete-time finite-state Markov chains with application in the operation of car-sharing services. The optimal control of probability distributions is the object of focus to ensure that the controlled distributions are as close as possible to the given ones. The problem is formulated using Wasserstein distances, which allows us to measure the difference among probability distributions and is suitable for the objective of this study. For the control problem, we provide a necessary condition for optimality in the control inputs and develop an algorithm for obtaining optimal control inputs. The developed algorithm is then applied to the probability distribution control of a one-way car-sharing service, which provides a rebalancing strategy to resolve car unevenness.