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

A Path-Dependent Variational Framework for Incremental Information Gathering