  This paper presents a novel convex optimization-based method for finding the
globally optimal solutions of a class of mixed-integer non-convex optimal
control problems. We consider problems with non-convex constraints that
restrict the input norms to be either zero or lower- and upper-bounded. The
non-convex problem is relaxed to a convex one whose optimal solution is proved
to be optimal almost everywhere for the original problem, a procedure known as
lossless convexification. This paper is the first to allow individual input
sets to overlap and to have different norm bounds, integral input and state
costs, and convex state constraints that can be activated at discrete time
instances. The solution relies on second-order cone programming and
demonstrates that a meaningful class of optimal control problems with binary
variables can be solved reliably and in polynomial time. A rocket landing
example with a coupled thrust-gimbal constraint corroborates the effectiveness
of the approach.
