  This paper shows how to find lower bounds on, and sometimes solve globally, a
large class of nonlinear optimal control problems with impulsive controls using
semi-definite programming (SDP). This is done by relaxing an optimal control
problem into a measure differential problem. The manipulation of the measures
by their moments reduces the problem to a convergent series of standard linear
matrix inequality (LMI) relaxations. After providing numerous academic
examples, we apply the method to the impulsive rendezvous of two orbiting
spacecrafts. As the method provides lower bounds on the global infimum, global
optimality of the solutions can be guaranteed numerically by a posteriori
simulations, and we can recover simultaneously the optimal impulse time and
amplitudes by simple linear algebra.
