  Information gathered along a path is inherently submodular; the incremental
amount of information gained along a path decreases due to redundant
observations. In addition to submodularity, the incremental amount of
information gained is a function of not only the current state but also the
entire history as well. This paper presents the construction of the first-order
necessary optimality conditions for memory (history-dependent) Lagrangians.
Path-dependent problems frequently appear in robotics and artificial
intelligence, where the state such as a map is partially observable, and
information can only be obtained along a trajectory by local sensing. Robotic
exploration and environmental monitoring has numerous real-world applications
and can be formulated using the proposed approach.
