Optimality and robustness in path-planning under initial uncertainty

Classical deterministic optimal control problems assume full information about the controlled process. The theory of control for general partially-observable processes is powerful, but the methods are computationally expensive and typically address the problems with stochastic dynamics and continuous (directly unobserved) stochastic perturbations. In this paper we focus on path planning problems which are in between -- deterministic, but with an initial uncertainty on either the target or the running cost on parts of the domain. That uncertainty is later removed at some time $T$, and the goal is to choose the optimal trajectory until then. We address this challenge for three different models of information acquisition: with fixed $T$, discretely distributed and exponentially distributed random $T$. We develop models and numerical methods suitable for multiple notions of optimality: based on the average-case performance, the worst-case performance, the average constrained by the worst, the average performance with probabilistic constraints on the bad outcomes, risk-sensitivity, and distributional-robustness. We illustrate our approach using examples of pursuing random targets identified at a (possibly random) later time $T$.Comment: 24 pages, 14 figures. Keywords: optimal control, path-planning, Hamilton-Jacobi PDEs, uncertainty, robustness, delayed information acquisitio