  Presented is a method to compute certain classes of Hamilton-Jacobi equations
that result from optimal control and trajectory generation problems with time
delays. Many robotic control and trajectory problems have limited information
of the operating environment a priori and must continually perform online
trajectory optimization in real time after collecting measurements. The sensing
and optimization can induce a significant time delay, and must be accounted for
when computing the trajectory. This paper utilizes the generalized Hopf
formula, which avoids the exponential dimensional scaling typical of other
numerical methods for computing solutions to the Hamilton-Jacobi equation. We
present as an example a robot that incrementally predicts a communication
channel from measurements as it travels. As part of this example, we introduce
a seemingly new generalization of a non-parametric formulation of robotic
communication channel estimation. New communication measurements are used to
improve the channel estimate and online trajectory optimization with time-delay
compensation is performed.
