  Optimal control remains as one of the most versatile frameworks in systems
theory, enabling applications ranging from classical robust control to
real-time safe operation of fleets of vehicles. While some optimal control
problems can be efficiently solved using algebraic or convex methods, most
general forms of optimal control must be solved using memory- expensive
numerical methods. In this paper we present a theoretical formulation and a
corresponding numerical algorithm that can find Pontryagin- optimal inputs for
general dynamical systems by using a direct method. Pontryagin-optimal inputs,
those satisfying the Minimum Principle, can be found for many classes of
problems using indirect methods. But convergent numerical methods to solve
indirect problems are hard to find and often converge slowly. On the other
hand, convergent direct optimal control methods are fast and founded on solid
theory, but their limit points are usually Banach-optimal inputs, which are a
weaker form of optimality condition. Our result, founded on the theory of
relaxed inputs as defined by J. Warga, establishes an equivalence between
Pontryagin- optimal inputs and optimal relaxed inputs. Then, we formu- late a
sampling-based numerical method to approximate the Pontryagin-optimal relaxed
inputs using an iterative method. Finally, using a provably-convergent
numerical method, we synthesize approximations of the Pontryagin-optimal inputs
from the sampled relaxed inputs.
