A simple variational problem for a moving vehicle

The problem of determining an optimal path of a moving vehicle, considered in Sherali and Kim (1992), is studied in this paper. We generalize their model by including a general cost structure in the problem and show that a specific, non-trivial cost function in this setting yields an easy solution. This follows from the observation that the variational problem that arises in the model with this particular cost function is the problem of geodesics on the Liouville surface. Through a comparative computational study, we also show that the optimal path for the model with this cost function better approximates the optimal path for the original model in Sherali and Kim than their second-order approximation approach does.