  This paper considers the optimal control for hybrid systems whose
trajectories transition between distinct subsystems when state-dependent
constraints are satisfied. Though this class of systems is useful while
modeling a variety of physical systems undergoing contact, the construction of
a numerical method for their optimal control has proven challenging due to the
combinatorial nature of the state-dependent switching and the potential
discontinuities that arise during switches. This paper constructs a convex
relaxation-based approach to solve this optimal control problem. Our approach
begins by formulating the problem in the space of relaxed controls, which gives
rise to a linear program whose solution is proven to compute the globally
optimal controller. This conceptual program is solved by constructing a
sequence of semidefinite programs whose solutions are proven to converge from
below to the true solution of the original optimal control problem. Finally, a
method to synthesize the optimal controller is developed. Using an array of
examples, the performance of the proposed method is validated on problems with
known solutions and also compared to a commercial solver.
