  We consider both discrete and continuous "uncertain horizon" deterministic
control processes, for which the termination time is a random variable. We
examine the dynamic programming equations for the value function of such
processes, explore their connections to infinite-horizon and optimal-stopping
problems, and derive sufficient conditions for the applicability of
non-iterative (label-setting) methods. In the continuous case, the resulting
PDE has a free boundary, on which all characteristic curves originate. The
causal properties of "uncertain horizon" problems can be exploited to design
efficient numerical algorithms: we derive causal semi-Lagrangian and Eulerian
discretizations for the isotropic randomly-terminated problems, and use them to
build a modified version of the Fast Marching Method. We illustrate our
approach using numerical examples from optimal idle-time processing and
expected response-time minimization.
