Measures and LMI for impulsive optimal control with applications to orbital rendezvous problems

This paper shows how to use semi-definite programming to find lower bounds on (and sometimes solve globally) a large class of nonlinear optimal control problems. This is done by relaxing an optimal control problem into a linear programming problem on measures, also known as a generalized moment problem. The handling of measures by their moments reduces the problem to a convergent series of standard linear matrix inequality relaxations. After providing a completely worked-through example, we apply the method to the linearized impulsive rendezvous problem between two orbiting spacecraft. As the method provides lower bounds on the global infimum, global optimality of the solutions can be guaranteed numerically by a posteriori simulations or by comparison with suboptimal local solutions obtained by other methods. On some problems, we can even recover simultaneously the actual impulse times and amplitudes by simple linear algebra. Finally, our approach can be readily implemented with standard software, as illustrated by numerical examples.