  Discrete-time robust optimal control problems generally take a min-max
structure over continuous variable spaces, which can be difficult to solve in
practice. In this paper, we extend the class of such problems that can be
solved through a previously proposed local reduction method to consider those
with existence constraints on the uncountable variables. We also consider the
possibility of non-unique trajectories that satisfy equality and inequality
constraints. Crucially, we show that the problems of interest can be cast into
a standard semi-infinite program and demonstrate how to generate optimal
uncertainty scenario sets in order to obtain numerical solutions. We also
include examples on model predictive control for obstacle avoidance with
logical conditions, control with input saturation affected by uncertainty, and
optimal parameter estimation to highlight the need for the proposed extension.
Our method solves each of the examples considered, producing violation-free and
locally optimal solutions.
