Convex relaxations for nonconvex optimal control problems

An optimal control problem with a nonlinear control system embedded is considered. Using the endpoint map of the control system, such problems can be written as nonlinear programs on the set of admissible controls. Though necessary optimality conditions exist for such problems, they are often nonconvex and such conditions are not sufficient. A relaxation procedure is outlined which generates a convex program whose solution value is a guaranteed lower bound on the solution value of the original problem. This result is a crucial step towards developing deterministic global optimization techniques for optimal control problems using a branch-and-bound framework. The major contribution is that, unlike other developments along these lines, the convex underestimating program derived here is valid on the original function space; i.e. there is no need to discretize the control.