  We study driver's optimal trajectory planning under uncertainty in the
duration of a traffic light's green phase. We interpret this as an optimal
control problem with an objective of minimizing the expected cost based on the
fuel use, discomfort from rapid velocity changes, and time to destination.
Treating this in the framework of dynamic programming, we show that the
probability distribution on green phase durations gives rise to a sequence of
Hamilton-Jacobi-Bellman PDEs, which are then solved numerically to obtain
optimal acceleration/braking policy in feedback form. Our numerical examples
illustrate the approach and highlight the role of conflicting goals and
uncertainty in shaping drivers' behavior.
